QP-PQ: Quantum Probability and White Noise Analysis Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy QP-PQ: Quantum Probability and White Noise Analysis VOl. 18:
Quantum Probability and Infinite-Dimensional Analysis From Foundations to Applications eds. M. Schurmann and U. Franz
Vol. 17:
Fundamental Aspects of Quantum Physics eds. L. Accardi and S. Tasaki
Vol. 16:
Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsui and A. Hora
Vol. 15: Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg Vol. 14:
Quantum Interacting Particle Systems eds. L. Accardi and F. Fagnola
Vol. 13:
Foundations of Probability and Physics ed. A. Khrennikov
QP-PQ VOl. 10:
Quantum Probability Communications eds. R. L. Hudson and J. M. Lindsay
VOl. 9:
Quantum Probability and Related Topics ed. L. Accardi
VOl. 8:
Quantum Probability and Related Topics ed. L. Accardi
VOl. 7:
Quantum Probability and Related Topics ed. L. Accardi
Vol. 6:
Quantum Probability and Related Topics ed. L. Accardi
Qp-pQ Quantum Probability and White Noise Analysis Volume XVIII
From Foundations to Applications Krupp-Kolleg Greifswald, Germany
22 - 28 June 2003
Editors
Michael Schurmann Uwe Franz University of Greifswald, Germany
NEW JERSEY
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EeWorld Scientific LONDON * SINGAPORE
BElJlNG * SHANGHAI * HONG KONG * TAIPEI
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QP-PQ: Quantum Probability and White Noise Analysis Vol. XVIII QUANTUM PROBABILITY AND INFINITE-DIMENSIONAL ANALYSIS From Foundations to Applications Copyright 0 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof;may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying,recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.
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INTRODUCTION
In Spring 2003, we organized the Special Semester “Quantum Probability: From Foundations to Applications” in Greifswald, Germany. The main events were 0
0
0
a school “Quantum Independent Increment Processes: Structure and Applications to Physics,” two workshops on “Non-commutative Martingales and Free Probability” and “Dilations, Endomorphism Semigroups, and their Classification by Product Systems,” and a n international conference “Quantum Probability and Infinite Dimensional Analysis.”
Furthermore, in Fall 2003 a workshop on “Quantum Independent Increment Processes” was held in Levico Terme, Italy, as a continuation of the Greifswald School. The goal of this Special Semester was to bring together scientists with different backgrounds who work in quantum probability and related fields, and to stimulate new collaborations. We would like t o thank all participants and lecturers for their hard work and for helping t o create a very stimulating atmosphere. The research papers collected in this volume represent the topics discussed during the Special Semester and reflect the active developments ranging from the foundations of quantum probability to its applications. The lecture notes of the school on “Quantum Independent Increment Processes” will be published in the Springer series Lecture Notes in Mathematics. Many people have been involved in the organization of the Special Semester, it would be impossible to name them all. We are particularly indebted t o the Volkswagen Foundation and the DFG, who supported the school on “Quantum Independent Increment Processes” and the international conference ‘Quantum Probability and Infinite Dimensional Analysis,” and thereby made the Special Semester possible. We also acknowledge the support by the European Community for the Research Training Network
V
vi
“QP-Applications: Quantum Probability with Applications to Physics, Information Theory and Biology” under contract HPRN-CT-2002-00279. Special thanks go to Mrs. Zeidler for taking care of the logistics of the Special Semester and for her help with the editing of these proceedings.
Michael Schiirmann Uwe Franz Greifswald, October 2004
CONTENTS
Introduction
V
Probability Measures in Terms of Creation, Annihilation, and Neutral Operators Luigi Accardi, Hui-Hsiung Kuo and Aurel Stan
1
Semi Groupes Associes B l’operateur de Laplace-Levy Luigi Accardi and Habib Ouerdiane
12
Stochastic Golden Rule for a System Interacting with a Fermi Field Luigi Accardi, R.A. Roschin and I.V. Volovich
28
Generating Function Method for Orthogonal Polynomials and Jacobi-Szego Parameters Nobuhiro Asai, Izumi Kubo and Hui-Hsiung Kuo
42
Low Temperature Superconductivity and the Stochastic Limit Fabio Bagarello Multiquantum Markov Semigroups, Interacting Branching Processes and Nonlinear Kinetic Equations. Finite Dimensional Case V.P. Belavkin and C.R. Williams A Note on Vacuum-Adapted Semimartingales and Monotone Independence Alexander C.R. Belton Quantum Stochastic Processes and Applications Mohamed Ben Chrouda, Mohamed El Oued and Habib Ouerdiane Regular Quantum Stochastic Cocycles have Exponential Product Systems B.V. Rajarama Bhat and J. Martin Lindsay
vii
56
67
105 115
126
viii
Evolution of the Atom-Field System in Interacting Fock Space P.K . Das Quantum Mechanics on the Circle through Hopf q-Deformations of the Kinematical Algebra with Possible Applications to LQvyProcesses V.K. Dobrev, H.-D. Doebner and R. Twarock
141
153
On Algebraic and Quantum Random Walks Demosthenes Ellinas
174
Dual Representations for the Schrodinger Algebra Philip Feinsilver and Rent! Schott
201
Harmonic Analysis on Non- Amenable Coxeter Groups Gero Fendler
216
A Limit Theorem for Conditionally Independent Beam Splittings K.H. Fichtner, Volkmar Liebscher and Masanori Ohya
227
On Factors Associated with Quantum Markov States Corresponding to Nearest Neighbor Models on a Cayley Tree Francesco Fidaleo and Farruh Mukhamedov
237
On Quantum Logical Gates on a General Fock Space Wolfgang Freudenberg, Masanori Ohya and Noburo Watanabe
252
The Chaotic Chameleon Richard D. Gill
269
On an Argument of David Deutsch Richard D. Gill
277
Volterra Representations of Gaussian Processes with an Infinite-Dimensional Orthogonal Complement Yuji Hibino and Hiroshi Muraoka
293
The Method of Double Product Integrals in Quantisation of Lie Bialgebras Robin L. Hudson
303
ix
On Noncommutative Independence Romuald Lenczewski
320
Lkvy Processes and Jacobi Fields Eugene Lytvynov
337
Ic-Decomposabilityof Positive Maps Wtadystaw A. Majewski and Marcin Marciniak
362
An Introduction to LBvy Processes in Lie Groups Ming Liao
375
Duality of W*-Correspondences and Applications Paul S. Muhly and Baruch Solel
396
Extendability of Generalized Quantum Markov States Hiromichi Ohno
415
White Noise Approach to the Low Density Limit Alexander N . Pechen
428
Asymptotics of Large Truncated Haar Unitary Matrices Jlilia Riffy
448
Quantum Optical Scenarios for Stochastic Resonance Vyacheslav Shatokhin, Thomas Wellens and Andreas Buchleitner
457
On a Classical Scheme in a Noncommutative Multiparameter Ergodic Theory Adam G. Skalski
473
LBvy Processes and Tensor Product Systems of Hilbert Modules Michael Skeide
492
Three Ways to Representations of B A ( E ) Michael Skeide
504
The Hamiltonian of a Simple Pure Number Process Wilhelm von Waldenfels
518
On Topological Entropy of Quotients and Extensions Joachim Zacharias
525
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PROBABILITY MEASURES IN TERMS OF CREATION, ANNIHILATION, AND NEUTRAL OPERATORS
LUIGI ACCARDI Centro Vito Volterm Facoltb di Economia Uniuersitd di Roma “Tor Vergata” 00133 Roma, Italy E-mail: accardi0uolterra.mat.uniroma2.it HUI-HSIUNG KUO Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA E-mail: kuo0math.1su.edu AUREL STAN Department of Mathematics University of Rochester Rochester, N Y 14627, USA E-mail: astan0math.rochester.edu Let p be a probability measure on Rd with finite moments of all orders. Then we can define the creation operator a + ( j ) , the annihilation operator a - ( j ) , and the neutral operator a o ( j )for each coordinate 1 5 j 5 d . We use the neutral operators ao(i) and the commutators [a-( j ) ,a + ( k ) ] to characterize polynomially symmetric, polynomially factorizable, and moment-equal probability measures. We also present some results for probability measures on the real line with finite support, infinite support, and compact support.
1. Creation, annihilation, and neutral operators
Let /A be a probability measure on Rd with finite moments of all orders, namely, for any nonnegative integers i l , i2,. . . ,id,
1
2
where x = (21,x2,.. . ,xd) E Pd.Let Fo = P and for n 2 1let F, be the vector space of all polynomials in x1,22,. . . , z d of degree 5 n. Then we have the inclusion chain
Fo C FI C ... C Fn C ... C L2(p)Next, define Go = R and for n 2 1 define G, to be the orthogonal complement of F,-1 in F,. Then the spaces G,, n 2 0 , are orthogonal. Define a real Hilbert space 3t by 00
3t = @ G,
(orthogonal direct sum).
n=O
For each n 2 0, let P, denote the orthogonal projection of 3t onto G,. Let X j , 1 5 j 5 d, be the multiplication operator by xj. Accardi and Nahni5 have recently observed that for any 1 5 j 5 d and n 2 0
XjG, I G k , Q k f n - l , n , n + l , where G-1 = ( 0 ) by convention. Then they used this fact to obtain the following fundamental recursion equality
XjPa = Pn+lXjPn
+ PnXjPn + Pn-1XjPn,
1 _ 0 ,
(1)
where P-1 = 0 by convention. When d = 1, this equality reduces to the well-known recursion formula
xPn(x) = Pa+l(x) + anPn(z)+ wnPn-l(z),
(2)
where P,(z)’s are orthogonal polynomials with respect to p , Pn(x)is a polynomial of degree n with leading coefficient 1, and {an,wn}’S are the Jacobi-Szego parameters of p . Now, for each n 2 0 and 1 5 j 5 d, define three operators by
D i ( j ) = Pn+lXjPn
:
Gn +Gn+lr
D,(j) = Pn-lXjPn : Gn +Gn-1, DO,(j) = P,XjP,
:
G,
+Gn.
Using these operators, we can define for each 1 5 j 5 d three densely defined linear operators a + ( j ) , u - ( j ) , and a o ( j )from 3c into itself by n
2 0,
= D,(j),
n
2 0,
aO(j)lcn= D%),
n
2 0.
a+(j)lc, = DRW, a-(j)Ic,
3
The operators a + ( j ) , u - ( j ) , and u o ( j )are called creation, annihilation, and neutral operators, respectively. It can be checked that u - ( j ) = a+(j)* and u o ( j ) = u o ( j ) * for each 1 5 j 5 d. The collection I1 I j I d )
{X,a+W, a - ( A , &)
is called the interacting Fock space of the probability measure p . For convenience, we will use the term “CAN operators” to call the creation, annihilation, and neutral operators. By using the multiplication and CAN operators, we can rewrite the fundamental recursion equality in Equation (1) as the equality in the next theorem. Theorem 1.1. For each 1 xj
j
5 d , the following equality holds
= u+(j)
+ 2 ( j )+ u - ( j ) .
(3)
We can use the equality in Equation (3) to extend the Accardi-Boiejko unitarity theorem’ to the multi-dimensional case. In this paper we will present some results to answer the following question. Question: What properties of p are determined by the associated CAN operators? 2. Polynomially symmetric measures Definition 2.1. A probability measure p on Rd is said to be polynomially symmetric if
for all nonnegative integers i l , i2,. . .id with il integer.
+ i2 + . . . + id being an odd
Note that if p is a symmetric measure with finite moments of all orders, then it is polynomially symmetric. But the converse is not true. Consider the function e(z) = e - ( ’ n ~ sin(2n )~ In $1,
z
> 0.
(4)
It is well-known that P“
zV(z) dz = 0, 10
V n = 0, 1,2, . . .
(5)
4
Define a function
f ( x )=
{
c8+(x), 0, c8-(-x),
if x > 0, if x = 0, if x
< 0,
where 8+ and 8- are the positive and negative parts of 8, respectively, and the constant c is chosen such that Jw - m f (x)d x = 1. By using Equation (5) one can easily check that the probability measure
dP(Z) = f dx is polynomially symmetric. Obviously, p is not symmetric. The next theorem has been proved in our paper2. Theorem 2.1. A probability measure p o n Rd with finite moments of all orders is polynomially symmetric if and only i f u o ( j ) = 0 f o r all j = 1,2,...,d. 3. Polynomially factorizable measures Definition 3.1. A probability measure p on Rd is said to be polynomially factorizable if
for all nonnegative integers il ,i 2 , . . .id. Obviously, if p is a product measure with finite moments of all orders, then it is polynomially factorizable. However, the converse is not true. Consider two modified functions of the function 8 in Equation (4):
el (x)= e-Qn e2(2) = e-('" ')'
2 ) '
[I
+ sin(2r In x ) ],
x > 0,
[I - sin(2r In x)], x > 0.
Define a function g ( x , y ) on R2 by
(0'
k[el(x)sin2 y
dX,Y) =
+ &(x) cos2y l e - ~ ,
if x
> 0, y > 0 ,
elsewhere,
where the constant k is chosen so that probability measure
d&,
JR2
g(x,y) d x d y = 1. Then the
Y ) = g(x,Y ) dXdY
5 can be shown to be polynomially factorizable, but not a product measure. The next theorem follows from Theorem 4.10 in our paper2. Theorem 3.1. A probability measure p on Rd with finite moments of all orders is polynomially factofizable if and only if for any i # j ,
the operators in {a+(i),a-(i), a o ( i ) } commute with the operators in {.+(A, a - w , a O ( j ) } . 4. Probability measures by means of the CAN operators
Let p be a probability measure on Rd with finite moments of all orders. We have the associated CAN operators a + ( j ) , u - ( j ) , and u o ( j ) . Define a d x 1 matrix A t and a d x d matrix A;>+ by
where [a,b] = ub - ba, the commutator of a and b. Definition 4.1. Two probability measures p and u with finite moments of all orders are said to be moment-equal if
for all monomial functions m ( x ) . The next two theorems have been proved in our paper3. Theorem 4.1. Two probability measures p and u on Rd with finite moments of all orders are moment-equal if and only if A t = A: and A;,+ = & I + .
6
Theorem 4.2. A probability measzlre p o n Rd with finite moments of all orders is the standard Gaussian measure o n Rd i f and only i f A: = Od and A;'+ = I d , namely, ao(i)= 0 f o r all 1 5 i 5 d and [a-(j),a+(k)]= d j , k I f o r all 1 5 j , k 5 d. The above discussion leads to the following problem for specifying a probability measure /I in terms of the matrices A: and A;*+.
Problem: Let V be the vector space of all polynomials on Rd. Let ai and a j , k be linear operators on V for 1 5 i , j , k 5 d. Find conditions on {~i}f and =~ {aj,k};,k=l so that there exists a probability measure p on Rd satisfying ai = ao(i)and a j , k = [ u - ( j ) , a + ( k ) ]for all 1 5 i , j , k 5 d. In the next section we will give some results on the solution to the above problem for the case when d = 1.
5. Probability measures on the real line Let p be a probability measure on R with finite moments of all orders. Let V be the vector space of all polynomials in 2 and let V,, be its subspace consisting of all polynomials of degree 5 n. Let F,, = Vn/-. Here the equivalence relation is given by p-almost everywhere, namely, f g if f = g holds p-a.e.
-
-
Assumption. In this section all linear operators T : V + V are assumed to satisfy the condition that T(V,) C V, for all n 2 0, namely, all subspaces Vn,n 2 0, are invariant under T . 5.1. Probability measures o n R with finite support Observe that if a probability measure p on R is supported by m distinct points, then
Fj=V,,
,...,m - 1 , j = m , m + l , ....
j=O,l,2
Fj =Vm--l,
The following theorem can be easily verified.
Theorem 5.1. Suppose p is a probability measure o n R supported by m distinct points. T h e n the following equalities hold: (1)
D(u:) , I
(2) TI-( [.;,a:]
= TI-(a:
), I
=
lvm-,)
f o r all k 2 m - 1.
o f o r ail k 2 m - 1.
7
Definition 5.1. Two linear operators S and T from V into itself are called trace equivalent on V , denoted by S T on V , if n-(slvb)
=Tr(TIVb))
vk_>o'
They are called trace equivalent on Vn, denoted by S
T on Vn,if
VOIkSn.
n ( s ( V ,= ) n(Tlvb),
The next theorem is from our paper3. It characterizes those measures supported by finitely many points in R in terms of the CAN operators.
Theorem 5.2. Let m 2 1 be afixed integer. Let a' and a-*+ be two linear operators f r o m V,-l into itself. Then there exists a probability measure p t o n R supported by m distinct points such that a' A a t and a-i+ [a;,.:] o n V,-l if and only if the following conditions hold:
-
(1) The spaces v k , 0 5 k
I,) (3) Q(a-*+ IvJ (2) Tr(a-9'-
5 m - 2, are invariant under a' and a->+. > 0 for all 0 5 k 5 m - 2. = 0,
5.2. Probability measums on R with infinite support Let p be a probability measure on R with infinite support, namely, the support of p contains infinitely many points. In this case, we have
Fn =Vn,
Vn_>O.
The next theorem has been proved in our paper3.
Theorem 5.3. Let a0 and a-i+ be two linear operators f r o m V into itself. Then there exists a probability measure p o n R with infinite support such that a' a! and a->+ [ a ; , a i ] o n V i f and only i f the following conditions hold: (1) The spaces Vn,n 2 0, are invariant under a' and a-i+. (2) Tr(a-i+
I,)
> o for all n 2 0.
Let E denote the set of all trace equivalent classes of ordered pairs (ao,a-i+) of linear operators from V into itself satisfying either one of the following conditions (a) and (b): (a) Q( a-$+
l v n)
> 0, V n 2 0.
8
(b) There exists m such that (1) n ( a OIVJ = n ( a Olvm-l), V k
(2)
n(a-,+
(3)
Tr(a-l+
lvk) lvk)
2m-
1,
> 0, vo 5 k 5 m - 2, = 0, V k 2 m - 1.
Theorem 5.4. There is a one-to-one correspondence between the set B and the set of all probability measures o n R with finite moments of all orders. 5.3. Probability measures on R with compact support
The Paley-Wiener type problem is to characterize probability measures with compact support. We have the next theorem from our paper3.
Theorem 5.5. A probability measure p o n R with finite moments of all orders has compact support if and only if the following two sequences of real numbers are bounded: (1) m
(2)
( 4), I
-*(a;
n ( q +I F = ) ,
n> - 1.
F l J7
n 2 1.
6. Classical probability measures on the real line Let p be a probability measure on R with finite moments of all orders. We have the associated orthogonal polynomials { Pn} and the Jacobi-Szego parameters {an,un}as given in Equation (2). The corresponding CAN operators are given by
UEP,
= pn+l,
aGPn = WnPn-1,
U:P~
= CX,P,, n 2 0,
where P-l = 0 by convention. Therefore, the commutator a;'+ is given by
= [a;,
]a :
Consider the following classical probability measures on the real line: 1 (1) Gaussian: dp(x) = -e - S dx, x E R (g > 0). &a
.=
(2) Poisson: p ( { l ~ }= ) e-'-,
Ak
k!
k = 0,1,2,. . . (A > 0).
9
(3) gamma:
Q
> 0. 1
d p ( x ) = -x a - l e -3 d x , x J3a) (4) Pascal: T > 0, 0 < p < 1.
> 0.
1
( 5 ) uniform: dp(x) = - d x , - 1 5 x 5 I. 2 1 1 (6) arcsine: d p ( x ) = - ___ d x , - 1 < x < 1.
di=?
(7) semi-circle: d,u(x) = Z d S d x , lr
(8) beta-type: j? > -1/2,
-1
1x 5 1.
,8 # 0.
For the above probability measures, the Jacobi-Szego parameters are given in the next table. By convention, wo = 1.
measure u Gaussian Poisson
+ n - 1)
Pascal
+ 2n ( 2 - p)n + ~ ( 1 p- ) P
p”
uniform
0
n2 (2n 4-1)(2n- 1)
arcsine
0
semi-circle
0
-
beta-type
0
n(n - 1 28) 4(n P)(n - 1 + P )
gamma
a!
n(Q n(n
+ T - 1)(1 - P)
-
ifn=l
1
4
+
+
10
Furthermore, the operator a: and the commutator a;’+ = [a,, a]: given in the following table.
are
measure p Gaussian
D 2Pn
Poisson gamma
APn (a!
+ 2n)P,
(a!
+ 2n)P,
Pascal uniform ifn=O arcsine
0
--PI,
(0,
semi-circle
0
beta-type
0
if n = 1 ifn>2
It is interesting to compare the above table with Theorems 2.1, 5.1, 5.2, 5.3, and 5.5.
Acknowledgments This research was initiated in May-June 2002 during the visits of H.-H. Kuo and A. Stan to the Centro Vito Volterra (CVV), Universita di Roma “Tor Vergata”. They are grateful to the CVV for financial support and would like to thank Professor L. Accardi and his staff members for the warm hospitality during the visits.
References 1. Accardi, L. and Boiejko, M.: Interacting Fock space and Gaussianization of probability measures; Infinite Dimensional Analysis, Q u a n t u m Probability and Related Topics 1 (1998) 663-670 2. Accardi, L., Kuo, H.-H., and Stan, A.: Characterization of certain probability measures by creation, annihilation, and neutral operators; Infinite Dimensional Analysis, Quantum Probability and Related Topics (to appear) 3. Accardi, L., Kuo, H.-H., and Stan, A.: Moments and commutators of probability measures; Preprint (2003) 4. Accardi, L., Lu, Y. G., and Volovich, I.: The QED Hilbert module and interacting Fock spaces; I I A S Reports No. 1997-008 (1997) International Institute for Advanced Studies, Kyoto 5. Accardi, L. and Nahni, M.: Interacting Fock space and orthogonal polynomials in several variables; Preprint (2002) 6. Chihara, T. S.: A n Introduction t o Orthogonal Polynomials. Gordon and Breach, 1978 7. Kubo, I.: Generating functions of exponential type for orthogonal polynomials; Inifinite Dimensional Analysis, Q u a n t u m Probability and Related Topics 7 (2004) 155-159 8. Szego, M.: Orthogonal Polynomials. Coll. Publ. 23,Amer. Math. SOC.,1975
SEMI GROUPES ASSOCIES A L’OPERATEUR DE LAPLACE-LEVY
LUIGI ACCARDI Centro Vito Volterra Facoltb d i Economia Universitd di Roma “Tor Vergata” 00133 Roma, Italy E-mail:
[email protected] HABIB OUERDIANE Dipartement de Mathimatiques FacultC des sciences de Tunis i060 Tunas, Tunisia
1. Introduction et prkliminaires
Soit H un espace de Hilbert skpkrable rkel et A un opkrateur positif auto adjoint sur H tel que son inverse A-’ soit de type Hilbert-Schmidt. Alors il existe une suite de nombres rkels positifs 0 6 A1 _< A2 5 ... 5 A,. . . et c Dom(A) tel que: une suite de vecteurs
La suite (en)n21 forme une base orthonormke de H . Pour tout rkel p E R posons:
n=l
oh (., -) est le produit scalaire dans H et I. 10 est la norme Hilbertienne sur H . Pour tout pour p 2 0, Ep = {[ E H , [Elp < oo} est un espace de Hilbert
muni de la norme [_o E-, est le dual topologique fort de E.
La
P++W
forme bilinkaire canonique sur E* x E est notde par < ., . > . L’ensemble typique de triplet de type (1) qu’on considbre en analyse du Bruit blanc, voir 191, [13] et [14] est le triplet:
E I S(R) c L2(R,dx)c E* = S(R) oh S(R) est l’espace de Schwartz des fonctions
B ddcroissance rapide sur R, L2(R,dx) est l’espace des fonctions de carrk intdgrable par rapport B la mesure de Lebesgue dx sur R et S ( R )l’espace des distributions tempdrkes. Soit l’op6rateur
et (e,),>lune d’HermiG
base orthonorm6e de L2(R,dx) formke par les fondions
Oh
+
est le polyndme d’Hermite de degr6 n. Alors Ae, = 2 ( n l)en, i. e. An = 2 n + 2, n = 0 ’ 1 ’ 2 , . . . sont des valeurs propres de l’opdrateur A et de plus W
1
< 00 si p > 1 / 2
n=O
Pour tout p 1 0, on d6finit lfl, = lAPflo oh de L2(R,dx). Alors l f l p est donnke par:
[.lo
est la norme Hilbrtienne
\n=o
If/,
Soit l’espace de Hilbert S,(R) = {j E L2(R,dx), < co} alors S(R) = nploS,(R) et S(R) est un espace de F’r6chet nucleaire. Soit SL(R) = S-,(R) le dual topologique de S,(R) muni de la norme Hilbertienne Ifl-, = IA-Pflo . On obtient alors les inclusions suivantes
S(R) C S,(R) C L2(R,dx) C S-,(R) C S ( R )= U,?oS-,(R)
14
1.1. Fonctions differentiables
Une fonctions F : E + R (ou C ) est dite de classe C 2 au point [ E E s’il existe un klkment F’([) E E* et PI’(O
qu'on munit de sa topologie limite projective. De m&mel'espace
G, (E,):=
(J
EZP(E,,,, 8, m )
(9)
p>O,m>O
muni de sa topologie limite inductive est appeld espace de fonctions entibres sur E, d'ordre de croissace 8-exponentiel de type arbitraire. Par dCfinition f E T,g(EE)et g E Ge(E,) admettent comme developpement en sdrie deTaylor B l'origine
f(z) = C(z"",fn,,z
E E,*
(10)
n>O
n>O
oh (., .) est la forme canonique de dualit4 entre (E,*)Bnx E,On et oii E,On
d4signe le produit tensoriel symbtrique d'ordre n de E,. En vue de caret Gg(E,) en fonction de leurs sdries formelles actkriser les espaces Fg(E,*) de Taylor, on intoduit les espaces de Fock interactifs suivants:
oh pour tout entier n
16
np>O,m>O Fe,m(Ec,p)muni de la topologie limite pro-
Soit alors Fo(E,) = jective. De mGme on pose
Ge(E,*)=
u
Gs,m(Ec,-p)
p>O,m>O
muni de la topologie limite inductive oh pour tout m
Gs,m(E,*,-p)=
{ 6 = (@n)n>o :
@n
> 0 et p > 0
E (Ec,-p)o",l1611e,m,-p < m}
avec n>O
I1 est facile de verifier que Fe(E,) est un espace de F'rCchet nucl6aire et que les espaces Fg(E,) et Ge(E,*)sont en dualit6, auterement dit que le dual fort de Fe(E,) est identifik B l'espace Ge(E,*).La forme de dualit6 entre Gs(E,*)et Fs(E,) est donn6e par m
n=O
L'application Skrie de Taylor 7 (A l'origine) associe B toute fonction entibre f(z) = Cn>o(z@p",fn) E Fe(E,*)ses coefficients de Taylor = (fn)n2~. Il est demo&r6 dans [ll] que l'application 7 Btablit deux isomorphismes topologiques:
f
1.3. Transformation de Laplace Soit FT*, (E:) le dual topologique fort de 3 0 (E,').La dualitk entre FT*, (E,*)et Gp(E,) est d6finie par la forme de dualit6 (11) entre les espaces de sdries formelles associks Fe (E,) et Gs (E:): 00
w E F:(E,*)et
f E F ~ ( E , * )i
((4,f)):= @,fj= C.!(dn,fn) n=O
De la condition
on dkduit que pour tout E E, la fonction exponentielle et : E,' + C , qui A tout z E E,* associe ec(x) = e("iE) est un Clement de Fs(E,*).Donc pour
17
toute fonctionelle
+ E 3:(E,*)sa transformke de Laplace est dkfinie par:
.~(+)(t) =
= (6, e') =
Pn = C(+n, C n!(+n,7) Pn) n.
(15)
On utilisea dans la suite un thkorbme de dualitk obtenu dans [ll]
Theorem 1.1. La transformation de Laplace L e'tablit un isomorphisme topologique
Fi(EE)+ Go* (Ec) 1.4. Reprdsentation intdgmle de distributions positives
Une fonction test cp de Fo(E,*) est dite positive ( et on note cp 10), si pour tout x E E , cp(x io) 1 0 . Une distribution 4 E Fi(E*)est dite positive si pour toute fonction test cp positive de Fo(EE)
+
(4,cp) 1 0 On dksigne dans la suite par Fi(E*)+l'ensemble de toute les distributions positives.
Theorem 1.2. [21] Pour toute distribution 4 E Fi(E,*)+, il existe une unique mesure de Radon p sur l'espace ( E * ,23) muni de sa tribu Borelienne telle que
vf
(4, f ) =
E Fe(E,*)
/
E '
f(x + io)dp(x)
(16)
De plus la transforme'e de Fourier de p est donne'e par:
fi(t)dp(z)
Pour montrer la convergence de cette dernikre intkgrale on utilise la condition (17) d'integrabilitk verifike par la mesure p associke a la distribution positive 4, i.e., 3 q > 0 , m > 0:
en effet
en posant alors
21
on a
e@”)dv(z)= (,cY)(0 etpEN*,
I
e'P'(ml"I-P)dpg(Z)
< co
%--P
D e plus la transforme'e de Fourier F ( p g ) de pg est donne'e par: F'(LLg)(t)
= 35) =
J
ei("9%g(z)
i
v t E Eo
EO,-p
Proof. En effet en utilisant le thdorhme de Bochner-Minlos, voir [14], il existe une unique mesure de probabilite pg sur le dual fort de Eo et qui est associke B la fonction caractdristique S I X , . Plus prbcisement, comme g E G,(Eo)alors
Alors on sait qu'il existe un entier q > p tel que I'injection i, : Eo,q + Eo,p est de type Hilbert-Schmidt et , L L ~ ( E O= , - 1. ~ )Pour montrer la condition d'intkgrabilitb vdrifide par la mesure pg on a besoin d'avoir une estimation et n E N des moments de pg. En effet on a V < E
En utilisant maintenant la formule de Cauchy, alors pour tout
,W)
en posant 9 k = infrBo7 , k E N on a:
T
>0:
23 En utilisant maintenant la formule d'identitk de polarisation et l'indgalitd (23) on obtient
1
lzlzqdpu,(z) 5 (11911q,p,mcpzn(2n)!)112( h m l l i q p l l H S ) n , \ ~ EnN
(25)
oh ~ ~ i P est qla ~ norme ~ ~Hilbert-Schmidt s de l'injection i p q .Pour compldter
la preuve de cette proposition on a pour tout t 2 0 et
E
> 0:
D'autre part on peut montrer facilement que 1
\Jt2 0 , e-q(t)tn < -(Pn
D'oh:
En utilisant maintenant l'estimation (24) des moments de la mesure puset le fait que: &12npn
7
\Jn€N:
on a
comme
d'oh en choissant 44mlliq,pllHS
0 e t p > 0,
I
ev*(mlzl-p)dpg(2)< 00
E O F P
On en d6duit alors que: 3 c > 0, m > 0 et p
> 0 tel que
e,*(mlzl-p)dpg(x) < o o , ~ Et
I(Ptg)(t)l< c / E0r-P
EO
25 et c = c(t, E). Finalement pour montrer que le Processus Pt est Markovien, il suffit de montrer que si g = 1, i.e., g(() = 1, VE, alors pg = So. En appliquant la formule explicite (27) du semi-groupe Ft,on a
d’autre part comme pour tout g dans G$(Eo):
En posant dvg(x):= e-tllzlltdpg(x) , i.e., dv9W = e-t11211:
4%(x) oiI
dP&l(X)
est la d6rivde au sens de Radon-Nikodym. Alors on a: ($‘g)(() =
]
ei(”z[)dv(z)= .T(vg)(E)
E,*
qui est par d6finition une fonction caract6ristique’ et donc le semi-groupe Pt transforme les fonctions definies positives en fonctions definies positives, i.e.:
Pt : G p o )
-+ G,+(Eo)
(28)
D’autre part si P t g = P t h alors 7 v g = Fvh, et comme la transformation de Fourier est injective ceci donne: vg = V h et donc e-tIIzI12dpg (x)= e-tIIzII:dph (z)
d’oh p g = p h et par suite F 1 ( p g ) = T-l(ph), donc g = h. Donc l’application ddfinie par (27) est injective. Montrons qu’elle est aussi surjective. En effet d’aprks la proposition prdcddente, pour tout G de G$ (Eo) alors il existe une mesure p telle que 3 ( p ~ =) G. Cherchons alors s’il existe un antbcedent g (3 G$(Eo)de G , i.e.,
P t g = G equivaut B - T ( ~ G=) Ptg = 3 ( u g ) d’oh p~ =
or
dug- - e-tllz112 - dPG dP9
dP9
26
d’oii puse t donc g par Laplace inverse.
0
Corollary 3.1. En combinant les ope‘rateursPt agissant sur Ees distributions positives et agissant sur les fonctions @(Eo) on a Ee diagramme suivant:
m%)+ -5Wa+
T
7-l
oi F est la transformation de Fourier. D’o% la relation :
References 1. L. Accardi, P. Gibilisco and I. V. Volovich: The Ldvy Laplacian and the Yang-Mills equations, Rendiconti dell’Accademia dei Lincei,( 1993). 2. L. Accardi :Yang Mills equations and LCvy-Laplacians, Dirichlet Forms and Stochastics Processes,(l995), 1-23, Eds.: Ma/Roekner/Yan.Walter de Gruyter and Co., New York . 3. L. Accardi and V. I. Bogachev: The Ornstein-Uhlenbeck process associated with the Lkvy Laplacian, and its Direchlet form, Probability and Math. Statistics , Vo1.17, Fasc.1 (1997), 95-114. 4. L. Accardi and 0. G. Smolyanov: On the Laplacians and traces, Rend. Sem. Mat. Bari, Ottobre 1993, Preprint- Volterra. 5. L. Accardi, P. Roselli and O.G. Smolyanov: The Brownian Motion Generated by the Ldvy-Laplacian, Mat.Zamatki, 54 (1993), 144-148 6. L. Accardi and N. Obata: Derivation Property of the Ldvy-Laplacian, White noise analysis and quantum probability, (N. Obata, ed.) RIMS Kokyuroku 874, Publ. Res. Inst. Math. Sci.(1994), 8-19. 7. L. Accardi and H. Ouerdiane: Fonctionelles analytiques assocides B I’opCrateur de LaplaceLdvy, Preprint No. 477 (2001), Centro Vito Volterra. 8. S. Albeverio, Y. L. Daletsky, Y. G. Kondratiev and L. Streit: Non-Gaussian infinite dimensional analysis. Journal of functional analysis 138 (1996), 311-350. 9. D. M. Chung U. C. Ji. and K. SaitB: Cauchy problems associated with the Ldvy Laplacian on white noise analysis. Infinite Dim. Analysis, Quantum Prob. and Related Topics. Vol. 2, No. l(1999). 10. W. G. Cochran., H-H. Kuo and A. Sengupta: A New Class of White noise Generalized Functions. Infinite Dim. Analysis, Quantum Prob. and Related Topics. Vol.1, No. 1(1998), 43-67. 11. R.Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui: Un thdorhme de dualitd entre espaces de fonctions holomorphes B croissance exponentielle. J. Funct. Anal., Vo1.171, No.1 (2000), 1-14.
,
27
12. R. Gannoun, R. Hachaichi, P. Kree and H. Ouerdiane: Division dc fonctions holomorphes ti croissance 8- exponentielle. Preprint, BiBos NO : 00-01-04. (2000). 13. I. M. Gel’fand and N. Ya. Vilenkin: Generalized Functions, volume 4, Academic Press. New York and London (1964). 14. T. Hida: Brownian Motion, Springer Verlag, Berlin-Heidelberg-New York, 1980. 15. T. Hida, H- H. Kuo, J. Potthoff and L. Streit: White noise, An infinitedimensional calculus, Kluwer Academic Publishers Group,l993. 16. P. Krbe and H. Ouerdiane: Holomorphy and Gaussian analysis. Prkpublication de 1’Institut de mathkmatique de Jussieu. C.N.R.S. Univ. Paris 6 (1995). 17. H- H. Kuo, N. Obata and K. SaitB: Lkvy Laplacian of generalized functions on a nuclear space, 3. Funct. Anal. 94 (1990) 74-92. 18. P. Lbvy: Leqons d’analyse fonctionnelle, Gauthier-Villars, Paris (1922). 19. N. Obata: White noise calculus and Fock space. L. N. Math. 1577 (1994). 20. H. Ouerdiane: Fonctionnelles analytiques avec conditions de croissance et application B l’analyse gaussienne. Japanese Journal of Math. Vol. 20, No. 1, (1994), 187-198 . 21. H. Ouerdiane: Noyaux et symboles d’opkrateurs sur des fonctionnelles analytiques gaussiennes. Japanese Journal of Math. Vol21. No.1. (1995), 223-234 22. H. Ouerdiane: Alghbre nuclkaires et kquations aux dkriv6es partielles stochastiques. Nagoya Math. Journal Vol. 151 (1998), 107-127. 23. H. Ouerdiane :Distributions gaussiennes et applications aux Bquations aux dkrivkes partielles stochastiques in Proc. International conference on Mathematical Physics and stochastics Analysis ( in honor of L. Streit’s 60 th birthday ) S. Albeverio et al.(eds.) World Scientific (2000). 24. H. Ouerdiane and A. Rezgui: Reprksentations integrales de fonctionnelles analytiques. Stochastic Process, Physic and Geometry : New interplays. A volume in honor of S.Albeverio. Canadian Math. Society. Conference Proceedings Series. Vol. 28 (2000), 283-290. 25. K. Sait6 and A. Tsoi: The Lkvy Laplacian as a self-adjoint operator, Proceeding of the first Int. Conf. Quantum Information. World Scientific (1999). 26. L. Schwartz: Radon measures on arbitrary topological spaces and cylindrical measures, Oxford Univ. Pr.,1973.
STOCHASTIC GOLDEN RULE FOR A SYSTEM INTERACTING WITH A FERMI FIELD
L. ACCARDI Centro Vito Volterra Universitd d i Roma "Tor Vergata" Via Columbia, 2, 00133, Roma, Italy E-mail: accardiQvolterra.mat.uniroma2.it R.A. ROSCHIN Steklov Mathematical Institute Russian Academy of Sciences Gubkin St. 8, 117966, GSP-1, Moscow, Russia E-mail: roschin0mi.ras.ru I.V. VOLOVICH Steklov Mathematical Institute Russian Academy of Sciences Gubkin St. 8, 117966, GSP-1, Moscow, Russia E-mail: volovich0mi.ras.m We consider the (causally) normally ordered form of the quantum white noise equation for a Fermi white noise. We find a new form of the normally ordered equation for some class of Hamiltonians and we obtain the inner Langevin equation for such Hamiltonians.
1. Introduction
The stochastic limit approach is a powerful method that allows to study the quantum dynamics of an open system. This method proved to be very efficient in study of various systems, interacting with Bose fie1ds.l In the present paper we concentrate on the Fermi case. In this case the superselection rules restrict the class of fundamental Fermi Hamiltonians. This restriction is discussed in Sec. 2. The main goal of the stochastic limit approach is to study the quantum dynamics of open systems. Such a dynamics is given by an evolution operator U ( t ) ,which satisfies the evolution equation in the interaction rep28
29
resentation:
& U ( t ) = -iAK,,t(t)U(t), U ( 0 ) = 1
.
(1)
Here x,t(t) is the evolution of the interaction Hamiltonian under the free evolution. However, even in the simplest non-trivial cases, there are no exact solutions of Eq. (1). Therefore, we have to use some approximations. One can express the solution of Eq. (1) using the iterated series and Feynman diagrams. The Feynman diagrams are a graphical representation of the combinatorial procedure, which is called "normal ordering". Usually, only a few terms of these series can be explicitly calculated. The stochastic limit approach provides another, more efficient and elegant way of solving Eq. (1).
Consider the evolution of an open quantum system in the rescaled time: In the stochastic limit approach we are interested in the limits of the rescaled evolution operator and of the rescaled interaction Hamiltoniama
t
+ t/A2.
In many physically important cases, one can prove that Ut satisfy the (Bose) quantum white noise equation:
&Ut = -ihtUt, Uo = 1 ,
(3)
where the limit of the interaction Hamiltonian, hi, can be expressed in terms of (Bose) quantum white noises. Although Eq. (3) seems to be very similar to its predecessor, Eq. (l),it can be explicitly solved. Again, we are going to find the normally ordered form of Eq. (3). It is called causal normal order to emphasize the fact that the commutation relations used to bring to normal order the terms of the iterated series, only have a meaning for time ordered products of creation and annihilation operators. The causally normally ordered form of Eq. (3) can be found explicitly. The technique that allows to find and bring to causal normal order Eq. (3) is called the stochastic golden rule. The aim of the present paper is to develop the stochastic golden rule for the cases when the interaction in (1) is of dipole type (cf. (8) below) and driven by a Fermi field. aThe exact sense of these limits is the subject of Statement 1
30
2. Hamiltonian of the model
Let 31 be a Hilbert space of the form
(4)
?I!=?I!S@%!R
where 7-Is is a finite dimensional Hilbert space, and 3 1 is~ an infinite dimensional Hilbert space. Suppose H is a Hamiltonian of the form
H =Hs @ 1
+ 1 @ HR + XVjnt
7
(5)
where X is a positive real parameter. We denote the first two terms of this Hamiltonian by Ha, and refer HO as ”free Hamiltonian”. Then, the pair (31,H) describes an open quantum system. In other words, an open quantum system is a ”small”, discrete spectrum quantum system, interacting with some infinite dimensional environment, or field. The simplest model that gives the Fermi white noise equation in the stochastic limit is a two-level system, interacting with a Fermi field. Let us introduce this model, using the notation of Eqs. (4,5). Let the Denote the elements of the orthonormal system Hilbert space be 31s = basis for this space by ti), i = 1,2. Let the environment Hilbert space be the Fermi (antisymmetric) Fock space
a?.
M
n=O
Denote the Fermi creation and annihilation operators in this space by ul and U k , k E R3,respectively. They satisfy anticommutation relations:b
{al,uk’) = 6 ( k - k ’ ) , { a k , a k f } = o . Let the free Hamiltonian be:
HO = H s 4-H R = E 11)(11 @ 1 + 1 8
s
w(k)aiuk dk .
(7)
Here E is a positive real number, w ( k ) = k2+m2, m 2 0. Let the interaction Hamiltonian be:
the following, {,} denote anticommutator, {x,y} = xy +yx, and [,] denote commutator, [x,y]= zy - yx.
31
where D is a bounded operator, acting on the system space, and g(k) is a smooth complex function with finite support. In the following, we omit the tensor product in our formulae. The superselection rules (see Ref. 2, require that fundamental Hamiltonians should be even in the Fermi operators.c If the Hamiltonian is even, then D should be odd in the Fermi operators. Hence, D and a,at should anticommute.d We will see that this case leads to a stochastic golden rule similar to the one known in the Bose case. Effective (for example, non-relativistic) Hamiltonians may be odd in Fermi operators, in these cases D and a,at may commute. An example of such Hamiltonian was considered in Ref. 4. If D and a,at commute, then we get the new kind of the stochastic golden rule. We will study both cases; first the ”commutative”, then the ”anticommutative” one. As mentioned in the introduction, there are two main components in the stochastic golden rule: the existence of the limit and the causal normal order form of the white noise equation. The first part, the existence of the limit, is quite similar for Bose and Fermi cases. Both cases were studied in Ref. 1. Here we just formulate the result for Fermi case. Statement 1. The limit (2) of the evolution operator in the model defined by Eqs. (6-8), exists in the sense of correlators. Moreover, the limit of the rescaled evolution operator Ut satisfies the white noise equation atUt = -ihtUt, with the white noise Hamiltonian of the form ht = D’bt
(9)
+ bfD,
where bt, bi are Fermi Fock white noise creation and annihilation operators. This means that they are operator valued distributions, acting on the Fock space r = F ( L 2 ( R ) )and satisfying the following relations:
{ bf, b”’} = y-6+(t
- t’) ; {bt, bt’} = 0 .
(11)
where d+(t) is the causal &function (see Ref. 1) and
‘An operator is called even, if it commutes with the parity operator, and odd, if it anticommutes. The parity operator in this case can be defined as in Sec. 3. dAn obvious generalization of Eqs. (4-8) is needed to include this case.
32 The relation between D and b, bt is the same as the relation between D and a,at: they either commute, or anti-commute.
3. The "Commutative" case The causally normally ordered form of the Fermi white noise equation and the Langevin equation in the "commutative" case are the main results of the present paper. Let us formulate these results as a theorem. First of all, let us give some definitions. Consider a monomial of the form:
bt: biz
. ..bi," ,
were, bii denote either bt, or bj. Denote the vacuum state in I? by 90. There exists a unique operator 0 , called parity operator, such that for any monomial
Ob;ibf,2 ...b:,"Qo = (-l)nb;:b;,2.. Note, that O2= 1, hence 0-l = 0 = O*. I f A is an operator, then the map: A automorphism. We will use the notation:
+
A := O A O .
.b;,"@o.
(13)
C3AO-l = OAO is a
*-
(14)
Theorem 2. Keep the notation and the assumptions of Statement 1. Suppose the operator D is such that
[D,bt]= [D,bl] = 0 . A
h
and define Ut U := OUO. Then, (1) The following relations are satisfied:
(15)
33
(2) The causally normally ordered form of (9) is the system
+ Dbf Ut) - y-DtDUt = i (DtUtbt + D b f c t ) - y - D t D c t
&Ut = -i ( D t ctbt
{
@t
(17)
h
with the initial conditions Ut = Ut = 1. (3) Let X be an operator on the system space. Consider the following matrices:
T:= Then,
(Jij
(;:),
S:=
(:_4)
(19)
( X ) )satisfies the following inner Langevin equation:
at ( ~ ( x=)-7-) ( J ( x D ~ D ) -) 7: ( ~ ( o t ~ x ) ) (7- + y ? ) J i i ( D t x D ) (-7-+ 7 T ) J i 2 ( D t X D ) + ((-7: + y - ) J 2 i ( D t X D ) (7- + f ) J 2 2 ( D t X D ) ) + ibfST ( & ( D X ) ) - ibfT ( J & ( X D ) )S
+iS ( A j ( D t X ) )Tbt - i ( , 7 , j ( X D t ) )TSbt . (20) -
(4) The master equation for the partial expectation (U,*XUt), = X is:
&X
= -y-XDtD - y-DtDX
+ 2 (!J?y-) D t X D .
(21)
The proof of Theorem 2 is given in the next section. Remark. The causally normally ordered form of the Fermi white noise This is the new feature of the equation is a pair of equations for U and Fermi case.
c.
4. Proof of Theorem 2. 4.1. Quasi-commutation mles for bt and U t .
Let us express the evolution operator as Ut = lim U,( N ) , N+a, N
Ir
u:N)= z(-i)" dtl f ' n=O
0
0
dt2.. . fn-' dt,
(fi
0
j=1
(Dtbtj + D b l j ) )
34
The time-consecutive principle (see Ref. 2) states that expanding the product btUr according to (22), any term of the form btb;: b;,2 . . . b::
with t 2 tl 2
t2
,
(23)
2 . . . 2 tn7 ~i E { ,t}, is equal to (-l)nb,E:bL,2.. .bE,"bt
+ { b t , b::} b:,Z . ..biz
,
where the anti-commutation relation for the fields bt is given by (11). Using this, we obtain:
'&-i)n
l' l dtl
tl
d t 2 . ..
Ln-'
dt,(-l)n j=1
n=l
(fi
{ bt, bfl}
+ (-i)(-i)n-lD
(D'bt,
j=2
= 6:N)bt - i D y - ~ [ ~ , ~ l ( t ) U .: ~ - ' ) Here X[,,~I is the characteristic function of the given interval [ O , T ] . In the limit N -+ 00 we obtain Eq. (16.1) A
btUt = Utbt - i r - D U t , For btU,* we obtain:
btUjN)*=
?(i)" n=l
IT Jo" Jd'=-' dtl
dtg
. ..
(h
+ (i)(i)n-l
j=n
(Dtbtj
+ Db!,))
D { bt, bf,}
j=n
and in the limit we get Eq. (16.2):
btU,* = @bt
+ iy-U,*D.
Let A , B be some operators. Since 0' = 1, observe that
P ( A B ) = OABO = OAG2B0 = P ( A ) P ( B ) .
35
and
(PA)* = P ( A * ) . l?rom the general formula (13) we get
(25)
Pbf = -bf .
Pbt = -bt, Let us apply P operator to bt6t and have:
bt6,*,
(26) and use Eqs. (16.1,16.2). We
(-Utbt - ir-DUt)
bt6t = -P (btUt) = -P
= - (-U& - iT-DUt) = Utbt + i y - D 6 t . 6
Hence, we get Eq. (16.3):
btUt = Utbt bt6,* = -P (btU,*)= -P
+ iy-D6t.
(6,*bt + ir-U,*D = - (-U,*bt
1
+ ir-U,*D - 1 = U,*bt- i ~ - 6 , * D
Hence, we get Eq. (16.4): A
btU,* = U,*bt - ir-U,*D. Eqs. (16.1*)-(16.4*) are adjoint of Eqs. (16.1)-(16.4). In Appendix A we prove the relations (16) using the integral form of the evolution equation, as done in Ref. 1 for bosons. 4.2. Normally ordered equation f o r Ut
.
Let us rewrite the evolution equation (9),using (16.1).
atUt = -i (Dtbt
+ Dbf) Ut = -i
(Dt6tbt
+ DbfUt) - T-DtDUt.
Eq. (27) is not close because it involves both Ut and operator to it, and using (24-26), we obtain:
(27)
6t.e Applying the P
&6t = i (DiUtbt + DbtGt) - T - D t D 6 t .
(28)
The system of Eqs. (27,28) is closed and causally normally ordered in the sense that all the bf operators are on the left hand side and all the bt are on the right hand side of the Ut system. eThe operators U and 6 are dependent, hence if one substitutes the definition of 6into (27), then the result will be closed. R.R. is grateful to Prof. Y.G.Lu for pointing this out.
36 4.3. Inner Langevin equation.
The inner Langevin equation is a result of a direct computation. The idea of this computation is to apply Eqs. (27,28) to express &Ut, then to use Eq. (16) to find the causally normally ordered form of the terms. Let us compute a t J l l ( X ) ( t ) :
atgll(x)(t) = at (u,*xut) = atu,*xut + u,*xatvt
+ U,*Dtbt) - f U , * D t D ) XUt + U,*X (-i(Dt6tbt + bfDUt)- r-DtDUt) = (i(bi6,*D
= ibf6,*DXUt +iU,*DtXbtUt - f U , * D t D X U t
- iU,*XDtGtbt - iU,*bfXDUt - r-U,*XDtDUt Using
iU,*DtXbtUt = iU,*DtX6tbt
+ r-U,*DtXDUt
and
-iU,*bfXDUt we obtain:
Let us compute
Using
and
-ibfG,*XDUt
1
(X)(t):
+ rZU,*DtXDUt
37
we obtain:
Let uss compute
(X)(t):
Using
and
W e ob tain :
The computation of (X)(t) is simila and therefore omitted. Combining all terms in the matrix equation, we find:
Using the matrices T and S, defined in (19), one can rewrite the last two terms as (in the notation (18))
+
i b f ( S T J ( D X )- T J ( X D ) S ) i ( S , 7 ( D t X ) T - , 7 ( X D t ) T S ) b t . this can be checked directly by matrix multiplication.
(33)
4.4. Canonical form of the Jangevin equation.
We would like to represent the following inner Langevin equation: at
( & ( X ) ) = -7- (Zj(XDtD)) - 7' ( & ( D t D X ) )
+
(-7-+ 7')&2(DtXD) (-7' + r - ) & @ X D ) (7- + 7')&2(DtXD) + ibfST ( s j ( D X ) )- ibfT ( A j ( X D ) )S is ( z j ( D t X ) )Tbt - i ( z j ( X D t ) )TSbt
(y- 7 ' ) J l , ( D t X D )
)
+
(34)
in the canonical form at
(zj(x))=
(Jkl(@iij(x))
+ blJkl(8Lij(x))+ Jkl(8,,j(X))bt)
*
kl=1,2
(35) Let us find structure maps 8O, 8%. The map Zj is linear, hence 8O, 8% can be obtained by rewriting the matrix multiplications in (34) in the index form: ( a i j ) ( b j l )= (Cjaijbjl). Thus, where
and
where
nij
:=
(0 ) , -1
(nij) = T
s = (Sij) = oz = (::1).
S,
39 4.5. The Master equation.
The master equation is the vacuum expectation of the inner Langevin equation (32). The last two terms, containing bt and b i , vanish. It is enough to take vacuum expectation only for one matrix element. The most interesting is A I ( X ) . Denote by X the vacuum expectation (,711(X))0.We obtain:
8tX = - y - X D t D - y - D t D X + 2 8 y - D t X D .
(36)
5. "Anti-commutative"case Theorem 3. Keep the notation and assumptions of Statement 1. Suppose the operator D is such that
{D,bt} = { D , b i } = O .
(37)
Then, the following relation is satisfied
Proof. Let us express the evolution operator as:
Using the time-consecutive principle (see (23)), we obtain: bt U i N ) =
c(-i)" /' dtl [' N
=
n=l
0
dtg . . .
0
rn-'
(fi
dt,(-1)2n
{b,,bil}
+ b i j D ) ) bt
j=1
0
+ (-l)(-i)(-i)n-lD
(Dtbtj
(fi
(Dtblj
+ bl,D
j=2
= U:N)bt Taking the limit N
+ iDy-~[o,~](t)U$~-l) (39)
-+ oi, of Eq. (39) we obtain: bt Ut = Utbt
which is (38). The theorem is proved.
+ iy- DUt
40
The relation (38) between b and U is similar to the Bose case.’ In the case of Bose quantum white noise, the causally normally ordered form of the white noise equation (3) can be obtained using only the relation between Bose white noise and the evolution operator. Using the proof for the Bose case, one can easily prove that replacing Bose by Fermi quantum white noise operators, the causally normally ordered form of the white noise equations is the same.
6. Conclusions We studied the Fermi quantum white noise equations with a dipole-type interaction Hamiltonian (10). In the ”commutative” case we obtain a new form of causally normally ordered white noise equation (17) and of inner Langevin equation (20). In the ”anti-commutative’’ case we found the commutation relation (38). From Eq. (38) we get that in the ”anticommutative” case the causally normally ordered form of a Fermi white noise equations is the same as in the Bose case.
Acknowledgements This research was carried out while one of the authors (R.R.) was visiting at Centro Vito Volterra. R.R and I.V. were partially supported by the RFFI grant 02-01-01084 and the grant 1542.2003.1 for scientific schools.
Appendix A. Proof of the relations (16) using integral equation The evolution operator Ut satisfies the following integral equation
i?t
Ut = 1- i
I’
dt’(Dbi, + Dtbr)Uti .
Gt = 1+ i
1
dt’(Dbi, + Dtbtl)ctj
satisfies
t
Consider the iterated series U ( N )for the solution of the integral equation with the initial condition u(0)=
and relation
1
41
and the same series for 6 with the initial condition 6p)= 1. The limit of the series (which exists under our assumptions) is the solution of the integral equation
We want to prove that for t 2 T h
btUT = UTbt - ~ ~ - D X [ O , T ] ( ~ ) U T .
(-4.1)
Let us prove the following relation for the iterated series (for t
btU$N)= @"bt
-iy-D~[o,~](t)U$~-~).
This equation clearly holds for N = 1. Suppose it holds for all N Let us proof it for N = M :
btU$M)= bt (1 - i
Jd'
2r):
5M
- 1.
dt'(Dbf, + Dtbtt)U,(,M-')
= bt - i
I'
)
+
dt' bt(Dbf, Dtbtt)U{?-"
- ir-DX[o,T](t)UT (M-1)
= G!M)bt
+ 0 - iy-D~[o,~](t)U$~-~) . (A.2)
The second term is equal to 0, because t 2 r. Taking the limit N -+ 00, we obtain (A.1). Substituting t for T in ( A . l ) , we get (16.1). References 1. L. Accardi, Y.G. Lu and 1.V. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, 2003. 2. Bogoliubov, N. N., Logunov, A. A., Oksak, A. I., Todorov, 1. T., General Principles of Quantum Field Theory, Kluwer, Dordrecht 1990 3. R.L. Hudson and K.R. Parthasarathy, Unification of boson and f e n i o n stochastic calculus, Commun. Math. Phys., 104, 457-470 (1986). 4. A.F.Andreev, Mesoseopic superconductivity in superspace, JETP Lett. 68, 673 (1998)
GENERATING FUNCTION METHOD FOR ORTHOGONAL POLYNOMIALS AND JACOBI-SZEGO PARAMETERS
NOBUHIRO ASAI Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan E-mail: asaiQkurims. kyoto-u. ac.jp IZUMI KUBO Graduate School of Environmental Studies Hiroshima Institute of Technology Hiroshima 731-5193, Japan E-mail: kubo Qcc.it- hiros hima. ac.jp HUI-HSIUNG KUO Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA E-mail: kuoOmath.1su. edzl Let p be a probability measure on R with finite moments of all orders. Suppose p is not supported by a finite set of points. Then there exists a unique sequence {Pn(z)}rz0 of orthogonal polynomials such that P,(z) is a polynomial of degree n with leading coefficient 1 and the equality ( 2 - an)P,(z)= P,+i(z) +w,P,-i(z) holds. The numbers {anr u,}?=~ are called the Jacobi-Szego parameters of p . The family {Pn(z),an,u,}:=~ determines the interacting Fock space of p . In this paper we use the concept 'of generating function to give several methods for computing the orthogonal polynomials Pn(z)and the Jacobi-Szego parameters a, and wn. We also describe how to identify the orthogonal polynomials in terms of differential or difference operators.
1. Accardi-Boiejko unitary isomorphism
Let p be a probability measure on R with finite moments of all orders. Assume that p is not supported by a finite set of points and that the linear span of the monomials (2";n 2 0 ) is dense in the complex Hilbert space L 2 ( p ) .Then we can apply the Gram-Schmidt orthogonalization procedure
42
43
to the monomials {1,x,x2,.. . , x n,...}, in this order, to get orthogonal polynomials {Po(x),P1(x),. . . ,P,(x), . . .}. Here P,(x) is a polynomial of degree n with leading coefficient 1. It is well-known (see, e.g., the books by Chihara5 and by Szego7) that these orthogonal polynomials satisfy the recursion formula
(x - a n ) P n ( x ) = pn+l(x) + wnPn-l(x),
n >_ 0 ,
(1)
where a, E R, w, > 0 and by convention wo = 1, P-1 = 0. The numbers a, and w, are called the Jacobi-Szego parameters of p. associated with the measure p by Define a sequence A, = wow1 .-.w,,
n 2 0.
(2)
It can be easily checked that A, = JRIPn(x)12dp(z). Assume that the sequence satisfies the condition that infn>O - A" :/ > 0. Define a complex Hilbert space rp by
with norm
I\ . 1) given by
+
Let @, = (0,. . ., O , l , O , . . .) with 1 in the (n 1)st component. Define the creation, annihilation, and neutral operators a+, a-, and ao acting on F p , respectively, by
a+@, =
a-Gn = wn@,-l,
a'@, = anan, n 2 0 ,
where @.-I = 0 by convention and an's and w,'s are the Jacobi-Szego parameters of p. It can be easily shown that the operators a+ and a- are adjoint t o each other. The Hilbert space F p together with the operators {a+, a-, a o } is called the interacting Fock space associated with the measure p. It has been shown by Accardi and Bozejkol that there exists a unitary isomorphism U : F p + L 2 ( p )satisfying the conditions: (1) U@O = 1, ( 2 ) Ua+U*P, = Pn+1, (3) Ua-U*P, = WnPn-1, (4) U(a+ a- a0)u* = X,
+ +
44
where the polynomials Pn(z)’s are given in Equation (1) and X is the multiplication operator by z. Note that the Hilbert space is determined only by the numbers w,’s, while the numbers an’s and the polynomials P,(z)’s are related to the unitary operator U . It is natural to ask the following question: Question: Given a probability measure p on B!, how to compute the associated orthogonal polynomials and the Jacobi-Szego parameters {Pn, *n, wn}?
In Section 2 we will explain the generating function method to derive the orthogonal polynomials. In Section 3 we will describe two ways for the computation of the Jacobi-Szego parameters. In Section 4 we will discuss the computation of the Orthogonal polynomials by differential and difference operators. In Section 5 we will list some important classical examples from the viewpoint of generating functions. 2. Pre-generating and generating functions
Let p be a probability measure on R satisfying the conditions mentioned in Section 1. In a series of papers2>324we have introduced the generating function method to derive the associated orthogonal polynomials {Pn(z)} and the Jacobi-Szego parameters {an,wn). A pre-generating function is a function cp(t,x) which admits a power series expansion in t as follows:
n=O
where gn(z) is a polynomial of degree n and limsup,,, ~ ~ g n ~ The technique that we have adopted is that of the stochastic limit approach (SLA), whose details are described in the m ~ n o g r a p h .In ~ particular, we will show here that the SLA allows us to compute the same value of the critical temperature T,when we consider the model in Ref. 1 but in a significantly simpler way, and it allows a simple control of T, also for different models, like the ones discussed in Ref. 4. The paper is organized as follows: In the next section we review in some details the model analyzed in Ref. 3. In Section 111, after some physical justification, we discuss the role of a second reservoir in the open BCS model and we show how this reservoir may affect the value of T,,reviewing some of the results contained in Ref. 4. 2. The physicals model and its stochastic limit
Our model consists of two main ingredients, the system, which is described by spin variables, and the reservoir, which is given in terms of bosonic 'This paper is dedicated to my beloved parents, which are always close to me
56
57
operators. It is contained in a box of volume V = L 3 , with N lattice sites. We define, following Ref. 1, 2
where the Pauli matrices satisfy the following commutation rules
bi*
[a;,o-] 3 =&joy,
7
4
- T 2dij0f.
We will use the following realization of these matrices:
If we now define the following bounded operators, -
N
Hps)
can be simply written as = N(ZS& - gRN) and it is easy to check that [S%,RN]= [H P ” , R j v ] = [H$”),S%] = 0, for any given N > 0. It is also worth noticing that the commutators [ S ~ , C $go ] to zero in norm as & when N + 00, for all j , a and p. Our construction of the reservoir follows the same steps as in Ref. 2, but for the commutation rules. In particular, we introduce here as many bosonic modes a3,j as lattice sites are present in V . This means that j = 1,2, ..., N . p’ is the value of the momentum of the j-th boson which, if we impose periodic boundary condition on the wave functions of the bosons, has necessarily the form p’ = %fi, where n’ = (121,122,723) with nj E 2. These operators satisfy the following CCR,
[ap;i,ag,jl = [at. P,%.,at. 923.I = 0, and their free dynamics is given by
[ag,i,at. 4 > .I 3 = dij+g
(4)
N
-a 47r2(4+?4+4) where A N = {p’= %n’, n’ E Z 3 } and €3 = L2m = 2mL2 ‘ The form of the interaction between reservoir and system is assumed to be
N
+
~ $ 1= C ( a j + a j ( f ) h.c.1, j=1
(6)
58
where aj ( f ) = f (p3, f being a given test function which will be asked t o satisfy some extra conditions, see equation (15) below and the related discussion. We would like to stress that, in order to keep the notation simple, we will not use the tensor product symbol along this paper whenever the meaning of the symbols is clear. The finite volume open system is now described by the following Hamiltonian,
H N = H&
+ AH;),
where H L = H P s J
+Hge8)
(7)
and X is the coupling constant. The free evolution of the interaction Hamili H g V " ) t + -iHg'")t i H g e " ) t ( I ) ( t ) = eiHktH$)e-iHkt = EEI ( e tonian, H N e aj e
+
a j(f)e-iHge"'t h.c.), can be easily computed using a semiclassical approximation . Defining
(8)
where w = g,/(So)2 Q = 0 , f, we get
+ 4S+S-,
Y
= 22
+ gSo and va(p3 = v - e,j + o w ,
N j=1 c r = O , f
The next step in the SLA consists in computing the following quantity
and its limit for X going to zero. Here the state wtot is the following product state wtot = wsyswp, where waysis a state of the system, while wp is a KMS state corresponding to an inverse temperature ,b = It is convenient here t o use the so-called canonical representation of thermal state^.^ For that we introduce two sets of mutually commuting bosonic operators {#}, y = a , b, as follows:
&.
a5,j = Jmlj)cgj
+&'j$c~~~+,
(11)
where
mm = 4 a , j , j a ; , j ) = 1-
1 e-pep>
e-pec nm = wp(ap!,ja,j,j)= 1 - e-pep. (12)
59
The operators c $ ! satisfy the following commutation rules rC$,j (a),cLT)t] g,b = 6J.k 6p-q4
(13)
while all the other commutators are trivial. Now, if we introduce the vacuum of the operators c g j , @o, $!Go >J = 0, Vfl E AN, j = 1, ..N, a = a, b, then wg can be represented as a vector state: wp(X,) =< @o,X,@o> for any observable of the reservoir, X,. Finally, if we define fm@J = and fnm = we get
m f m
mfm,
cc N
H$)(t) =
{ph ( c ~ ) ( f m e i t ” -+) c ~ ) ’ ( f n e i t V a )+ ) hx}.
(14)
j=1 a=O,f Finally, if we require that the following integral exists finite:
where f,(jJis fm@J or fn@J and v,@J
is given above, we find that
(16) where the two complex quantities
(17) both exist because of the assumption (15). This suggest to define the following stochastic limit Hamiltonian N
H F ) ( t )=
c {d
(c&)(t)
j=1 a=O,f
+ c a j ( t )) + h.c } ,
(18)
where the operators ch‘j‘(t) are assumed to satisfy the following commutation rule,
[C$)(t),cg)t(tr)l= sj,
6,,6(t
-
t/)rp),
for t
> t’,
since, as it is easily checked, the following quantity
J ( t ) = (-i)2 [ d t l 0
[’ 0
dtzRtot(H~)(tl)H~‘)(t2))
(19)
60
coincides with I ( t ) . Here Otot = wsvsO = wsys < Qo, QO >, where QO is the vacuum of the operators c$)(t): c$)(t)Qo = 0 for all a , j , y and t , cf. Ref. 5 . Following the SLA5 we now use H g ) ( t )to compute the generator of the theory, which is found recalling that Otot(dtjt(X 81I,)) = O t o t ( j t ( L ( X ) ) ) , where j t ( X @ It,) = U j ( X @ Il,)Ut and Ut is the wave operator, which satisfies the following differential equation: at Ut = - i H g ) ( t ) U t . For all self-adjoint observables X we get
L ( X )=
+L2(X),
(20)
where
cf. Ref. 3. As discussed in Ref. 1, 2, we need to find the dynamics of S& and RN to get some insight about the phase structure of the model and, in particular, to compute the value of the critical temperature corresponding to a transition from a normal to a superconducting phase. These intensive operators are both self-adjoint, so that we can use equations (20) and (22). It is now a direct computation to deduce that
where it is necessary to introduce the F - strong topology instead of the uniform t ~ p o l o g y ,and ~ , ~we have defined
while
61
The phase structure of the model is now given by the right-hand sides of Equations (23) and (25),see Ref. 1, 2, and, in particular, from the zeros of the functions
where x = So and y = S+S-. In particular, the existence of a superconducting phase corresponds to the existence of a non trivial zero of fl and f 2 , cf. Ref. 1, 2, that is to a non trivial zero of the function h: h(xO,yo) = 0 with (x,,y,) # (0,O). In order to find such a solution, it is first necessary to obtain an explicit expression for the coefficients Rrp). This is easily done:
lf,rn~~s(~*rn),
Erg) = =
xrf) = r
C IMPI~S(Y*(P~). IXAN
$€AN
(27) It is now almost straightforward to recover the results of Ref. 1,2.Following Buffet and Martin’s original idea, we look for solutions corresponding to Y = 0. This means that x = -2Z/9, v+@j = w - €5, which is zero if and only if w = €5, and v- (p3 = -w - €5, which is never zero. For these reasons we deduce that Rr?) = 0, y = a,b, while the sums in (27) for Rry) are restricted to the smaller set, EN c AN, of those values of $such that, if 4‘ E EN then 6; = w. Therefore, recalling the expression of m(P3 and n(p3 in (12),we find
From Definition (24), finally, we get the following equation
or
which, as we have proven in Ref. 3, is equivalent to the one obtained in Ref. 1, 2, g tanh = w. For this reason, then, we deduce the existence of a critical temperature, T, := &, coinciding with that found by Martin and Buffet, such that, when T < T,, the system is in a superconducting phase. The values of the order parameters also are recovered.
% 0
62
3. Possible generalizations In this section we discuss some possible generalizations of the model which may produce higher values of T,. The idea is very simple and is well emphasized using the SLA: suppose that the free evolution of the annihilation operator of the reservoir, ag,i(t) = ap-, z.e--iept , is replaced, for some reason, which we will investigate later, by ag,i(t) = ag,,ie-iTept, y being some real constant less than one, y < 1. Then we have shown in Ref. 4 that Equation (29) is replaced by following one:
eDw/r = 9 +W 9-W’
which admits a non trivial solution in 10, g [ if gD/y - 2 > 0, that is under a new critical temperature = = which is larger than T, since y is smaller than 1 by construction. Therefore, this very easy mechanism makes the value of the critical temperature to increase. It is worth stressing that a similar conclusion was by no means evident in Ref. 1, 2. The main point, therefore, is to find a mechanism able to change the free evolution of the boson operators, possibly as shown above. For that, a possibility consists in switching on an interaction between the boson reservoir in Ref. 3, which we will call R1 and mother reservoir, Rz,which only communicates with R1 and not with the system S. This will be our point of view: we will discuss now a single model which generalizes the one discussed in Section I1 and the consequences of the presence of this second reservoir on the value of T,. More models are considered in Ref. 4. Let
‘TP)
5,
where H g Y S )is given in (l),H$) in ( 6 ) and
(32) Here both the reservoirs satisfy a bosonic statistic and they are mutually independent:
63
With these definitions it is clear that the free time evolution of H E ) , H g ) ( t ) = e i H g t H g ) e - i H s t ,depends on R2 only through its interaction with R1.We have, using the definitions in (8), e i H ~ t o ~ e - i H= ~ te i H g “ ) t
+e - i H g u ” t -eiutd oj
+ ei(u+w)tpj+ + e i ( u - w ) t
j
P-
>
while
e - i H & t - e i H $ e S ) t a,;ie- i H g e S ) t aP,Z,(t)= eiNa t U 6 , i - e-icpt [a,-,i -
cos(pt) - ib,j,i sin(pt)] ,
as it can be easily derived. If we now introduce the following function:
vapO=v-€$++QIW+Pp, + = O , f ,
P=&,
(34)
we get
+bj(feitva-)
+ h.c),
- bj(feitua+)]
(35) a- . - b - .
Let us now define the operators Ag,j = a5d&bb3j, B s , ~= p ~ l p 3 1 . The only non trivial commutation rules are [ A g , i , A > , . ] = [ B ~ , i , B i = , ~dijdcf, l and we can write H $ ) ( t ) as
which producess, as in the previous section
The only difference is in the coefficients defined as
Here we have introduced
and
which are now
64
and we recall that m ~ = oW A ( A ~ , ~ Am $ ,~~@ ) ,= 3w ~ ( B ~ , ~ B nA(p3 $ , ~=) , W A ( A $ , ~ Aand ~ , ~nB(p3 ) = w ~ ( B $ , ~ B c ,Wj A) . and WB are the KMS-states of respectively the A and the B-operators. The total state is wtot = wsYs @ WA
@WB.
Remark:- For all our results to be meaningful, we have to require that all these integrals exist finite. This gives a condition on f(p3, which extends the analogous one previous given in the previous section, due to the appearance of two different reservoirs. Due to the fact that I ( t ) in (37) coincides formally with the one in (16), it is easy to check that all the previous steps can be repeated and the conclusion is again the same: the system undergoes a phase transition, from a normal to a superconducting phase, if the function h(x,y) defined in analogy with (24) as
has a non trivial zero (20,yo). Find such a zero may be very hard, in general. First we observe that
i
= f &AN = f &AN sry)= ;
c
lf(P3l2(mA@36(Y+-(p3) + mB(p3d(Y++(p3)) lf(P3l2( m A ( p 3 6 ( Y - - ( p 3 ) + mB(p36(Y-+(p3)) > (41) (nA(p36(Y+-(p3) + nB(p36(Y++(p3)) 7
sr- -- 57r Cc,=,nN lf(p3I2 (nA(p36(V--(p3)+ nB(p36(v-+(p3)) (n)
7
which are much more complicated than the expressions in (27). We have discussed in Ref. 4 that choosing Y = 0 does not allow us to find any new result: the value of the critical temperature obtained in Ref. 1, 2 and Ref. 3 is recovered. For this reason we now assume that Y # 0 and, in particular, we look for solutions such that only u+-(PT assume, for some 3, the value 0, while u++(fl,u--(p3 and v+-+(p? are always different from zero. For such a solution t o exists it is enough that the following inequalities are all satisfied: Y
{
+ w + p < 0,
Y -
Y
w-p
< 0,
- w + p < 0,
v+w-p>o.
65
A trivial solution surely exists if we fix Y = p as far as w E ]2p,-2p[. This means, because 0 5 IwI 5 f i g , that the coumust be negative and less than - g g . pling constant p in However, it is not hard to check that with this choice =
%rim)
%r(”) - = 0, while %I$?
= I ~ ( J $ ~ ~ ~ ~ ( J $ ~ ( Y +and - ( J%I??) $) = If(p312n~(p36(~+-(P3). Now, as an easy consequence, we recover the same equation as in the previous section, epAw = g+w which implies 9 --w ?r
that the critical temperature is not affected in this case. More interesting is the situation when the System (42) holds true without having v = p. This is possible: the choice p = -w, v = is an example of this situation. If (42) is satisfied we deduce that
-:
In order t o check whether this equation admits non trivial solutions for some w €10, g[, we consider three different situations: (i) if Y = p then we go back to the usual ~ o n d i t i o n and , ~ to our previous analysis, and we deduce the existence of a critical temperature which coincides with the usual one. (ii) if v > p then the situation is different: since the function F ( w ) := e B A ( w + Y - p ) - @ is such that F ( 0 ) = e P A ( ” - p ) -1 > 0 and lim-w+g-F ( w ) = g--w -00, and since F ( w ) is continuous, then we surely have a solution F(w,-,) = 0 with wg ~]0,g[,for all the values of PA. This suggests the existence of a superconducting phase for all values of the temperature! (iii) if Y < p then F ( 0 ) = e f i A ( V - - I 1 ) - 1 < 0 and we cannot conclude that a non trivial solution of the equation F( w) = 0 does exist even in this case. A deeper investigation need to be carried out in this case but, since, it is not relevant for our present purposes, we will omit it here. The conclusion of this analysis is therefore that, at least for those values for which System (42) holds true (if Y # p), the equation h(z,g) = 0 has always a non trivial solution independently of the temperature. This seems quite promising and we hope to get a deeper understanding of this fact in a close future, since it suggests the possibility of using this model in a first approach to high temperature superconductivity. We also would like t o mention that more models, sharing with the one considered here a sort of control on the critical temperature depending on the value of the parameters, are discussed in Ref. 4. Remark:- As discussed in Ref. 3, the operators of both R1 and R2 are
66
in general unbounded. For this reason a special care is required to make everything rigorous. This can be done using the framework discussed in Ref. 7, but we will not do it here to avoid a useless complication of the procedure. References 1. E. Buffet, P.A. Martin, Dynamics of the Open BCS Model, J. Stat. Phys., 18,NO.6, 585-632, (1978) 2. P.A. Martin, ModBles en MCcanique Statistique des Processus IrrCversibles, Lecture Notes in Physics, 103, Springer-Verlag, Berlin, (1979) 3. F. Bagarello The stochastic limit in the analysis of the open BCS model, J. Phys. A, in press 4. F. Bagarello The role of a second reservoir in the open BCS model, submitted to IDAQP 5. L. Accardi, Y.G. Lu, I. Volovich, Quantum Theory and its Stochastic Limit, Springer (2002) 6. W.Thirring and A.Wehr1, On the Mathematical Structure of the B.C.S.Model, Commun.Math.Phys. 4, 303-314 (1967) 7. F. Bagarello, Applications of Topological *-Algebras of Unbounded Operators, J. Math. Phys., 39,6091-6105, (1998)
MULTIQUANTUM MARKOV SEMIGROUPS, INTERACTING BRANCHING PROCESSES AND NONLINEAR KINETIC EQUATIONS. FINITE DIMENSIONAL CASE.
V.P. BELAVKIN AND C.R. WILLIAMS School of Mathematical Sciences University of Nottingham University Park Nottingham NG?’ 2RD, U.K.
1. Introduction
This paper introduces an algebraic approach to the theory of systems consisting of random numbers of particles of the same type. In this paper we will consider those systems such that each particle is described as a finite dimensional quantum state, with an associated algebra of (finite dimensional) observables. The multi-particle systems are described by a decomposable algebra of observables compatible with the total number operator on Fock space. The equivalence is shown between multi-particle states and complex valued positive definite contractions defined on a unit ball of matrices and analytic within this ball. A similar correspondence is shown for transformations of multi-particle states, by extension to a family of matrix valued functions. We consider semi-groups of operators on an infinite dimensional space, which are represented as an infinite block matrix of transformations between the finite particle states. Such semi-groups are defined by a suitable family of matrix valued analytic functions of the unit ball of single particle observables which we call generating functions for the semi-group. We define a Markovian multi-particle process in a broad sense by introducing conditions on these matrix valued generating functions and determine necessary and sufficient conditions on the infinitesimal generator of these generating functions for the existence of a Markovian multi-particle process. We consider the class of all possible non-interacting systems and a particular class of interacting systems having an overall interaction strength. A family of generating functionals indexed by the positive real line is given 67
68 which describe the evolution of a multi-particle state for non-interacting systems. We introduce a pair of equations called a canonical pair of quantum kinetic equations and give, in the case of interacting systems having an overall interaction strength, a formal asymptotic solution to the family of state generating functionals in the case where the canonical pair have a solution satisfying a mixed boundary value condition, as the interaction strength tends to zero. At least in the case of pairwise interaction it is possible to show that the asymptotic solution tends to an actual solution in such a limit. 2. The algebraic theory of finite dimensional multi-particle
systems Let A be a C*-algebra of finite dimensionality a, which we will call the algebra of observables for a single particle system. For every such algebra there exists a Ic E N and a set
such that A has the structure k
A3A=@Ai i=l
for some collection of matrices Ai E M (nj). We therefore represent A as a *-subalgebra of M ni having the block diagonal form:
&, 0
‘M(n1) 0 * - . 0 M(n2).
0
..
0
, o
0M
(nk)
For each 1 5 i 5 k let {&,k : 1 5 k 5 n f } be an orthonormal basis for M (ni) with respect to the usual matrix inner product:
( A , B )= T r { A * B } For each 1 5 i 5 k and 1 5 k 5 nf let h i & E A be the matrix; k Ei,k
= @6i&,k j=1
(1)
69
and let {Ei : 1 5 i
I a } be the orthonormal basis set for A on M
(with respect to the inner product (1)) given by Uf=:=,{&,k
:1
x:=l 0 5k5 ni
n:}.
k We define by %(’) the Hilbert space @ ( l ) where d(1) = Ci=:=,ni. We define A’ to be the commutant of A, namely the algebra of all matrices in M (EL1 n . which commute with any single particle observable A E A. Specifically A’ is the algebra of diagonal matrices of the form
.)
‘AIInl
0
0 hIn2
.
.. . . ..
0
..
0
.. .
, o
0 L,In,,
where X I , . . .,A,, E C‘ and Inis the identity matrix on M (n). A single particle state is defined as a positive linear functional having norm at most one (we call such a state stable if it has norm one). Note that any linear functional p on A can be represented by a matrix e = : ei E M (ni) in the sense k i= 1
for any A = $f=lAi E A. It immediately follows that p 2 0 if and only if ei 2 0 for every 1 5 i 5 t. Of course the representation e = is not unique amongst all density matrices on M ni) , but it is the unique A valued element in the equivalence classes defined by the relation:
(‘&
el
N
ez iff Tr { A e l } = Tr { A Q ~VA } EA
Z:=l BT A } where { O BT is the transpose of B , which obviously has the same structure as A. Note Let dT be the transposed algebra B E M
ni
:
E
that positivity is invariant under matrix transposition. We define a pairing
( A ,B ) = Tr { A B T } between A and AT, so that any functional p is uniquely associated (via this paring) with an element in AT. We define a norm on dT as;
AT
3Q
I---)
Ilell, = SUP { ( A ,e> : A E A, IlAll I 11
70
where JJAJ1 is the usual operator norm, so that any single particle state is uniquely represented by a positive matrix in AT having trace at most one. We call AT the algebra of single particle states. is The algebra of n particle observables, which we will denote by defined as the symmetrical nthtensor power, defined as the algebra of finite linear combinations of operators;
with X@O = 1 for all X E A. Such an algebra has a representation in the finite dimensional algebra M (EfZl n,)"> with a basis given by the symmetric tensor products:
(
E ~ @ . . . @ E ? :~n -l +...+TI,=n can also be repreHowever, being finite dimensional, the algebra sented as an algebra of block diagonal matrices. Then any A(") E A(") has the form A!") for some Ic (n) E N, Ain) E M (mi"') for some
@z)
mjn) E N and we define d ( n ) = ~ ~ ~ ' r n { " We ) . define to be the Hilbert space of this representation. We denote by A(")' C M (d ( n ) ) the commutant of the algebra represented on C?(n). We define A:) as the transposed algebra to A(").We define n-particle ( states as the positive matrices en E A;) having trace at most one. We use the collection of algebras A(") : n E N to construct, for 6 > 0, (infinite dimensional) algebras .Ct as the Banach spaces of E summable sequences e = [en]:=o with entries en E A$) such that:
n=O
The multi-particle states are defined as those and sub-normalised in the sense that en =
e E ,C1 which are both positive
aand:
M
n=O
M
n=O
If e is normalised, llell = 1, then we say that the system has a random but finite number of particles with a probability distribution given by p , = Tr{en} for the events that the system consists of n particles. Thus a normalised n-particle state en has the physical interpretation of a multiparticle system which contains (with certainty) exactly n particles.
71
The more relaxed condition of sub-normalisation allows for the description of unstable systems having a non-zero probability 1 - llell of existing in a state having an infinite number of particles. The dual to the space C1, denoted by CT, is the space of all multiparticle observables, which are defined as sequences
satisfying the boundedness condition:
For a given e E Lc we can define a generating functional R (similar to those introduced in Ref. 1) on the unit ball Bt = { X E M : 1JXJI5 [} as the absolutely convergent sum:
The generating function R is an analytic function of a variables in The following proposition establishes the converse.
Proposition 2.1. If R : I3c e E Cg such that:
+ C! is analytic
c
Bt.
then there exists a sequence
W
R (X)=
Tr { $x@n}
n=O
Proof. Such maps R can be thought of as analytic functions in a variables. These agree with their Taylor series locally, giving W
nt=l
W
n,=l
where we denote a
i= 1
and:
The set of complex numbers
72
define a linear functional pn on the algebra d(n)via the linear extension of pn
(Epn'@. .@E?"-) = r (721,. . . ,na)
let en E A$) be the implementation of this form in the algebra A:), in the sense:
We note that the sums nl+...+n,z =n
can be written as ( X @ n , ~ ndefining )
R (X)=
C Tr { e:XNn> n=O
where: 00
C
n=O
{en) = R (61)
1:
m
en ( t )=
m=l k l +
...+k,=n
( t )@ * * * @ @k,
(@kl
( t ) )e m
Such a system has a clear interpretation as describing a multi-particle system whose evolution consists of independent branching. We now show that evolutions of the type typified by the above example are the only Markovian multi-particle processes n(t): t 0 which correspond t o single particle independent branching processes. Due to the condition that the multi-particle states have n particle components in the symmetric algebra A(") it must be that any such process is described at time t 2 0 by a multi-particle state e ( t ) of the type eo ( t )= eo
>
m
m=l kl+
where @ k ( t ):
...+k , = n It follows that
-$
c
Ilk)( t )=
@k1
Ilk)= 6r and that;
( t )@ * * ' @ @ k , ( t )
kl+...+k,=m
and P("-)(t,X) = T ( t ,X)@'". By taking n = 1 we see that T ( t ,X ) is positive definite, and that T ( t ,I ) 5 I for t > 0 from which follows T ( t ,X) E B1 for all X E B 1 . The requirement that ( t )be a semi-group means that for all n E Z+ we have;
IIk)
m
n
.
=m k=O k l + . .+k, =k j=1
MI
from which it follows (by taking n = 1) that
and hence that:
T (t
+ T , X ) = T ( t ,T
(T, X
))
90 By considering the case n = 1 we see that ;di i T ( t , X ) l = F ( l ) ( X ) t=o
which is necessarily conditionally positive definite and satisfies F ( l )(X)*= F ( l ) (X*) hence there exist A , (i) and B such that:
c
00
d(m)d(l)
m=O
i=l
F(l)( X ) =
A , (i)* X B m A , (i) - X B - B * X
Hence we see that; d 1 -T (t,X) = lim - (T (st,T (t,X)) - T (0,T (t,X))) dt 6t-0 6t =
C C
A , (i)* T ( t , X ) B ' m A m ( i )
m=O i=l -T (t,X ) B
- B*T(t,X
)
showing that the above example does indeed characterise all Markovian multi-particle systems of independent branching.
Example 5.3. Let G ( X ) : B1 + C and N ( X ) : B1 + d(l)be analytic in the unit ball 23' and positive definite. Let c E I&.+ be such that G ( I ) 5 c , let B E be such that H ( I ) 5 B B*. Let Qr (t,X) : B1 + be analytic throughout B1 and a solution to the non-linear inverse Heisenberg Kolmogorov equation defined by d -Qr ( t , X ) = Qr ( t J ) B + BQr ( t , X ) dr
+
m=O
i=l
for 0 5 r < t with boundary condition Qt ( t , X ) = X and where the matrices A , (i) are determined by the dilation of H in Proposition 3.3. As before we define T (t,X ) to be the analytic map satisfying T (t - r ,X ) = Qr (t,X ) , which necessarily has generator H ( X ) XB B * X and satisfies T ( t ,X) 5 I for all X E 23'. Define V (t,X ) : B1 + U2 as the exponent:
+
+
)
v ( t , X )= e x p ( l G ( T ( s , X ) ) d s - c t Now consider the family of analytic maps P('l)(t,X ) : B1 -+ d('l)defined by:
P(")(t,X ) = T (t,X)@'lV (t,X )
91
These have unbounded generator F(n) ( X ) given by n
F(n)(X) = G (X) - cX
+ C X @ j - l @ ( H (X)- X B - B * X )@ X@("-j) j=1
which are obviously conditionally positive definite and satisfy F ( n )( I ) 5 0 , F(") (X)*= F(n)(X*).Hence P(n)( t , X ) describe a Markovian multiparticle process II (t)which satisfies m.
k=O k l + . . . + k , = k
(t)]EOare
defined by T ( t , X ) using Proposition 3.1 and [Wk This process defines an evolution of the.state @ ( t E) L1 of a multi-particle system having initial state e E Ll and generating functional R ( t ,X ) via the equation
where
[@k
@)IF0.
from which it follows that;
which is well defined as the form generated by the trace of products of the matrix c a n
(in the sense that the functionals coincide on A(")). Such a system has a clear interpretation as describing a multi-particle system whose evolution consists of independent branching and birth from the vacuum. We now consider all the Markovian multi-particle processes II ( t ): t 2 0 which are non-interacting. Due to the condition that the multi-particle states have n particle components in the symmetric algebra A(") it must be that any such process is described at time t 2. 0 by a multi-particle state ~ ( tof)the type
m=O k=O k l + . . .+k,=k
92
[&]Eo
where O k : M(") -+ M ( l ) ,e = E .C1 is some initial particle space and 'uk ( t )describes the state of the birthed particles. It follows that; m k=O k i + . . . + k , = k
and therefore P(") (t,X ) = T (t,X)@n' V (t,X ) . By taking n = 0 we see that V ( t ,X ) is positive definite and satisfies P (t,I) I for t > 0. It then follows that T ( t , I ) I (since otherwise P(") @,I) > I for n 2 N E N). The requirement that IIk) ( t )be a semi-group means that for n = 0 we have;
0.
Example 6.1. Consider the generator described by the family F(n)( X ) = 0 for 0 5 n 5 1 and for n > 1
where
G ( X )=
w
4m)W
m=O
i=l
C
c
for some Am (i) : 3t(2)
A , (i)* XmmAm(i) - X @ 2 K- K * X B 2
X ( m ) and B
cc 03
E A(2)satisfying:
am+'
A , (i)* A , (i) IK
+ K*
m=O i=l
jFrom Theorem 4.1 such a family determines a Markovian multi-particle process. The equation of motion for the associated state generating functional RE( t ,X ) must satisfy, at least for positive X
where G ( X ) =
xiXiGi ( X ) @ ' . We write this in the form:
95
Let us assume that the initial state of the system is described by a state generating functional of the form
{: 1 {: 1
R, ( 0 , X ) = A ( X )exp - B ( X ) and look for solutions of the form
RE(t,X ) = A ( t ,X ) exp - B ( t ,X )
where B ( X ) is a conditionally positive definite dissipative analytic functional in B1 and A ( X ) is a positive definite contractive analytic functional in Bl. Under these assumptions the equation of motion for the state generating functional can be defined as the solution to;
equating powers of
E
we have:
Hence up to a term of order
E
we find that the state generating function is
96
determined by the solution to the equations
d -B d t ( t 7 X )= 2 ( G ( X ), (A8 (t,x))@ 2 ) with the initial conditions A ( 0 , X ) = A ( X ) , fi ( t , X ) = B ( X ) . These equations are recognised as the transport equation and Hamilton- Jacobi equation for a Hamiltonian system with Hamiltonian 1 2
H ( X ,P ) = -- (G ( X ),P@') where:
a
P ( t , X ) = -B dX
(t,X)
We now assume the existence of solutions Q t , Pt to the Hamiltonian system
satisfying the boundary conditions QO = Y and Po = &B ( Y ) for some Y E B1. Equivalently we assume Q,, P, are the solutions to the equations
97
due to the fact that, for example, for every
eE
We make the further assumption that not only is the Hamiltonian system solvable, but for each fixed t E I&+. and X E 23l there is a unique Y E 23l satisfying X = Qt. In finite dimensions this corresponds to the Jacobian
not vanishing. Due to this assumption we can write the Hamilton Jacobi equation in the form;
ddtB
(t,X )= --B (t,Q t ) dt .. /Qo=Y(t.X) =
((
&t,
a
a h (t,Q t ) Q.=Y(t,X)
the second equality being the explicit form of the total derivative &B (t,Qt). Then:
Likewise for
a (t,X) we have
d d - A (CX) = -A (t,Q t ) / dt dt Q,=Y(t,X)
98
Hence
and we have a solution:
Then subject to the stated assumptions being satisfied, there exists an approximate solution to the family of state generating functional R, (t,X) which describe the state of a multi-particle system undergoing independent pairwise interaction which can be written in the form:
ii, ( t , ~=)A ( t , x ) e x p We now investigate how closely RE(t,X ) approximates the exact solution R, (t,X ) . An exact solution of the form
R, (t,X ) = A (t,X ) exp must satisfy the equation
:
whereas our
=LA(t,X)
f& ( t ,X ) satisfies:
99
We use the Du Hammel principle to write
which can be written in the operator form
-@) 2
A ( X )2 ( t , X ) = ( A ( X ) - E where
@
A (t,X)
is the operation defined by:
Then, provided the right hand side converges, we can write;
(
A ( t , X ) = 1---@) iA;X)
-1
A(t,X)
the right hand side is given by
from which follows the upper bound
showing that the asymptotic solution does indeed converge to the true solution as E + 0. The above example illustrates the method used in the proof of the following theorem
Definition 6.2. We call the following Hamiltonian system a canonical pair
100
of quantum kinetic equations, Qt = X E B1,PO= Y E M ( l )
a ( K ( ~ ) * , ( P ~ Q T )p+ @~)) pr+ca(dPQT T
O0
m=O
Theorem 6.1. If the family
Fin): B1 + M(")describes the generator of
a Markovian multi-particle system with interaction strength E > 0 , and the initial state of the system is described by a generating function of the f o r m R C
c
1
( 0 , X ) = A ( X )e ~ p-B ( X )
where B ( X ) is a conditionally positive definite dissipative analytic functional in B1 and A ( X ) i s a positive definite contractive analytic functional in B', then there exists a solution RE( t , X ) of the state generating function at any t > 0 for which the canonical pair of quantum kinetic equations have solutions QT ( t , X ) : Qt = X , PT ( t , X ) : PO= &B (Y)ly=q,(t,x) f o r any X E B1.These solutions are given formally by the expressions R € ( t , X ) = e x p { f B ( t , X ) } (l--@)1 A (XI where
-1
A(t,X)
101
and 0 is the operation defined by:
Proof. The proof is a generalisation of the method illustrated by the previous example. We look for a solution to the equation of motion
-RE d ( t , X )= 1 dt
&
25(
( t , X ) ,G(") ( X ) )
&RE
m=O
satisfying the required initial condition, by looking for solutions of the form:
{: 1
R E ( t , X )= A ( t , X ) e x p - B ( t , X ) We find that under such an assumption
and throw away the higher coefficients of
and:
E,
looking for solutions to
102
We again associate the above equations with the Hamilton-Jacobi and transport equation for a Hamiltonian system with Hamiltonian
H ( X ,P ) = -
c m! * 1
( G ( m )( X ) ,Porn) m=O with Pt = &B (t,X ) satisfying the boundary conditions A (0, X ) = A ( X ) and B ( 0 , X ) = B ( X ) . The resulting Hamiltonian system can then be written in the form
from which it follows that
gdt B (t,X ) = dt
(t,Qt (t,X ) )
= (Qt(t,W,
a
( t , Q t ( t , X ) ) )- H (Qt ( t , X ),Pt ( t , X ) )
the second equality being the explicit form of the total derivative $ B (t,Qt). Then:
B (t,X ) = B (Qo (t,X I )
Likewise for
( t , X ) we have
103
where the right hand side takes the form:
In this case we know that Qt(t,X ) = and so we have
hence
and therefore:
To complete the proof we note that
c:=, 5 &(G(m)(Qt(t,X ) ) ,P R m )
104
whereas an exact solution for RE(t,X ) of the assumed type must satisfy
the result then follows from the Du Hammel principle as described in the previous example. 0
References 1. V. P. Belavkin, in: Mathematical Models of Statistical Physics [in Russian], Izdat. TGU, Tyumen', 1982: Dokl. Akad. Nauk SSSR 293, 18 (1987) 2. W. F. Stinespring, Proc. Am. Math. SOC. 6, 211 (1995) 3. G. Lindblad, Comm. Math. Phys. 48, 119 (1976) 4. J. T. Lewis and D. E. Evans, Dilations of Irreversible Evolutions in Algebraic Quantum Theories, Comm. Dublin Inst. for Adv. Studies, Ser. A, Vol. 24, 1997
A NOTE O N VACUUM-ADAPTED SEMIMARTINGALES AND MONOTONE INDEPENDENCE
ALEXANDER C. R. BELTON Lady Margaret Hall, Oxford OX2 6QA, United Kingdom E-mail: beltonamaths. ox.ac.uk A class of vacuum-adapted regular quantum semimartingales, with integrands which act as the conditional expectation on Fock space, are proved to possess increments which are monotone independent. The vacuum-adapted analogue of the Poisson process is shown to have increments which are distributed according to the monotonic law of small numbers.
1. Introduction
The vacuum-adapted analogue of Brownian motion (i.e., the solution to the quantum stochastic differential equation Z(0) = E(O), d Z = EdA + E d A f ) has increments which are distributed (in the vacuum state) according to the arcsine law [2, Section 4.41 (the result therein for Z ( t ) generalizes simply t o Z ( t ) - Z(s) for all t 3 s 2 0). The monotone Brownian motion of Muraki also has increments which are distributed according to the arcsine law [5, Section 4.21 and this observation suggests a connexion between monotone and vacuum-adapted processes. In Section 2 it is proved that if M is a regular vacuum-adapted quantum semimartingale, with integrands which act as the conditional expectation on Fock space, then M satisfies the identities
( M ( t )- M(s))’E(t) = ( M ( t )- M(s))’ = E ( t ) ( M ( t )- M(s))’ (an immediate consequence of vacuum-adaptedness) and
IE(s)(M(t)- M ( S ) ) ’ I E ( S ) = (a,( M ( t )- M ( s ) ) ’ W ( s )
0, we denote by E x p ( N P , 8 , m )the space of entire functions f on the complex Hilbert space N p such that
Then the spaces .Fe(N’) and Gs(N) are represented as
3e(N’) = p o j lim Exp(N-,, 8, m ) PEN m>O
Ge(N) = ind lim Exp(N,, 0, m). PEN m>O
The space .Fe(N’) and its dual F;(N’) equipped with the strong topology are called the test functions space and the distributions space respectively. We denote by > the dual pairing between .Fh(N’) and .Fe(N’). It is easy to see that for every 5 E N , the exponential function eg : z I--) e(’*g) , z E N’ where ( , ) denote the dual pairing between N’ and N , belongs to the test space .Fe(N’) for any Young function 0.Then we define the Laplace transform of a distribution q5 E Fh(N’) by
L ($ )(J ):=< 4,eF
>> , J E N .
In Ref. 3, the authors prove the important duality theorem: the Laplace transform realizes a topological isomorphism of .Fh(N’) on (N). In this paper we establish the analytic characterization theorem of continuous linear operators from Fe(N’) into .Fi(N’) in terms of their symbols, and we give a criterion for the convergence of sequence of operators. Finally, as an application, we solve some quantum stochastic differential equations and characterize their solutions. We refer to Ref. 11 in the particular case where B(x) = x k .
117
2. Characterization theorems
2.1. The Operator symbol
In this section we characterize the space of continuous linear operators from .F,e(N’)into .FL(N’) using spaces of holomorphic functions with exponential growth. Let Ce = C(.Fe(N’),.F,L(N’))be the space of .F;(N’)-valued continuous linear operators on .Fe(N’). For p E lN and m > 0 we put
(((EII(e,p,m = SUP{(+C EYJ,$J
1 ; ( I q ( ( e , p , mI 1 and ll$lle,p,m I 1) , E
E
Leo
Then Ce,p,m = { E E Ce , IIIEllle,p,m< +m} becomes a Banach space with norm Ill.llle,p,m and we obtain
In the general t h e ~ r y , * J ~ ~ if ’we ~ ~take ’ ~ two nuclear F’rechet spaces X and V then the canonical correspondence E t)E K given by
( E u , v ) = ( E K , u @ v ) ,u
E
X,VE V ,
yields a topological isomorphism between the spaces C ( X , V ’ ) and (XBD)’. In particular if we take X = 2) = .Fe(N‘) which is a nuclear F’rechet space, then we get
C(.Fe(N’),.F,L(N’)) (.Fe(N’)8 .Fe(N’))‘.
(2)
We define the symbol of an operator E E Ce denoted by ,!? as follows
E(t,r))= 0,m > 0 and p E JV such that
IF((, r ] ) l 5 c e"(ml~lp)+e'(ml~lp) ,
t,r] E N .
Remark This theorem is a generalization of the characterization theorem of operators given in Refs. 10, 11, 12 in the particular case 6(z) = xk and given in Ref. 1 using C.K.S spaces. 2.2. Convergence of operators
Recall that ( N ) @ 0s. ( N ) equipped with the r-topology is a nuclear F'rechet space, one of the properties of nuclear spaces is that the closed bounded sets are compacts15. Now we describe the convergence in 6s. ( N ) @ Be* ( N ) .
Theorem 2.2. Let ( f n ) be a sequence in G p ( N ) €3 G,g*(N). Then the following assertions are equivalents: (1) (f n ) converges in G e e ( N ) @ Go* ( N ) . ( N ) and is pointwise convergent. (2) (f n ) is bounded in 60. ( N ) @ (3) ( f n ) is bounded in G,g*(N)8 G p ( N ) and it has a unique cluster point in Be* ( N ) @ G,p ( N ) .
Proof ( N ) i.e. there exist p 2 0 Assume that (f n ) converges to f in ( N )@ and m > 0 such that limn++oo 11 f n - f II,g*,p,m = 0. In particular (f n ) is bounded in Ge*(N)@ Ge*(N)and is pointwise convergent. This proves 1) 2). Assume that ( f n ) has two cluster points f and g i e . , there exists two subsequences (f n l ) and (f n 2 ) of (f n ) which converge in Be* ( N ) 8 0s. ( N ) to f and g respectively. In particular the subsequence (f n l ) ( r e s p .f n 2 ) are pointwise convergent t o f (resp.g). Hence the sequence (f n ) does not pointwise convergent which is not true by assumption. This proves 2) 3). Finally, P ut S = {fn, n 2 0 } , S is a bounded set in ( N ) @ Go* ( N ) and ( N ) becomes a compact by consequence the closure 3 of S in ( N )@ set which contains the sequence ( f n ) . It is known15 that if a sequence has a unique cluster point in a compact set, then it converges. This completes the proof. w
*
+
As a consequence of theorem 2.2 we characterize the convergence of operators by the convergence of its symbols.
119
Corollary 2.1. Let (E,),>o be a sequence in Lo. Then ( E n ) converges in Lo if and only if ( 0 1 ) there exist p 2 0 , m > 0 and c 2 0 such that for every n E N Ign(J,q)I 5 c ~ z P ( ~ * ( ~+ oI *[ (Im~~ )q ~ p >E,V ), E N ; (02) f o r every
[, q
E N , limn++, &([, q) exists in C.
This result was proved in Ref- 11 with 8(x) = xk , x 2 0 , k ~ ] l , 2 ] . Remark A similar analytic characterization theorem for operator symbols and convergence of operators can be established if we replace the space Lo by the space Lo,,,, of continuous linear operators from Fe(N’) into FG(N’),where 8 and cp are two Young functions on R+. 3. The Convolution product of operators
If the Young function 8 satisfies limZ.++,
< +co,we get3
Fe(N’) L) L ~ ( x ’ ,L) ~ )F ~ ( N ’ ) ,
(5)
where y is the standard gaussian measure on X’. Consequently, to include the space L(Fe(N’),Fe(N’)) into Le we assume from now on that 8 satis< +co. In this case the function ( J , q ) t--) e-(tsq), J , q E fies lim+, N belongs to Bo*(N)8 Be*(N) N Go+,s*(Nx N ) see Ref. 14. Since Be*(N)8 Ge*(N)equipped with the pointwise multiplication is an algebra, theorem 2.1 shows that for every E1,E2 E Lo there exists E E Lo uniquely determined by g ( [ , q )= e - ~ ( t + q l € + q ) g l ( J , q ) ~ ~ ( sJ ,,qq )E, N .
(6)
The operator E defined in (6) is denoted by El * E2 and is called the convolution product of El and E2. We denote by E*, = E * E * ... * E , n times. Theorem 3.1. Let y be a Young function o n R+ which does not necessarily = +co and let f ( z ) = CnEN fnzn E E z p ( C , y , m ) f o r satisfy lim+, fnE*n E some m > 0. Then V E E Lo, the operator f * ( E ) := CnEN L(roee* )* .
Proof It is easy t o see that f ( k )E BToee*( N ) @ Groee* ( N ) . Then we conclude by theorem 2.1 that f * ( E )E C(,,,e*)*.
120
Corollary 3.1. Let E E Lo, t h e n the convolution exponential of the operator E defined by e*E := C $E*n as a n element of E L(,P)..
Remark 1) If an operator E E Lo is of degree k E N, k 2 2, i.e. e($,q) is a polynomial in t and q of degree k, then e*E E L, with cp(x) = x”-’. This result was proved in Ref. 11 with Wick product. 2) In infinite dimensional complex analysis2, a convolution operator on 3o(N’) is an operator which commutes with translation operators. Let x E N’, we define the translation operator T-, on To(”) by 7-xcp(~ =) cp(x
+Y) , 9 E N’ , cp E -7e(N’).
It was proved in Ref. 4 that T is a convolution operator on Te(N’) if and only if there exists 4 E FA(”) such that
T(cp) =
4 * cp >
vcp E To(”).
(7)
Proposition 3.1. Let E E Lo, we denote by E+ its adjoint operator, then the following assertions are equivalents: ( I ) T * E = T E , V T E L(3;,3;). (2) E+ * S = E+S , V S E L(T,9,3e). (3) E is a convolution operator. Proof The equivalent between 1) and 2) is obvious by duality. 1) j 3) For x E N’ we denote by T, the translation operator on 3,9(N’). Its symbol satisfies
?,(t,q) = a T,e> = e(xic) ec,e, >>
- e(X>c)&( c + , , e + r ) ) . For every x E N’ we have
( E * T,)(t,q ) = e - ~ ( ~ + ~ ~ e + q ) ~ x ( t=, eq( )x i~e () E~( ,t ,qq ) ,
(8)
and
( E T x ) ( tq, ) == e(,%c)>=e(xit),!?([,q ) .
(8) and (9) imply that
E * T, = ET,, V x
E N’.
(9)
121
On the other hand we know by hypothesis that
E*Tx=TxE,Q x E N ‘ . Then for every x E N‘ we get ET, = T,E, i.e. E is a convolution operator. 3) 2) Let E be a convolution operator, then by (7) there exists @ E FL(N’) such that E(cp) = @ * cp, V cp E .FO(N’), and we get
+
Z+( = > = 3(q)>=3(q)ei(c+qzc+,).
Thus
-
(E+T)( = >=a Tee,@* e , >> = $(q) >=$(q)?(t+,) = e-+(c+,L+,)S( in particular it is bounded in 0 and Ct > 0 such that for every 5,q E N p we have
(t,q)I 5 Ct ee’
(mlclp)+e’
(mlqlp)
,v s E [o, t].
In particular I&(o - of bimodule M-unitary maps satisfying the obvious compatibility conditions, namely that the following diagram commutes:
Es O Et AsaAtl
EL 0El
-
Es+t
E:+,
By M-isometric, (respectively M-unitary), for a linear map C : FI + F2 between Hilbert W*-modules over M we mean a module map (resp. isomorphism) satisfying
(equivalently, by polarisation, (CE,Cq)= (E, q)), similarly we shall say that an element is M-normalised if l l t j l ~= 1 ~ . A unit of a product system is a family (wt)t>o such that
O, where Fk,l denotes the symmetric Fock space (or exponential Hilbert space, in another parlance) over L 2 ( I ;k) for a Hilbert space k and subinterval I of !&, with Fk,[O,s+t[
F k , [ O , s [ @ Fk,[s,s+t[
Fk,[O,s[
@ Fk,[O,t[
(1)
giving the requisite identifications. For such systems multiples of certain exponential vectors:
provide suficiently many units in the sense that the following collection of vectors is total in F k := &$+ ([Arl]): & ( d l l [ t o , t l [ ) ~ . . . ~ ~ ( d_l,t,[) ,l[t,
nEN,di,...& E k,O=to 5 . . * i t , .
130
Analogously, an exponentid system of Hilbert W*-bimodules over M , is based on a Hilbert W*-bimodule F over M , and is the family (FF,[o,t[)t>o defined as follows ([BhS]). For each subinterval 1 of R+ and n E Z+define
rI := { a c I : #a < m>and rp) :=
cI:
=n>
so that for n 2 1 there is a natural bijection
rp)
+)
A?) := {s E P
:s1
.
Write I'for rl[p+Lebesgue . measure on A p ) thereby induces a measure on I??) and so, by stipulating that 0 is an atom of unit measure, a measure on Un20 I?) = rI. This is the Guichardet measure (called symmetric measure in [Gui]), integration with respect to which is conveniently denoted J . . . do. Now
f i , I:=
FQn@FJn), where FYI := L 2 ( r p ) ) ,
(3)
n20
with the convention Fo0 := M , may be viewed as a subspace of
and thereby inherits pre-Hilbert W*-bimodule structure from the interior tensor products:
.1cI . a2 := Ul$(
$9
:=
/- ($W>
$'(a))do. rr The product structure may be described in two ways, corresponding to the two views (3) and (4). Thus for cp E G,[o,a,[ and $ E PF,[,,,[, cp 0$ should be identified with the element of PF,[o,s+t[ defined by a1
0
*
)a2,
($3
++ cp( (a - t ) fLO, l4)@ $(a f l [O, tr),
where a - t := { s - t : s E a}. Or, in picture (3), for simple tensors x @ f E FonC&;![ and 71 €3 g E. FQm€3 F{$, the product element (x@ f ) 0( q @g) is identified with
(x@ 7)€a ( S t f @ 9 )
r[ti+,r
where St is the second quantised shift: (Stf)(a) = f(a- t ) for a E Notice that time is running to the left. (FF,[o,t[)t>ois a product system Letting FFJ be the completion of PF,I, of Hilbert W*-bimodules, all contained in the Hilbert W*-module FF := FF,w+.We call it the exponential system based o n F ; in [BhS] it is called
131
the time-ordered Fock module - to distinguish it from earlier experiments in module analogues of symmetric Fock space. The basic unit of this system the collection of all units of such a system which is ( d @ ( . ) lE~ Fp,[O,t[)t,O; are norm continuous (& maps JR++ FF)is indexed by M x F as follows. For each continuous unit w there is lc E M and q E F such that
where pt = etk ([LiS]), in clear generalisation of the situation with exponential systems of Hilbert spaces - see (2). By an E-semigroup on a von Neumann algebra ([Arz]) is meant a pointwise weak*-continuous semigroup of normal *-homomorphisms of the algebra (older terminology: eo-semigroup). If each map is also unital then it is called an Eo-semigroup. Following Skeide, who treated the unital (Eo) case ([Skz]), we next describe the product system of an E-semigroup 4 = ( $ t ) t > o on Ba(E) where E is a Hilbert W*-module having an M-normalised eLment:
which we fix. Set
Q = K ( ~ M I [)=( ) < I where
K
:M
: 77
e o , p : M + B(k)@ M is a *-homomorphism and S : M 3 Ik) @ M is a pderivation, with St defined by St(a) = S(a*)*. (Lk
135
In passing we mention that these stochastic dilations fit into the general picture described earlier, in the following sense. Lemma 2.2. The element [ = 1~ ‘8 I n k ) of the Hilbert W*-module E := M ‘8 I F k ) satisfies assumption (7) above, for the E-semigroup (Jt := 5 o fJt)t 2.0 : Jt(1-
Q) 5 I -Q
for Q = IE)(tI E Consider stochastic generators of the form
for a normal *-homomorphism p : M -+ B(k) €3 M . In case k = C it is well-known ([Hud]) that the stochastic cocycle generated by (13) has the simple form j t = pNt , in the sense that (jt(a)f)(fJ)= P#(‘”[o’tl)(4
(m),
for a E M and f E Ij @ F = L 2 ( r ;Ij). The cocycle continues to have a simple explicit form in the case of multidimensional k as the following result confirms. Proposition 2.3. Let j be the quantum stochastic cocycle o n M with generator (13). Then, under the identifications
b ‘8F
k
= @ (kBn €3
b)
‘8 F:;j[ ‘8 Fk,[t,m[,
n2.0
j is given by
.’?,(a)= @ pn(a) ‘8 I;;;[ €3 I k , [ t , o o [ n>O
where pn : M
+ B(kBn)€3 M
is defined by
with p ( i ) := idB(kB(i-1))@ p
2 1.
This proposition is key; its proof is a simple verification.
136 3. Product system of a stochastic cocycle
The strategy here is to view the stochastic cocycle as a perturbation (in the sense of [EvH] and [GLW]) of the stochastic cocycle generated by the number/exchange component of the cocycle generator, and thereby reduce the problem to pure number/exchange cocycles, by appealing to Theorem 1.l. In turn the explicit form of pure number/exchange cocycles given in the previous section leads us to the product system of such stochastic cocycles. Given a normal *-homomorphism p : M + B(k ) @ M , define a Hilbert W*-bimodule as follows:
Fk>j':= p ( l ) ( l k ) @ M) with a . p ( l ) q . b := p(a)qb, and standard M-valued inner product Now we need two Lemmas.
(t,q )
I+
(15)
("q.
Lemma 3.1. Let F = Fk,p, for a normal *-homomorphism p : M + B(k) @ M . In the notation (14) define Hilbert W*-bimodules FO := M and, for n 2 1,
Fn := p n ( 1 ) ( ( k @ n )@ M ) With u * pn(l)q . b := pn(a)qb. Then (a) The correspondence p n ( 1 ) (1x1 8.. . @ X n ) @ l M )
* p ( 1 ) ( 1 X 1 ) @ 1 M )0.. * @ p ( l )( I Z n ) @ l M )
extends to an M-unitary bimodule map Fn + Fan. (b) Under the resulting identifications, pn(1))(I.
x * Pn+m(l)(lX)@ pm(a)X)
Q a) 0
for x E kBn, a E M and
(16)
x E Fam = F,.
Lemma 3.2. Let k , p and (Fn)>o be as above. Then, with the identifications there, the product structure of the exponential product system for F = F k i pis determined by ( P n ( l ) ( l X ) @ a )@ f ) @ ( P m ( l ) ( I d@ b ) 8
9)
*
for x E kBn, y E kBm, a, b E M , f E F{$[ and g E F[$, where (St)t>o - is the semigroup of right shifts on F.
137
Using these we are able to identify the product systems of quantum stochastic E-semigroups. Proposition 3.3. Let j be a quantum stochastic cocycle on M with generator of the f o r m (13). Then the product system of its associated E-semigroup is (isomorphic t o ) the exponential system for the Halbert W*-bimodule F k @ .
Now consider a *-homomorphic stochastic cocycle j with general (bounded) generator (12). It follows from the Christensen-Evans Theorem ([ChE]) that there is d E Ik) @ M and h = h* E M such that .(a) = d * p ( ~ ) d h{d*d, U }
+ i[h,u ] , and
d(a) = da - p(a)d. Letting j o denote the quantum stochastic cocycle generated by [I p!Lk] i h - i d ' d d' and setting 1 = where p = p(l), the quantum stochastic -pd P~ differential equation
]
[
dWt = (Ic @ Wt)(idBci;,@ j f ) ( l ) d h t ; W O= I has a unique strong solution ([GLW]). Setting q5 = and q5' = J , where J o and J are the E-semigroups on B, ( M @ I&)) = M @ B(&) corresponding to j o and j respectively, it may be shown ([BLl]) that W satisfies (8). This establishes the following result. Proposition 3.4. Let J be the E-semigroup associated with a *homomorphic stochastic cocycle o n M with bounded ultraweakly continuous generator 9 = p-Lk . Then J is cocycle conjugate to the E-semigroup
[.
1
associated with the stochastic cocycle with generator
[I
In view of Theorem 1.1 these two propositions entail the following theorem. Theorem 3.5. The product system of a regular normal *-homomorphic quantum stochastic cocycle on a uon Neumann algebra is exponential. SpecificaEly, i f the cocycle generator is ;;p6_tLk] then the product system
[
is (-TF,[o,~[)~~o where F = Fk%p. 4. Irreducible quantum stochastic dilations
Let P = (Pt)t>o - be a Markov semigroup on M . A *-homomorphic quantum stochastic cocycle j on M , with noise dimension space k, is a dilation of P
138
if its Markov semigroup is P , that is lE& 0 j t = Pt, where & := idM @Uk. If P is norm continuous with generator T then j has bounded generator of the form [i In fact every norm continuous P has such a quantum stochastic dilation ([GoS]), but typically the dilation is not (and cannot be) the minimal dilation of P ([BL2]). This motivates the following definition. A quantum stochastic dilation j of a Markov semigroup P is irreducible when there is no other quantum stochastic dilation j' which is dominated by j in the sense that the corresponding E-semigroups satisfy (Jt - Ji) is
completely positive for all t 2 0.
Then the following results may be proved.
Theorem 4.1. Let P be a n o r m continuous Markov semigroup o n M . Each of its quantum stochastic dilations j dominates an irreducible quantum stochastic dilation. Specifically, i f [i is the stochastic generator of j then there is a projection q E B(k) 8 M such that the stochastic cocyop(p;-ck is t-homomorphic, is dominated by j, and is cle generated by irreducible.
LtLk 1
[
1
For the second result we assume unitality of the Markov semigroup.
Theorem 4.2. Let j be a n irreducible quantum stochastic dilation of a Markov semigroup P o n M . If P is norm continuous and unital then the product system of j coincides with that of the minimal dilation of P . The generic nonminimality of quantum stochastic dilations appears to us t o be a defect of quantum stochastic calculus in its current form. This might be rectified by refounding the calculus on more general Hilbert W*(and C*-)bimodules than the cut-down standard modules p(lM) (I k) @ M ). We view this as a fruitful direction for future research.
Acknowledgements We are grateful to Michael Skeide and to Adam Skalski for their comments on a preliminary draft. BVRB was supported by a fellowship under the Exchange Agreement between the Indian National Science Academy and the Royal Society of London, and by Visiting Fellowship GR/S57099/01 from the U.K. Engineering and Physical Sciences Research Council. The EU Research and Training Network Quantum Probability with Applications to Physics, Information Theory and Biology has faciliated useful discussions with fellow researchers.
139 References [Arl]. W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. SOC. 80 (1989)no.409.
[Arz]. W. Arveson, “Noncommutative Dynamics and E-semigroups,” Springer Monographs in Mathematics, Springer, New York 2003. [Bl]. B.V. Rajarama Bhat, Cocycles of CCR flows. Mem. Amer. Math. SOC.149 (ZOOl), no. 709. [Bz]. B.V. Rajarama Bhat, Product systems of one-dimensional Evans-Hudson flows, in “Quantum Probability Communications X” (eds. R.L. Hudson and J.M. Lindsay) World Scientific, Singapore 1998,pp. 187-194. [BLl]. B.V. Rajarama Bhat and J.M. Lindsay, The product system of a stochastic E-semigroup, in preparation. [BLz]. B.V. Rajarama Bhat and J.M. Lindsay, Irreducibility for quantum stochastic dilations, in preparation. [BhS]. B.V.Rajarama Bhat and M.Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top 3 (2000)no. 4,519-575. [Ble]. D.P.Blecher, A new approach to Hilbert C*-modules, Math.Ann. 97 (1997),pp. 253-290. [ChE]. E. Christensen and D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London Math. SOC.20 (1979) no. 2,
358-368. [EvH]. M.P. Evans and R.L. Hudson, Perturbations of quantum diffusions, J. London Math. SOC.41 (1990)no. 2,373-384. [GLW]. D. Goswami, J.M. Lindsay and S.J. Wills, A stochastic Stinespring theorem, Math. Ann. 319 (2001)no. 4,647-673. [GoS]. D. Goswami and K.B. Sinha, Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra, Comm. Math. Phys. 205 (1999)no. 2,377-405. [Gui]. A. Guichardet, “Symmetric Hilbert Spaces and Related Topics,” Lecture Notes in Mathematics 267,Springer, Heidelberg 1970. [Hud]. R.L. Hudson, Quantum diffusions on the algebra of all bounded operators on a Hilbert space, in “Quantum Probability and Applications IV” , (eds. L. Accardi and W. von Waldenfels), Lecture Notes in Mathematics 1396, Springer, Heidelberg 1989,pp. 256-269. [Lan]. E.C. Lance, “Hilbert C*-modules, A toolkit for operator algebraists,” London Mathematical Society Lecture Note Series 210,Cambridge University Press, Cambridge 1995. [LiS]. V. Liebscher and M. Skeide, Units for the time-ordered Fock module, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001)no. 4,545-551. J.M. Lindsay, Quantum stochastic analysis, in “Lectures in the Spring [L]. School on Quantum Independent Increment Processes,” Lecture Notes in Mathematics, Springer, Heidelberg (to appear). [LWl]. J.M. Lindsay and S.J. Wills, Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory
140
Related Fields 116 (2000) no. 4, 505-543. [LWz]. J.M. Lindsay and S.J. Wills, Markovian cocycles on operator algebras, adapted to a Fock filtration, J. Funct. Anal. 178 (2000) no. 2, 269-305. (LWs]. J.M. Lindsay and S.J. Wills, Existence of Feller cocycles on a C*-algebra, Bull. London Math. SOC.33 (2001) no. 5, 613-621. [LW4]. J.M. Lindsay and S.J. Wills, Homomorphic Feller cocycles on a C*algebra, J. London Math. SOC.(2) 68 (2003) no. 1, 255-272. [Mey]. P.-A. Meyer, “Quantum Probability for Probabilists,” 2nd Edition, Lecture Notes in Mathematics 1538,Springer, Heidelberg 1993. [Par]. K.R. Parthasarathy, “An Introduction to Quantum Stochastic Calculus,” Monographs in Mathematics 85,Birkhauser Verlag, Basel 1992. [Ski]. M.S. Skeide, “Hilhert Modules and Applications in Quantum Probability,” Habilitation Thesis, Cotthus, Germany 2001. [Skz]. M.S. Skeide, Dilations, products systems and weak dilations, Math. Notes 71 (2002) 914-923.
EVOLUTION OF THE ATOM-FIELD SYSTEM IN INTERACTING FOCK SPACE
P.K.DAS Physics and Applied Mathematics Unit Indian Statistical Institute 203,B.T.Road, Kolkata- 700f08, Indaa e-mai1:daspkOisical. ac.in Here we discuss interaction of a single two-level atom with a single mode of interacting electromagnetic field in the Jaynes-Cummings model with the rotating wave approximation.
1. Introduction
Light is absorbed and radiated by atoms and one of the most fundamental problems in quantum optics is the interaction between the quantized electromagnetic field and an atom. But the real atoms being complicated systems it is often desirable to approximate the behaviour of a real atom by that of a much simpler quantum system. Sometimes only two atomic energy levels play a significant role in the interaction with the electromagnetic field and in many theoretical treatments it has become customary to represent the atom by a quantum system with only two energy eigenstates. Jaynes and Cummings considered a system consisting of a nonrelativistic two-level atom coupled to a single quantized mode of the electromagnetic radiation field under the dipole and rotating wave approximation. This simple model in quantum - optics is one of the few exactly solvable quantum mechanical models describing the interaction of matter with an electromagnetic field. The wave length of the field mode is assumed to be so long, compared with the atomic dimension that the dipole approximation can be made. Originally, Jaynes and Cummings wanted to study the QED predictions for an ammonia maser. But they failed to find an exact solution for this application with the help of simplifying assumptions described above. They had to make the additional approximation by removing transitions that correspond to processes which do not conserve energy. This approximation is called
141
142
the rotating wave approximation. In the rotating - wave approximation this model can be solved exactly. Exact solutions describing the dynamical behaviour of expectation values of variables such as the population inversion, the atomic dipole - correlation function and the mean photon number can be obtained in this case in the form of an infinite series. Starting with a cavity field in a coherent state and with the atom in its upper state it was found that repeated decays and revivals of the Rabi oscillations occur and that for certain times the field mode shows squeezing. The predicted collapses and revivals of the inversion oscillations are in agreement with the experiments done with Rydberg atoms in a microwave cavity. In recent years there have been several generalizations of the Jaynes - Cummings Hamiltonian in which the interaction between the atom and the radiation field is no longer linear in the field variables and are only particular cases of JCM in which the creation and annihilation operators of radiation field are replaced by deformed harmonic oscillator operators with the given commutation relations. Since the JCM is one of the basic models in quantum optics, its extensions in different directions are generally very interesting [6].In this paper, we study JCM in which field variables are taken as the actions on an interacting Hilbert space with the specified commutation relation. The work is organized as follows. In section 2, we have given definitions and preliminaries that will be used throughout the paper. In section 3, we discussed the interaction of a single two-level atom with a single mode of interacting field. In section 4, we described the evolution of the atominteracting field system by probability amplitude method. In section 5 , we gave a conclusion. 2. Definitions and Preliminaries
As a vector space one mode interacting Fock space I?(@) is defined by 00
r(@) = @@in >
(1)
n=O
for any n E N where @In> is called the n-particle subspace. The different n-particle subspaces are orthogonal, that is, the sum in (1) is orthogonal. Thenorm of the vector In > is given by
< nln >= An
(2)
{An} 2 0 and if for some n we have (A,} = 0, then A{}, = 0 for all m 2 n. The norm introduced in (2) makes r(@) a Hilbert space. where
143
An arbitrary vector f in r(C) is given by
f
= COlO > +Clll
> +c212 > +... +c,ln > +.. . llfll = (C,"==, < 00.
(3)
for any n with We now consider the following actions on r(C) :
A* is called the creation operator and its adjoint A is called the annihilation operator. To define the annihilation operator we have taken the convention o/o = 0. We observe that An < nln >=< A*(n- l ) ,n >=< (n - l ) ,An >= < n - 1,n - 1 >= . . . A,--1
(5)
and
By (2) we observe from (6) that A. = 1. The commutation relation takes the form AN+1
AN
AN
AN-1
[A,A*]= -- where N is the number operator defined by Nln >= n(n >. In a recent paper [l]we have proved that the set { n = 0 , 1 , 2 , 3 , . . .} forms a complete orthonormal set and the solution of the following eigenvalue equation
$$,
Afa = Qfa is given by (9)
T.
= Cr?o where $J(laI2) Now, we observe that
We call
fa
a coherent vector in r(C).
AN+1 AN AA* = , A*A = AN
We further observe that AA*.
(e
- &)
AN-1
commutes with both A*A and
144
3. Interaction of a single two-level atom with a single mode field
The interaction of a interacting single-mode quantized field of frequency Y with a single two level atom is described by the Hamiltonian in the dipole approximation
H = HF
+ H A + gHI
(10)
+
+
where H F = hvA*A and H A = ;twoz with HI = h(a+ a - ) ( A A * ) . Here A , A* are the interacting annihilation and creation operators for the photons at frequency v. The two - level atom is described by the usual spin - operators and the inversion operator az with w as atomic transition frequency. Also g is the coupling constant. The interaction energy HI consists of four terms. We drop the energy nonconserving terms corresponding to the rotating - wave approximation and obtain the simplified Hamiltonian as
H = hvA*A
+ -21h a , + hg(a+A + A*u-).
(11)
At this stage , for simplicity, we take h = 1 and consider the exact resonance case v = w . The resulting simplified Hamiltonian is 1 2
H = v(A*A+ -az) + g(a+A + A*o-).
(12)
H = Ho + H i
(13)
We write
where 1 2
(14)
+ A*o-)
(15)
Ho = Y ( A * A+ - a z ) and Hi = g(a+A
The Hamiltonian, given by equations (13), (14) and (15) describes the atominteracting field interaction in the dipole and rotating-wave approximation. It is convenient t o work in the interaction picture. The Hamiltonian, in the interaction picture, is given by
v = ,iHotHle-iHot
(16)
145
Then
v = ei[vA*A+1/2vuz]t~le-i[vA*A+1/2vur]t
- eivA'At~ei/2vuzt~le-i~A*At~e-i/2vuzt - eivA'At.ei/2vu,tg(a+A + ~ * ~ _ ) ~ - i v A ' A t . ~ - i / 2 v u = t = ge ivA'
At .ei/2vu,ta+Ae-ivA'
+geivA' At .ei/2vu,
- gei/2uuzta+e -i/2vuZt +geivA'AtA*e-ivA'At
Using
we get
At.e-i/2vu,t
tA*O-e-ivA'
At .e-i/2vuz t
eivA'AtAe-ivA'At .ei/2vaz ta-e-i/2vo,
t
(17)
146
ei/2vazto +e-i/2vazt = o+ + i / 2 v t [ o z , o + ] + ~ [ o z , [ o z , o + +... ll = Is+ i / 2 v t ( 2 a + ) -220+ .. . = o+ ivto+ @ g o + . ..
+
+ +
+
+ + = o+[l+ivt+ (zvt)2 2! +...I =o + p
(22)
From (17) we get
v = S[o+eivt e-iv( hx-N + ) t
N--l
A +A*eiv(e-*)to-e-ivt
I
(23)
4. Evolution of the atom-field system
In this section, we present probability amplitude method to solve the evolution of the atom-field system described by the Hamiltonian (23). The Pauli spin operators are:
We may define the following operators: 0-
= 1/2(o,
+ io,)
(::)
and
The o- operator takes an atom in the upper state into the lower state whereas o+ takes an atom in the lower state into the upper state. We begin with the following spin matrices:
o-=
(;;),
(;;),
ff+=
uz=
(
-1 0 l).
They satisfy the following commutation relations: [o+,Is-]=
oz, [o,,o+]= 2o+,
[Isz,o-]
= -%-.
Then we proceed to solve the equation of motion for 111, >, that is,
-i
at
’
= Vl11,>
(25)
147
At any time t , the state vector I+(t)> is a linear combination of the states la, > and Ib, >. Here la, > is the state in which the atom is in photons. A similar description the excited state la > and the field has is given for the state Ib, >. As we are using the interaction picture, , C~b , n . The state we use the slowly varying probability amplitudes c ~and vector is therefore
e
-&
-&
-&
k
Now,
and
Now we observe the following facts:
a+la > = la
>< bla > = 0
o+p > = la >< b[b> = la > a-la
a-lb We see the followings:
> = Ib >< ala > = Ib > > = Ib >< alb > = 0
148
From equations (29), (31), (32), (33) and (34) we observe that
Now comparing equations (26), (28) and (35) we get
149
From (36) we get
Or
Or
with
for
Now we assume a trial solution to get
or
S2 - A S - g2(-)An+l = 0 An whose solutions are
S= In (42) we take
fli
Af
4 A 2 + 4g2( h) An
(42)
2
A2 + 4g2(*)
i A+n t ( 2")
and write the solution as
i A-nn)t ( 2
ca,n(t) = A l e +A2e =A l e F . e W + A
~
~
F
.
~
~
= [ A l e w + Aze--i"]e% -in
t
+ i sin y}+ Az{cos y - i sin ?}ley = [(Ai+ A2) cos y + i(A1 - A2) sin
(43)
= [Al(cos
To find the constants
A1
and A2 we observe that Ca,n(O)
= A1
+ A2
(44)
150
Also we see that
Hence we see that
-ig/?Cb,n+l(O)
= A1 .i/2(A
+ 0,) + A2.i/2(A - Rn)
(46)
On solving (44) and (46) for A1 and A2 we find that
(47) and
(48) and
Subsstituting the values of
and
from above in (43) we find
finally
In a similar manner we find the value of ~ b , ~ + l (tot ) be
151
If initially the atom is in the excited state la > then c ~ , ~ ( O=) cn(0) and = 0. Here cn(0) is the probability amplitude for the field alone. We then obtain qn+1(O)
-
ca,n(t) = cn(o)[cos(?)
sin(F)]ey
(52)
or
Equations (52) and (54) give us all the relevent informations relating to the quantized interacting field and the atom. I ~ . , ~ ( tand ) l ~ I ~ b , ~ ( t are ) l ~ the probabilities that, at time t , the interacting field has n photons present and the atom is in levels la > and Ib > respectively. To obtain the probability p(n)that there are n photons in the interacting field at time t we take the trace over the atomic states: P(n) =IC~,~(+ ~ )I I C ~ ~ , ~ ( ~ ) I ~ 2 n t = I~n(0)l2[COS + (&I2 sin2(*)nI +Icn-1(0)l2
4g2(&)
a:_,
t
2 %-It
sin (7)
(55) The inversion W ( t )is given by
152
5. Conclusion
In conclusion, we have studied the atom-interacting field evolution in the dipole and rotating wave approximation by probability amplitude method and for simplicity we considered only resonance case. References
1. P. K. Das, Coherent states and squeezed states in interacting Fock space, International Journal of Theoretical Physics, vol 41, no. 06, (2002), MR. No. 2003e: 81091 (2003). 2. P. k. Das, Quasiprobability distribution and phase distribution in interacting Fock space, International Journal of Theoretical Physics, vol 41, no. 10, (2002), MR. NO. 2003j: 81098 (2003). 3. J. S. Peng and G. X. Li, Phase fluctuations in the Jaynes-Cummings model with and without the rotating wave approximation, Phys. Rev. A, vol. 45, no. 5, (1992), 3289 - 3293. 4. M. 0. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press,(1997). 5. Yariv, Amnon. Quantum Electronic, John Wiley and Sons,Inc. NY.(1967) 6. Crnugelj, J., Martinis, M., Mikuta-Martinis, V., Properties of a deformed Jaynes- Cummings model. Phys. Rev. A, vol. 50, no. 2, (1994), 1785-1791.
QUANTUM MECHANICS ON THE CIRCLE THROUGH HOPF Q-DEFORMATIONS OF THE KINEMATICAL ALGEBRA WITH POSSIBLE APPLICATIONS TO LEVY PROCESSES
V.K. DOBREV1t2, H.-D. DOEBNER3, R. TWAROCK4 School of Informatics, University of Northumbria, Newcastle-upon-Tyne N E l 8ST, UK, vladimir. dobrev0unn. ac.uk, Permanent address: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 7.2 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria, dobrev0inrne.bas.bg Department of Physics, Metallurgy and Material Science, Technical University Clausthal 38678 Clausthal-Zellerfeld, Germany
[email protected] Centre f o r Mathematical Science, City University Northampton Square, London EC1 V OHB, UK r.twarockQcity.ac. uk A formulation of quantum mechanics on S' or its N-point discretisation Sh based on different types of q-deformations of subalgebras of the kinematical algebra of the system was discussed ( [l])in the framework of Bore1 quantisation. We review this method and introduce new Hopf q-deformations of the full kinematical algebra, i.e. q-deformations of both the subalgebras of position and of momentum observable. The implications of this deformed approach to the dynamics and the resulting evolution equations is assessed and compared with previous results including the non-deformed case. The presented algebraic method for q-deformations can be translated to LBvy processes on algebraic structures and the related evolutions; possible applications are outlined.
Key words: Hopf q-deformation, kinematical algebra, quantum dynamics,
153
154
Ldvy processes 1. Introduction
The quest for a discrete quantum mechanical theory is of interest both for conceptual and computational reasons, and different arguments in support of this have been discussed in [l]. There are various possibilities to construct discrete analogs to quantum mechanics, which rely on different guiding principles for the replacement of differential operators by suitable difference operators. The method adopted in [l]is of an algebraic nature, and is based on the kinematical algebra that represents position and momentum observables in the framework of Bore1 quantisation [2]. Via a q-deformation of this algebra, expressions of the quantum mechanical position and momentum operators are obtained in terms of q-difference rather than differential operators, i.e. expressions of the form
where e.g. f E C(R). Furthermore, the quantum dynamics based on these operators and a suitable generalisation of the first Ehrenfest theorem is given in terms of qdifference operators and corresponds to a discrete, nonlinear q-Schrodinger equation. The algebraic guiding principle based on q-deformations of the kinematical algebra leads to different discrete quantum mechanical theories for different choices of q-deformations. While a q-deformation of the subalgebra related to the momentum observable is necessary in order to obtain a quantum mechanical theory in terms of difference operators, a deformation of the subalgebra related to the position observable is not necessary for this purpose, which is minimal in the sense that it involves only a deformation of the subalgebra related to the momentum observable [3]. Therefore, initial attempts to construct a discrete quantum mechanical theory based on a q-deformation of the kinematical algebra have focused on a q-deformation of the kinematical algebra. However, from an algebraic point of view, this does not use the full potential of the algebraic guiding principle for the construction of a discrete quantum mechanical theory, and different options for this have been discussed in [l].Among these, an approach based on a Hopf q-deformation of the full kinematical algebra (see also [4])plays a distinguished role. It is the aim of this contribution to consider this case in further detail and to
155
compare the resulting quantum mechanical theory with the results in the minimal approach. We will show that the approach based on a Hopf q-deformation of the full kinematical algebra has interesting implications in the continuous limit. In particular, we discuss the quantum dynamics, which is given as a family of nonlinear q-Schrodinger equations, in greater detail and compare it with the family of Doebner-Goldin type nonlinear Schrodinger equations [5-81 that arise in the continous approach. The interesting feature of the ansatz based on a q-deformation of the full kinematical algebra is that both real part and imaginary part of the nonlinear Schrodinger equation are completely determined. This is in contrast to the situation in the undeformed case, where only the imaginary part of the nonlinearity follows from the Borel quantisation formalism, and the real part is inferred by some additional plausibility arguments related to the shape of the imaginary part of the nonlinearity [9].We show that in the continuous limit of our approach based on a q-deformation of the full kinematical algebra one obtains a nonlinear Schrodinger equation, in which the imaginary part of the nonlinearity coincides with the one in the Doebner-Goldin formalism and the real part falls in one of their proposed classes. In this way, this study may be understood also as a justification for the Doebner-Goldin models. Finally, we outline possible applications of these results to LBvy processes. Their investigation on noncommutative algebraic structures (Lie groups, quantum groups, bialgebras,. ..) is often intricate and involves representation theory, c.f., e.g., [lo-171. In this approach L6vy processes are characterized by their generators. Classifying such generators originating in our algebraic framework would be one of the main challenges. Another reason for the study of L6vy processes on algebraic structures is their relation t o evolution equations, c.f., e.g., [13-151. The paper is organised as follows. After a review of Borel quantisation in section 2, we discuss the Hopf q-deformation of the kinematical algebra and its implications on the quantum kinematics in section 3. In section 4 a corresponding quantum dynamics is derived and compared with the dynamics obtained via the minimal ansatz in [l].In section 5 we outline the possible applications to L6vy processes.
2. Short review of Borel quantisation with application to
S1
We consider (non relativistic, point-like) systems S moving and localized on a smooth manifold M (with measure p ) and specialize later to S1. The
156 exposition follows [l]. 2.1. The kinematics
To model possible localization regions of S on M and possible infinitesimal movements of the regions we choose for the regions Borel sets B from a Borel field B ( M ) and for the movements (smooth, complete) vector fields X E Vecto(M). These two geometrical objects are the building blocks of the kinematics K ( M ) of S:
-
I v
K ( M ) = (f?(M),Vecto(M)).
-
(2)
Borel sets are displaced through X by its flow @: as B' = {m'lm' = aT%),m E B , X E Vecto(M),7 E [0,1)}. For a quantisation of K ( M ) one has to construct a map Q = (Q, P) which maps the blocks in K ( M ) into the set SA(7-l)of self-adjoint operators on a Hilbert space 3t. It is reasonable to interpret the matrix elements of Q ( B ) ,i.e. ($,Q(B)$),$ E 3-1, as the probability t o find the system localized in B in a state $. The properties of B ( M ) and further physical requirements (e.g. no internal degrees of freedom) show that Q ( B )acts on 3-1 as the characteristic function x ( M ) of M , if 3t is realized via square integrable functions over M , i.e. as L 2 ( M ,d p ) . From the spectral theorem and from Q ( B ( M ) )we infer a quantisation map for C" ( M ,IR)
Q : C"(M,R)
-+ SA('W,
Q(fM = f$-
(3)
Hence we can use in the kinematics C"(M,IR) instead of B ( M ) , that is the kinematics can be viewed as an infinite dimensional Lie algebra, more precisely as a semidirect sum of the abelian algebra C" ( M ,IR) and a subalgebra of the Lie algebra of vector fields, and we denote it as K ( M ) (without tilde) in the following:
K ( M ) = C"" ( M ,R)EVecto ( M ).
(4)
To construct the quantisation map P for Vecto(M) we need further assumptions, which we will call P-assumptions following [l]: 0
0
The partial Lie structure (in connection with complete vector fields) of K ( M ) is conserved. The operator P ( X ) are - in analogy to the canonical quantisation in Rn - local differential operators.
157
With these P-assumptions we have the following result [2]: The P(X)are differential operators of order one with respect to a differential structure on the set M x C. We characterize this assumption (up to isomorphism) through Hermitian line bundles L over M with compatible flat connection V. Wave functions are sections ~ ( min )the bundle and L2( M ,p ) can be viewed as a space of square integrable sections. Unitary equivalent irreducible maps Q(..*”) - quantisations - are given by a bijective mapping onto the set
( a , D ) E 7 q M ) x R.
(5)
$ ( M ) denotes the dual of the first fundamental group of M , a topological quantity. D is connected with the algebraic structure of K ( M ) and characteristic for Bore1 quantisation. (a,D) are quantum numbers in the sense of Wigner. Q is labeled by these numbers, i.e. = (Q(a*D),P(criD)) and one has (m E M ) Q(a3D)
Q(”>”)(fb(m)= f(m)~(m)
+
P(*’”)(X)o(m)= (40% (-ii
(6)
+ D )div,g)
o(m),
(7)
which are self-adjoint operators on a common dense set. Here, V% denotes the connection in the line bundle L ( M ) over M and div, denotes divergence. Note that the quantum number D appears as a real factor in front of div,g and that the nontrivial topology yields the a dependence of ‘7%. 2.2. The dynamics
States of S are modeled via density matrices W , i.e. through trace class operators with Tr(W) = 1. We introduce a time dependence for W (in the Schrcdinger picture), which is based on Q(..*”), through a quantum anaof time dependent log to the classical relation between time derivatives functions f ( m ( t ) )and momenta, i.e. for M = R’
6
d
-f(z(t)) dt
-
PVf .
(8)
One can show [6] that (in the Schrijdinger picture) one has the following relation for expectation values (Exp,(A) = T r ( W A ) ) :
d -Tr(W(t)Q(*’”’(f)) = Tr(W(t)P(”’D) (Xgradf)) ,Vf E Cm(M,R) (9) dt
158
This is a restriction for the evolution of W ( t ) . For pure states it implies, under the condition that pure states evolve into pure states [6], the following generalized version of the first Ehrenfest relation
.
.
with a scalar product (., .) in L2( M ,dp). 2.3. The kinematical algebra K(S1) and a family of evolution equations
We consider now an application of Bore1 quantization to the case that the configuration space is S1. S1 is topologically nontrivial with n;(S1) = [0,27r), and we denote elements in 7r;(S1) as a. The flat line bundles over S1 are trivial, the vector fields are X = X(I$)-& E Vecto(S1) and the Hilbert space is L2(S1,dq). In these coordinates K(S1) is given by the generators
Q(UID)(f)$(I$)
= f(dM4)
P'"'D)(X)$(I$)= (-iX(I$)-&
(11)
+ (+
(y) +ax($)) $(I$X12)
+D)
To analyse the structure of K(S1) we use a Fourier transform F with z = &4: m
n=-w
fn
= f-n, X n = X - n . For the F-transformed quantum kinematics we find w
With the operators
Tn = zn
159
(14) can be expressed as
n=-w
c x, 00
P(*ID)(X)=
(LE
+ iDnT,)
(16)
.
,=-w
The generators T, are an Abelian Lie algebra which we denote as T,and for fixed cr E [0,27r) the L, LE fulfill the commutation relations
=
[Tm,T,I = 0
(17)
[L,, Tm] = mTm+n [Lm,Ln3 = ( n - m)Lm+n
and span an inhomogenisation of the Witt algebra W through T . This [18], where we use the index z to gives the algebraic structure of Kz(S1) indicate that it is given in terms of the variable z as opposed to the angle variable 4. We have from (17) (we have dropped the upper index ( a , D ) for convenience)
[Q(f),Q(g)1 = 0,
[P(X), P ( Y ) l = -.1’P([X,YI). Now we introduce the time dependence for pure states $($) E L2(S1,d4) [P(X),Q ( f ) l = -iQ(Xf),
and we evaluate the restriction (10) with
xgradf
= f’(4)”
d4
(I
G A): d4
This implies a generalized continuity equation of Fokker-Planck type for p = $$:
i
p = &j$” - $5))+ Dp” - crp’ = --(jt)’ + Dp”, d
(19)
where
corresponds for vanishing cr to the usual quantum mechanical current density on S1. This can be derived also by other methods based on Q(a*D)[5], [19],[9]. We use this information in (19) for a general ansatz for a Schrodinger equation of the type
i&$
= H$
+ G[$,$I$
160
in which H is a linear operator and G[$,$] can be written (formally) as a nonlinear complex function G[$,$] = GI[$,$] iG2[$,$] depending on $,$, their derivatives and explicitly on 4 and t . Hence G acts as a multiplication operator. This Ansatz leads to a family Fp of Schrbdinger d4) with G2 enforced by (19) equations [3], on L2(S1,
+
The real part G1 cannot be determined by Borel quantization. Hence a set of (natural) assumptions for G1 motivated by the form of the imaginary part Gz,has been introduced [5], [9]: (1) G1 is proportional to D , i. e. vanishing for D = 0. (2) G1 is a rational function with derivatives no higher than second order and occurring in the numerator only. (3) G1 is complex homogeneous of order zero, i. e. Gl[a!$,&$] = GI [$,$1 for all a! E C. These assumptions restrict G1 in the family Fp to the Doebner-Goldin family (DG-family) 3 D G [5] on S':
with free real parameters Dk, k = 1 , . . . ,5.
3. A discrete quantum kinematics based on a Hopf q-deformation of the kinematical algebra In [l]various q-deformations of the kinematical algebra in Borel quantisation have been discussed. As mentioned earlier, our focus in this contribution is on that version among these (see also [20,21]) which involves a q-deformation of the full kinematical algebra and hence exploits the maximal freedom. Since the corresponding operators are difference operators, it is possible to restrict the configuration space to an N-point discretisation of S1,i.e.
161
3.1. A Hopf q-deformation of the kinematical algebra
We discuss q-deformations of the subalgebras of the kinematical algebra related to the position and the momentum observable separately, and then consider the coupling between these two algebras in a next step. The q-deformation leads to a Hopf algebra, and is therefore called a Hopf qdeformation. 3.1.1. The position observables In the undeformed setting, the subalgebra of the kinematical algebra that corresponds to the position observables in the framework of Bore1 quantisation is given by the generators {T,} in (17). In the q-deformed setting, we augment this basis to {Tn,K,"},where the Abelian subalgebra K := {K,"} with s E Z corresponds to q-shift operators, i.e. operators that act by rescaling the arguments of a function f(x) according to
where x,q E S1.They have the property:
[K,",K;,] = 0,
(24)
with limq+l K," = 1. Moreover, the coupling to the generators {T,} is given by
K:T, = qsnTnK,Q
(25)
The q-analog to the generators T, is given via a restriction of the universal enveloping algebra of the algebra generated by the basis {T,, K,"}to monomials of the form T2,5:= T,K,". The generators in the q-deformed setting are hence related to the undeformed generators via the mapping MZ, which restricted to the subalgebra related to the position observables acts as follows:
The generators T& form a non-commutative algebra with the following commutation relations:
162
3.1.2. T h e m o m e n t u m observables The subalgebra of the kinematical algebra related t o the momentum observables is given by the Witt algebra with basis {L,} and commutation relations as in (17). We consider here a q-deformation, where the q-Witt algebra is generated by the basis {C&} with the following commutation relations:
As expected for consistency, these commutation relations reproduce the commutation relations of the Witt algebra in the continuous limit q + 1. The extra parameter j , which has been introduced during the deformation, vanishes in the limit q + 1. The generators in the q-deformed setting are hence related to the undeformed generators via an extension of the mapping M i to the subalgebra related to the momentum observables M i :
M:,i : Lz
Cz,i, (29) where n,j E Z and a E [0, 1). We hence have a mapping Mq = { M i ,MB,j} that acts on the full kinematical algebra. H
3.1.3. Coupling between the subalgebras: the Hopf q-deformation of the kinematical algebra As a first step, we consider the coupling between the q-Witt algebra in (24) and the algebra of shift operators in (28). One obtains a quadratic Lie algebra with basis {Cz,,, K & } ,with
K,QCZli = q""C"n93.Kf
(30)
It is a noncommutative and co-commutative Hopf algebra - Hopf-q-Witt algebra - 3tWq with coproduct A, counit c and antipode y given as follows:
A ( C g , j )= Cg,j 8 K&
+ K& 8 L$,j , A(KZ)= KZ 8 KZ
E ( C ; , ~ )= 0 , e(KZ) = 1 y(Ck,j) = -(K&)-'Ck,j(K&)-' , y(K;) = (KZ)-'
(31)
Further, we should couple the position observables {T,} . The latter have trivial co-algenra structure: A(T,) = T, 8 1 1 8 T, , e(Tn) = 0, y(T,) = -T, . Thus, one obtains the inhomogeneous Hopf-q-Witt algebra
+
163
with basis { L z , k K , L , Tz,,} and co-algebra relations of Tz,, : A(Tz,,)= T$,,@Kj+Kj@Tz,,, €(Ti,,)= 0, ~ ( 7 2 = , ~- ()K j ) - " T & ( K j ) - ' . The q-deformed kinematical algebra is then given in terms of the basis { L z , k , via a restriction of the monomials in the universal enveloping algebra of { K k ,T,} to {Tz,,},and the commutation relations are as follows:
x}
, is quadratic. Hence the deformed algebra with basis { L z , k Tz,,} We note that a restriction to {L&,Tm} would lead to the results discussed in [l],and it is the purpose of this contribution to discuss the changes in the quantum kinematics and quantum dynamics implied by the occurrence of the shift operators K L in the subalgebra related to the position observable. 3 . 2 . The quantum kinematics on S1
In this subsection we discuss possibilities to associate position and momentum operators to the q-deformed kinematical algebra in the previous subsection. Since on S1the position operators Q ( " > D ) (inf )(16) depend on the generators Tn and the momentum operators P("vD)(X)in (16) depend on the generators Tn and LE,j, a q-deformation of the quantum kinematics can be achieved based on the mapping M Q in (26) and (as),because it relates the basis of the kinematical algebra in the undeformed setting with the basis of the q-deformed kinematical algebra. In particular, M8,j induces the following mapping M b K from the undeformed quantum kinematics in (16) to the Hopf q-quantum kinematics induced by the q-deformation of the kinematical algebra.
M ~ : KQ ( f ) ++ 0;C.f)= C:='=_, fnM:(Tn) P ( X ) ++ P i ( X ) = Cz='=_, Xn (M:,j(LE) i w D M p ( T n ) ) .
+
(33)
{ M $ K ( Q ( f ) )M, b K ( P ( X ) ) }will be called the Hopf q-quantum Borel kinematics induced by Mq = { M : , M:,j}. Indeed, the mapping M $ K depends on two further parameters, j , s E Z, and expressions for the position and momentum operators in the Hopf q-quantum Borel kinematics are given by
164
where K,, respectively K,, is short-hand notation for the shift operators K,“ and K;. For each different choice of the deformation parameter q one hence obtains a different quantum kinematics. Of particular interest for applications are the cases where q is a root of unity, because in this case, the Hopf q-quantum Borel kinematics leaves the N-point discretisation Sh of the configuration manifold S1 invariant. In particular, in this discrete setting the parameter j in Lila) and s in K , have the following interpretation: contains - dependent on j - differences between different points of Sh,e.g. between next nearest neighbours for j = 2 or even further points, i.e. it measures how coarse-grained the discretization is. Similarly s is a measure for how coarsegrained the jumps initiated by the shift operators K , are. 4. The quantum dynamics with Hopf q-structure
In this section we derive a quantum dynamics that is compatible with the Hopf q-quantum Borel kinematics in (34). As an important building block in the derivation, an appropriately defined q-version of the first Ehrenfest theorem is necessary. In order to derive it, we introduce a symmetrization of T,K, and K,T, as follows: 1 Sm,s := -(TmK.g 2
+ KsTm).
(35)
It is given in terms of the shift operators K , and leads to the generators T , in the limit q + 1 as expected for consistency. Based on the symmetrised coordinates, we furthermore introduce the following symmetrised scalar product ( @ , S 4 ) s g r n :=
1 z{(@, S4) + ( S @ 4)) ,
7
(36)
where (,) denotes the usual scalar product as in section 2, S denotes a shift operator, S is its complex conjugate. This symmetrisation of the dependence on the shift operator S is sufficient to ensure that the quantum mechanical probability density p = is a real quantity in the deformed setting; without this symmetrisation, this would not be the case. With this definition of a symmetrised scalar product, the Hopf q-version of the first Ehrenfest relation reads:
165
where
Note that the coefficients r and s, that define the shift operators, are independent. 4.1. Derivation of a nonlinear q-Schrodinger equation
In order to derive the Hopf q-quantum dynamics, the left and right hand side of (37) have to be evaluated explicitly. 4.1.1. The right hand side of (37)
With N , as short-hand notation for zdZ,q-shift operators are given as qaNz and act on functions f(z) via
(40) Then the right hand side of (37) can be evaluated to yield an expression of q a N z f ( 4= f(q."z).
the form ( f , B )in the usual scalar product with
166
In order to obtain the last term in (41) we relate K, and K , such that K,Ks = 1. This ensures that the last term is a function of p = In the following we restrict ourselves to the case of a = 0 to keep the argument transparent. Furthermore, the a-dependence is not essential for the derivation of the nonlinear parts, so that it is not crucial for the main result, which is the derivation of possible real parts. In terms of $1 and $2, where $ = $1 i $ ~ , one finds for the part not depending on the parameter D in (38) the following expression:
4$.
+
4.1.2. The left hand side of (37) The left hand side of (37) can be computed with (13) to be of the form
at ($(t,z ) , Qi(f)$(t,z
)
) =~ at~($(t, ~ 21, f (z)Ks$(t,z))sym
~&($,f(~S+ d )((KS$Lf4)1 (fJt;(4FS$) + (KS4M)) =: (f,4 = =
(43)
where the real-valued functions f are time independent and hence are not affected by the operator at. The last line in (43) contains the usual scalar product.
167
In order to determine an expression for the quantity A in (43), one needs an ansatz for a nonlinear q-Schrodinger equation. We use an ansatz with H i linear in $ and Gi[$,$]nonlinear in $, 4:
i(&$)$ = (Hi$)(S$) + (G{[$,$l$)(R$)
(44)
S and R are shift operators like in (40),with parameters a in (40) given as a, and a,, respectively. Such shift operators typically occur in a q-deformed theory. (44) reduces to the corresponding expression for the evolution equation given in section 2 in the limit q + 1, if H! and Gi[$, $1 are such that they give H and G[G,$] in this limit. With (44) we get
4.2. The q-deformed evolution equation
Fulfilling the Hopf q-version of the first Ehrenfest relation (37) means equating the expressions for A and B in (42) and (45). We will consider linear and nonlinear terms independently.
The t e r n s connected with H; Assuming now that the linear operator H i , denoted by H , and the corresponding shift S are real, and using $ = i q ! ~ we ~, obtain that the terms on the right hand side of (45) involving the linear terms H are given by :
+
-(HK,$1 ) ( W 2 ) . (46) Equating this expression with B' in (42), we find uniquely ( H Q 2 )(SKS$JI)+(HK,$J2) (S$l)- (H$1)(SK,$2)
where E = f l . In particular, since K , = q s N z ,the parameter s is fixed in dependence on j . Note that these results differ from the nonHopf case considered in [3]. The formalism to derive a q-Schrodinger
168
equation corresponding to the ansatz (44) hence requires the selection s = Reinserting this into the Hopf q-quantum Bore1 kinematics in (34) means that the freedom in the position and momentum operator is not independent and linked precisely by this condition on s via the dynamics. The resulting part of the q-SchrBdinger equation is hence ( j E 2N) :
%.
(
f (qNz + q - N z ) 4.
(Hi$)(S$) = [ j N z ][ j % ]
(48)
The terms connected with Gi We focus now on the nonlinear terms Gi. In order to derive them, the terms in (45), which contain a dependence on G{[$, $1 := G1+iG2, are collected. The imaginary terms cancel as expected and one obtains with the R = R1 + iR2, $ the following real quantity:
GIhl(d1, $2,
+ G2h2($1, $2 ,R1,R2, Ks)
R1, R2, Ks)
(49)
where h1($1,$2,Rl,Rz,K,)
+ (Ks$2)(R2$2)
= (KS$l)(R2$1)
+(KS$2)(Rl$l)
- (KS$l)(R1$2)
+(41)(R2&$1) + ($2)
+($2 1(R1Ks$1)
(50)
(R2Ks$2)
- ($1 ) (R1Ks$2 )
and h2($1,42,Rl,R2,KS)
= (Ks$l)(Rl$l)
+ (KS$2)(&$2)
+(KS$l)(R2$2) +($l)(R~Ks$l)
+($1
(R2Ks$2
- (KS$2)(R2$1)
+ ($2)(R1Ks42) - ($2
(51)
(R2Ks$l).
It has to be equated with the terms proportional to D in (41). The corresponding identity implicitly contains GI, G2, as well as the shifts R1 and R2. Depending on the shifts R1 and R2, the components of the nonlinearity G1 and G2 can be obtained from the nonlinear terms:
FNL 3 (Gi[$,$]$)(R$)
= ((GI
+ iG2)$)(R1 + iR2)4.
(52)
In particular, the nonlinear q-Schrijdinger equation is then given as ( j E 2N, € 1 , €2 = f l ) :
i(&$)4= (q$)(s4) +FNL
(53)
169
with H{ and S as in (48). It is important to stress that the nonlinear term FNL depends on the choices for R1 and R2, and indeed for different choices of R1 and R2 different types of nonlinear terms are obtained. 4.3. The continuous limit
In this subsection we consider the continuous limit of our family of nonlinear q-Schrodinger equations. As before, linear and nonlinear case are treated separately.
The linear terms in the continuous limit A restriction to the linear terms in the q-evolution equation leads for all j in the limit q + 1 to the Hamiltonian obtained in section 2. It resembles the Hamiltonian obtained with the more restricted deformation in [l] up to a factor q e z j % , which now is compensated by the operator K,. The nonlinear terms an the continuous limit The limit q + 1 of the linear part of the evolution equation (48) has already been discussed above. In order to obtain the nonlinear part in the limit q + 1, it is necessary to expand the functions hl($l, $2 R1 ,R2, K,) and h2 ( $ 1 , $ 2 , R1 ,R2 K,) as well as the nonlinear terms in (42) in leading orders of h, where h is given implicitly by q = eh. For the nonlinear terms in (41) one finds the sum of the following two terms: B(1) = & $ / $ , / / I - $,/$///)h2 + O(h3) (54) B(2)= -Dp” + O(h)
-is
+
+
+
Moreover, using the ansatz R = R1 iR2 = (aqQNz bqBNz) i (cqTNZ dqdNZ)( a , b, c , d, a , ,f?, y, S real constants), one derives the expression
+
hi ($1
,$2, Ri, R2, Ks)= 2(c + d)($
+ $2) + 2h
((c + dk2; f CY +a) ($l$i + $ 2 4 ) +(aa f W ) ( $ 2 $ , ; - $I$;) + O ( h 2 ) . (55) A similar expression for h2 ( $ 1 , $ 2 , R1, Rz, K,) follows from (55) using hz ($19 $2, Ri ,Rz, Ks) = hi ($1, $ 2 , R2, Ri ,Ks). Based on these expansions one finds that in the limit q + 1 one always obtains the same imaginary part for the nonlinear functional.
170
It coincides with the expression that has been derived without qdeformation in the framework of Bore1 quantisation. In addition, one obtains results on the real part of the nonlinear functional. For nontrivial shift operators R, that is in the case R # 1, the following class of real parts occurs: Dp" 2P . Otherwise, if R = 1 is the trivial shift operator, the real part remains undetermined like in the undeformed setting as expected for consistency. 4.4. Comparison with previous results
The results derived for the real part of the nonlinear functional in the limit q + 1 is sensitive to the choice of shift operators in the q-deformed setting. In particular, q-deformations of the momentum subalgebra of the kinematical algebra as considered in [l]lead to results different from the ones that arise under a Hopf q-deformation of the full kinematical algebra and a related symmetrisation of the first Ehrenfest relation. The reason for this discrepancy lies in the fact that here one has obtained nonlinear terms in leading orders of h2 in (54), whereas the counterpart to this formula in [l]is given in leading orders of h, which have cancelled each other here due to the contributions from the symmetrisation. In this way, the formalism presented here contains more information and leads to more specific results. It is interesting to note that the real part derived here lies in the DG-class of real parts introduced in section 2. In contrast to this, there are two nontrivial classes of real parts for the non-Hopf q-deformation in [l],one of which coincides with the class derived here, and another one which does not fall into the DG-classes. The Hopf q-deformation of the full kinematical algebra together with a related symmetrisation of the first Ehrenfest theorem has, on the contrary, led to a specification of only one type of real part for the nonlinear functional which coincides with one of the classes in the DG-family of nonlinear Schriidinger equations. Moreover, it has led to the fact that the coordinates, that is the operators pn,are no longer commutative, and the Hopf q-deformation has hence resulted in a noncommutative theory. A method to derive the quantum kinematics on Sy through difference operators which is based on spectral triples is given in [22].
171
A further interesting feature of an approach related to a Hopf qdeformation of the full kinematical algebra is the fact that the q-SchrBdinger difference equations (53) adopt a more symmetric form due to the symmetrisation procedure. In particular, one obtains the following family of difference equations in dependence of j :
For comparison with the results related to a q-deformation of the momentum subalgebra of the kinematical algebra, see (96) in [l]. 5. Applications to LQvy processes Here we outline how to apply our formalism to Levy processes. Our starting point is the relation between the algebraic structures of the formalism developed here and results on Levy processes on compact and noncompact quantum groups obtained in [16,17]. First we note that in both formalisms there are Abelian subalgebras. In our case this is the infinitely generated Abelian Hopf algebra K . This could be related to the Cartan subalgebras K of quantum groups generated by group-like elements Ici, ki', (i = 1 , . . . ,n ) , in the formalism of Jimbo [23]. Since Cartan subalgebras are sub-Hopf-algebras of quantum groups G, the restriction of a Ldvy process on 6 to K is still a Ldvy process. We further restrict t o any of the group-like elements and relate the resulting Lkvy process with the subalgebra generated by any of our generators K j . For group-like elements one can use [12] to get explicit expressions without having t o solve any quantum stochastic differential equations, cf. [16]. Ldvy Processes on V,(G) have been considered for G = d ( 2 ) [16] and this may be applied to the subalgebra of (17) generated by L1,L-1,L0, which is isomorphic to the algebra sZ(2). Furthermore, there should be some relation between the real forms of sZ(2) and their quantum group deformations with real forms and deformations of subalgebras of the Witt algebra. Acknowledgments
V.K.D. and R.T. would like to thank the Alexander von Humboldt Foundation for financial support in the framework of the Clauthal-Leipzig-Sofia
172
Cooperation. V.K.D. was also supported in part by the Bulgarian National Council for Scientific Research grant F-1205/02 and R.T. by an EPSRC Advanced Research Fellowship.
References 1. V.K. Dobrev, H.-D. Doebner and R. Twarock, “Quantum Mechanics with Difference Operators” , Rep. Math. Phys. 50, 409-431 (2002). 2. B. Angermann, H.-D. Doebner and J. Tolar, , “Quantum Kinematics on
3.
4.
5. 6.
7.
8. 9.
10. 11.
12. 13. 14. 15. 16.
17.
Smooth Manifolds”, in: Lect. Not. Math. vol. 1037 (Springer, 1983) pp. 171208. V.K. Dobrev, H.-D. Doebner and R. Twarock, “A discrete, nonlinear qSchrodinger equation via Bore1 quantization and q-deformation of the Witt algebra”, J. Phys. A: Math. Gen. 38,1161-1182 (1997). R. Twarock, “A q-Schrodinger equation based on a q-Hopf deformation of the Witt algebra”, J . Phys. A: Math. Gen. 32,4971-4981 (1999). H.-D. Doebner and G.A. Goldin, “On a general nonlinear Schrodinger equation admitting diffusion currents” , Phys. Lett. A162, 397-401 (1992). H.-D. Doebner and J.D. Hennig, “A quantum mechanical evolution equation for mixed states from symmetry and kinematics”, in: Symmetries in Science VIII, eds B Gruber (Plenum Publ., New York, 1995) pp. 85-90. H.-D. Doebner and P. Nattermann, ,“Bore1 quantization: Kinematics and dynamics”, Acta Phys. Pol. B27, 2327-2339 (1996). P. Nattermann, W. Scherer and A.G. Ushveridze, “Exact solutions of the general Doebner-Goldin equation”, Phys. Lett. A184, 234-240 (1994). H.-D. Doebner and G.A. Goldin, “Properties of nonlinear Schrodinger equations associated with diffeomorphism group representations ” , J . Phys. A : Math. Gen. 27, 1771-1780 (1994). L. Accardi, M. Schiirmann and W.V. Waldenfels, Quantum independent increment processes on superalgebras, Math. 2. 198,451-477, 1988. M. Schiirmann, “A class of representations of involutive bialgebras”, Math. Proc. Camb. Philos. SOC.107,149-175 (1990). M. Schiirmann, White Noise on Bialgebras, Lecture Notes in Math. Vol. 1544 (Springer-Verlag, Berlin, 1993). U. F’ranz and R. Schott, “Diffusions on braided spaces”, J. Math. Phys. 39, 2 7 4 ~ ~ (1998). 6 2 U.Franz and R. Schott, “Evolution equations and Lkvy processes on quantum groups”, J . Phys. A: Math. and Gen. 31, 1395-1404 (1998). U. Franz and R. Schott, Stochastic Processes and Operator Calculus on Quantum Groups. (Kluwer Academic Publishers, Dordrecht , 1999). V.K. Dobrev, H.-D. Doebner, U. Franz and R. Schott, “Lkvy processes on U,(g) as infinitely divisible representations”, in: Probability on Algebraic Structures, eds. G. Budzban, Ph. Feinsilver and A. Mukherjea, Contemp. Math. vol. 261 (American Math. Society, 2000) pp. 181-192; [math.PR/9907016]. V.K. Dobrev, H.-D. Doebner, U. Franz and R. Schott, “Lkvy Processes on
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U,(G)”, in: Proceedings of the International Workshop ”Lie Theory and Its 18.
19.
20. 21.
22.
23.
Applications i n Physics 111, (Clausthal, 1999); eds. H.-D. Doebner et all (World Scientific, Singapore, 2000; ISBN 981-02-4421-5), pp. 280-292. H.-D. Doebner and J. Tolar, “Infinitedimensional symmetries” , A n n , Phys. (Leipzig) 47, 116-122 (1990). H.-D. Doebner and G. Goldin, in: Proceedings of the First Geman-Polish Symposium on Particles and Fields, (World Scientific, Singapore, 1999) p.115. H. Hiro-Oka, 0. Matsui, T. Naito and S. Saito, “On the q-deformation of the Virasoro algebra”, TMUP-HEL-9004, (1990). S. Saito, “q-Virasoro and q-strings in: Quarks, Symmetries and Strings, eds. M. Kaku, A. Jevicki and K. Kikkawa (World Scientific, Singapore, 1991), pp. 231-240. H.-D. Doebner and R. Matthes, “Remarks on Spectral Triples Related to Difference Operators” in: Proceedings of the Fifth International Workshop Lie Theory and its Applications in Physics V, eds. H.-D. Doebner and V. K. Dobrev (World Scientific, Singapore, 2004), to appear. M. Jimbo, “A q-difference analogue of U ( g ) and the Yang-Baxter equation” Lett. Math. Phys. 10,63-69 (1985). ” )
ON ALGEBRAIC AND QUANTUM RANDOM WALKS *
DEMOSTHENES ELLINAS Technical University of Crete Department of Sciences, Division of Mathematics, GR-731 00 Chania Crete Greece E-mail: ellinasOscience.tuc.gr
Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are investigated and related. Based on minimal data provided by the underlying bialgebras of functions defined on e. g the real line R, the abelian finite group Z N , and the canonical Heisenberg-Weyl algebra hw, and by introducing appropriate functionals on those algebras, examples of ARWs are constructed. These walks involve short and long range transition probabilities as in the case of R walk, bistochastic matrices as for the case of Z N walk, or coherent state vectors as in the case of hw walk. The increase of classical entropy due to majorization order of those ARWs is shown, and further their corresponding evolution equations are obtained. Especially for the case of hw ARW, the diffusion limit of evolution equation leads to a quantum master equation for the density matrix of a boson system interacting with a bath of quantum oscillators prepared in squeezed vacuum state. A number of generalizations to other types of ARWs and some open problems are also stated. Next, QRWs are briefly presented together with some of their distinctive properties, such as their enhanced diffusion rates, and their behavior in respect to the relation of majorization to quantum entropy. Finally, the relation of ARWs to QRWs is investigated in terms of the theorem of unitary extension of completely positive trace preserving (CPTP) evolution maps by means of auxiliary vector spaces. It is applied to extend the CPTP step evolution map of a ARW for a quantum walker system into a unitary step evolution map for an associated QRW of a walker+quantum coin system. Examples and extensions are provided.
1. Introduction
Random walks formulated in an algebraic f r a r n e ~ o r of k ~finite ~ ~ groups, ~~~~ bialgebras and operator algebras as well as in the framework of Quantum Mechanics,5~6~7i8~g~10i11~12~13 and references therein), are investigated. Min'Based on talk given in: Volterra-CIRM-Grefswald International Conference on Classical and Quantum Levy Processes: Theory and Applications, Trento Italy 27 Sept. - 3 Oct. 2003.
174
175
imal data for such constructions consist of a bialgebra14 and an integral (functional) defined on it, or alternatively of some Lie algebra, and two quantum systems modelling the walker and the coin system, together with a map modelling the coin tossing, that decides probabilistically the stepping of the walker. Examples of ARWs treated in the following subsections are walks on algebras of functions on R, on ZN, and on the canonical algebra HeisenbergWeyl hw15(section 2). For those walks we show how to define the entropy functional of their respective integral and/or Markov transition operator, and how to deduce that these are entropy increasing random walks by using arguments based on the interrelations relations between majorization bistochastic matrices and e n t r ~ p y . ' ~Moreover, ~ ' ~ ~ ~ ~ ~ ~ ~theoretic dea number composing of ARW on ZN is analysed, that is called prime decomposition,20 and refers to its factorization into products of similar smaller ZN-walks. Mathematically this decomposition is based on the Chinese Remainder Theorem and the co-associativity property. Also for the ARWs on R, in addition to the usual case of short range walk with nearest neighbor (NN) transitions (Polya walk),21 we discuss in our algebraic framework the cases of i) the NN centrally biased random walk (Gillis walk)22 and the case of symmetric random walk with exponentially distributed steps (LinderbergShu1er:LS walk).23 Finally, for the h w ARW, where its functional is constructed by means of the eigenstates of the annihilation operator of that algebra i.e the family of coherent state v e c t ~ r s ,the ~ ~continuous ,~~ time, or diffusion like limit, is obtained.15 This limit results into a trace preserving quantum master e q ~ a t i o n ~ for~ the 3 ~ density ~ matrix of a quantum boson system, which physically is identified with the evolution equation of an open quantum boson system interacting coherently with a classical electric field and incoherently (dissipative interaction) with a bath of quantum oscillators rigged initially into a squeezed vacuum state (squeezed white noise) .28,29,30 Section 3, gives a concise prescription of the concept of QRW, using the example of QRW on integers as paradigm.13 It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and coin systems, and the entropy increase without majorization effect of its probability distributions (pd). This section ends with a group theoretical scheme of classification of various known QRWs. In section 4, a relation connecting ARW and QRW is put forward. The
176
connection is grounded on the theorem due to Naimark that asserts the possibility of implementing in a unitary way a CPTP map operating on e.g the density operator of a quantum system.31 This unitary extension is realized in the original vector space of the density operator augmented by an auxiliary vector space, the ancilla space in the terminology of Quantum Information theory.32 Applied in the context of CPTP of a ARW,the ancilla space is identified with the quantum coin state space of a QRW.The Kraus generators determining the CPTP map of a ARW serve to built, albeit in non unique way, the unitary evolution operator of the associated QRW. This section concludes with the example of an explicit construction of a Q R W associated to the h w ARW,of section 2. Finally, section 4, summarized some of the results and gives some prospected applications of the ARW-QRW concepts and formalism.
2. Algebraic Random Walks
2.1. The case of R
=
Proposition 1. Let the bialgebra of real formal power series H Fun(R) generated by the coordinate function X , and let the positive definite functional 4 : H + R, defined as q5 = CiEZ pi&, with 0 5 pi 5 1, CiEZ p; = 1, where q5ad(f(x)) = f(cq), the functional that evaluates any function f E H , at the point cui = icu, defined for some step cu E R+, for i E 2. The n-step convoluted functional becomes
iEZ
where p(") = Dn-'p('), with the stochastic column vectors p ( k ) = (pi ( k )) i E ~ , = k 1 , 2 , 3,... and initially p ( l ) p = ( p i ) i E z . Also D = ( D i j ) = ( p ; - j ) , i,j E 2 a bistochastic infinite matrix (delta matrix), and 4 = ( 4 , ; ) i ~azcolumn vector. Majorization ordering among pd's is valid at each step i.e p("+l) 4 p("),and consequently the ARW is entropy increasing, namely, S(C$*"+') 2 S(q5*"), S(Tt+') 2 S(T;), where S($*") S(T;) S@")), with S(z) any Shur-convex function e.g the classical entropy i.e S(z) = 2i log 2;. Proof: Operating with convoluted functionals on some function f E
-
=
xi
177
H
yields
By induction we obtain the aimed relation $*" = $Tp(") = $*D"-lp('). Two important properties of the delta matrix are: i) bistochasticity, i.e the column and row sums is one, which is expressed by means of the column vector of units e = (. . . , 1 , 1 , 1 , . . .), that is left and right eigenvector of D, i.e De = e, eT = eTD, and ii) the shift property i.e D i j = pi-j = pi+l-(j+l) = Di+l,j+l. This property in the case of ARW in 2 governed by a pd with finite support, or in the case of a N C 2 finite dimensional walk (see remark below), amounts to a bistochastic matrix D. The functional of the walk at each time step c#P = CiEZpin)$ai, is characterized by the pd p(") = (pin))iEz, which in turn is determined by the bistochastic matrix D i.e p(") = Dp("-l) = D"-'p('). Let us assume that the pd's are of finite support (but see remark below), then by invocation of the theorem stating that two discrete pd's x = ( z ~ ) ~ , y = ( Y k ) k , that are connected by a bistochastic matrix D i.e x = Dy, are ordered by majorization i.e x 4 y, we conclude that the sequence of pd's { ~ ( ' ) , p ( ~ ) , .p. (.}~ ,) ,of site occupation probabilities resulting at each time step during the evolution of the walk is partially ordered by majorization i. e p(") 4 p(n-11, n = 1 , 2 , 3 , . . . . Let us adopt now the definition of the entropy of a functional to be the entropy of the pd that determines that functional i.e we set S($*") z S(T$) S(p(")),or more generally we do so for any convex function S : R + R of the type of the so called Shur-convex functions e.g the classical Shannon entropy S(p) = x i p i l o g p i , or the functions F ( p ) = x i p t , for any constant k 5 1, or F ( p ) = - nipi.17 By virtue of the theorem stating that x 4 y implies F(x) < F(y) , where F ( x) = x i f ( x i ) , for any convex function f : R + R, or otherwise said that the convex functions isotonic to maj~rization,~' we conclude that for the pd resulting from the random walk the majorization ordering is valid at each step i.e. p(") 4 p("-'), n = 1 , 2 , 3 , . . . , and this implies ordering for e.g their entropies i.e. ,903") > S(p"-l), and similarly for their functionals and transition operators. As majorization order implies entropy increase, it is considered as a measure of disorder, and this allow us to con16917318119
=
178
clude the ARW are getting more disordered in the course of time with respect to their site-visiting pd’s, which are getting more entropic, approaching, if left uninterrupted, to the uniform distribution of maximal entropy.
Remarks: 1) The assumption in the previous proof about the support of the pd’s been finite is not actually necessary. In fact in the proof given by Hardy et. a141 is stated that given p 4 q for finite sequence of q , p pd’s, use of Muirhead’s algorithm leads to a bistochastic matrix A , such that q = Ap, and then the proposition: H ( A p ) 2 H ( p ) for H information/entropy or more generally a Shur convex function, is applied. This proof is constructive and builds A in a number of steps not greater than the lengths of p , q , therefore for infinite pd’s a theorem not using bistochastic operators for the characterization of majorization is needed. Such a theorem is provided in Ref. 42. 2 ) For the above proposition the corresponding Markov transition operator defined as T+= (4 @id) o A, is equal to T4 = C i E Z p i e a i A For . the simplest case of p* = ( p , 1 - p ) , a&= f a , and all other p’s and a’ s been zero, the continues limit limn--too = (T4)n T r has been obtained that leads to a diffusion equation.33s334s15 3) The above general algebraic setting implies that the stepping probability matrix D = ( D i j ) = (pi-j), i , j E Z, would be expanded in the enveloping algebra U ( e ( 2 ) ) of the Euclidean Lie algebra e ( 2 ) M is0(2), spanned by monomials of its generators { E+,E- ,L } , that satisfy the defining commutation relations43
An irreducible matrix representation of those generators in the Hilbert space 31 Zz(Z) spanned by the eigenvectors of the ”distance operator’’ L, is useful in expressing the D matrix for various random walks, and look for solutions by means of e.g the Fourier method. This irrep in the canonical basis of 31, and using the same symbol for abstract generators and their matrices reads:
=
mEZ
mEZ
Next we present three different random walks in R, that can be used as show cases of the scheme of ARW presented here. In these concrete examples the defining the walk transition probability matrix D ,is written as an element of the U ( e ( 2 ) ) algebra. No attempt will be made t o give an
179
algebraic solution for the problem of finding the n th-step site occupancy probability distribution, as this can be solved by other means. The examples include: i) the simplest case of symmetric nearestneighbor (NN) random walk (Polya-walk2’); one of its deformations, ii) the NN centrally (site n = 0) biased random walk (Gillis-walk22), which refers t o a solvable case of a walk with no translational invariance, and stepping probabilities with a bias which has power law decay, or more specifically which decays in proportion from the origin of coordinates. The &-deformation parameter is chosen so that when E > 0, the walk is biased to enhance returns to the origin, while if E < 0, escape from the origin is enhanced, and iii) a symmetric random walk with non- nearest-neighbor transitions with transition step length decaying according to an exponential law ( L S - ~ a l k ~Explicitly ~). we have: 1 ) Polya-walk: symmetric nearest-neighbor ( N N ) random walk with Markov transition operator 1
D p = -(E2
+ E+)
(5)
with matrix elements the inter-site transition probabilities 1 1 DP(l,l ) = p l J I - 1
1
+ 24,,I+l.
(6)
2 ) Gillis-walk: nearest-neighbor centrally (site n = 0) biased random walk with Markov transition operator 1 D G = D G ( E )= -2( E - + E + ) + ( E - - E + ) z N E p , ,l -1> l ) , growth values are m a 4c >ff5 QRW2
turn out to be
-d
w
0,".
4. Relation of Algebraic and Quantum Random Walks
This section will put forward a relation between the two types of random walks under investigation so far i.e the ARQ and QRW.The relevant theory here is Naimark's extension theorem that allows to express in a non unique manner a positive trace preserving map, operating by means of its Kraus generators on a density matrix describing the state of some quantum system in a certain Hilbert space, by a unitary operator acting on a extension of the original space. Stated in the language of random walks the extension theorem assumes a ARW described by a CPTP map E(e,,,) = Cm=fPmSme,,,SA, operating on the density matrix of walker system with its Kraus generators (&S+, S S - ) , defined to act on Hilbert space H,. It is further assumed as usually for ARWs,that the step generators S* are related to and algebra of operators that needs to be specified. Then a unitary operator V is considered acting on H , @ H,,, i.e an extension of the original space by an extra or ancilla space H , M C2 =span(l+ >, 1- >), which in the context of QRW stands for the coin system. Let a pure density matrix in the coin system pc = I@ >< @I. Then the extension theorem provides a unitary representation of the CPTP i.e Ev(ew) =
C PmSmezus;
= Trc(V@c8 e w v + ) .
(58)
m=f
The unitary operator provides the Kraus generators as G S , = (ml V I@), up to a local unitary operator W, i.e the transformation V -+
194
W €3 1, V,provides the same generators. For the case of ARQs the unitary operator V is specifically expressed by means of the coin states projections
P&,and the unitary matrix U(p*) =
Em=*
6 6 of the coin space, as (6-6)
V = PmU €3 Sm.In particular the step operators are unitary and inverse to each other i.e S+ = (S-)t.This unitary representation can be extended t o products of CPTM maps by first defining the unitaries
C
v@'
PmU
QD PnU QD SmSn.
(59)
m,n=&
Then we obtain
&(ew)
=
C
PmPn
SmSnew(SmSn)t= TTcQDTTc(V~Vec~ecQDewVt€3Vt),
m,n=f
(60) This unitary extension of E', requires a double ancilla space or two coin quantum systems coupled, so the total space is Hc 63Hc @ H w .In the general case the kth power of the CPTP map of an ARW can be implemented unitarily by extending the original walker space by k anchillary coin systems, so the total space becomes HFk QD H,, and the total unitary operator is
v@L"
C
PmlU
€3
... B PmkU
QD Sml...Smk
...,mk=f
ml,
(61)
with unitarity condition V@'"VBkt = lyk€3 1,. Then we obtain for the kth step of the QRW as described by k successive actions of its CPTP map, a unitary realization which involves tensoring of quantum walker t o k quantum coin systems, followed by a coupling of them by a unitary evolution operator on the space of coins+walker composite system, and finally a decoupling of coins from the walker system, taken by partially tracing with respect t o the coin Hilbert spaces. The partial tracing corresponds to coin tossing in an ordinary random walk, and results into a density matrix for the quantum system of the walker, which further may provide statistics of various quantum observables of the walk. Explicitly the unitarization of ARW reads Et(ew)
=
C
pml...pmkSm,...Smkew(Sm,...Smk) t
...,mk=f
ml,
= TrFk((V@"~Fk @ e,V@'"'),
and can be identified with a QRW.
(62)
195
Let us remark at this point that an equivalent decomposition of the unitary Vmkwould be
n k
VB'" =
~
i
where ,
~i
= ( P + u )@ ~(s+)k+l+ ( P - u ) ~ @(s-)k+1,
i=l
and the subindex denotes the position of the embedding of the respective operator into the k-fold tensor product. In fact each of these operators Wi E End(Hpk @ H w ) , provide a new decomposition of the C P T P map i.e ~k = E W ~ W ~ . . . ,Wwhich ~ is equivalent to a nonstationary QRW with k different unitary evolution operators empoyed in order to construct the k - t h step. As a matter of fact this new decomposition helps to account for the type of quantum entanglement involved between coin and walker systems. Let us take the simplest k = 2 case, where Vm2= WlW2, with
Wl = (P+ @ 1, @ s++ P-
@ 1, @ s-) u @ 1, @ 1, ,
W2= ( ~ c @ ~ + @ s + + l c @ P - @ s - ) l c @ u @ l w . The action of these operators on the product density matrices pc @ pc @ pw is akin t o theaction of some control-control-S* type of non-local operator, which uses the two coin states as control spaces and the walker state as the target space, preceded by the local unitary operator U which acts on the control spaces and creates appropriate superposition of coin states. These actions generate quantum entanglement and can be described by the quantum circuit of Fig.1.c below. For purpose of comparison in Fig.l.a, we have included the corresponding circuit that generates the four entangled bipartite Bell states upon action of the composite operator UCNH @ 1 on the four orthogonal product qubit states, and on Fig. l b the circuit that corresponds to the unitary V = (P+@ S+ P- @ S-)U @ 1, of the EV map.
+
Though the topic of the entanglement in the QRWs will not be investigated further here, it should be obvious from the above analysis that the CPTP map that implements some discrete time-step of a QRW,also generates quantum entanglement that can be studied by appropriate circuits and evaluated by effective measures, as usually is done in other cases of coupled quantum systems. Although quantum correlations have been generally accepted to be the common cause of all novel effects in the QRW performance, the exact evaluation of the entanglement resources needed in the course of a QRW is still an open problem.
196
Fig.1 .a
Fig. I .b
Fig. 1.c Figure 1. Fig.1 (a) Circuit generating entangled Bell states from product states using local Hadamard and non-local control-not gate; (b) Circuit generating the unitary evolution of the 1-step map E V ; (c) Circuit generating entangled coin-walker states from corresponding factorized ones using local unitary (e.g Hadamard gate in the case of the homonymous QRW), and non-local control-control-& gates, that form the map of the 2-step QRW E $ = E W ~wZ.
To establish further the connection among ARQs and QRWs we give four particular examples: a) the Euclidean QRW13 [iso(2)-QRW]: with the distance operator L Im) = m Im) , with its eigenspace HE =span{ Im) ,m E Z}, and its dual phase operator @ Icp) = cp 19) , with its eigenspace H,” =span{Icp) ,cp E [0,27r); related by a Fourier transform with H i . Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the distance random walk on Z, with C P T P map constructed with Kraus generators been the step operators in the distance operator eigenstates i.e S& Im) E& Im) = e f i 6 Im) = Im f 1), and the phase random walk on the circle S, with CPTP map constructed with Kraus generator been the step operators in the phase operator eigenstates
g},
=
197
i.e S* 19) e f i L 19 f I) ; ii) the Canonical Algebra QRW [ hw-QRW]: with the position operator Q Iq) = q Iq) , with its eigenspace H,& =span{ Iq) ,q E R; dq}, and its dual m o m e n t u m operator P Ip) = p Ip) , with its eigenspace H,' =span{ Ip) , p E R;d p } , related by a Fourier transform with H,&. Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the position random walk on R, with CPTP map constructed with Kraus generators been the step operators in the position operator eigenstates i.e Sk Iq) e f i P lq) = Iq f 1), and the m o m e n t u m random walk on R, with CPTP map constructed with Kraus generator been the step operators in the momentum operator eigenstates i.e Sk lp) ehiQ lP) = IP f 1); iii) the M - dimensional Discrete Heisenberg Group QRW [~M-QRW]: with the action operator N In) = n In) , with its eigenspace H," =span{ In > , n E ZM}, and its dual angle operator 0 IS,) = 29, IS,) , with its eigenspace H," =span{lS,), 19, E &ZM}, related by a finite Fourier transform with H,". Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the action random walk on ZM, with C P T P map constructed with Kraus generators been the step operators in the action operator eigenstates i.e S* In) K e *ie In) = h*l In) = In f 1), and the angle random walk on & Z M , with CPTP map constructed with Kraus generator been the step operators in the momentum operator eigenstates i.e S* 18), gfl = e * w IS,) = IS,*,) ; iv) the Coherent State QRW15[ CS-QRW]: with the annihilation operator a la) = a la) , with its eigenspace H: =span{la) ,a E C; %}.Two ARWs and its associated QRWs (modulo local unitary operators in coin spaces, as explained above), can be constructed: the annihilation random walk on C, with CPTP map constructed with Kraus generators been the step operators in the annihilation operator eigenstates i.e Sh la) = eJ(*') la) = D A la) ~ = la f 1).The step operators here identified as special case of the canonical coherent state displacement operator D*g = e*DatFBa = eJ(*g), have based the indicated step property on the following operator identity D,Dg = e-iaxfiDa+g, applied for the case of co-linear a,,B vector on complex plane. To give an explicit identification of the ARW based on the hw algebra constructed in Ref. 15, as a quantum random walk, and in particular as a CS-QRW, we make the following choices: the transition probabilities are p+ = p , p - = 1 - p , the coin state is pc = 10 >< 01, the U operator is U ( p , l - p ) , and the step operators are CS displacement operators with
=
IS,)
198
steps k/3 E C. Then the total 1-step evolution operator in the coin+walker system is V = PU , 18 S, = fiD+P c p D + f i ) , a n d t h e G D - P -@-s reduced walker evolves in 1-step by the CPTP &v(e,,,) = p D + g ~ ~ D i ~
Em=*
(
+
(1 - P)D-aeWD!P = T T c ( V Q c 63 e w v ' ) . For n steps the evolution of the walker has been chosen in Ref. 15 to be +((e,,,). It is important to notice that this is a choice based on the ARW construction methodology, and that our present treatment of the same walk as a QRW, sees the $(e,,,) type of evolution to result from a partial tracing of the coin system at every step. Our previous discussion of other types of tracing schemes motivates the study of CS-QRWs with delayed tracing, in order t o investigate phenomena such as enhanced or anomalous diffusion in ARWs. This problem will be taken up elsewhere. 5 . Discussion
We have outlined a mathematical framework where the conception of random walk and its associated statistical notions, and equations of motion, can both be studied in an algebraic and quantum mechanical manner. ARWs and QRWs appear to be two aspects of the same mathematical device, so their interconnection serves to conceptually clarify the common ground between them and to enrich the heuristics of formulating new problems and methodically searching for their solutions. Quantum random walks are important both as quantum algorithms to experimentally be realized and as modules in a general quantum computing algorithm-devise that could outperform some classical rival. The connection ARW-QRW could serve to generalize, unify and compare such algorithms. Also Quantum Information Processing concepts and tools, could be developed for ARW-QRWs. The step taken here is only a preliminary one towards developing such a theory. Finally, ARW-QRWs come with lots of free choices for its constituting parameters. To mention only one expected application in the field of Open Quantum Systems, we should emphasize the importance of choosing the functional in e.g the hw ARW. Various choices of functionals in terms of types of coherent state vectors, combined together with various choices of ordering the operator basis in the enveloping algebra U(hw), i.e normal, antinormal, symmetric etc, could serve as a guiding rule for constructing quantum master equations for open boson systems interacting with various types of quantum mechanical baths.
199
6. Acknowledgments
I wish t o thank the organizers of the Volterra-CIRM-Grefswald Conference for the opportunity to give a talk. Discussions with L. Accardi, U. F’ranz, R. Hudson, M. Schurmann, and with my collaborators A. Bracken and I. Tsohantjis, are gratefully acknowledged. I am also grateful to the anonymous referee for suggesting eq.(63). References 1. P. A. Meyer, Quantum Probability for Probabilists (Lect. Notes Math. 1538), (Springer, Berlin 1993). 2. M. Schiirmann, White Noise on Bialgebras (Lect. Notes Math. 1544), (Springer, Berlin 1993). 3. S. Majid, Foundations of Quantum Groups Theory (Cambridge Univ. Press, 1955), ff. chapter 5. 4. U. Franz andR. Schott, Stochastic Processes and Operator Calculus on Quantum Groups, (Kluwer Academic Publishers, Dodrecht 1999). 5. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Proc. 33rd Annual Symp. Theory Computing (ACM Press, New York, 2001), p.37. 6. D. Aharonov, A. Ambainis, J. Kempe and U. Vasirani, Proc. 33rd Annual Symp. Theory Computing (ACM Press, New York, 2001), p.50. 7. A. Nayak and A. Vishwanath, arXive eprint quant-ph/0010117. 8. J. Kempe, Proc. 7th Int. Workshop, RANDOM’OJ, p.354 (2003). 9. A. M. Childs, E. Farhi and S. Gutmann, Quantum Information Processing 1,35 (2002). 10. B. C. Travaglione and G. J. Milburn, Phys. Rev. A 65, 032310 (2002). 11. B. C. Sanders, S. D. Bartlett, B. Tregenna and P. L. Knight, Phys. Rev. A 67,042305(2003). 12. J. Kempe, Contemp. Phys. 44, 307 (2003). 13. A. J. Bracken, D. Ellinas and I. Tsohantjis, J. Phys. A : Math. Gen. 37, L91(2004). 14. E. Abe, Hopf Algebras (CUP Cambridge 1997). 15. D. Ellinas, J. Comp. Appl. Math.133, 341 (2001). 16. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic Press, New York, 1979). 17. R. Bhatia, Matrix Analysis (Spinger-Verlag, New York , 1997). 18. P. M. Alberti and A. Uhlmann, Stochasticity and Partial Order: Double Stochastic Maps and Unitary Mixing (Dordecht, Boston, 1982). 19. M. A. Nielsen, An Introduction to Majorization and its Applications to Quantum Mechanics (unpublished notes). 20. D. Ellinas and E. Floratos, J . Phys. A : Math. Gen. 32,L63 (1999). 21. B. D. Hughes, Random Walks and Random Environments Vol. I, (Clarendon Press, Oxford 1995) 22. J. Gillis, Quarterly J. Math., (Oxford, 2nd series)7, 144 (1956). 23. K. Lakatos-Lindenberg and K. E. Shuler, J. Math. Phys. 12,633 (1971).
200 24. J. R. Klauder and B.-S. Skagerstam, Coherent States (World Scientific, Singapore (1986) 25. A. Perelomov, Generalized Coherent States and their Applications, (Springer - Verlag, Berlin 1986). 26. E. D. Davies, Quantum Theory of Open System, (Academic, New York, 1973). 27. G. Lindblad, Non-Equilibrium Entropy and Irreuersibility, (Reidel, Dordrecht 1983). 28. M. 0. Scully and M. S. Zubairy, Quantum Optics, (Cambridge Univ. Press, Cambridge 1997), p. 255, 448, 453. 29. S. Stenholm, Phys. Scripta T12(1986). 30. J. Gea-Bauacloche, Phys. Rev. Lett. 59,543 (1987). 31. K. Kraus, States, effects and operations (Springer-Verlag, Berlin, 1983). 32. M. A. Nielsen and 1. L. Chuamg, Quantum Computation and Quantum Information, (Cambridge Univ. Press, Cambridge 2000). 33. S. Majid, Int. J . Mod. Phys. 8 , 4521-4545 (1993). 34. S. Majid, M. J. Rodriguez-Plaza, J . Math. Phys. 33,3753-3760 (1994). 35. P. Feinsilver and R. Schott, J. Theor. Prob. 5,251 (1992). 36. P. Feinsilver, U. F’ranz and R. Schott, J . Theor. Prob. 10, 797 (1997). 37. U . Franz and R. Schott, J . Phys. A: Gen . Math. 31 , 1395 (1998); 38. U. Franz and R. Schott, J. Math. Phys. 39, 2748 (1998). 39. D. Ellinas and I. Tsohantjis, J . Non-Lin. Math. Phys.8Suppl. 93(2001). 40. D. Ellinas and I. Tsohantjis,Inf. Dim. Anal.- Quant. Prob. 611 (2003). 41. G. H. Hardy, J. E. Littlewood and G. Polya, Messenger Math. 58, 145 (1929). 42. H. Sikic and M. V. Wickerhauser, Appl. Comp. Harm. Anal. 11, 147 (2001). 43. R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications, (Wiley, New York 1974). 44. D. Ellinas, et. al, work in progress. 45. P. J . Davis, Czrculant Matrices, (Wiley, New York 1979). 46. M. R. Schroeder, Number Theory in Science and Communication, (Springer, Berlin 1997). 47. C. D. Cushen and R. L. Hudson, J . Appl. Prob. 8, 454 (1972). 48. This second line of the equation has the form a Lindblad type master equation taken in the large temperature limit, where for the average number of thermal photons we have A NN A + 1 = (yI,c.f. Ref. 28. 49. H. Risken, The Fokker-Planck Equation, (Springer, Berlin 1996). 83 (1990).
DUAL REPRESENTATIONS FOR THE SCHRODINGER ALGEBRA
PHILIP FEINSILVER Department of Mathematics Southern Illinois University Carbondale, IL. 62901, U . S . A . RENE SCHOTT IECN and LORIA UniversitC Henri PoincarC-Nancy 1, BP 239, 54506 Vandoeuvre-lks-Nancy, fiance. Starting from the multiplication table for a basis of a Lie algebra, we show how to find a realization as vector fields in the case there is a Lie flag. Then we show how this is connected to finding coordinates of the second kind. We find the explicit form for coordinates of the second kind for the Schrodinger algebra in a particularly useful basis. These coordinates are related to dual vector fields that form an abelian Lie algebra generating an associated family of polynomials. Families of polynomials in quantum variables arise by specialization of the parameters of the first kind corresponding to quantum observables.
201
202
1. Introduction
The Schrodinger Lie algebra plays an important role in mathematical physics and its applications. It has been introduced and investigated as the algebra of symmetries of the free Schrodinger e q ~ a t i o n . The ~~~ re-J sulting structure of the semidirect product of the Heisenberg algebra and sl(2) was investigated in Ref. 11. Recently this algebra has appeared in the context of the square of white noise.’ And G. Pap is using it to compute the Fourier transform of Brownian motion on Heisenberg groups.15 This article is composed of two parts. In Part I, we present the calculation of dual representations of a Lie algebra using Lie flags and illustrate its application to the Schrodinger algebra. In Part 11, we recall the Splitting Lemma for coordinates of the second kind. For the Schrodinger algebra, these coordinates are found in two steps. First using a basis that reflects the semidirect product structure of the algebra the appropriate equations are solved. Since the “standard basis” is a permutation of that first basis, the coordinates of the second kind for the standard basis can be found using group theory. These coordinates are an essential part of a generating function for a family of polynomials giving a representation of a n interesting associated abelian Lie algebra. 2. Dual representations
First, dual representations, particular realizations of a Lie algebra in terms of vector fields, are defined along with the corresponding .rr-matrices. Then we recall the definition of Lie Aag and how it is used to find a dual representation. This approach is carried out for the Schrodinger algebra. The main advantage is that the coefficient matrices of the vector fields can be computed directly in terms of “partial adjoint matrices” which allow for fast symbolic calculations. 2.1. Dual representations and r-matrices
For a given Lie algebra 8, with basis {[I,. . . ,&}, we have the corresponding PBW-basis for the universal enveloping algebra, namely ordered monomials 0, IC > 0 such that for all f with supp f c {u : 1(u) = n}:
I1 f * h 112 < C ( n+ 1)“11f llzll h 112,
h E Z2(G).
(3)
There are equivalent: o G satisfies a Haagerup inequality o There exist CI> O , I C ~> 0 such that for all finitely supported
f :G +
C:
>0
, >~ 0 such that for all finitely supported nonnegative symmetric f : G + R+:
o There exist C2
Here f (2k) denotes the 2k-th convolution power of f ,further one may suppose in the last statement that 11 f 111 1. That is, f is a symmetric probability measure on G and f(2k)(e)is the probability that the random walk with transition probabilities p(g, h) = f(g-lh), starting at e, will return to the identity in 2k steps.
0, we find that the first component of all states limb+G Qi,A(qY @ $'), 1z1' > a, is a coherent state with fixed mean intensity Ic11' = a. If further PZ = this holds for the second component as well. This is the result of interest mentioned in the introduction. Clearly, we can apply the above theorem with this P unless 1z1' = a. The next results study the limiting behaviour of the left hand side in (8) for this and similar situations. First, we weaken the continuity condition and get a mixing effect.
d
m
,
Theorem 3.2. Suppose z E R and 6 > 0 are such that all Pi have left and right limits as r --+ 1z1' and
231 0
in the internal (1zI2- 6, 1zI2],the functions Pt,
0
are continuously differentiable. in the interval [ ! . I 2 , 1zI2 6 ) , the functions P;,
+
are continuously differentiable. Then
with
,,(@ @ ~ O ) A , A . By definition, for any f E L2(G, v),
W$f c2 $O)((Pl,
c
-
(P2)
$01 ( 7 1 A ( P l ) $ b (71 A (P2)$03 ( 7 2 A (Pl)$fk(72 A (P2)$f(71)$0(72)
71+72=v1+v2
- $01 (v1+v2) ((P1)%2(v1+p2) ((P2)$f((P1 - ~01(v1+vz)f((P1)~02(vpl+vz)f((P2).
+ Cp2)
This yields p ( ' ~ 1 (, ~ 2( , ~ 3( , ~ 4 = ) e-llzxAAlla$p(
vi+v2)zxA((~1)$ba(vi+vz)zxA((~2)
$b: (v3++'4)zXA ((P3)$bt(p3+v4)zX~ (p4)
where XA denotes the indicator function of the set A. We find that the state G&(V @ + O ) K has again property ( 7 1 ) [17,15],i.e. its kernel po depends only on particle numbers: pO((p1, ( ~ 2( ,p 3 , p 4 )
c
=
e-lz12totk+k'z'+l' ck,kt,Z,l' t
k,k',l ,I'EN
(l(Pll)X{k'})( ~ ' P 2 ~ ) x { Z } ) ( I ~ 3 ~ (1941) )x{l~}) where [(PI = ( P ( K )We . find from (for a precise definition of this formula in the general case see 114,161) x{k})
po((Pli(P2,(P3,(P4)
= 1~~\,(d[@l,d2])p((P1+dl,lpz+d2,(P3+@1,(P~+@2)
that (denoting @(k) = & ( k / C ( A ) ) ) t
Ck,kr,l,l'
=
-1zIZt
Fi\K(d[+l,@21) k+lGlI
+ 1+11+
1@21)
P,^@ + k'
k'+ Ik2 I
+ 1@11+ 1@21) + k' + l@l)+ J @ 2 J ) Z + v 3 ; ( 1 + 1' + 1@11 + 1@21)"+Id1 1 I421@1+82I
P:'@ + k' P:'(k
1
Now is it enough t o prove convergence of ck,k!,l,l, as t + 00 to a suitable limit. The reason is the following lemma which can be found in [3]. Lemma 4.1. If w,, w are normal states on B(3-1) such that for a total set
T23-1 wn(P+,+)n~mw(P+,+~)i ( 1cI,1cI' E T )
(10)
233
then ( w , ) , ~ ~ converges an n o r m t o w . We apply this lemma with the total set consisting of vector being in the span of {&xK : z E C} resp. orthogonal to it. For the orthogonal ones, on both sides (10) there is zero, for the others we use convergence of c : , k , , l , / , We find from definition of F and Pi for k,k‘, 1,l‘
We see for random variables X t , X t ck,k’,l,ll
~
1
~
that 1 2
~
~
=
(P(
+ty;xt
) , P ( l +l’+X,))X’
t
+ to
Additionally, we know X t / t +%(zI2. Upto here only the distribution t-w matters, but we can assume for convenience (due to the Skorohod r e p resentation Theorem, cf. e.g. [4]), that the convergence is actually almost surely on some artificial probability space. Due to (7) we find pz(r)pz(r’)( 5 1, such that we can apply the dominated convergence theorem. We want to find the limit by Taylor expansion. First, let p be one of Pi and apply the expansion
I~/?I(T’) +
P(2
+
E)
= P(2)
+
EP‘(Z)
+
O(E)
(11)
234 in an interval around \z(’ leading to
-
!l?Zpi(r)p:(r)= 0. This leads
Differentiating the first line in (7) yields to
k + k’ + X ” , $ ( P ( t+to
=1
+ 1’ + X t ) ) t
+to
+ o(t-1)
and thus
k+k’+Xt l+l’+Xt )) (@( t to ),p( t +to
+
xt =1
+ o ( X t / t ) = 1 + o(1).
This Shows
Since the left hand side gives the correct matrix element for (q5f11(lz12)z q5flz(lz12)z) K this completes the proof.
@
Proof. (of Theorem 3.2) We apply the same method, but derive the limit of c ; , ~ , , ~differently. ,~, It holds by the central limit theorem even
for some random variable Y and again we assume almost sure convergence on some artificial probability space. Rewriting
xt = ( Z l 2 t + (21JiY + o ( J i ) ,
(t
--+
00
)
we derive that
P ( X t / t > 1z12 eventually ) = 1/2 = lim P ( X t / t < 1z12 eventually t+m
1.
On both sets we may apply the above scheme to derive the result. Proof. (of Theorem 3.3) Here it is crucial to estimate for some 0 E C P2(12I2
+ E ) = m +4&)
(12)
235 what leads to (separately on {Y
since consideration of leads to
> 0)
and {Y
< 0))
1 for small
References 1. 0. Bratteli and D. Robinson Operator Algebras and Quantum Statistical Mechanics I 2nd edition Springer Berlin Heidelberg New York 1987 2. D. Carter and P. Prenter Exponential spaces and counting processes Z. Wahrscheinlichkeitstheor. Verw. Geb. 21:l-19 1972 3. E. Davies Quantum Theory of Open Systems Academic Press New York 1976 4. S. N. Ethier and T. G. Kurtz Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics John Wiley & Sons New York etc. 1986 5. K.-H. Fichtner and W. Freudenberg Point processes and the position distribution of infinite boson systems J . Stat. Phys. 47:959-978 1987 6. K.-H. Fichtner and W. Freudenberg Characterization of states of infinite boson systems I. On the construction of states of boson systems Commun. Math. Phys. 137:315 - 357 1991 7. K.-H. Fichtner and W. Freudenberg Remarks on stochastic calculus on the Fock space In L. Accardi, editor, Quantum Probability and Related Topics VII World Scientific Publishing Co. Singapore, New Jersey, London, Hong Kong 1991 pages 305 - 323 8. K.-H. Fichtner, W. Freudenberg, and V. Liebscher On Exchange Mechanisms for Bosons submitted 1998 9. K.-H. Fichtner, W. Freudenberg, and V. Liebscher Time Evolution and Invariance of Boson Systems Given by Beam Splittings Infinite Dimensional Analysis, Quantum Probability and Related Topics 1(4):511-531 1998 10. K.-H. Fichtner, W. Freudenberg, and M. Ohya Recognition and teleportation pages 85-107 11. K. Fichtner and M. Ohya Quantum teleportation with entangled states given by beam splittings Commun. Math. Phys. 222(2):229-247 2001 12. K. Fichtner and M. Ohya Quantum teleportation and beam splitting Commun. Math. Phys. 225(1):67-89 2002 13. W. Freudenberg On a Class of Quantum Markov Chains on the Fock
14.
15. 16.
17. 18. 19.
20. 21.
22.
Space In L. Accardi, editor, Quantum Probability & Related Topics IX World Scientific Publishing Co. Singapore 1994 pages 215 - 237 V. Liebscher Diagonal Integration and Diagonalized Versions Forschungsergebnisse der Fakultat fiir Mathematik und Informatik Jena Math/Inf/96/38 1996 V. Liebscher Beam Splittings, Coherent States and Quantum Stochastic Differential Equations Habilschrift FSU Jena 1998 V. Liebscher Diagonal Versions and Quantum Stochastic Integrals on the Symmetric Fock Space with Nonadapted Integrands Prob.Th.Rel.Fields 112:255-295 1998 V. Liebscher Using weights for the description of states of boson systems in preparation 2001 J. Lindsay Quantum and Noncausal Stochastic Calculus Prob. Th. Re1. Fields 97:65-80 1993 J. Lindsay and H. Maassen An integral kernel approach to noise In L. Accardi and W. uon Waldenfels, editors, Quantum Probability and Applications I11 volume 1303 of Lecture Notes in Mathematics Springer-Verlag Heidelberg 1988 pages 192 -208 J. Lindsay and H. Maassen The stochastic calculus of Bose noise Preprint 1988 H. Maassen Quantum Markov processes on Fock space described by integral kernels In L. Accardi and W. von Waldenfels, editors, Quantum Probability and Applications I1 volume 1136 of Lecture Notes in Mathematics Springer Berlin, Heidelberg, New York 1985 pages 361 - 374 P. Meyer Quantum Probability for Probabilists volume 1538 of Lecture Notes in Mathematics Springer Berlin Heidelberg New York 1993
ON FACTORS ASSOCIATED WITH QUANTUM MARKOV STATES CORRESPONDING TO NEAREST NEIGHBOR MODELS ON A CAYLEY TREE
FRANCESCO FIDALEO Dipartimento di Matematica Universitd di Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Roma, Italy, E-mail: fidaleoOOmat.uniroma2.it FARRUH MUKHAMEDOV Department of Mechanics and Mathematics National University of Uzbekistan Vuzgorodok, 700095, Tashkent, Uzbekistan, E-mail: far75mOOyandex.r~ In this paper we consider nearest neighbour models where the spin takes values in the set @ = {q1,v2,.,.,qq} and is assigned to the vertices of the Cayley tree .?I The Hamiltonian is defined by some given A-function. We find a condition for the function A to determine the type of the von Neumann algebra generated by the GNS - construction associated with the quantum Markov state corresponding t o the unordered phase of the A-model. Also we give some physical applications of the obtained result. 2000 Mathematical Subject Classification: 47A67, 47L90, 47N55, 82B20. Keywords: von Neumann algebra, quantum Markov state, A- model, Cayley tree, unordered phase, Gibbs measure, GNS - construction.
1. Introduction It is known that in the quantum statistical mechanics concrete systems are identified with states on corresponding algebras. In many cases the algebra can be chosen to be a quasi-local algebra of observables. The states on this algebra satisfying the KMS condition, describe equilibrium states of the quantum system. Basically, limiting Gibbs measures of classical systems with the finite radius of interactions are Markov random fields (see e.g. Ref. 8, 22). In connection with this, there arises a problem to construct analogues of non-commutative Markov chains. In Ref. 1 Accardi explored
237
238 this problem, he introduced and studied noncommutative Markov states on the algebra of quasi-local observables which agreed with the classical Markov chains. In Ref. 3, 15,2, modular properties of the non-commutative Markov states were studied. In Ref. 10 Fannes, Nachtergale and Werner showed that ground states of the valence- bond- solid modles on a Cayley tree were quantum Markov chains on the quasi-local algebra. In the present paper we will consider Markov states associated with nearest neighbour models on a Cayley tree. Note the investigation of the type of quasi-free factors (i.e. factors generated by quasi-free representations) has been an interesting problem since the appearance of the pioneering work of Araki and Wyss5. In Ref. 21 a family of representations of uniformly hyperfinite algebras was constructed, which can be treated as a free quantum lattice system. In that case factors corresponding to those representations are of type I&, X E (0,l). More general constructions of product states were considered in Ref. 4. Observe that the product states can be viewed as the Gibbs states of Hamiltonian system in which interactions between particles of the system are absent, i.e. the system is a free lattice quantum spin system. So it is interesting to consider quantum lattice systems with non-trivial interactions, which leads us, as mentioned above, to consider the Markov states. Simple examples of such systems are the Ising and Potts models, which have been studied in many papers (see, for example, Ref. 24). We note that all Gibbs states corresponding to these models are Markov random fields. The full analysis of the type of von Neumann algebras associated with the quantum Markov states is still an open problem. Some particular cases of the Markov states were considered in Ref. 11,14,17,18,19. The present paper is devoted t o the type analysis of some class of diagonal quantum Markov states, which correspond to a X-model on the Cayley tree, in which spin variables take their values in a set @ = {ql, ..., q q } ,where r]k E k = 1,.. . ,q. Observe that the considered model generalizes a notion of X-model introduced in Ref. 23, where the spin variables take their values f l .
2. Definitions and preliminary results
The Cayley tree rk of order k 2 1 is an infinite tree, i.e., a graph without cycles, such that each vertex of which lies on k 1 edges. Let I? = (V,A), where V is the set of vertices of rk,A is the set of edges of I?. The vertices x and y are called nearest neighbor, which is denoted by I =< x , y > if
+
239 there exists an edge connecting them. A collection of the pairs < z, z1>, . . . , is called path from the point z to the point y. The distance d(z,y), z, y E V, on the Cayley tree, is the length of the shortest path from z to y. We set Wn = {Z E v J ~ ( x2'), = TI},
Ln = ( 1
=< Z , Y >E L ~ z , YE Vn},
for an arbitrary point xo E V. Denote 1x1 = d(x,xo), x E V. Denote S(Z) = {y E Wn+l :d ( ~ , y = ) I}, x E Wn. This set is made of the direct successors of x. Observe that any vertex x # xo has k direct successors and xo has k + 1. Theorem 2.1. There exists a one-to-one correspondence between the set V of vertices of the Cayley tree of order k 2 1 and the group Gk+l of the free products of k 1 cyclic groups of the second order with generators
+
al,a2,.**,ak+l.
Consider a left (resp. right) transformation shift on follows. For go E Gk+l we put
T g o h= goh (resp. T g o h= hgo,), h E
Gk+l
defined as
Gk+l.
It is easy to see that the set of d l left (resp. right) shifts on Gk+l is isomorphic t o the group Gk+l. Let @ = { q l , q 2 ,...,qQ}, where ql,q2,..., qq are vectors in RqP1, such that
We consider models where the spin takes values in the set @ = { q l , q 2 , ...,q q } and is assigned to the vertices of the tree. A configuration o on V is then defined as a function x E V + o(x) E @; the set of all configurations coincides with R = &. The Hamiltonian is of an A-model form : HA(0)
=
c
<X,Y>
A(+),
4Y); J ) ,
(2)
240
where J E Iwn is a coupling constant and the sum is taken over all pairs of neighboring vertices < z,y >, a E R. Here and below A : @ x @ x Rn 3 R is some given function. We note that A-model of this type can be considered as a generalization of the Ising model. The Ising model corresponds to the case q = 2 and X(Z, y; J) = -J X Y . We consider a standard a-algebra 3 of subsets of R generated by cylinder subsets, all probability measures are considered on (0,3).A probability measure p is called a Gibbs measure (with Hamiltonian H A )if it satisfies the DLR equation: for n = 1,2, ... and a, E GVn:
where
Y $ ~ ~ + , is
the conditional probability
(0,) YWIW,+l vn
= 2-l (w IWn+l) e ~ ( - P ~ ( IaIwsIwn+1)1.
where /3 > 0. Here snlvn and WI W , + ~ denote the restriction of a,w E R to V, and Wn+l respectively. Next, a, : 2 E V, 3 a,(z) is a configuration in V, and H(a,,llwl~,,+,)is defined as the sum H(o,) +U(~,,WIW,,+~) where
c
H(an) =
<X,Y>€L,
U(an,WIwn+l) =
A ( ~ n ( ~ ) , % b J), );
c
A(aTa(z),w(y); 4.
<x,y>:X€Vn,yEWn+l
Finally, Z ( w l ~ , + ~stands ) for the partition function in V, with the boundary condition w I : Z(wlwn+l) =
C
e~(-PH(~~IIwIwn+A. cnEOVn Since we consider nearest neighbour interactions, the Gibbs measures of the A-model possess a Markov property: given a configuration w, on W,, random configurations in V,-l and in V \ Vn+l are conditionally independent. It is known (see Ref. 26) that for any sequence dn) E R, any is a Gibbs measure. Here limiting point of measures
52,,)
IWn+l
is a measure on fl such that Vn’
0,
otherwise.
> n:
52 ,wn+l n,
24 1
We now recall some basic facts from the theory of von Neumann algebras. Let B ( H ) be the algebra of all bounded linear operators on the Hilbert space H ( over the field of complex numbers C). A weak (operator) closed *-subalgebra N in B ( H ) is called von Neumann algebra if it contains the identity operator 1. By Proj(N) we denote the set of all projections in N . A von Neumann algebra is a factor if its center
Z(N := {x E N : xy = yx,
Vy
EN}
is trivial, i.e., Z ( N )= { A 1 : X E C}. The von Neumann algebras are direct sum of the classes I (In,n < 00, Im), I1 (IIl,II,) and 111. Further, a factor is of only one type among these listed above, see e.g. Ref. 30. An element x E N is called positive if there is an element y E N such that x = y*y. A linear functional w on N is called a state if w(x*x) 2 0 for all x E N and w(1) = 1. A state w is said to be normal if ~ ( s u p x , ) = supw(x,) for any a
a
bounded increasing net {xa} of positive elements of N . A state w is called trace (resp. faithful) if the condition w(xy) = w(yx) holds for all x,y E N (resp. if the equality w(x*x) = 0 implies x = 0). Let N be a factor, w be a faithful normal state on N and aW , be the modular group associated with w (see Definition 2.5.15 in Ref. 6). We let I?(&) denote the Connes spectrum of the modular group a; (see Definition 2.2.1 in Ref. 6). For the type I11 factors, there is a finer classification. Definition (Ref. 9). The type I11 factor N is of type (i) 1111,if r(aw) = (ii) IIIx, if r(aw) = {nlogX,n E Z}, X E (0,l); (iii) I&, if r(aw) = (0); see, e.g. Ref. 6, 28 for details of von Neumann algebras and the modular theory of operator algebras.) 3. Construction of Gibbs states for the A-model
In this section we give a construction of a special class of limiting Gibbs measures for the A-model on the Cayley tree. Let h : x + h, = (hl,,, h2,z, ...,hq-l,z) E be a real vector-valued function of x E V . Given n = 1 , 2 , ... consider the probability measure ~ ( on aVndefined by /Jn)(an)= Z,-l exp{-PH(an)
+
c
XEW,
hzo(x)}
7
(3)
~
1
242
Here, as before, a, : x E V, partition function:
+ on(x) and
2, is the corresponding
X€W,
e n EQv,
The consistency conditions for p(,)(o,), n 2 1 are ~ p ( ~ ) ( a , - l , a ( n=) p(n-l)(a,-l), )
(4)
a(")
where a(n) = { a ( x ) , xE Wn}. Let V1 c V2 c ... Ur?, V, = V and p 1 , p 2 , ...be a sequence of probability measures on aV1, Ovz, ... satisfying the consistency condition (4). Then, according to the Kolmogorov theorem (see, e.g. Ref. 25), there exists a unique limit Gibbs measure jJh on R such that for every n = 1,2, ... and on E aVnthe equality holds
0
P {alv, = an}
= P(%,).
(5)
Further we set the basis in RQ-l to be ql, 72,..., qq-l. The following statement describes conditions on h, guaranteeing the consistency condition of measures p(,) (a,). Theorem 3.1. The measures ,u(,)(a,), n = 1,2, ... s a t i s b the consistency condition (4) if and only if f o r any x E V the following equation holds:
h', =
c
F(h',,X) >
(6)
l/ES(X)
Here, and below ha stands f o r the vector *hx and F : RQ-' function is F(h;X)= (Fl(h;X),..., Fq-l(h;X)),with
+ RQ-' (7)
Fi(h,,h2,...,hq-,;X)
i = 1,2, ...,Q - 1, h = ( h l ,...,hq-1). The proof uses the same argument as in Ref. 18,19. Denote
V = { h = (h, E EXq-'
:x E
F(h,,X), V X E V } .
V ) : h, = l/ES(X)
According to Theorem 3.1 for any h = ( h x , x E V ) E V there exists a which satisfies the equality (5). unique Gibbs measure
243
If the vector-valued function ho = (h, = (0, ...,0 ) , x 6 V)is a solution, i.e. ho E 2) then the corresponding Gibbs measure p p ) is called the unordered phase of the X-model. Since we deal with this unordered phase, we have to make an assumption which guarantees us the existence of the unordered phase. Assumption A. For the considered model the vector-valued function ho = (h, = (O,O, ...,0 ) , x E V) belongs to 2). This means that the equation (6) has a solution h, = ho = 0,x E V. According to Theorem 2.1 any transformation S of the group Gk+l induces an automorphism ,$ on V. By &+I we denote the left group of shifts of Gk+l. Any T E G k + l induces a shift automorphism 5! : + by (Fa)(h)= a ( T h ) , h E Gk+i, 0 E R . It is easy to see that p e l o = p o('1 for every 5! E G k + l . AS mentioned above, the measure p c ) has a Markov property (see, Ref. 27). Assumption B. We suppose that the measure p p ) enables a mixing property, i.e. for any A, B E 3 the following holds
Note that the last condition is satisfied, for example, if the phase transition does not occur for the model under consideration.
4. Diagonal states and corresponding von Neumann algebras
Consider C*-algebra A = @lykMq(c),where Mq(C!)is the algebra of q x q matrices over the field C! of complex numbers. By e i j , i , j E {1,2, ...,q} we denote the basis matrices of the algebra M q ( C ) . We let CM,(@)denote the commutative subalgebra of Adq(@)generated by the elements eii i = {1,2, ...,q } . We set CA = @ykCMq(C). Elements of commutative algebra CA are functions on the space R = { e l l , ..., eqq}". Fix a measure p on the measurable space (0,B ) , where B is the 0-algebra generated by cylindrical subsets of R. We construct a state w p on A as follows. Let P : A + CA be the conditional expectation, then the state w p is be defined by wp(x) = p ( P ( z ) ) ,x E A, here p(P(x)) means an integral of a function P ( z ) under measure p, i.e. p(P(x)) = J, P(x)(s)dp(s) (see Ref. 29). The state is called diagonal.
244
By w c ) we denote the diagonal state generated by the unordered phase pt;"). The Markov property implies that the state up)is a quantum Markov state (see 3). On a finite dimensional C*-subalgebra Avn = @vnMq(C) c A
we rewrite the state
up) as follows
where tr is the canonical trace on Av,. The term X(a(x)o(y);J) in (4) is given by a diagonal element of Mq(C) @ Mq(C) in the standard basis as follows B(1) 0 . . .
P X o ( x ) ,4 4 ) ;
4=
(10)
. .. Here, B ( k )= ( b i j , k ) : , j = l , k = l,dots,q are q x q matrices, and
Consequently, using (9) and from (10),(11) (cp. Ref. 26, Ch.1, $1)the form of Hamiltonian fi(Vn) in the standard basis of Av, (i.e. under the basis matrices) is regarded as
0 Denote M = 7r (X)(A)",where T WO
(A)
WO
. ... . . . .B('3)
0
is the GNS - representation asso-
ciated with the state w c ) (see Definition 2.3.18 in Ref. 6). Our goal in this section is to determine a type of M . Remark. In Ref. 29 general properties of a representation associated with diagonal state were studied, but there concrete constructions of states were not considered. In Ref. 20 a deep classification of types of factors generated by quasi-free states has been obtained. For translation-invariant Markov states the corresponding type analysis has been made in Ref. 13.
245
The investigation of the type of factor arising from translation invariant or periodic quantum Markov states on the one dimensional chains is contained in Ref. 11. Now we define translations of the C*-algebra A . Every T E !&+I induces a translation automorphism rT : A + A defined by
Z€V"
XEV,
Since measure 111"' satisfies a mixing property (see (8)), then we can easily obtain that w?) also satisfies the mixing property under the translations { T T } T ~ B ~,+ i.e. , for all a, b E A the equality holds
According to Theorem 2.6.10 in Ref. 6, the algebra M is a factor. We note that the modular group of M associated with is defined by
up)
(A) CTYO
(2)=
where R(A) =
lim exp{itfi(A)}zexp{-itl?(A)},
Atrb
C
x
EM
,
(12)
It is well know that the last limit exists
<x,y>EA
if a suitable norm of the potential l? is finite (see Theorem 6.2.4. in Ref. 7). First of all, we recall the definition of the norm of a potential Q = CxCrb Q(W
where d > 0. Here @(X)E Ax = @ x M q ( C ) . NOW we compute 1lRlld:
m~ w,k 1 lOgpij,kI < co .
Hence the norm of l? is finite, therefore the limit in (12) exists. By M u one denotes the centralizer of w:'), which is defined as (A)
M" = {z E M
:
op
(z) = 2,
t E R} .
Since up)is Gibbs state, according to Proposition 5.3.28 in Ref. 7, the centralizer M u coincides with the set M ( A ) = {z E M : w F ) ( z y ) = ~ F ) ( y z ) y, E M } , (13) wo
246
where we denote by w r ) also the normal extension of the state under consideration t o all of M . By II[n]we denote the group of all permutations y of the set V, such that $22)
= 2, 2 E
w,
Every y E D[n] defines an automorphism ar : M
.rcn,”,,
a,) =
+M
by
n,,, 8
Mp(@) = id,
where id is the identity mapping. Denote
so = UIarIY E II[nl}. Simply repeating the proof of a proposition in Ref. 16, we can prove the following Lemma 4.1. T h e group
Go = {a E SOI wr’(a(x))= w p ) ( x ) ,
acts ergodically o n M , i.e. the equality a(.)
2
EM},
= x, a E Go implies x = 8 i ,
e E c. Lemma 4.2. T h e centralizer M u i s a factor of type IIl.
Proof. From the definition of the automorphism a7 (see (14)), it is easy to see that every automorphism a E Go is inner, i.e. there exists a unitary u, E M such that a(.) = u,xuL,x E M . From the condition up)o a = we find
up)
It follows from (13) that u, E M ( A ) . According to Lemma 4.1 the group *O Go acts ergodically, this means that the equality u,x = xu, for every a E Go implies x = 81,B E C. Hence, we obtain {u,la E Go}’ = Cl. Since Mu’ c {u,}’ we then get
M‘‘nM=ci. In particular Mu’ n M u = @I.This means that M“ is a factor.
I7
247
NOW we are able to prove main result of the paper (compare with the analogous result in Ref. 11). Theorem 4.1.
(i) If the fraction
is rational f o r every i, j , m, 2, k , p , u , v E 1,2, ...,q, whenever the denominator is different f r o m 0, then the von Neumann algebra M associated with the quantum Markov state corresponding t o the unordered phase of A-model (2) o n a Cayley tree is a factor of type IIIe . (ii) If M is a type of 1111factor, then all the fraction cannot be rational. Proof. It is known (see Proposition 2.2.2 in Ref. 9) that Connes' spectrum r(a)of group of automorphisms Q: = { Q : ~ } ~of~von G Neumann algebra M has the following form
r(Q)= n { S p ( a e ) l e E P r o j ( Z ( M " ) ) , e # O},
(15)
where a e ( z )= Q:(eze),z E e M e and Z ( M " ) is the center of subalgebra
M" = {z E M : a g ( z )= z,
9 E G}.
Here, Sp(a) be the Arveson's spectrum of group of automorphisms a: (see for more details Ref. 9,28). By virtue of Lemma 4.2 we have Z ( M " ) = Cl. The equality (15) (a) implies r(&))= sp(ow0 1. We now consider the operator H(V,) = C CP,,. We let ELn
Sp(fi(Vn)) denote the spectrum of the operator R(Vn).Setting (A)
oyo 'n(z) = exp{itfi(V,)}a:exp{-itfi(V,)},
a: E M
,
we obtain S p ( a W ~ ' l= n )Sp(fi(V,))-Sp(fi(Vn)) = {A-p : A,p E Sp(fi(V,))}. (16)
It is clear that ,BA(qi, q k ; J ) E Sp(&(Vn)),V i , k E 1 , . . .,q. Formula (16) then implies that Sp(oWr)9")is generated by elements of the form A ( r l ~ , ~ ~ ; J ) - A ( r l r C , r l ~ ; iJ,)j , k , Z E l , . . . q *
248
Since XX ((V9 ~b~,Tp;J)-X('lu 9j'J)-~(Vm~91'J) is a rational number, then there is a number t9u ; J ) y E (0,l) and integers mi,j,k,l E 2, (i,j,k,1 E {1,2, ...,a}) such that X(77i,qj;J) - X(77k,77l3 J ) = mi,j,k,llOgy*
(17)
Hence we find that an increasing sequence { E ( n ) }of subsets Z such that E(-n) = -E(n) and Sp(H(V,)) = {mlogy}mEE(n)is valid. It follows that Sp(o"b*)) c (nlog7)nEz. Hence there exists a positive integer d E Z such that we have (A)
qawo
) = { n log y d } n G z .
This means that M is a factor of type 1110, 8 = rd.
0
5. Applications and examples 5.1. Potts model
We consider the Potts model on the Cayley tree regarded as
rk whose Hamiltonian is
where J E R is a coupling constant, as usual < 2, y > stands for the nearest neighbor vertices and as before a ( . ) E @ = (71,q 2 , ...,qP}. Here 6 is the Kronecker symbol. Equality (1) implies that
for all z,y E V. The Hamiltonian H ( a ) is therefore
H ( a )= -
c
J'a(z)a(?/) ,
(18)
EL
where J' = q-lJ. 9 Hence the A-model is a generalization of the Potts model, that is in this case the function X : @ x @ x R + R is defined by A(z, y; J ' ) = -J'(z,y). Here, z, y E Rq-' and (2, y) stands for the scalar product in Rq-'. From (1)it is easy to see that
249
From (19) and (6) we can check that the assumption A (see section 3) is valid for the Potts model. So there exists the unordered phase. From (19) we can find that the fraction A(qi ,qj ;J’)- Vqrn,qi; J’) takes values A ( q k , q p ; J ’ ) - A(qzt,%J; J’Y k l and 0. So by Theorem 4.1 a von Neumann algebra M is a I&-factor.
{(if?)}.
From (17) we may obtain that I#J = exp -
Hence we obtain the
following
Theorem 5.1. The von Neumann algebra M corresponding t o the unordered phase of the Potts model (18) o n a Cayley tree is a factor of type III@h,f o r some k E Z, k
> 0, where
# IJ
= exp
Remark. If 4 = 2 the considered Potts model reduces to the Ising model, for this model analogous results were obtained in Ref. 17. For a class of inhomogeneous Potts model similar result has been also obtained in Ref. 18. 5.2. Markov random fields
In this subsection we consider a case when A ( x , y) function is not symmetric and the corresponding Gibbs measure is a Markov random field (see Ref. 27). Let P = ( ~ i j ) & =be ~ a stochastic matrix such that p i j > 0 for all i,j E 1,d. Define a function A ( x , y ) as follows: m i , q j ) = - 10gpij 7
(20)
for all i, j E { 1,.. . ,d } . From now on, we will consider the case /3 = 1 and 4 = d. It is easy to verify that Assumptions A and B, for the defined function A, are satisfied. By p we denote the corresponding unordered phase of the A-model. Observe that if the order of the Cayley tree is k = 1 then the measure p is a Markov measure, associated with the stochastic matrix P (see Ref. 27). By w,, one denotes the diagonal state corresponding to the measure p on C*-algebra A = @ , r b n / i d ( c ) .
Theorem 5.2. Let P = (pij)&=l be a stochastic matrix such that pij > 0 for all i, j = 1,. . .d and at least one element of this matrix is different from 112, and w,, be the corresponding Markov state. If there exist integers rnij i, j E (1,. . . ,d } , and some number 01 E (0,l) such that
250 then T,,,,(A)” , is a factor of tppe 1110 for some 6 E (0’1).
In order t o prove this theorem it should be used the rationality condition of Theorem 4.1. Namely, if the condition (21) is satisfied then using (20) one can see that the rationality condition holds, so we get the assertion. Remark. If all elements of the stochastic matrix P equal to 1/2 then the corresponding Markov state up is a trace and consequently T+ (A)” is the unique hyperfinite factor of type 111. Remark. There is a conjecture (see for example, Ref. 31) that every factor associated with GNS representation of a Gibbs state of a Hamiltonian system having a non-trivial interaction is of type 1111. Theorems 4.1 and 5.2 show that the conjecture is not true even if the Hamiltonian has nearest neighbour interactions. Acknowledgements The final part of this work was done within the scheme of NATO-CNR Fellowship at the Universita di Roma ’Tor Vergata’. We are grateful to L. Accardi and E. Presutti for useful comments and observations.
References 1. L.Accardi, On noncommutative Markov property, f i n k t . anal. i ego proloj. 8(1975), 1-8. 2. L.Accardi,F. Fidaleo, Non homogeneous quantum Markov states and quantum Markov fields, J . finct. Anal. 200 (2003), 324-347. 3. L.Accardi, G.Frigerio, Markovian cocycles, Proc. R.Ir. Acad. 83A(1983), 245273. 4. H. Araki, E.J. Woods, A classification of factors Publ. R.I.M.S. Kyoto Univ. 3(1968), 51-130. 5. H. Araki, W. Wyss, Representations of canonical anticommutations relations. Helv. Phys. Acta 37(1964), 136-159. 6 . 0. Bratteli, D. Robinson, Operator algebras and Quantum Statistical Mechanics I., Berlin: Springer-Verlag, 1979. 7. 0. Bratteli, D. Robinson, Operator algebras and Quantum Statistical Mechanics II. Berlin: Springer-Verlag, 1981. 8. R.L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity, Theor. Probab. Appl. 1 3 (1968), 197-224. 9. A. Comes, Une classifications das facteurs de type 111, Ann. Ec. Norm. Sup. 6(1973), 133-252. 10. M.Fannes,B. Nachtergaele, R.Werner, Ground states of VBS models on Cayley trees, J.Stat. Phys. 66(1992), 939-973.
251 11. F. Fidaleo F., F.Mukhamedov, Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras, preprint, 2004. 12. N. Ganikhodjaev, A group representations and automorphisms of Cayley tree, Dok1.Akad.nauk Rep. Uzb. 1990, 3-6. 13. N.N.Ganikhodjaev, F.M.Mukhamedov, Markov states on quantum lattice systems and its applications, Methods of f i n c t . Anal. and Topology, 4(1998), n.3, c. 33-38. 14. V.Ya.Golodets, S.V.Neshveyev, Non-Bernoullian K-systems, Commun. Math. Phys. 195( 1998), 213-232. G.N.Zholtkevich, On KMS-Markovian states, 15. V.Ya.Golodets, Theor.Math.Phys. 56(1983), 80-86. 16. W. Kriger, On the finitary isomorphism of Markov shifts that have finite coding time, 2.Wahr.uerw. Geb. 65 (1983) 323-328. 17. F.M. Mukhamedov, Von Neumann algebras corresponding translation - invariant Gibbs states of Ising model on the Bethe lattice, Theor. Math. Phys. 123(2000), n.1, 489-493. 18. F.M.Mukhamedov, U.A.Rozikov, Von Neumann algebra corresponding t o one phase of inhomogeneous Potts model on a Cayley tree, Theor.Math.Phys. 126(2001), n.2, c. 206-213. 19. F.M. Mukhamedov, U.A. Fbzikov, On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras. J. Stat. Phys. 114(2004), 825-848 20. T.Murakami, S.Yamagami, On types of quasi-free representations of Clifford algebras, Publ. R.I.M.S. Kyoto, Uniu. 31(1995), 33-44. 21. R. Powers, Representation of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. Math. 81(1967), 138-171. 22. C. Preston, Gibbs states on countable sets (Cambridge University Press, London 1974). 23. U.A. Rozikov: Description of limiting Gibbs measures for A-models on the Bethe lattice, Siberian Math. Jour. 39(1998), 427-435. 24. C.M. Series, Ya.G. Sinai: Ising models on the Lobachavsky plane, Commun. Math. Phys. 128(1990), 63-76. 25. A .N.Shiryaev, Probability, M.Nauka, 1980. 26. Ya.G. Sinai, Theory of phase transitions: Rigorous results, Pergamon, Oxford, 1982. 27. F.Spizer, Markov randon fields on trees, Ann. Prob. 3(1975), 387-398. 28. S. Stratila, Modular theory in operator algebras, Bucuresti, Abacus Press, 1981. 29. S.Stratila, D.Voiculescu: Representations of AF-algebras and of the group U(m), Lec. Notes Math. 486(1975), Springer-Verlag. 30. StrtitilB S., Zsid6, L. Lectures on uon Neumann algebras, Abacus press, Tunbridge Wells, Kent, (1979). 31. M. Takesaki, Automorphisms and von Neumann algebras of type 111. Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 111-135,Proc. Sympos. Pure Math., 38, Amer. Math. SOC.,Providence, R.I., 1982.
ON QUANTUM LOGICAL GATES ON A GENERAL FOCK SPACE
WOLFGANG FREUDENBERG Brandenburgische Technische Universitat Cottbus, Institut f i r Mathematit, Postfach 101344, 03013 Cottbus, Germany, E- Mail:
[email protected] MASANORT OHYA Department of Information Science, Tokyo University of Science, Noda City, Chiba 278, Japan, E-Mail:
[email protected] NOBURO WATANABE Department of Information Science, Tokyo University of Science, Noda City, Chiba 278, Japan, E-Mail: watanabe0is. noda.tus. ac.jp In this paper we investigate quantum logical gates of Fkedkin type. The information 0 and 1 will be encoded by coherent states on a general Fock space, the interaction between the inputs will be a general beam splitting procedure. The interaction in the control gate is given by a unitary operator. The aim of the paper is to give examples for quantum channels transmitting some information such that the gate will fulfill the truth table. For the output we get simple explicit expressions. Our aim is to construct a gate using the well-known splitting procedures and coherent states.
1. Introduction In [15]Fredkin and Toffoli proposed a logical conservative gate on the base of which Milburn [25] constructed a quantum logical gate using a Kerr medium. The gate is composed of two input gates 11, 12, and one control gate C. Two inputs come t o a first beam splitter and produce two outputs. One of them passes via a control gate to the second beam splitting
252
253 apparatus, the second output (of the first beam splitting procedure) passes directly to the second beam splitter. After the second beam splitting we will have two final outputs 01. 0 2 .
Figure: Model of a Redkin-Tofloli-Milburn gate The aim of the paper is to give an example for quantum channels transmitting some information such that the gate will fulfill the truth table given below. Hereby, the signals 0 and 1 will be encoded by two different coherent states. One of these states also could be the vacuum state. The control gate if switched on will change the information ( 0 , l ) into (1,O) but leave the informations (0,O) resp. ( 1 , l ) unchanged. Input 11 0 0 1 1 0 0
1 1
Input 0
1 0 1 0 1 0
1
I2
Control State 0 0 0 0 1 1 1 1
Output O1 Output O2 0 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1
We will consider quantum channels for states on the symmetric Fock space over a general metric space G equiped with a measure u. In most concrete models G will be the Euclidean space Rd, and u will be the Lebesgue measure on Rd. The splitting rates a,@ must not be constants but arbitrary measurable functions from G to C. satisfying 1a(z)I2 I@(z)12 = 1 for all z E G. It turns out that for this model the whole process can be described in an explicit way. For the output state we get simple expressions. In [26,
+
254 281 the channel is described by a mapping V having on coherent vectors expe@exp, with 6, y E C. the simple form
= exPcre+pr -Be+x,. (1) A similar description is given there for photon number states. In the present paper we get formally the same description but with 6, y being functions from G t o C. So the description given in [26] fits into our context in the case that G consists of one point. The interaction in the control gate is given by an operator of multiplication and seems to be rather an artificial one. With respect to this control gate the results presented in this paper should be considered only as a first step to realize such quantum logical gates. In all calculations we have to assume nothing about the two exponential vectors (coherent states) carrying the information 0 or 1. For instance one can take the vacuum state and a coherent (non-vacuum) state. Another possibility is to choose two exponential vectors generated by functions being orthogonal or with disjoint support. So there is a big choice of representatives one can choose for building the logical gate. The present paper is an enlarged and generalized version of [18, 171. Another model where the action in the control gate is given by beam splitting procedures will be given in [16]. Ve-0
Be-,
2. The General Model Let d be the von Neumann-algebra of all bounded linear operators on some separable Hilbert space. By S(d) we denote the set of normal states on the algebra A. In the following wo E S(d)and 70 E S(d)will denote the input states, w1 E S(d)and r)l E S(d)will be the output states after the first beam splitting, Q E S(d)will be the control state and G; the output after passing the control gate. W; E S(d)and 771 E S(d)will be the inputs for the second beam splitting, and finally w2 E S(d)and r)2 E S(d)are the final outputs.
Definition 2.1. A mapping E' : S(d@d)+ S(d@d) we call a completely positive compound channel if the dual map E : d@d+ d@d given by the relation J(E(C))= E*("
l
Gn
(2CLj)
l.44zl,...,zn])
(YEtW.
j= 1
(8) Hereby, xy denotes the indicator function of a set Y and o is the empty configuration in M , i.e. o(G) = 0. Observe that F is a a-finite measure. Since Y was assumed to be diffise one easily checks that F is concentrated on the set
M , := {'p E M : 'p({z}) 5 1 for all
2
E G}
of so called simple counting measures (i.e. without multiple points).
(9)
257 Definition 3.1. M := L2(M,m,F) is called the (symmetric) Fock space over G.
Remark: Usually one defines the symmetric Fock space I'(31) over a Hilbert space 31 as the direct sum of the symmetrized tensor products 31$$Lrnof the = ,; 7-lgm As,we . remarked already underlying Hilbert space 31, i.e. I?(%) = @ above we introduced the Fock space in a way adapted to the language of counting measures especially to avoid symmetrization procedures. A further advantage of the definition given above is that the Fock spwe over a L2-space again will be a L2-space. It is very easy to show that I'(7-l) and M are isomorphic (cf. [12]). For details we refer to [13, 14, 191 but also e.g. to [21]where a similar definition of the Fock space is given. Observe that M is again a separable Hilbert space. Now, for each n 2 1 let M" := M@" be the n-fold tensor product of the Hilbert space M . Obviously, M" can be identified with L2(Mn,F"). Basic for the proofs in this paper will be that integration in M n (with respect to Fn)can be replaced by an integration with respect to F.
Lemma 3.1. Let f : M" Then
+ C be integrable with respect to F"
(or 2 0).
Hereby, q3 5 cp means that q3 is a subconfiguration of cp, i.e. cp - q3 E M . The proof of the first part in (10) for the case n = 2 one can find e.g. in [ll 1, in [13 ] or in [22, 241 where the above lemma is called
simple induction shows that (10) is valid for all n (10) is just a reformulation of the first one.
-lemma. A $ 2 2. The second part of
258
Definition 3.2. For a given function g : G M + CC defined by exp,(cp) =
{
+ C the function
exp, :
if cp= o,
1
n g(z) if
=ED
cp
#o
(11)
V(I.))>O
is called exponential vector generated by 9. We make use of the following well-known properties of exponential vectors: Lemma 3.2. Let f and g be functions from G to C and ( p , c p ~ , c p ~be elements from M . Then we have exPf(cpl+
92)
= expf('p1) . expf('p21,
(12)
c
e x P f ( a . exp,(cp - $1, G5v exPf.,(cp) = exPf(cp) * exp,(Cp).
expf+,('p) =
llexpg[l& = el1g1lt,(G,y) (expf, exp,)M = e(f9 9
) ~ 2 ( ~ 8 ~ )
(13)
(14)
(9 E L 2 G 41,
(15)
(f, 9 E L2(G,v)).
(16)
Observe that exp, E M if and only if g E L2(G,v). Further, it is wellknown that the linear span of exponential vectors from M is dense in M . The von-Neumann algebra L ( M ) of all bounded linear operators on the Fock space M we will denote by d and the tensor product d@d of the vonNeumann algebras A by d2.Obviously, we may identify d2and L ( M 2 ) . The identity in d we denote by 11. For g, h E L,(G,v) we denote by exph@exh,i. e. Bg,hQ(cp) =
Bg,h
the integral operator with kernel
J ,F(dg)exP,(cp)exp,(~)Q(~)
(9E M , cp E M ) .
(17) Observe that for g, h E L2(G,v) we have Bg,h 6 A. Immediately from (16) we get B,,hexpf = e(,>f).exph
(1,9, h E L z ( G , v ) ) .
(18) Operators of the type Bg,hdetermine a normal state on A completely, i. e. we have the following result:
259
Lemma 3.3. ([6, 81). Let wj, j = 1, 2 be normal states o n A. Then = w2 if and only if
w1
4. The Quantum Channel
Let a, /3 : G
-+
CC be measurable mappings such that
We define a linear operator and 91, 9 2 E M
Va,p : M 2 + M 2 by setting for CP E M 2
The above defined operator plays a basic r61e in the definition of the quantum channel we want to consider. First we will show that on exponential vectors the operator Va,s has the same form as the operator (1) mentioned in the introduction and describes (with (Y and /3 being constants) the state change of the usual beam splitter model of quantum optics (cf.inst. [2]). The results below in this section are contained at least partially in [8, 7, 9, lo]. However, for completeness and readability of the present paper we will present all proofs.
Proposition 4.1. Let Va,p be the operator defined b y (21) with a and satisfying (20). For all g, h E L2(G,u ) one has
260
Proof: For arbitrary 9, h E &(G, v) we have expg@exphE M 2 . Applying several times Lemma 3.2 we get for pl, 9 2 E M
cc
expa($l)expp(pl - $1) dl591 92592 -exp-~($2)e%(c2 - $2)expg($1 $2)exph(w 9 2 - $1 - $2) (Va,p
e q g @
exph >(91,@) =
+
=
+
expa($l)eqg(dl)expp(pl - $l)exph(ql - $1) dl591 d25'f'Pa 'exp-8($2)expg($2)e~(p2 - $2)exph(@ - $2)
eqcxg($l)expph(pl - $1) exp-Bgenh(p2 - $2) dl591 925v2 = eqag+ph(pl) .eq-?3g+,h(p2) = exPcxg+ph @ exp-8g+zh('P1 ,@) what ends the proof. =
More general splitting models were considered in [7, lo]. Observe that for all 9 E L2(G,v) Va,p exp,@expo = expag €3 e x p q g
(23) Splitting procedures of this type (considered as operators from d to d@d) were studied in detail for instance in [8]. Similar models were considered in [4, 53.
Proposition 4.2. The operator Va,a defined by (21) with 01 and /? satisfying (20) is an isometry. Proof: For 9, h E &(G, v) we get 1IVa,a m
g @ eqh11%f2 =
2
~ ~ e x p a g + ~'hllexp-~g+Ehllh ~~M
= exp{Il~9+/?h1I2}.exP{ll- O 9 + W I 2 } = exp{142119112 + IP1211~112 +w?(9lh)
+d(h,9)}
+ 1~12119112- W ( 9 , h) - olS(h9)} ~exp{l~1211~112 = eq{llgl12 + llhl12} = I l e q g @ exphIl%f2. Since the linear span of all tensor products exp, @ exp, of exponential vectors is dense in M 2 this proves that Va,p is isometric. 0 The operator V& adjoint to Va,p is of the same type as Va,p.
Proposition 4.3. Let Va,p be given by (21) with 01 and /? satisfying (20). Then
v:,p = %,-p
(24)
261
Proof: It is sufficient to show (24) for exponential vectors. For g, h E L2( G, v )we get
%,-avcY,sexpg
@
exph = e w a(ag+Bh)-B(-$g+Zh
-
@
- e~l~lag+ijiSh+lBI'p-~Bh @
= expg
@
e w ( a g + B h ) + a ( -B'g+Zh
expa~g+JB12h-aB'g+lalzh
exph
what ends the proof.
0
+
From (20) we get lE(z)I2 I - P(z)I2 = 1 for all z E G. So we conclude from Proposition 4.2 that V& = V Z , - ~: M 2 + M 2 is an isometry too. Consequently, we obtain
Proposition 4.4. The operator Va,a : M 2 + M 2 with a and (20) is unitary.
satisfying
From Proposition 4.4 we get immediately that the mapping E : C ( M 2 )+ L ( M 2 )defined by
(C E L ( M 2 ) is completely positive and preserves the identity.
Definition 4.1. The completely positive channel €& : S(d2)+S(d2) the dual map of which is given by (25) (i. e. E&B e ~ - ~ g + ~ h ) e - ” g ” 2 - l l h l l Z
-
= Ul(A) ql ( B ) we see (as remarked already in Section 2) that for coherent states the ’independence’ of input and noise ’survives’, i. e. (cf. (6)) wl@ql = wo@qo 0 €a$.
(29)
5. Control Gate Let U : M @ M + M @ M be unitary and let the control state normal state on A. We have a ( A ) = wi@@(U*(A@X)U)
(A E A).
e
be a
263
Problems : Which U are reasonable (from technical point of view)? Which e can serve as control states? Which U and Q are so that the truth table will be satisfied? Define U : M @ M c)M @ M by ~ ~ 2 ~( (~ ~( c1 p uq(cpl, cp2) := ( - 1 ) ~ 9 ~ ~ ~,cp2) E~ ~ MI ~ 2 Q E M @ M ) (30) where 191 = cp(G).
Lemma 5.1. U is se2f-adjoint and unitary. For g E L2(G,u) and q E M we have U ( ~ X P ~ @ Q ) ( P I= , V~~Q) ( - ~ ) I ~ ~ I ., Q(c~2)(vI)
(31)
Proof: Since U is just a multiplication operator with a real-valued function it will be self-adjoint. Because of ((-l)m)2 = 1 for all m E N one has U2 = I[. For g E L2(G, u ) and Q E M we get using Lemma 3.2
Now, let e be a pure normal state on A, i.e. there exists a Q E M , Q such that
#0
The output from the control gate will be W; = ul@e(U*((.)@I[)U). Proposition 5.1. Let wo = u9, qo = uh,and 1 Gi(A) = - F(dcp)(9(cp)(2u( - l ) ’ u l ( c r g + B h )
iiw
JM
Q
be given by (32). Then (A)
( A E A).
(33)
Proof: Because of Lemma 3.3 it is enough to show (33) for operators A being of the type B,, ,92. Using (31) we obtain for all 91, cp2 E M , 91, g2, f E L2(GI 4 ( B 9 ~,92@x)
U ( e q f @ Q ) ( v lI 9 2 ) = ( B 9 ~,gn@~)exp(-l)lV21 f(’P1) . Q(cp2) = B,,,,,(exP(-l)lV,If)(cpl)
. Q(cp2)
Consequently, setting f = crg 6 1(
4 1 ,.!I2
+ ph we get for all gl, g2 E L2(G, v)
1
where we used in the last line (27) and 11 f 112 = 11 (-1)l~lf 112 for all
(P
E M.
Denote by Modd resp. Meven the configurations with odd resp. even total number of particles, i.e. Modd = {(P E M : /(PIodd), M even = {(P E M : ](PI even). As a direct consequence of Proposition 5.1 we obtain
Proposition 5.2. Let
Q
satisfy (32). Then
w;( A ) = Xlw
-("g+Dh) + XZW ffg+ph
with
So, if for instance the control state Q is the vacuum state (Q= wO)then LS; = wl = w w+Bh . If Q is an one-particle state (or some state concentrated on odd configurations) then &c = w -("g+ph). The choice of the control state allows us to switch from the coherent state w "g+ph to w -("g+ph).
6. Final Output AEter Second Beam Splitting Let wo = wg, 710 = wh be coherent normal states on A (the inputs Il and I2) and Q = (Q, (-)*)/11Q1(2 a pure state (the control state). Further, let
265 -
XI, A2 be given by (34). Then W; (cf. Proposition 5.2) and q1 = w - P ~ + ~ ~ represent the inputs for the final second beam splitting the rates of which we denote by a1, 81, i. e. a1, PI : G C ) C and Ia1(2)I2 IP1(2)12 = 1. Consequently, the final outputs 01 and 0 2 are given by the states
+
= ~ l @ q l ( ~ ~ ~ , P l ( ( . ) @ ~ ) ~ a l , P lrh ) , = G@v1( G 1 , p lP @ ( . ) ) V a 1 , P 1 ) (35) Directly from Theorem 4.1 we obtain w2 = AlW(-a"l-P1a)g+(-alP+EPl)h + )(2W(UPl-P1p)g+(alP+EPl)h (36) and q2 = A1 ( & - E p ) g + ( B z + a a l ) h + x2 (- az-7Tip)g-f (-p&+acrl) h . (37) w2
We consider two cases. Case 1: Let e be a pure state concentrated on even configurations, i. e. A1 = 0, A2 = 1. For example let e be the vacuum state wo. Then we obtain w2
= ,(ual-P18)g+(alP+~Pl)h
(38)
and q2
= ,(-a&-EP)g+(-PS;+m)h.
(39) Case 2: As second case let us consider e being concentrated on odd configurations, i. e. A1 = 1, A2 = 0. For example let e be an one-particle state: Q(cp) = 0 for cp $! Ad1 = {&. 2 E G}. Then we obtain w2
=,(-aal-P18)g+(-alS+~Pl)h
(40)
and q2
= ,(aE-(.1P)g+(PZ+aal)h
(41) satisfied if
truth tables of our logical gate will be incase 1: w2 = wg and q 2 = wh and in case 2: w2 = wh and q 2 = wg. So the gate has to be constructed in such a way that in the control gate there is the possibility to choose between two control states: The control gate is switched off with a control state el of the form considered in case I and is switched on with a control state e2 of the form considered in case 2. An easy calculation shows that this will be satisfied if and only if the splitting constants a, p, a1, 81 fulfil The
a = -p = p1 =
(42)
266
Let us remark that a1 = 5, = -p are the splitting constants of the operator adjoint to Va,p. So this assumption is already necessary to get the original input back if there is no disturbance in the control gate. Summarizing, if the splitting constants satisfy (42) the full information even represented by the original input states - will be obtained at the end after the second beam splitting. So if the information "0" resp. "1" are represented by coherent vectors g and h and the beam splitting constants fulfill (42) the truth table will be satisfied. All calculations above have been done for arbitrary g and h from L2(G, v). We obtain the right truth values also in the (trivial case) that both vectors are equal. If one does not demand that the output will be the original input 11resp. 12 but only that the output has to belong to a certain class of coherent states we may drop the assumptions about the splitting constants and find more interesting control gates.
References 1. L. Accardi and M. Ohya. Compound channels, transition expectations and liftings. Applied Mathematics & Optimization, 39:33-59, 1999. 2. R.A. Campos, B.E.A. Saleh, and M.C. Teich. Quantum-mechanical lossless beam splitter: m(2) symmetry and photon statistics. Phys.Rew. A , 40(3):1371-1384, 1989. 3. D.J. Daley and D. Vere-Jones. A n Introduction to the Theory of Point Pro-
cesses. Springer Series in Statistics. Springer-Verlag, New York, Berlin, Heidelberg, 1988. 4. Karl-Heinz Fichtner and Masanori Ohya. Quantum teleportation with entangled states given by beam splittings. Comm. Math. Phys., 222:229-247, 2001.
5. Karl-Heinz Fichtner and Masanori Ohya. Quantum teleportation and beam splitting. Comm. Math. Phys., 225:67-89, 2002. 6. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Beam Splittings and Time Evolutions of Boson Systems. Forschungsergebnisse der Fakultit f i r Mathematik und Informatik, Math f Inff 96f 39:105 pages, 1996. 7. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Non-independent Splittings and Gibbs States. Mathematical Notes, 64(3-4):518 - 523, 1998. 8. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Time Evolution and Invariance of Boson Systems Given by Beam Splittings. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(4):511 - 531, 1998. 9. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. Characterization of Classical and Quantum Poisson Systems by Thinnings and Splittings. Math. Nachr., 218:25-47, 2000. 10. K.-H. Fichtner, W. Freudenberg, and V. Liebscher. On exchange mechanisms for bosom. to appear in: Random Operators and Stochastic Equations, 2004.
267 11. K.-H. Fichtner and W. Freudenberg. Point processes and normal states of boson systems. Naturwiss.- Technisches Z e n t m m N T Z , Leipzig, 1986. 56 p. 12. K.-H. Fichtner and W. Freudenberg. Point processes and the position distribution of infinite boson systems. J. Stat. Phys., 47:959-978, 1987. 13. K.-H. Fichtner and W. Freudenberg. Characterization of states of infinite boson systems I. On the construction of states of boson systems. Comrnun. Math. Phys., 137:315 - 357, 1991. 14. K.-H. Fichtner and W. Freudenberg. Remarks on stochastic calculus on the Fock space. In L. Accardi, editor, Quantum Probability and Related Topics VII, pages 305 - 323, Singapore, New Jersey, London, Hong Kong, 1991. World Scientific Publishing Co. 15. E. Fredkin and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21:219-253, 1982. 16. W. Freudenberg, M. Ohya, N. Turchyna, and N. Watanabe. Quantum logical gates realized by beam splittings. Preprint, 2004. 18 pages. 17. W. Freudenberg, M. Ohya, and N. Watanabe. On beam splittings and mathematical construction of quantum logical gate. Surikaisekikenkyusho Kokyuroku, 1142:23-35, 2000. Mathematical aspects of quantum information and quantum chaos. 18. W. Freudenberg, M. Ohya, and N. Watanabe. On beam splittings and quantum logical gate on Fock space. Surikaisekikenkyusho Kokyuroku, 1139:113123, 2000. New developments in infinite dimensional analysis and quantum probability theory. 19. W. Freudenberg. Characterization of states of infinite boson systems 11. On the existence of the conditional reduced density matrix. Commun. Math. Phys., 137:461 - 472, 1991. 20. W. Freudenberg. On a class of quantum Markov chains on the Fock space. In L. Accardi, editor, Quantum Probability and Related Topics, volume IX, pages 215-237, Singapore, New Jersey, London, Hong Kong, 1994. World Scientific Publishing Co. 21. A. Guichardet. Symmetric Halbert Spaces and Related Topics, volume 231 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1972. 22. J.M. Lindsay and H. Maassen. An integral kernel approach to noise. In L. Accardi and W. von Waldenfels, editors, Quantum Probability and Applications 111, volume 1303 of Lecture Notes an Mathematics, pages 192 -208, Heidelberg, 1988. Springer-Verlag. 23. K. Matthes, J. Kerstan, and J. Mecke. Infinitely Divisible Point Processes. Wiley, Chichester, 1978. 24. P.A. Meyer. Quantum Probability for Probabilists, volume 1538 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, 1993. 25. G.J. Milburn. Quantum optical Fredkin gate. Physical Review Letters, 62:2124-2127, 1989. 26. M. Ohya and H. Suyari. An application of lifting theory to optical communication process. Reports on Mathematical Physics, 36:403 - 420, 1995. 27. M. Ohya and N. Watanabe. Construction and analysis of a mathematical
268 model in quantum communication processes. Electron. Commun. Jpn., Part 1, 68(2):29-34, 1985. 28. M. Ohya and N. Watanabe. On mathematical treatment of Fredkin-ToffoliMilburn gate. Physica D, 120:206-213, 1998. 29. M.Ohya. Some aspects of quantum information theory and their applications to irreversible processes. Reports on Mathematical Physics, 27:19 - 47, 1989.
THE CHAOTIC CHAMELEON
RICHARD D. GILL Mathematical Institute, University of Utrecht, Netherlands, €4 E URANDOM, Eindhoven, Netherlands gzllOmath.uu.nl http ://www. math.uu.nl/people/gill Various local hidden variables models for the singlet correlations exploit the detection loophole, or other loopholes connected with post-selection on coincident arrival times. I consider the connection with a probabilistic simulation technique called rejection-sampling, and pose some natural questions concerning what can be achieved and what cannot be achieved with local (or distributed) rejection sampling. In particular a new and more serious loophole, which we call the coincidence loophole, is introduced.
1. Introduction
It has been well known since Pearle (1970) that local realistic models can explain the singlet correlations when these are determined on the basis of post-selected coincidences rather than on pre-selected event pairs. These models are usually felt to be unphysical and conspiratorial, and especially that they simply exploit defects of present day detection apparatus (hence the name “the detection loophole”). However, Accardi, Imafuku and Regoli (2002) (“the chameleon effect”),Thompson and Holstein (2002) (“the the chaotic ball effect”),and others have argued that their models could make physical sense. Further examples are provided by Hess and Philipp (2001a,b, 2004), Kracklauer (2002), Sanctuary (2003), in many cases unwittingly. Already, Gisin and Gisin (1999) show that these models can be simple and elegant, and should not be thought of as being artificial. Accardi et al. (2002) furthermore insist that their work, based on the chameleon effect,has nothing to do with the so-called detection loophole. Rather, they claim that the chameleon model is built on a fundamental legacy of measurement of quantum systems, that there is also indeterminacy in whether or not a particle gets measured at all, and when it gets measured. Furthermore, they focus entirely on perceived defects of the
269
270 landmark paper Bell (1964), where the incompatibility of the singlet correlations with local realism was first established. Now Bell himself became well aware of imperfections in his original work and in Bell (1981) (reprinted in Bell, 1987), taking account of one and a half decades of intense debate, he explicitly elaborated on the experimental protocol which is necessary, before one can conclude from an experimental violation of the Bell-CHSH inequality, that a local realistic explanation of the observed phenomena is impossible. That protocol is not adhered to by Accardi et al. (2002),nor (of course) by any of the previously cited works in which local realistic violations of Bell-CHSH inequalities are obtained. It is a mathematical fact that “chameleon model” of the type proposed by Accardi et al. (2002) can be converted into a “detection loophole model”, and vice-versa. This result has been independently obtained by Takayuki Miyadera and Masanori Ohya, and by the present author (unpublished). In this paper I do not want to continue the philosophical debate, nor address questions of physical legitimacy of these models. Instead I would like to extract a mathematical kernel from this literature, exposing some natural open problems concerning properties of these models. Possibly some answers are already known to experts on Bell-type experiments and on distributed quantum computation. I would especially like to pose these problems to experts in probability theory, since the basic renormalization involved both in the chameleon model (under the name of a “form factor”) and in detection-loophole models, is well known in probability theory under the name of rejection-sampling. From now I will use the language of Applied Probability: simulation, rejection-sampling, and so on; and avoid reference to physics or philosophy. The main new contribution of this paper is the discovery of a new loophole, which we call the coincidence loophole, which occurs when particle pairs are selected on the basis of nearly coincident arrival times. It has recently been shown by Larsson & Gill (2003) that this loophole is in a certain sense twice as serious as the well-known detection loophole. 2. The Problem
Suppose we want to simulate two random variables X, Y from a joint probability distribution depending on two parameters a, b. To fix ideas, let me give two key examples:
X,Y are binary, taking the values f l . The parameters a, b are two directions in real, three dimensional space. Case 1 The Singlet Correlations.
271
W e will represent t h e m with two unit vectors in @I (two points o n the unit sphere S2). The joint density of X, Y (their joint probability mass function) is 1 P r a , b{X =x ,Y = y } = p ( x , y ; a , b ) = -4( l - x y a . b ) ,
(1)
where a . b stands f o r the inner product of the unit vectors a and b and x,y = f l . Note that the marginal laws of X and Y are both Bernoulli (i) on {-1, +l}, and their covariance equals their correlation equals - a . b. In particular, the marginal law of X does not depend o n b nor that of Y o n a. Case 2 The Singlet C o ~ l o t i o n Restricted. s This is identical t o the previous example except that we are only interested in a and b taking values in two particular, possibly different, finite sets of points o n S2.
Next I describe two different protocols for “distributed Monte-Carlo simulation experiments”; the difference is that one allows rejection sampling, the other does not. The idea is that the random variables X and Y are going to be generated on two different computers, and the inputs a, b are only given to each computer separately. The two computers are to generate dependent random variables, so they will start with having some shared randomness between them. The programmer is allowed t o start with any number of random variables, distributed just how he likes, for this purpose. Cognoscenti will realize that it suffices to have just one random variable, uniformly distributed on the interval [0,1], or equivalently, an infinite sequence of fair independent coin tosses. There is no need for the two computers t o have access to further randomness-they may as well share everything they might ever need, separately or together, from the start. The difference between the two protocols, or two tasks, is that the first has to get it right first time, or if you prefer, with probability one. The second protocol is allowed to make mistakes, as long as the mistakes are also “distributed”. Another way to say this, is that we allow “distributed rejection sampling”. Moreover, we allow the second protocol not to be completely accurate. It might be, that the second protocol can be made more and more accurate at the expense of a smaller and smaller acceptance (success) probability. This is precisely what we want to study. Success probability and accuracy can both depend on the parameters a and b so one will probably demand uniformly high success probability, and uniformly good accuracy.
272
Task 1 Perfect Distributed Monte- Carlo. Construct a probability distribution of a random variable 2, and two transformations f and g of 2, each depending o n one of the two parameters a and b, such that
f(Z;a),g(Z;b)
N
X,Y
f o r a l l a , b.
(2 )
The symbol ‘N’ means ‘is jointly distributed as’, and X , Y on the right hand side come from the prespecified (or target) joint law with the given values of the parameters a and b.
Task 2 Imperfect Distributed Rejection Sampling. As before, but there are two further transformations, let me call t h e m D = S(2,a) and E =~ ( 2 b),; such that S and E take values 1 and 0 or if you like, ACCEPT and REJECT, and such that
f(Z;a),g(Z;b)I D = l = E
A
X,Y.
(3)
The symbol ‘1’ stands for ‘conditional on’, and ‘A’ means ‘is approximately distributed as’. The quality of the approximation needs to be quantified; in our case, the supremum over a and b of the variation distance between the two probability laws could be convenient (a low score means high quality). Moreover, one would like t o have a uniformly large chance of acceptance. Thus a further interesting score (high score means high quality) is inf,,bPr{D = 1 = E } .
3. The Solutions By Bell (1964) there is no way to succeed in Task 1 for Case 1. Moreover, there is no way to succeed in Task 1 for Case 2 either, for certain suitably chosen two-point sets of values for a and b. Consider now Task 2, and suppose first of all that there are only two possible different values of a and b each (Case 2). Let the random variable 2 consist of independent coin tosses coding guesses for a and b, and a realization of the pair X , Y drawn from the guessed joint distribution. The transformations 6 and E check if each guess is correct. The transformations f and g simply deliver the already generated X , Y . One obtains perfect accuracy with success probability 1/4. It is known that a much higher success probability is achievable at the expense of more complicated transformations. Now consider Task 2 for Case 1. So there is a continuum of possible values of a and b. Note that the joint law of X , Y depends on the parameters
273
a, b continuously, and the parameters vary in compact sets. So one can partition each of their ranges into a finite number of cells in such a way that the joint law of X , Y does not change much while each parameter varies within one cell of their respective partitions. Moreover, one can get less and less variation at the expense of more and more cells. Pick one representative parameter value in each cell. Now, fix one of these pairs of partitions, and just play the obvious generalization of our guessing game, using the representative parameter values for the guessed cells. If each partition has k cells and the guesses are uniform and independent, our success probability is l / k 2 , uniformly in a and b. We can achieve arbitrarily high accuracy, uniformly in a and b, at the cost of arbitrarily low success probability. However, Gisin and Gisin (1999) show we can do much better in the case of the singlet correlations: Theorem 1 Perfect conditional sirnulation of the singlet cornlations. For Case 1 and Task 2, there exists a perfect simulation with success probability uniformly equal to 112. See Gisin and Gisin (1999) for the very pretty details. Can we do better still? What is the maximum uniformly achievable success probability? The joint laws coming from quantum mechanics always satisfy n o action at a distance (“no Bell telephone”), i.e., the marginal of X does not depend on b nor that of Y on a. This should obviously be favourable t o finding solutions t o our tasks. Does it indeed play a role in making these simulations spectacularly more easy for quantum mechanics, than in general? Does “no action at a distance” ensure that we can find a perfect solution to Task 2 with success probability uniformly bounded away from O? Am I indeed correct in thinking that one find probability distributions p with action at a distance, depending smoothly on parameters a, b, for which one can only achieve perfection in the limit of zero success probability? It would be interesting to study these problems in a wider context: arbitrary biparameterized joint laws p ; extend from pairs to triples; . . . 4. Variant 1: Coincidences
Instead of demanding that S and E in Task 2 are binary, one might allow them t o take on arbitrary real values, and correspondingly allow a more rich acceptance rule. Suggestively changing the notation t o suggest times, define now S = d ( 2 ; a ) and T = ~ ( 2 b).; Instead of conditioning on the
274
separate events D = 1 and E = 1 condition on the event (S-2’1 < c where c is some constant. Obviously the new variant contains the original, so Variant Task 2 is at least as easy as the original. Accardi, Imafuku and Regoli (2002) suggest that they tackle this variant task claiming that it has nothing t o do with detector efficiency, but on the contrary is intrinsic to quantum optics, that one must post-select on coincidences in arrival times of entangled photons. By Heisenberg uncertainty, photons will always have a chance t o arrive (or to be measured) at different times. In those cases their joint state is not the singlet state. Therefore, if we were to collect data on all pairs (supposing 100%detector efficiency) we would not recover the singlet correlations. In actual fact the mathematical model of Accardi et a2. (2002) applies to the original task, not the variant. Still, in many experiments this kind of coincidence post-selection is done. Its effects (in terms of the loophole issue) has never yet been analysed. The common consensus is that it is no worse than the usual detection loophole. I convert this consensus into a conjecture:
Conjecture 1 No improvement from coincidences. There is no gain from Variant Task 2 over the original. Amazingly, this conjecture turns out to be false. In quantitative terms the “coincidence loophole” is about twice as serious as the detection loophole; see Larsson and Gill (2003). Fortunately, modern experimenters are moving (as Bell, 1981, stipulated) toward pulsed experiments and/or to event-ready detectors. In such an experment the detection time windows axe fixed in advance, not determined by the arrival times of the photons themselves. There seems t o be a connection with the work of Massar, Bacon, Cerf and Cleve (2001) on classical simulation of quantum entanglement using classical communication. After all, checking the inequality I S - 2’1 < c is a task which requires communication between the two observers. 5. Variant 2: Demanding More
Instead of making Task 2 easier, as in the previous section, we can try to make it harder by demanding further attractive properties of the simulated joint probability distribution of D ,XI E , Y.For instance, Gisin and Gisin (1999) show how one can achieve nice symmetry and stochastic independence properties at the cost of an only slightly smaller success probability
275 4/9 = (2/3)2. In fact, this solution has even more nice properties, as follows. One might like the simulated X to behave well, when D = 1, whether or not E = 1, and similarly for Y. Suppose we start with a joint law of X , Y depending on a, b as before. Let q be a fixed probability. Modify Task 2 as follows: we require not only that given D = 1 = E , the simulated X , Y have the prespecified joint distribution, but also that conditional on D = 1 and E = 0, the simulated X has the prespecified marginal distribution, and also that, conditional on D = 0 and E = 1, the simulated Y has the prespecified marginal distribution, and also that D and E are independent Bernoulli(q). Another way to describe this is by saying that under the simulated joint probability distribution of X , D, Y ,E , we have statistical independence between D, E , and ( X ,Y ) ,with ( X ,Y ) distributed according to our target distribution and D and E Bernoulli(q), except that we don’t care about X on {D = 0 ) nor about Y on { E = 0} Gisin and Gisin (1999) show that this Variant Task 2 can be achieved for our main example Case 1, with q = 2/3. It is known from considerations of the Clauser and Horne (1974) inequality that it cannot be done with q > 2/(1+ M 0.828. It seems that the precise boundary is unknown. In fact, for some practical applications, achieving this task is more than necessary. A slightly more modest task is to simulate the joint probability distribution just described, conditionally on the complement of the event {D = 0 = E } , i.e. conditional on D = 1 or E = 1. This means to say that we also don’t care what is the simulated probability of {D = 0 = E } . Gisin and Gisin (1999) show that this can be achieved with a variant of the same model, and with success probability 100% (i.e., the simulation never generates an event {D = 0 = E}),and q = 2/3.
a)
Acknowledgments
I am grateful for the warm hospitality and support of the Quantum Probability group at the department of mathematics of the University of Greifswald, Germany, during my sabbatical there, Spring 2003. My research there was supported by European Commission grant HPRN-CT-200200279, RTN QP-Applications. This research has also been supported by project RESQ (IST-2001-37559) of the IST-FET programme of the European Commission.
276 References
L. Accardi, K. Imafuku, and M. Regoli (2002). On the EPR-chameleon experiment. Infinite Dimensional Analysis, Quantum Probability and Related Fields 5,1-20. quant-ph/O112067. J . S. Bell (1964). On the Einstein Podolsky Rosen paradox. Physics I , 195-200. J. S . Bell (1981). Bertlmann’s socks and the nature of reality. Journal de Physique 42, C2 41-61. J. S. Bell (1987). Speakable and Unspeakable in Quantum Theory. Cambridge: Cambridge University Press. J . F. Clauser and M. A. Horne (1974). Experimental consequences of objective local theories. Phys. Rev. D I U , 526-535. N. Gisin and B. Gisin (1999). A local hidden variable model of quantum correlation exploiting the detection loophole. Phys. Lett. A 260,323-327. quant-ph/09905158. K. Hess and W. Philipp (2001a). A possible loophole in the theorem of Bell. Proc. Nat. Acad. Sci. USA 98,14224-14227. quant-ph/0103028. K. Hess and W. Philipp (2001b). Bell’s theorem and the problem of decidability between the views of Einstein and Bohr. Proc. Nat. Acad. Sci. USA 98, 14228-14233. quant-ph/0103028. K. Hess and W. Philipp (2004). Breakdown of Bell’s theorem for certain objective local parameter spaces. Proc. Nat. Acad. Sci. USA 101,17991805. quant-ph/0103028. A. F. Kracklauer (2002). Is ‘entanglement’ always entangled? J. Opt. B: Quantum Semiclass. Opt. 4, S121-Sl26. quant-ph/O108057. J.-A.-Larsson and R.D. Gill (2003). Bell’s inequality and the coincidencetime loophole. To appear in Europhysics Letters. quant-ph/0312035. S. Massar, D.Bacon, N. Cerf and R. Cleve (2001). Classical simulation of quantum entanglement without local hidden variables. Phys. Rev. A 63, 052305 (9pp.). qua11t-~h/0009088. P. Pearle (1970). Hidden-variable example based upon data rejection. Phys. Rev. D 2, 1418-1425. B. C. Sanctuary (2003). Quantum correlations between separated particles. quant -ph/0304186. C. H. Thompson and H. Holstein (2002). The “Chaotic Ball” model: local realism and the Bell test “detection loophole”. quant-ph/0210150.
ON AN ARGUMENT OF DAVID DEUTSCH
FLICHARD D. GILL Mathematacal Institute, University of Utrecht, Netherlands, & E URANDOM, Eindhouen, Netherlands gill0math.uu.nl http://www.math.uu.nl/people/gill We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born’s law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born’s law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason’s theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using diflerent assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.
1. Introduction: are quantum probabilities fixed by
quantum determinism? Quantum mechanics has two components: a deterministic component, concerned with the time evolution of an isolated quantum system; and a stochastic component, concerned with the random jump which the state of that system makes when it comes into interaction with the outside world, sending at the same time a piece of random information into the outside world. The perceived conflict between these two behaviours is ‘the measurement problem’ as exemplified by Schrodinger’s cat. Here we do not resolve this problem but just address the peaceful coexistence, or possibly even the harmony, between the two behaviours. We will show that some classical deterministic quantum mechanical assumptions, together with the assumption that the outcome of measuring an observable is random, uniquely determines the probability distribution of
277
278 the outcome-harmony indeed. More specifically, two generally accepted invariance properties of observables and quantum systems determine the shape of the probability distribution of measured values of an observablenamely, the shape specified by Born’s law. The invariance properties are connected to unitary evolution of a quantum system, and to functional transformation of an observable, respectively. This work was inspired by Deutsch (1999). There it is claimed that a still smaller kernel of deterministic classical quantum theory together with a small part of deterministic decision theory together force a rational decision maker to behave as if the probabilistic predictions of quantum theory are true. In our opinion there are three problems with the paper. The first is methodological: we do not accept that the behaviour of a rational decision maker should play a role in modelling physical systems. We are on the other hand happy to accept a stochastic component (with a frequentist interpretation) in physics, so we translate Deutsch’s axioms and conclusions about the behaviour of a rational decision maker into axioms and conclusions about the relative frequency with which various outcomes of a physical experiment take place. The second problem is that it appears that Deutsch has implicitly made use of a further axiom of unitary invariance alongside his truely minimalistic collection, and needs to greatly strengthen one of the existing assumptions concerning functional invariance, from one-to-one functions also to many-to-one functions. Neither addition nor strengthening is controversial from a classical deterministic quantum physics point of view, but both are very substantial from a mathematical point of view. The third problem is that the strengthening of the functional invariance assumption puts us in the position that we have assumed enough to apply Gleason’s (1957) theorem. Thus at best, Deutsch’s proof is an easy proof of Gleason’s theorem using an extra, heavy, assumption of unitary invariance. The fact that Deutsch’s proof is incomplete has been observed by Barnum et al. (2000). However these authors did not attempt to reconstruct a correct proof. In the concluding section we relate our work to theirs. Wallace (2002, 2003a,b) has also studied Deutsch’s claims at great length and from a rather philosophical point of view. I did not attempt to relate his work to mine. The same goes for Saunders (2002). The paper is organised as follows. In Section 2 we put forward functional and unitary invariance assumptions, which are usually considered consequences of traditional quantum mechanics, but are here to be taken as axioms from which some of the traditional ingredients are to be derived, turning the tables so to speak. One would like to make the axioms
279
as modest as possible, while still obtaining the same conclusions. Hence it is important to distinguish between different variants of the assumptions. In particular, we distinguish between (stronger) assumptions about the complete probability law of outcomes of measurements of observables, and (weaker) assumptions about the mean values of those probability laws. An invariance assumption concerning a class of functions, is weaker, if it only demands invariance for a smaller class of functions, and in particular we distinguish between invariance for all functions, including many-to-one functions, and invariance just for one-to-one functions. In Section 3 we prove the required result, Born’s law, for a special state (equal weight superposition of two eigenstates). This case is the central part of Deutsch (1999), who only sketches the generalization to arbitrary states. And already, it seems an impressive result. We prove the result, f o r this special state, in two forms-in law, and in mean v a l u e t h e former being stronger of course; using appropriate variants of our assumptions. Deutsch’s proof is incomplete, since he only appeals to unitary invariance, while it is clear that a functional invariance assumption is also required. The strengthening of the functional invariance assumption can also be used t o derive probabilities as well as mean values, and it is moreover useful from Deutsch’s point of view of rational behaviour, if one wants to extend in a very natural way the class of games being played. Roughly speaking, we extend from the game of buying a lottery ticket t o a game at the roulette table. In the former game the only question is, how much is one ticket worth. In the latter game one may make different kinds of bets, and the question is how much is any bet worth. However, so far we have only been concerned with a rather special state: an equal weight superposition of two eigenstates. As mentioned before Deutsch only sketches the extension to the general case of an arbitrary, possibly mixed, state. He outlined a step-by-step argument of successive generalizations. In Section 4 we follow the same sequence of steps, strengthening the assumptions as seems to be needed. In Section 5, we look back at the various versions of our assumptions, in the light of what can be got from them. We also evaluate the overall result of completing Deutsch’s programme. From a mathematical point of view, it turns out that we have done no more, at the end of the day, than derive the same conclusion as that of Gleason’s theorem, while making stronger assumptions. The payoff has just been a much easier proof. Gleason’s theorem only assumes functional invariance, we have assumed unitary invariance too. We argue that unitary invariance corresponds to a
280
natural physical intuition, while functional invariance is something which one could not have expected in advance. It is supported by experiment, and is theoretically supported in special cases (measurements of components of product systems) by locality. We conclude with the conjecture that unitary invariance together with a weakened functional invariance assumption is sufficient to obtain the same conclusion. 2. Assumptions: degeneracy, functional invariance, unitary invariance
Recall that a quantum system in a pure state is described or represented by a unit vector in a Hilbert space, supposed to be infinite-dimensional, and that an observable or physical quantity is described or represented by a self-adjoint (perhaps unbounded) operator X on that space. I shall assume that X has a discrete and nondegenerate spectrum; thus there is a countably infinite collection of real eigenvalues x and eigenstates IX=x), so that one can write X = C,x IX=x )(X=x 1 , while = C A,IX = x ) where A, = (X=xI$). Throughout the paper we make the following background assumption:
I$)
I$)
Assumption 0 Random outcome, in spectrum. The outcome of measuring X is one of its eigenvalues x-which one, is random. Its probability distribution (law) depends o n X and o n
I$).
I write meas+(X) for the random outcome of measuring observable X on state and law(meas+(X)) for its probability distribution, i.e., the collection of probabilities Pr{meas+(X) E B } for all Bore1 sets B of the real line. Deutsch’s paper has the more modest aim just to compute the mean value of this probability law, E(meas+(X)), though as I shall argue before, even under his own terms (computing values of betting games) the whole probability law is of interest. Throughout the paper I will be playing with three main assumptions, though sometimes in stronger and sometimes in weaker forms. Here are the three, in their strongest versions:
I$),
Assumption 1 Degeneracy in eigenstates.
Pr{meas
I-)
(x) = x} = 1.
281 In an eigenstate of an observable, the corresponding eigenvalue is the certain outcome of measurement. Assumption 2 Functional invariance.
Pr{f(meas+(X)) = y ) = Pr{meas+(f(X)) = y).
(2)
Measuring a function f of an observable is operationally indistinguishable from measuring the observable, and then taking the same function of the outcome. Parenthetically remark that this indistinguishability is only as far as the outcome is concerned; as far as the new state of the quantum system is concerned there will be a difference, if the function is many-toone. Parts of Deutsch’s proof only need this assumption for one-to-one functions. In fact he only explicitly used this assumption for the afine functions f(z)= uz + b, but implicitly other functions, including many-toone functions, are involved too. Assumption 3 Unitary inuariance.
We will see that, at first instance, we only require this assumption to hold for a special class of unitary operations U , namely those which permute eigenstates of X . There is then a one-to-one correspondence u on the eigenvalues of X with inverse u* such that UXU* = u(X), U*XU = u*(X), and U I X = z ) = IX=u(z)). In the special case that is a n eigenstate IX =z), Assumption 3 follows from Assumption 1 (degeneracy-ineigenstates). Later we dso need Assumption 3 for unitary operations, diagonal in the basis corresponding to X. Since in the above assumptions, x and y are arbitrary, one could restate the three main assumptions as:
where law denotes the probability law of the random variable in question, so that in particular law(z) denotes the probability distribution degenerate at the point 2. An apparently weaker still set of assumptions would only
282
restrict the mean values of the distributions in Assumptions 2 and 3:
As mentioned above, one can weaken the assumptions by restricting the class of functions f or unit&ies U for which the relevant equalities are supposed to hold.
3. The first part of the proof
I return to a discussion of the assumptions after an outline of the proof of my main result:
I will make use of Assumptions 1-3 in their original form, postponing discussion of how one might reach the same conclusion from weaker versions of the assumptions. In this section, following Deutsch, I only prove the result in the special case (a)
for which I am going to obtain the probabilities 1 / 2 for x = z1and x = x2, and zero for all other possibilities. After this, Deutsch attempts to generalize, first (b) to equal weight superpositions of a binary power of eigenstates of X, next (c) to an arbitrary number, then (d) to dyadic rational superpositions, next (e) to arbitrary real superpositions, and finally (f) to arbitrary superpositions. The proofs he gives of these steps are similarly incomplete. I will complete the proof by an alternative and rather short route in the next section, but return to Deutsch’s completion in the section after that. Suppose u, a one-to-one correspondence on the eigenvalues of X , maps 21 to x2 and vice-versa, and, after we have labelled the other eigenvales as x1,, n r i Z,maps x1, to z;+~. Let U denote the unitary which performs the same permutation of the eigenvectors. Let u* denote the inverse of u. Exploiting the relationship between u and U , and their relationship to X
283
and $, as well as our other assumptions, we find,
Pr{meas+(X) = q } = Pr{measv+(X) =
21)
= Pr{meas+(U*XU) = q } = Pr{meas+(u(X)) = q }
= Pr{u(meas+(X)) =
XI}
= Pr{meas+(X) = u*(z1)}
(6)
= Pr{meas+(X) = 22).
Replacing z1by an eigenvalue xk, i.e., any other than 2 1 or 22, and running through the same derivation, we see that all other eigenvalues have equal probabilities. Since there are an infinite number of them, and since according t o our background assumption the outcome of measuring X lies in its spectrum, we have obtained the required result: the probabilities of z1and 22 must both equal 1/2, all the other eigenvalues 2; must get zero probability. We used Assumptions 2 and 3 (functional and unitary invariance), not Assumption 1 (degeneracy in an eigenstate). However, this assumption is needed t o deal with the case o f . . . an eigenstate. The proof method allows us t o deal with an equal weight superposition of any positive finite number of eigenstates of X . We only used functional invariance for one-to-one functions. Deutsch was only interested in mean values of the probability distributions of outcomes, since the fair value of the game: measure X on and receive the value of the outcome in euro’s (g), is precisely g E(meas+ (X)). (Here we are assuming that the utility of having some number of euro’s is equal to that number. The reader may replace euro’s by dollars, camels, or whatever else he or she prefers). In a moment I will also add a new game and receive g 1 if the outcome 20 is to the discussion: measure X on found. The value of this game should be g ~ ( Z O 12. Let us assume that the spectrum of X consists of all the integers (negative and non-negative). Then for given z1 and 2 2 there is an affine map u(2) = u z b = 2 1 x2 - 2 which defines a unitary transformation U as above. For these U , X and the same $ as before we rewrite the argument
I$)
I$)
+
+
I$)
284
before as E(meas+(X)) = E(measv+(X)) = E(meas+(U*XU)) = E(meas+(W))) = E(4meas+(X)))
+
= z1 z2- E(meas+(X))
(7)
yielding the required equality, E(meas+(X)) = i(z1
+ z2).
(8)
Deutsch’s proof was a cryptic version of the argument I have just given, except that he did not mention the unitary invariance assumption. He writes 21 for value, instead of E. In my opinion, without the extra (unitary invariance) assumption, his proof fails. The degeneracy Assumption 1 is not used at this stage. However one may note that Assumption 1 (degeneracy) implies that Assumption 3 (unitary invariance) holds when the state is an eigenstate of the observable X. One could therefore consider Assumption 3 as a natural interpolation from Assumption 1. I return to this later. As has been shown by de Finetti and by Savage, a rational decision maker who must make choices when outcomes are ‘indeterminate’ (I must avoid all terminology suggestive of probability theory, since the words ‘random’, ‘probability’ and so on, are not allowed to be in our vocabulary) behaves as if he (or she) has a prior probability distribution and indeed updates it according t o Bayes’ law when new information (outcomes) becomes available. Thus it seems to me that whether one starts with utilities and assumes rationality, or with probability and the frequency interpretation, is very much a matter of taste. In my opinion the latter is closer to physical experience and indeed we know that casinos and insurance companies make good money from the frequency interpretation of chance. I consider the many repetitions in the frequency interpretation to be no more and no less than a thought experiment. When one claims that the probability of some event is some number, one is asserting that the situation in question is indistinguishable from a certain roulette game or lottery. This allows me also to talk about probabilities of outcomes of once-off experiments. For instance, a certain physical experiment might have some chance of producing a black hole which would swallow the whole universe. The probability that this would indeed happen, if the devilish experiment were actually carried out, would be computed by doing real
I$)
285 physics in which one would imaginarily set the chain of events into motion, many many times, in which uncontrolled initial conditions would vary in all kinds of ways from repetition to repetition. How they would vary, and what possibilities could be considered equally likely, should be a matter of scientific discussion. This may appear circular reasoning or an infinite regress or just plain subjectivism, but this does not bother me: it works, and it is not subjective, since we may rationally discuss the probability modelling. When I use the mathematical model of probability, I am only claiming an analogy with something familiar, like a casino, lottery, or coin toss. I think that it is the same in the rest of physics, when we talk about mass, electric charge, or magnetic field: we might think or we might hope that we are talking about real things in the real world but we can only be certain that we are talking about ingredients of mathematical models which are anchored t o the real world by analogies with familiar down to earth daily experience. My frequentistic position is perhaps better labelled “Laplacian counterfactual frequentism” and though one might collapse this label to “subjectivism”, I believe it is as instrumentalistic or as operationalistic as anything else in physics. 4. Completing the proof
More can be got out of the functional invariance assumption, by considering other functions f, and most crucially, certain many-to-one functions. In my opinion we must do this anyway, in order to complete the proof on the lines indicated by Deutsch (see next section). It is an open question, whether we can do without. With the choice f = I{z},and writing [X = x] for the projector onto the eigenspace of X corresponding to eigenvalue x (and later also for the eigenspace itself), since I,,}(X) = [ X = z ] ,we read off Pr{meas$(X) = x} = ~ r { m e a s $ ( ( [ ~ = x ]= ) I}.
(9) Indeed, if we only assume the mean value form of the functional invariance assumption, we can read off the same conclusion, since the random variables I{,}(meas~(X)) and meas$([X=x]) are both zero-one valued. Till this point we had dealt with nondegenerate observables and equal weight superpositions of eigenstates. Now we can add to this, also degenerate observables (since these can always be written as functions of nondegenerate observables). Moreover, even if we start with the assumptions in their weaker mean value form, we can still obtain the stronger conclusion about the whole probability law of the outcome.
286
In fact, with brute force we arrive now very quickly at the most general result (it remains, namely, to consider arbitrary states). From functional invariance (whether in terms of probability laws or whether in terms of their mean values) we have shown that a probability can be assigned to each closed subspace of our Hilbert space, countably additive over orthogonal subspaces, and equal to 1 on the whole space. Now we can invoke Gleason’s theorem to conclude that the probability of any subspace is of the form tr{pA} for some density matrix p. It remains to show that p = but this follows from our first axiom that measuring an observable on an eigenstate yields with certainty the corresponding eigenvalue: consider the observable X = itself, and subspace A = [$I (the one-dimensional subspace generated by Deutsch’s extension of his results to the most general case (see next section) is very hard to follow. He repeatedly invokes substitutability, whereby an outcome of one game may be replaced by a new game of the same value. He does not say which substitutions are being made. However he is clearly thinking of substitutions, leading to composite games with composite quantum systems, product states, and observables on each subsystem. During these constructions and substitutions, the observables being measured and the states on which they are being measured, keep changing, while the Spartan notation v(x) in which the symbol x refers to an observable, an eigenvalue, and an eigenstate simultaneously, begs confusion. The mere construction of product systems implies that more is being assumed above the structure so far (so far we only spoke of observables and states on one fixed quantum system). As I will indicate below, it appears that the extra assumption of unitary invariance and the strengthened functional invariance assumption involving many-to-one functions as well as one-to-one functions, together with a natural assumption about measuring separate observables on a product system in a product state, enable one to fill the gaps. If the repair job is not too difficult, one finishes with a relatively easy proof of Gleason’s theorem, under the supplementary condition of unitary invariance. The construction of product systems will also help us extend results from infinite-dimensional quantum systems to finite dimensional, including 2-dimensional-the case not covered by Gleason. Functional invariance assumptions on product systems, or more generally, for compatible observables, play a key role in many foundational discussions of quantum mechanics. Recall that observables X , Y commute (or are compatible with one another) if and only if both are functions of a
l$)($l
1$)($1
I$))!
287
third 2; and the third can be chosen in such a way (with a minimal set of eigenspaces) t o make the mapping 2 I+ (X, Y)a one-to-one correspondence in the sense that we can write X = f(Z), Y = g ( Z ) , Z = h(X,Y) where h i s the inverse of ( f , g ) . In other words, two (or more) commuting observables can be thought of as components of a vector-valued observable, or equivalently as defining together one ‘ordinary’ observable. Whether one thinks of them together as a vector or as a scalar observable is merely a question of how the eigenspaces are labelled. One can define joint measurement of compatible observables in several equivalent ways. Assuming Liiders’ projection postulate for how a state changes on measurement, the sequential measurements, in any order, of a collection of compatible observables, are operationally indistinguishable from one another. One may therefore think equally well of ‘one-shot ’ measurement of 2, sequential measurement of X then Y ,and sequential measurement of Y then X. This leads to a further extended functional invariance assumption: = law (meas+(f(-f)I), law (f(meas$ (2)))
(10)
where 2 = (XI ,... ,X,) is a vector of mutually compatible observables and f : Rk + Iwm. Apparently weaker is the mean value form of this: E ( f ( m e W 2 ) ) ) = E(meas$/,f(-f)));
(11)
though as I showed above, by playing around with indicator functions, the two are equivalent. We can recover from the assumption the fact that the probability law of a measurement of X alone is the same as the first marginal of the joint law of the two outcomes of a joint measurement of commuting X ,Y.As I have argued in Gi11(1996a,b), these consequences of the standard theory form a crucial though often only implicit ingredient in many of the famous no-go arguments against hidden variables in the literature. Somewhat irreverently I have dubbed (11) ‘the law of the unconscious quantum physicist’. Deutsch’s approach is similar to that of some probabilists, in that he would prefer to make Expectation central, and have Probability a consequence (in fact, he would prefer to do without the word Probability altogether). This is fine, and indeed many probabilists do take this approach (Whittle in his textbook on Probability argues that one should do the same for quantum probability, too). Now in our situation we want to start with hypothesizing existence of mean values, and by making some structural assumptions about them. From this we want to derive the form of the mean values. As I have noted above, since l ~ , ~ ( X is )both an observable itself,
288
and a function of the observable X, it would appear that fixing all mean values of (outcomes of measurements of) all observables, fixes all probability laws of (outcomes of measurements of) all observables. The point I want to make, is that this indeed works, provided we have the functional invariance assumption (for mean values only, if you like, but we must have if for a very large class of functions). Do we need to consider many-toone functions? If our assumptions are only about expectations, I think we do need many-to-one functions. However, with modest distributional input, one need further only consider one-to-one functions, as follows. Suppose we know the mean value of meas+(exp(it arctanx)), and suppose we assume functional invariance, in law, for all one-to-one functions; in particular, the functions f(z)= exp(itarctanx), for each real t. Then we know law(meas+X). It is possible to avoid complex-valued functions, try for instance f(z)= scos($(arctanz+.rr/2)) +tsin(i(arctanz+w/2)) for all r e d s and t. Let me return to the contrast between Deutsch’s and Gleason’s argument. Deutsch’s proof, on completion, seems a little simpler and more direct. His assumptions are much stronger: he needs unitary invariance. His assumptions are more representative of classical quantum mechanicsunitary evolution has to be considered an essential part of this. In the first stages of his argument, deriving mean values for some rather special observables and rather special states, he moreover only needed to consider functional invariance under one-to-one transformations. This assumption is close to tautological (the apparatus for measuring a bX is not going to be essentially different from that for measuring X).However, even from the point of view of deriving fair values of games, probability laws as well as mean values are equally relevant. For instance, what is the fair value of the game: measure X and receive ~91 if the outcome z o is obtained? The easiest way to deal with this game too, is to include functional invariance for the indicator functions too, and then one need not work any more but simply appeal to Gleason’s theorem.
+
5. Discussion
Later in this section I will run through Deutsch’s steps to complete his proof. The aim will be to see whether, with weaker versions of our main assumptions, not strong enough to give us Gleason’s assumptions so easily, we could also arrive at the desired conclusion. (The answer is that at present, I do not know). But first I would like to discuss what grounds one
289
could have for the functional and unitary invariance assumptions, against the background assumptions that measuring an observable yields an eigenvalue, and that in an eigenstate, the outcome is certain. Functional invariance for one-to-one functions seems to me more or less definitional. For many-to-one it is much less definitional, also less empirical, since there will vary rarely truely exist essentially different measurement apparatuses for ‘doing’ X and doing f ( X ) . Just occasionally there will be empirical evidence supporting functional invariance: for instance, when X and Y do not commute, but for some many-to-one functions, one has f ( X ) = g ( Y ) ,there might be empirical (statistical) data supporting it, based on the quite different experiments for measuring X and for measuring Y , and finding the same statistics (or mean values) for f of the outcomes of the first experiment, g of the outcomes of the second. There is one very strong empirical fact supporting the assumption (in its form for vector observables): when we simultaneously measure observables on separate components of a product system (even if in an entangled state) we have the same marginal statistics, as if only one component was being measured. Altogether, the nature of this assumption would seem to me to be: we extend a definitional assumption concerning a smaller class of functions f-the affine functions-to a much larger class, by mathematical analogy, trusting that the world is so elegantly and mathematically put together, that the ‘obvious’sweeping mathematical generalization of an indubitable fact is usually correct; we are supported in this by some empirical (statistical) evidence for some special cases. Similarly the assumption of unitary invariance seems to be largely a leap of faith, since there will be little empirical (statistical) evidence to support it. But again, one might prefer to think of the leap of faith as a natural mathematical generalization. Our first assumption-that measuring an observable on an eigenstate produces the eigenvalue-tells us law (measuG( X ) ) = law (meas+(u*xu)),
(12)
whenever U permutes eigenspaces and II, is an eigenvector! Extending this to arbitrary states can be thought of as an interpolation, in harmony with ideas of wave-particle duality. It seems to me that waveparticle dualitythe very heart of quantum physics-essentially forces probability on us, since it is the only way to get a smooth interpolation between the distinct discrete behaviours at different eigenstates of an observable. We just have to live with smoothness at the statistical level, instead of at the (counterfactual) level of individual outcomes.
290
I would now like to discuss the remaining steps of Deutsch’s proof. As we saw, functional invariance in its strongest form implies the conditions of Gleason’s theorem, which makes all further conditions and further work superfluous. Now the reason functional invariance is so powerful, is that we assumed it to hold for all functions f , in particular, many-to-one functions. In the spirit of the first part of Deutsch’s proof it would make sense to demand it only for one-to-one functions. It seems to me a reasonable conjecture that Deutsch’s theorem is true under the three assumptions: functional invariance for one-to-one functions, unitary invariance, and the degeneracy assumption. As was stated earlier, after (a) the two-eigenstate equal weight superposition, Deutsch extends this (b) to binary powers, (c) to arbitrary whole numbers of equal weight superpositions, (d) to rational superpositions, (e) to real and finally (f) to arbitrary. As we saw, steps (b) and (c) can also be dealt with by his own method for the two-eigenstate case. Deutsch’s argument for (d) involves completely new ingredients and assumptions. He supposes that an auxiliary quantum system can be brought into interaction with the system under study, thus yielding a product space and a product state. The observable of interest X is identified with X 8 1, and this is considered as one of a pair (X8 1 , 1 8 Y )where the observable Y is cleverly chosen, so that in the product system, and with this product observable, we are back in an equal weight superposition of eigenstates. He then makes the assumption: measuring X on the original system is the same as measuring ( X , Y ) on the product sytem and discarding the outcome of Y. Uncontroversial though this may be, we are greatly expanding on the background assumptions. Moreover we are actually assuming functional invariance for a many-to-one function: namely, the function which delivers the 2-component of a pair (2,~). By the way, Deutsch’s proofs of steps (b) and (c) similarly involve such constructions. Step (e) is an approximation argument which can presumably be made rigorous, though perhaps differently to how Deutsch does it. Step (f) as presented by Deutsch involves yet another new assumption: measuring an observable can be represented as a unitary transformation on a suitable product system, so that after a new unitary transformation mapping 12) to e+4 12) one can remove complex phases from a superposition of eigenstates. This argument seems to be unnecessarily complicated. Our unitary invariance assumption together with the unitary transformation just described, takes care of extending results from real to complex superpositions. The work of Deutsch has been strongly criticised by Finkelstein (1999)
291
and by Barnum et a1 (1999). They also point out that the first step of Deutsch’s proof is incorrect, however, do not recognise that it can be repaired by a supplementary, natural, condition. They also point out that Gleason’s theorem does the same job as Deutsch purports to do, but do not see the very close connection between Gleason’s and Deutsch’s assumptions. They point out also that the later steps of Deutsch’s proof depend on various appeals to the substitutability principle, without stating which games were to be substituted for which. I must admit that it took me a long email correspondence with David Deutsch, before I was able for myself to fill in all the gaps. Finally they also point out that the work of de Finetti and Savage implies that rational behaviour under uncertainty implies behaviour as if probability is there. It is therefore just a question of taste whether or not one adds a probability interpretation to the ‘values of games’ derived by Deutsch. My conclusion is that Deutsch’s proof as it stands is valid, though the author is implicitly using unitary as well as functional invariance. All his assumptions together imply the assumptions of Gleason’s theorem, and much more. Consequently the proof as given does not have a great deal of mathematical interest. However the fact that distributional conclusions could already be drawn for some states and some observables, at a point at which only functional invariance for one-to-one functions had been used, and in my opinion, with a most elegant argument, justifies the conjecture I have already mentioned:
Conjecture 1 Deutsch’s theorem is true under the three assumptions: functional invariance f o r one-to-one functions, unitay invariance, and the degeneracy assumption. Unitary invariance alone tells us that the law of the outcome of a measurement of X only depends on the absolute innerproducts l ( ~ l. So the task is to determine the form of the dependence.
l$~)
Acknowledgments I am grateful for the warm hospitality and support of the Quantum Probability group at the department of mathematics of the University of Greifswald, Germany, during my sabbatical there, Spring 2002. My research there was supported by European Commission grant HPRN-CT-200200279, RTN QP-Applications. This research has also been supported by
292
project RESQ (IST-2001-37559) of the IST-FET programme of the European Commission. References
H. Barnum, C. M. Caves, J. Finkelstein, C. A. F’uchs and R. Schack (2000), Quantum Probability from Decision Theory? Proc. Roy. SOC. Lond. Ser. A 456, 1175-1182. D. Deutsch (1999), Quantum Theory of Probability and Decisions, Proc. Roy. SOC.Lond. Ser. A 455, 3129-3137. R.D. Gill (1996a), Discrete Quantum Systems, www.math.uu.nl/people/gill/Preprints/chapter2.pdf.
R.D. Gill (1996b), Hidden Variables and Locality, www.math.uu.nl/people/gill/Preprints/chapterl4.pdf. A. Gleason (1957), Measures on closed subsets of a Hilbert space, J. Math. Mech. 6 , 885-894. S . Saunders (2002), Derivation of the Born Rule from Operational Assumptions, quant-ph/0211138. D. Wallace (2002), Quantum Probability and Decision Theory, Revisited, quant-ph/0211104. D. Wallace (2003a), Everettian Rationality: defending Deutsch’s approach to probability in the Everett interpretation, quant-ph/0303050.To appear in Studies in the History and Philosophy of Modern Physics, under the title “Quantum Probability and Decision Theory, Revisited”. D. Wallace (2003b) ,Quantum Probability from Subjective Likelihood: improving on Deutsch’s proof of the probability rule, q~ant-ph/0312157.
VOLTERRA REPRESENTATIONS OF GAUSSIAN PROCESSES WITH AN INFINITE-DIMENSIONAL ORTHOGONAL COMPLEMENT
W J I HIBINO Faculty of Science and Engineering, Saga University, 84 0-8502, Saga, JAPAN e-mail: hibinoyOcc.saga-u.ac.jp
HIROSHI MURAOKA Research Center of Computational Mechanacs, Inc., 142-004I , Tokyo, JAPAN e-mad: muraokaOrccm.co.jp
We consider whether the noncanonical Volterra representation may have an infinite-dimensional orthogonal complement or not by the use of the method of the stationary processes.
Keywords: Gaussian processes; Noncanonical representations; Volterra representations. AMS Subject Classification: 60G15,60G10
1. Introduction
-
It is easy to see that, for a given Brownian motion B = { B ( t ) t; 2 0}, a Gaussian process B, = {E,(t);t 2 0} defined by &(t) =
I” (--+ 2q
9
1 UQ IY
’) dB(u)
-iq
is again a Brownian motion for any nonzero q > -1/2. Moreover, P. L6vy [9] pointed out that there were infinitely many polynomials P so that
Z(t) =
Jd”
P(u/t)dB(u)
293
294
is again a Brownian motion. Among these many representations of a Brownian motion, there is the only one special representation: so-called a canonical representation [5].
For a Gaussian process X = { X ( t ) ;t 2 0) defined by
the representation (2) is said to be canonical with respect to B, if & ( X ) = & ( B ) for each t , where & ( X ) is the o-field generated by { X ( s ) ; s5 t}. As & ( X ) can be interpreted as past information of X , we can say that in canonical representation theory past information of B can be completely acquired by that of X . It is noted that, in a Gaussian case, & ( X ) = & ( B ) is equivalent to H t ( X ) = H t ( B ) , where H t ( X ) is a closed linear hull of { X ( s ) ;s It>. For example, the representation (1) is noncanonical since B, satisfies the property
{
H t ( B ) = H t ( E q )a3 LS I ' u ' d B ( u ) }
,
where LS{. . .} is a linear span of {. . .}. Let g1,92,. . . ,QN E L&,[O,cm) be linearly independent in (0, t ) for each t > 0. In the joint work [3],the authors have found how to construct the noncanonical representation of a Brownian motion having the N-dimensional orthogonal complement whose basis is g = ( 9 1 ~ 9 2 ,...,g ~ } :
is a Brownian motion, and is noncanonical with respect to B satisfying
H t ( B ) = H&)
@ LS
S j ( U ) d B ( U ) , j = 1 , 2 , . ..
where
We remark that the Grammian matrix r(t)is invertible since g is linearly independent in (0, t).
295 The form (3) is a Volterra representation. (This terminology is due to [2].) Volterra representations are closely related to the famous innovation theorem, and so on. (see e.g. Hida and Hitsuda [S]) In this article, we shall consider whether the noncanonical Volterra representation may have an infinite-dimensional orthogonal complement by the use of the knowledge of the stationary processes. In Section 2, we review canonical representation theory for stationary processes, and then we give the result in [4] that there exists a noncanonical representation having an infinite-dimensional orthogonal complement. In Section 3, we discuss various conditions for Volterra representation to have an infinite-dimensional orthogonal complement. 2. Stationary processes
Suppose that F ( . ,.) in the representation (2) is a homogeneous function of degree cr i.e. F(at,au) = aQF(t,u)for any positive a. By using the transformation
the process X is transformed into the stationary process Y ( s )=
L
F(1,e-2(8-"))e-(8-")dW(u), s E R,
where dW(u) = $e-"dB(e2") is a Wiener measure. In these representations, each canonical property corresponds as follows: X is canonical with respect to B , if and only if Y is canonical with respect to W ;on the other hand, X is noncanonical with respect to B satisfying
I'
H t ( X )I
ugdB(u), t > 0,
for q > -1/2, if and only if Y is noncanonical with respect to W satisfying
H 8 ( Y )I
+
/'
ePudW(u), s E R,
-W
for p = 2q 1 > 0. Concerning stationary processes, canonical representation theory is well developed in deep connections with functional analysis. In this section, we shall give a brief review of the theory. Let a stationary process Y = { Y ( s ) ;s E R} be represented as
G(s- u)dW(u).
(5)
296
Due to the Paley-Wiener theorem [l],the Fourier transform 6 of G lives in the Hardy class H2+ in the upper half-plane C+ = { z E C ;Sz > 0) since G belongs to L2[0,00). It is also known that c E H2+ has a unique decomposition: c(X) = CCO(X)CI(X) with c ~ ( x= ) &exp cr(X) = n(X)exp
* 1+wX
d,B(w)
+ ihX} ,
where C is a constant of a unit modulus, II is a Blaschke product, f is a spectral density function of the process Y ,h is a nonnegative constant, and p is a nondecreasing function of bounded variation whose derivative vanishes almost everywhere. Here, co and CI are called an outer function and an inner function, respectively. Both are analytic in C + . The outer function never vanishes in C+.All the zero-points there are undertaken by the inner function. Though the representation kernel G satisfies 1 -16(X)lz = f(X), X E R, 2w the outer function co also satisfies 1
-1c0(X))~ = f (A), X E R. 21r Thus the inner function has a unit modulus on the real-axis. Related to canonical representation, the following fact in [8] is important: The representation (5) is canonical with respect to W , if and only if the inner function of 6 is absent. This tells us that the outer function determines the law of the process and that the inner function causes the noncanonical property.
Example 2.1. For the representation (1) of a Brownian motion, using the transformation (4) we obtain a stationary Brownian motion (OrnsteinUhlenbeck process) Y having the representation (5) with
The Fourier transform c of G is c(X) =
( 1 (=X)' +) , where p = 2q + 1. 1-iX ip
297 The first factor is the outer function, and the second is the inner one. As we have remarked, it is noncanonical satisfying
/'
H,(Y) I
ePudW(u), s E R.
-m
The zero-points in C+ are related to the orthogonal complement of the noncanonical representation:
J-00
if and only if the Fourier transform of G in (5) has a zero-point XO in C+. As the authors have pointed out in [4], there exists a noncanonical representation having an infinite-dimensional orthogonal complement: The stationary process Y defined by the representation ( 5 ) with
where 00
< ca and pn > 0,
(7)
n=l
satisfies
H,(Y)I
{ 100 ePnudW(u); n
EN
1
in H , ( W ) ,
sE
R.
(8)
The condition (7) guarantees the convergence of the infinite product in (6). By using the inverse transform X ( t ) = &?Y(log&) of (4), we can prove the following theorem. Theorem 2.1. There exists a noncanonical representation of a Gaussian process X satisfying
for the sequence
satisfying (7).
If we take co(X) = 1/(1- i X ) in (6),we can easily see that there exists a noncanonical representation of a Brownian motion having an infinitedimensional orthogonal complement.
298
3. Volterra representations We call a Volterra representation for the representation
where k ( s , .) belongs to L2(0,s) for any s
I" (la
k ( s , u)2du)
> 0, satisfying ds < 00.
It is well-known in [7] that X is equivalent to B (i.e. the distributions of X and of B are mutually absolutely continuous), if and only if X admits a Volterra representation with k E L2((0,t ) 2 )for any t > 0, namely,
I' la
k(s,u)2duds< oo for any t
> 0.
In this case, the representation (10) is canonical with respect to B . As we have seen in Section 1, the noncanonicd representation (3) of a Brownian motion is always a Volterra representation. Therefore, the Volterra kernel
is not square-integrable. Needless to say, we can check it by a direct calculation. In order that the representation kernel of (10) is a homogeneous function of degree zero, we shall restrict to the case of k ( s , u ) = (l/s)cp( u / s ) , where cp belongs to L2(0,1). Then the representation (10) turns to
By using the transformation (4) the stationary process Y is obtained as in the form
Y ( s )=
/'
--co
e-(8-u) (1 -
)
e"$(v)du dW(u).
(12)
Here we put $(v) = 2e-"cp(eM2") E L2(0,oo), for short.
Proposition 3.1. If a stationary process Y of the form (12) satisfies ( 8 ) f o r p , > 0, then supp, < 00.
299
Proof. If the property
H 8 ( Y )I/' eP"dW(u) -cQ
is satisfied, then
for any s E R. It is reduced to fcQ
By using the Schwarz inequality,
fiL lI+llL2(0,rn)
~1~ Though they were originally defined in a different way,15 double product integrals can be described as iterated simple products with initial or system algebra. There are two ways of doing this; that they produce the same result can be regarded as a kind of multiplicative Fubini theorem or as a continuous analog of the equality
which holds whenever the X j , k have the property that Xj,k commutes with xjlkl whenever both j # j ' and k # k', but not necessarily for example when j = j' but k # k'. Quantum stochastic double product integrals sometimes have rather poor convergence properties. For example, for the double product
where A is the usual number process20 and z is a complex parameter, it = dA, that the exponential can be shown,15 using the multiplication
306
matrix element
where 21 = j1g1 and 2 2 = S,’j.g, and pm,n is the number of m x n incidence matrices such that every row and every column contains at least one entry 1. Since pm,n can be approximated by 2mn for large m,n, since a random choice of 1’s and 0’s will usually have a 1 in each row and each column, it can be seen that this double series diverges when z = 1 far dl nonzero 2 1 and 22. A mare painstaking argument’ shows that the corresponding power series in z has radius of convergence zero. Thus it is fortunate that, not only can the theory of double products be abstracted away from quantum stochastic calculus, but also, in applications to quantisation, only formal power series occur. 2. The It6 Hopf algebra.
Let C be a complex vector space. The vector space T ( C ) = @,“==, (Bn L) of all tensors over L becomes a commutative unital associative algebra when equipped with the shufle product defined by linear extension of the rule that, for arbitrary L1, L2,. . . ,Lm+n E L,
(L1@3L2@*. .@Lm)(L1@L2@‘. . *@Ln) =
c
~ ( 1@)~ 7 r ( 2 @. )
*
*gL7r(m+n)
XES,,,
where Sm, is the set of (m,n)-shufles,that is permutations T of {1,2,. . . ,m n } such that ~ ( 1 < ) ~ ( 2 )< . . < ~ ( mand ) ~ ( m1) < ~ ( m2) < ... < ~ ( mn). Equivalently, for arbitrary a,B E T(L), a/3 = y, where the homogeneous components of y are given in terms of those of a and ,B by
+
+
+
+
c
AuB={1,2,...,n},AnB=B
ahl
Here and elsewhere we use the place notation that, for example, denotes that the [A[-thrank homogeneous component al~lof a is placed in the tensor product of the \A\ copies of L within @n L labelled by the elements all of A. BEl is defined analogously, so that in the combination
afil@,l
307
C aren occupied exactly once. The unit element is ~ T ( L = ) ( 1 @ , 0 , 0 ., ..) E 7 ( C ) . The commutative shuffle product algebra T ( C ) becomes a noncocommutative Hopf algebra when equipped with the coproduct defined by linear extension of n copies of
m
A ( & @ L2 @ .. .L,) = c ( L i @L2 @ ...L j ) @ ( L j + l @Lj+2
...L,)
j=O
where the summand is regarded as an element of T ( C ) @ 7 ( C ) . The counit is defined by 4Q:o,a1,a2,. . .)
(& C1 @ (Bm-j' C 1 C
= Qo,
and the antipode is the linear extension of
S(L1 @ L 2 @ . . . L m = ) (-I)m(Lm@Lm-l @ . * . L 1 ) . Now let 2 be a not necessarily unital complex associative algebra. Then we may define the It6 shufle product in T ( C ) by the equivalent formulas
( L I@L2@ * * * Lm)(L1@L2 8. * Ln) -,=(PI
c
L,, @ L P 2 @ * * * L P L
,P2>...Pb)EPm,n
AUB={1,2,...,n} Here, in the first formula,13 Pm,, is the set of It6 shufles (sticky shuffle would perhaps be a more descriptive term) consisting of partitions p = ( p 1 , p 2 , . . .p k ) of { 1 , 2 , . . . ,m n } in which each p j is either a pair (s, t ) with 1 5 s 5 m 5 t 5 m n or a singleton and in which the natural orders of the subsets { 1 , 2 , . . .,m} and {m 1 , m 2 , . . . ,m n } are preserved in the ordered set ( p l, p 2 , ... p k ) . Each LPj is defined to be L , if p j = {s} and the product L,Lt if p j = (s,t ) . In the second formula we use place notation as before and reduce the double occupancies of C which occur when A n B # 0 by using the product in C . The It6 shuffle product makes T ( C ) into a unital associative algebra, with the same unit element as before, which is noncommutative if C is noncommutative. When C is the algebra of It6 differentials of quantum stochastic calculus then the map which sends each ct E 7 ( C ) into the sum of iterated stochastic integral processes whose integrators are the homogeneous components of Q: is multiplicative.
+
+
+
+
+
308 The map L 3L (0, L, 0, 0, .. .) E T(L)is a Lie algebra homomorphism when both associative algebras are equipped with the commutator Lie bracket. It can be shown16 that the universal extension to the universal enveloping algebra U ( L ) of the Lie algebra L is an isomorphism of unital associative algebras from U ( L ) onto the subalgebra S(L)of T(L)formed by the symmetric tensors. Remarkably17, the coproduct A and counit E introduced above for the shuffle product algebra T(L)remain multiplicative for the It6 shuffle algebra T(L)and equip it with a Hopf algebra structure in which the antipode is a deformation4 of that for the shuffle product of the form L1
€3 L2 €3
. . . L,
* (-l),(L,
€3 L,-1
€3
.L1) + terms of lower rank.
When the associative algebra L is noncommutative, T(L)is thus a noncommutative and noncocommutative Hopf algebra containing a cocommutative sub-Hopf algebra S(L)isomorphic to U ( L ) . We call it the It6 Hopf algebra.
3. Calculus in the It6 Hopf algebra.
d
We define the right and left differential maps : T(L)+ T(L)€3 L and t d : T(L)+ L @ T(L)by linear extension of the actions on homogeneous product tensors
d(L18L2 8.. .L,)
= (L1 €3 ’ . . @ L,-1)
€3
L,,
t
d(L1€3L2€3.*.L,) =L1€3(L2€3.-.Lm).
Alternatively for arbitrary a E T ( L ) ,
a4
= (idT(L) €3
t
d ( a )=
(Q
@)
( A ( 4 - 1T(L)€3 a ) ,
€3 idT(L)) (N4 -
€3 1 T(L))
where Q is the associative algebra homomorphism a + a1 from the ideal T(L),in T(L)consisting of elements whose zero rank homo eneous components vanish, equivalently the kernel of E , to L. and d satisfy the Leibniz-It0 formulas
d
&QB) t
= &4B
8
+ J(P) + d(a)&B),
+ ad(D)+ %(a)%@)
d (aB) = %(a)B
in which T(L)€3 L and L €3 T(L)are regarded as two-sided T(L)-modules using the multiplicative right and left actions of T(L)on itself as well as associative algebras using the tensor product multiplication.
309
By comparing actions on homogeneous product tensors we see that the coproduct A is recovered from either of the differential maps as 00
M
where we make the natural identifications W
00
7 ( L )€4 T ( L )= @ (7(L)€4C?ln ( L ) )= @ ( @ y L )€4 7 ( L ) ) n=O
n=O
and the iterated differential maps are defined by t t = d ('1 = d , and for n > 1
d(') 2,
den) = (2€4 id Bn-IL)
%ten)
$(n-'),
d(O)= td
= (idBn-lL €4
(O)
= idT(L),
2)
%(n-l).
Alternatively we may describe A as the unique solution of either of the differential equations (idT(L) @
2)A = (A @ idL) 2,(idT(q @
E)
A = idT(L),
which exhibit A in each case as a flow of which the corresponding differential map is the generator. Combining the counit properties ( E €4 idT(L)) A = id T(L) = (idT(q 8 E ) A with (2) we obtain the right and left TaylorMaclaurin expansions idqq =
(E
@ idT(L)) n=O
4. Simple product integrals with system algebra.
Let d be a n associative algebra. When equipped with the convolution product / M
\
/ w
\
/ w
N
the space d[[h]] of formal power series in an indeterminate h with coefficients in d becomes an associative algebra, which is unital if d is so. We will find the following well known theorems useful (for proofs, see Ref. 14 for example).
31 0
Theorem 4.1. Let a[h]E hd[[h]] be a formal power series in which the zero-order coeficient vanishes. Then there exists a unique two-sided quasiinverse for a[h]in hd[[h]], that is an element a'[h] such that
+
a[h] a"h] -ta[h]a"h]= a"h]
+ a[h]+ a'[h]a[h]= 0.
Theorem 4.2. Let d be unital and let A[h] E d [ [ h ] be ] a formal power series in which the zero-order coeBcient is 1 A. Then there exists a multiplicative inverse A-l[h] which is of the same form. Let 7 ( C ) be the It8 Hopf algebra over the associative algebra C and let A be a unital associative algebra which we call the system algebra. Let Z[h]= Zj[h]€3 L(j) E h ( A €3 C)[[h]]. The infinite sums
'&
m
m
Y [ h ]=
C C...,
n=Ojl,j2,
Zj,
[h]Zj,-l[h]. . .Zj, [h]€3 L(jl) €3 L(j2)€3 . . . €3 L(jn) (4)
j.,
can be rearranged algebraicly as well-defined formal power series belonging which respectively satisfy the algebraic identities to (d 63 T ( C ) )[[h]] (idd 8 A) X[h]= X[h]132X[h]1'3, (ida €3 E ) X [ h ]= 1 A , (idd €3 A) Y [ h ]= Y[h]113y[h]1'2, (idd €3 E ) Y [ h ]= 1 A. Here the coproduct and counit act on formal power series coefficient-wise and we likewise apply place notation to coefficients in (-4 63 7 ( L )€3 7 ( C ) )[[h]]. It can be shown" that all solutions of these identities are of this form for some Z[h]E h (-4 €3 L ) [[h]]. Also, X [ h ]and Y [ h ]are solutions, unique in each case, of the differential equations
(ids €3 2)x [ h ]= x[h]"2z[h]1'3, (idd 63 (idd €3 (idd €31'd X[h]= Z[h]122X[h]1*3, (idd 63 2)Y [ h ]= l[h]113Y[h]1j2, (idA 63
E)
X[h]= 1 A ,
E)
X [ h ]= 1 a,
E)
Y [ h ]= 1 a,
31 1
We call X [ h ]and Y [ h ]the right and left directed product integrals generated by l[h]and denote them by
+
t
+
X [ h ]=d n(1 dl[h]),Y [ h ]=d n(1+ d [ h ] ) respectively, where the subscript indicates that the system algebra A is laced to the left of T(L)in each case. Product integrals dl[h]), Ifid(1 dl[h])in which the system algebra is on the right, generated by an element l[h]of h (C €3 -4) [[h]], are defined analogously and characterised algebraically as solutions of
+
+
(A €3 idd) X [ h ]= X[h]1’3X[h]2’3, ( E €3 idd) X[h]= 1 A ,
(A @ idd) Y [ h ]= Y[h]2’3Y[h]1’3, ( E @ idA) Y [ h ]= 1A , respectively, and differentially as solutions of
(2€3 idd) X [ h ]= X[h]1’3i[h]2’3,€3 idd) x [ h ]= 1A, (E
(2€3 idd) X [ h ]= l[h]1’3x[h]2’3,@ idd) x [ h ]= 1 A, (E
(2€3 idd) Y [ h ]= l[h]1’2Y[h]2’3,€3 idd) Y [ h ]= 1A. (E
-
When the system algebra is nonunital we define decapitated product h
h
integals A f f ( l + d [ h ] )A, f i ( 1 + d [ h ] )ffd(1 , +dl[h])and f i A ( 1 +dl[h])by omiting the initial unital term in the expansions (3) and (4). For example 03
+dl[h])=
lj, [h]2j2 [h].. .lj,
A n (1
[h]@L(jl)@L(j2) €3 * . .€3 L(jn)
n=l j 1 , j 2 , ...,j,,
and such a decapitated integral is characterised algebraicly as a solution of
+
(idd €3 A) X[h]= x[h]1’21%(L) x[h]’j31 ?j-(L)
+ x[h]112X[h]193,
(idd €3 E ) k [ h ]= 0 A , and differentially as that of (idd €3
2)x [ h ]= x[h]’121[h]173+ 1+(Lll[h]1’3,(id
€3 E ) x [ h ]= 0 A,
31 2
or of
as is easily proved by adjoining a unit to A.
5. Double product integrals. Now let r[h]be a n element of h (C@J C)[[h]]. Regarding the first copy of C in C €4 C as a left system algebra we may form the decapitated directed product integral ~ z ( +1 dl[h]).This is an element of h (C€9 ‘T(C))[[h]]. Hence, regarding 7(C) as a unital right system algebra we may form the directed product integral
f i ~ ((1~+)d ( ~ 3 (+1 dr[h])))which is
an element of (7-(C) @ 7 ( C ) )[[h]].Similarly we may form the element
~ ( , q f(1i
+ d ( E L ( l + dr[h]) of ( 7 ( C )€9 7 ( C ) )[[h]]. It is proved in Ref. 15
that
We define the common value to be the forward-backward directed double
n
-+e
+
product integral (1 dr[h])generated by r[h].By reversing arrows we can also define the backward-forward directed double product integral
+-+
n
F
(1 + dr[hl) =
1 + 4 L r - p + dr[hI))
Two other directed double products, forward-forward and backwardbackward, can be defined similarly but seem to be of less interest. It follows from the algebraic characterisations of directed simple product integrals that the double product integrals
=n
R[h]
(1
+ dr[h]),R”h]
=n + (1
dr[h])
satisfy
(A €4 idd) R[h]= R[h]193R[h]233, (idd €9 A) R[h]= R[h]1z3R[h]112, (5) (A €9 idd) R’[h]= R’[h]293R’[h]1,3, (idA 63 A) R’[h]= R’[h]192R[h]133,(6)
313
together with
It is not difficult t o show that these relations characterise forward-backward and backward-forward directed double product integrals (see Ref. 17 for the forward-backward case). (5) and (7) are known as the quasitriangularity satisfies the quasitrianrelations. Thus an element of (7(L)8 7 ( L ) )[[h]] gularity relations if and only if it is a forward-backward directed double product integral. The first stage of Enriquez’ quantisation procedure for Lie bialgebras is to construct solutions of the quasitriangularity relations. Theorem 5.1. Let r[h] and r’[h] be mutually quasiinverse elements of
n
-+t
(1 + dr[h])and h ( L @ L) [[h]].Then verses in ( T ( L )@ 7 ( L ) )[[h]].
n (1 + dr’[h])are mutual in-
t+
Proof. Setting
we have
where
Using the Leibniz-It6 formula, it follows that
314
Since
( z @ i d ~ ) q [ h= ] (z@idr)
-
(n,( + 1
dr' [h ] )
using differential characterisations of decapitated product integrals, again using the Leibniz-It6 formulawe have
since r[h]and r'[h]are mutually quasiinverse. Also by multiplicativity of E,
by characterisation of decapitated products. It follows that p[h] p[h]q[h]= 0 and hence by (8) that ( i d q ~@ )
2)(P[hlQ[hl) = 0.
+ q[h]+
31 5
Since by multiplicativity of E and characterisation of directed simple product integrals,
id^(^) €3 E ) (P[hlQ[hl)= id^(^:) €3 E) WI (&-(L)
@E)
(Q[hl)
= 1 T(L)
it follows that P[h]Q[h]=
~ T ( L ) ~ T ( Lthat ),
is
n
-tt
dr'[h])= 1 T ( L ) ~ ~ ( LAC )similar . argument shows that ( 1 + W h l ) = 1T ( L ) @ T ( L ) 0 -
+ dr[h]) n (1 + n ( 1 + dr'[h]) n t-i
(1
t-t
-it
A corollary t o Theorem 3 is that, for mutually quasiinverse r [ h ] ,r'[h] E h ( L €3 L ) [ [ h ] ]the , map J[h]from 7 ( L )€3 7 ( L )to ( T ( L )8 7 ( L ) )[[h]]given by
is multiplicative. Since A is multiplicative, it follows that the map A[h] = J [ h ] Afrom 7 ( L )t o ( T ( L )€3 T ( L ) )[[h]]is also multiplicative. Notice also that, for arbitrary E T ( L ) ,by multiplicativity of E , characterisation of product integrals and counitality of E ,
. Substituting in (l),we find t h a t
6(L) = [Tl
- 7(2,1)T1,
( L €3 1
+ 1 8L ) ] .
Thus we have a coboundary Lie algebra4 determined by the skew-symmetric part of the tensor T I . Thus commutator Lie bialgebras of this type can be quantised by t h e method of double product integrals.
References 1. R Bacher, private communication. 2. B Enriquez, Quantisation of Lie bialgebras and shuffle algebras of Lie algebras, Selecta Math. (N.S) 7, 321-407 (2001). 3. P Etingof and D Kazhdan, Quantization of Lie Bialgebras I, Selecta Math. 2, 1-41 (1996), 11, ibid 4, 213-231 (1998), 111, ibid 4, 233-269 (1998). 4. P Etingof and 0 SchifFman, Lectures on quantum groups, International Press (1998). 5. A S Holevo, Time-ordered exponentials in quantum stochastic calculus, pp 175-182 in Quantum Probability VII, ed L Accardi et al, World Scientific 1992. 6. A S Holevo, Exponential formulas in quantum stochastic calculus, Proc Royal Society of Edinburgh 126A,375-389 (1996). 7. A S Holevo, An analogue of the It8 decomposition for multiplicative processes with values in a Lie group, Sankya A 53, 158-161 (1991). 8. R L Hudson, Algebraic stochastic differential equations and a Fubini theorem for symmetrised quantum stochastic double product integrals, pp 75-87 in Quantum Information 111, ed T Hida et all World Scientific (2001). 9. R L Hudson, Calculus in enveloping algebras, Jour. London Math. SOC(2) 65, 361-380 (20020
31 9 10. R L Hudson, P D F Ion and K R Parthasarathy, Timeorthogonal unitary dilations and noncommutative Feynman-Kac formulas, Commun. Math. Phys. 83, 261-280 (1982). 11. R L Hudson and K R Parthasarathy, Quantum Ito formula and stochastic calculus, Commun. Math. Phys. 93, 301-323 (1984). 12. R L Hudson and K R Parthasarathy, Construction of quantum diffusions, pp173-198 in Quantum Probability I, ed L Accardi et all Springer LNM 1055 (1984). 13. R L Hudson and K R Parthasarathy, The Casimir chaos map for U ( N ) ,Tatra Mountains Math. Jour. 3, 1-9 (1994). 14. R L Hudson, K R Parthasarathy and S Pulmannovd, The method of formal power series inquantum stochastic calculus, IDAQP 3, 387-401 (2000). 15. R L Hudson and S Pulmannovd, Symmetrized double quantum stochastic product integrals, J Mathematical Phys 41,8249-8262 (2000). 16. R L Hudson and S Pulmannovd, Chaotic expansions of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus, Proc. London Math. SOC.(3) 77, 462-480 (1998). 17. R L Hudson and S PulmannovB, Double product integrals and Enriquez quantisation of Lie bialgebras I, to appear in J Mathematical Phys. 18. R L Hudson and S Pulmannovd, Double product integrals and Enriquez quantisation of Lie bialgebras 11, Nottingham Trent preprint, submitted to Lett. Mathematical Phys. (2003). 19. R L Hudson and S Pulmannovb, Explicit universal solutions of the quantum Yang-Baxter equation constructed as double product integrals, pp 289-296 in ICMP X I I I (2000) Proceedings, ed A Grigoryan et all International Press (2002) 20. K R Parthasarathy, An introduction to quantum stochastic calculus, Birkhauser (1992). SCHOOL OF COMPUTING AND
UNIVERSITY, BURTONSTREET, BRITAIN .
MATHEMATICS, NOTTINCHAM TRENT NOTTINGHAM NG1 4BU, GREAT
ON NONCOMMUTATIVE INDEPENDENCE *
ROMUALD LENCZEWSKI Institute of Mathematics Wroctaw University of Technology Wybrzeie Wyspiariskiego 27 50-370 Wroctaw, Poland E-mail: 1enczewOim.pwr.wroc.pl
We examine the main notions of noncommutative independence, namely tensor, free, boolean and monotone independence. We collect the results on unification of these notions from the point of view of reducing them to tensor independence. We also show how to reduce A-freeness to tensor independence and demonstrate that in a similar way one can construct analogous mixtures of tensor independence and boolean independence as well as tensor independence and monotone independence.
1. Introduction
In noncommutative probability there is no single notion of independence. In addition to the so-called tensor independence, which has similar features as classical independence, there are other, more noncommutative notions of independence like, for instance, freeness, boolean independence and monotone independence (or, equivalent to it, anti-monotone independence) each associated with some new type of probability theory. The axiomatic theory, see Refs. 1-2, singles out these notions of independence as such which are associated with the so-called 'natural' products of states satisfying a given set of axioms. On the other hand, it has been shown in Ref. 3 that there exists a discrete interpolation between boolean independence and freeness. If (dl, q 5 1 ) l ) l E ~ is a family of noncommutative probability spaces, then the hierarchy of freeness is a sequence of noncommutative probability spaces (A("),d m ) ) m -> l , such that
*This work is supported by KBN grant No 2P03A00723 and by the EU Network QPApplications Contract No. HPRN-CT-2002-00279.
320
321 in the sense of convergence of mixed moments, where j(") :UlELdl
+ A(")
are suitable *-homomorphisms. Here, U I E L d l denotes the free product of d l ' s without identification of units and * I ~ L + Iis the free product of 41's. (units are identified in the limit). Thus, the states dm) o j ( " ) , called mfree products, approximate the free product of states in the weak sense. What is important, the whole sequence (A("), dm))m>l - is embedded in a noncommutative probability space (.&$) of the form
IEL
IEL
and therefore the construction reduces the free products of states to the tensor products of states in the weak sense. Moreover, the first-order approximation m = 1 gives the boolean product of states. Next, F'ranz observed in Ref. 4 that a construction similar to that for the boolean product of states can be done to recover the monotone product of states. Finally, in Ref. 5 , it was shown that the free product of states can be reduced to a tensor product of states in the strong sense. We have constructed a suitable 'closure' of the tensor product of unital *-algebras 31 by adding certain 'affiliated' operators called monotone closed operators. Then we can embed free random variables represented by infinite series of simple tensors in a noncommutative probability space (A, $), where now
IEL
[EL
with ?3 denoting the 'closed' tensor product called monotone tensor product, where the procedure of taking the closure is similar to that in the von Neumann algebra tensor product. These results lead to the following theorem. Theorem 1.1. Let ( d l , 4 1 ) be ~ noncommutative ~~ probability spaces. For every 'natural' product of states 0 1 ~ ~ 4there 1 , exist noncommutative probability spaces ( d l , $ ~ ) L E Land a (in general, non-unital) *-homomorphic embedding A
h
j : UIELdl
+A
(1.3) h
such that J o j = 0 1 ~ ~ and 4 1 $ o j l d ~= 41 for every 1 E L, where d and are given by ( I . 2).
A
322 In other words, every ‘natural’ product is equivalent to a restriction of j ( U I E L d l ) , which means that every ‘natural’ product of states can be reduced to a tensor product of states. In this context, it would be interesting to determine whether a similar treatment is possible for a larger class of models, especially those related to the Gaussianization on the interacting Fock spaces studied in Ref. 7 by Accardi and Boiejko. For each of the ‘natural’ products we show in this paper how to explicitly realize this reduction. Thus we collect the results of Refs. 3-5, where we refer the reader for details. We also show that the A-free product of states introduced in Ref. 6 can be reduced to a tensor product of states. Finally, we introduce new ‘mixed’ products, which may be called A-boolean and A-monotone. If it suffices to use algebraic tensor products, we use the notation of (l.l),whereas if we need the monotone tensor product, we use the notation of (1.2). In all theorems we refer to the setting of Theorem 1.1 and, for notational simplicity, we assume that L = N.
2to the *-subalgebra
2. Boolean Independence
Assume that ( d i ) i E L is a family of unital *-algebras and ($l)lE= are states on these algebras.
Definition 2.1. By the boolean product of states state on U i E L d l given by the recursion
3
where XI, E d l ( k ) and Z(1) # Z(2) # ... variables come from different algebras.
$1
we understand the
# Z(n), thus the neighboring
2
In fact, one can show that the functional obtained in this way is a state, i.e. is positive and normalized. For the proof of positivity in the more general case of the conditionally free product of states, see Ref. 8. The mathematical roots of the boolean product of states come from the regular free product of functions on discrete groups of Boiejko given in Ref. 9. The basic tool which enables us to reduce the boolean product of states to the tensor product is that of the boolean extension (see Ref. 3 ) of a state $ on a unital *-algebra A. Namely, let’s extend d freely by a single projection P to get
323
and then extend $ to a state
$ on x by the recursive formulas
&wPv) = &J)IJ(v), & P ) = 1
7
for any v ,w E Then is called the boolean extension of 4 and 3 is called the boolean extension of A. In other words, is the boolean product of $ and the state h on C [ P ]given by the linear extension of h ( 1 ) = h ( P ) = 1. Note that the projection P plays the role of the separator of words from A. Namely
2.
6
&P*X1PX2P.. . PXnPP) = $ ( X l ) $ ( X 2 ) .. . $ ( X n ) where a,/3 E ( 0 , l ) and X I , . . . ,X , E A. In the easiest case of two noncommutative probability spaces, ( A , $ ) and (f?,$), it is not hard to see that the mapping
j:AUB+x@# where
x = A * @[PIand # = 23 * @[P’],given by j ( X )= x
@ P’,
j(Y)= P
€3Y
where X E A, Y E B,is a *-homomorphism such that ($8$) o j agrees with the boolean product of $ and $. It is also instructive to observe that one can associate a *-bialgebra structure with a unital free *-algebra A and produce boolean independent copies of random variables from A by iterating the coproduct. For instance, if A = @[XI,the coproduct is given by
A ( X )= X @ P + P @ X where the projection P is group-like, i.e. A(P) = P @ P. Succesive iterations of the coproduct give a finite number of boolean independent random variables. Namely, the sequence
-, ,
X i = P @ . . @ P @ X @ P @ . .@ P 1 < 1 < N , N-1 times
1-1 times
is a sequence of boolean independent random variables w.r.t. the state $’N. The general case is now straightforward. To conform with the statement =2 1 and 81 = 51. For simplicty we of Theorem 1.1, it is enough to put take L = N. For details, see Ref. 3.
A^1
Theorem 2.1. Let j : U n E N d n + BnEN An be the *-homomorphism given by
-
A
j ( X ) = P @ .. . @ P @ X @ P 8.. . n-1 times
324
where X E An. Then
3
0
j agrees with the boolean product of states
&.
Proof. It is an easy consequence of the definition of the boolean extension of a state. For instance, in the case of the boolean product of two states, $1 and I&, we have ( J o j ) ( x Y ..) . = &(XI? ..)&( PY...) = d l ( X )$1
(P.. .) $2 (Y.. .)
=+l(X)(&Jj)(Y...) for any X E A1 and Y E A2, which conforms with Definition 2.1.
0
3. Monotone Independence
In this Section we show how to realize the statement of Theorem 1.1 for monotone independence in the sense of Muraki and Lu, see Refs. 10-11.
Definition 3.1. By the monotone product of states the state on UIELAl given by the recursion
3
( ~ I ) I ~ weL understand
&xlx2.--xn) = ~ l ( k ) ( X k ) 3 ( X l . . . X k - l X k +... l
xn)
(3.1)
whenever Z(k) is a local maximum in the tuple (Z(l), Z(2), . . .,Z(n)), i.e. l k - 1 < l k and Ik+l < l k for Some 1 5 k 5 72, where xk E d l ( k ) and we adopt the convention that if k = 1 or k = n, then only one of these inequalities holds. Note that our definition is equivalent to that given by Muraki in Ref. 10. In a similar way we define monotone independent subalgebras of a given algebra. It was observed in Ref. 4 that one can reproduce monotone independent random variables by taking the coproduct which is ‘half-boolean’ and ‘halfclassical’. Namely, if we take the coproduct on A given by
A(X)= X @ P + I @ X
--
where X E A, with P being group-like, then the variables obtained by iterating this coproduct, namely
X n = l @... @ l @ X @ P @... @ P , 1 L n S N n-1 times
N-1 times
are monotone independent w.r.t. the state J B N . It is then straightforward to extend this result to any linearly ordered index set L. For notational
325 A
simplicity, we take L = N below. As in the boolean case, we set A
4n =
A,
2,
=
-
4n.
+ BnEN A, be the *-homomorphism given A
-
Theorem 3.1. Let j : UnENdn
by
j ( X ) = 1 8 . ..@I 1 €3x€3 P €3.. . n-1 times
where X E A,. Then
o j agrees with the monotone product of states.
Proof. Suppose Z(k) is a local maximum, i.e. Z(k) Z(k + 1) < Z(k). Then
(60 j ) ( ...Xk-lXkXk+l
> Z(k + 1)
and
-
...) = ...l#ll(k)(... P X k P...)... = f$[(k)(Xk) x ...&)(... PP...)...
= 4 l ( k )( X k where Xl E Ai for Z = k - 1, k,k shown in the calculation.
)(6 j )(...Xk-l 0
Xk+l. ..I
+ 1 and only the most relevant &(k)
is 0
Let us give another realization of the monotone product of states which will turn out useful in Section 5 . For each unital *-algebra dl we create free products of copies dl(k)of Al, extend these free products by an increasing sequence of projections (q,), i.e. such that Qmqn = qmhn and q: = Qn (for simplicity, we use (qm) for each algebra dl) and then take
-& = k N - A ( k ) * c [ q i ,~ 2 , 4 3 7 .. -I/ J (we identify the empty word in the product Uk,Nd(k) with the unit of the algebra @[q1,q 2 , 4 3 , . . .I), where J is the two-sided ideal generated by commutation relations
q m X ( n ) = X ( n ) q m for m
> n,
with X ( n ) denoting the n-th copy of X E di. Let A b e then given by (l.l), where, for simplicity we assume L = N.
2, of the above form, let j *-homomorphism given by
Theorem 3.2. For
-
: UnENdn
j(X)=qn+l €3**.€3qn+1€3X(n)@qn+1€3.-n-1 times
+ A^
be the
326
where X E An. Then there exist states with the monotone product of states &.
&,
Proof. It is enough to construct states
on
i n such that 4o j
agrees
&, which satisfy the conditions h
w ( q m x ( n ) q m - 4n(X)qm)v E kerdn for m In
w - q1wqi E ker&, for any X E An and w,'u E
An, together with ?n(qm) = 1
for any m E N. Note that these conditions can be satisfied by the states lifted from the tensor product of Boolean extensions of &'s of the form
Jn=
lJ$"
oq
where q : i n 3 22" is the *-homomorphism given by
q ( x ( n ) )= I@(~-') 8 x 8 I@" e(n-1) p @ ~ qI(qn) = 1 for any X E A and n E N. This finishes the proof.
0
Example 3.1. We illustrate the theorem with a diagram. The vertical
1
2
3
4
5
6
7
Figure 1. A diagram for monotone independence.
axis corresponds to n and thick lines are associated with projections qn+l.
327 Solid arcs represent connections between elements from the same algebra which produce joint moments, whereas dotted arcs represent connections between elements from the same algebra which produce factorized moments since there is a projection between the elements which acts as a filter (for a general framework with projections acting as filters, see Ref. 12). The diagram in Figure 2 corresponds to monotone independent random variables for Z(1) = Z(3) = Z(6) = 2, Z(2) = Z(5) = 4 and Z(4) = Z(7) = 1. The corresponding moment is given by J o j ( x l . -.x7) = 91(x4x7)92(x1x3)92(x6)94(x2)44(x5)
since the connections 3 - 6 and 2 - 5 cannot be realized due to the fact that the projections associated with X, and X4, namely 43 and q2, respectively, make the moments on level 2 2 factorize. The same result is obtained by using Definition 3.1 and succesive elimination of singletons associated with local maxima. 4. F’reeness
A reduction of the free product of states of Avitzour and Voiculescu, see Refs. 13-14, to a tensor product of states (or, equivalently, a reduction of freeness to tensor independence) is more complicated. In Ref. 3 we have done it in the weak sense of convergence of mixed moments by constructing the hierarchy of freeness and in Ref. 5 we have strenghtened this result as stated in Theorem 1.1. For limit theorems and the GNS construction, see Refs. 15-16. For notational simplicity we now describe the most important points of the construction of Ref. 5 for two algebras. Definition 4.1. By the free product of states state on * l E L d l given by the recursion
6
J(XIX2 *
a .
& a )
(41)lE~ we
= #l(l)(XI)J(X2* . . X,)
understand the
(4.1)
whenever XI E dl(l)and Xk E Alp) n ker$l(k) for 2 5 k 5 n with Z(1) Z(2) # .. . # Z(n).
#
A discrete interpolation between the boolean product of states and the free product of states is given by the hierarchy of m-free products of states given in Ref. 3, whose definition is given below. Note that for each m E N the m-free product of states has to be defined on the free product of dl’s without identification of units.
328
Definition 4.2. By the m-free product of states ( ~ ! J I )we ~ ~understand L the state $ on U I E L d l given by the recursion (4.1) under the assumption that x k E d l ( k ) for all k and x k E ker$l(k) for 2 5 k 5 m. This implies that mixed moments of the associated m-free random variables agree with the moments of free random variables if n _< 2m. In particular, if we take two noncommutative probability spaces ( d , 4) and (a,$), then the m-free random variables can be written as sums
k= 1 where X(k)’s and Y ( k ) ’ s are copies of X E d and Y E L? which are tensor independent w.r.t. some tensor product state $8 with 4, being extensions of 4 and $, and (Pk)’S, wk)’s are suitably defined orthogonal projections, some of which do not commute with X ( k ) ’ s and Y ( k ) ’ s . If m = 1, we obtain the boolean product of states. In turn, in the limit m + co,the mixed moments of the variables j(”)(X) and j(”)(Y) tend to the moments of free random variables. Thus the hierarchy of mfree products gives a discrete interpolation between the boolean product and the free product. Nevertheless, it is desirable to represent free random variables as series of the above forms, with m replaced by m. This requires us to introduce a suitable closure on a *-algebra level. We sketch below how this can be done. For details, see Ref. 5 . There are three essential steps in the construction. Step 1. For given unital *-algebras d and f3,we construct countable free products (without identification of units) of their copies, namely k=l
6,
%(A) = UkEd(k),
. . A
$J
%(a)= UkENa(k)
(we allow empty words which can be viewed as units) and then extend each such product freely by a sequence of increasing projections, namely %o(d) =%(A) *@[ Q I , ~ ~ , E I , - . . ] %iio(f3)
=GCi(a)*@[Q:,Q:,Q~,...]
where the empty words mentioned above are identified with the units of q q t , q 2 , 4 3 , . . .] and @[a:,qi, q$,. . .I, respectively. Note that we have qmqn = qm/\n and q; = qm (similar equations hold for the q;). Step 2. We want now the sequences (qm) and (qA) to play the role of approximate units in our products. For this purpose, we form quotients
3to(d) = Go(d)/I,3to(L?)
Go(B)/J’
329
where I and I’ are two-sided ideals generated by *-relations of the form
X ( k ) = q,X(k), k
< rn and Y ( k ) = q L Y ( k ) , k < m
respectively, where X E A and Y E a. In other words, for given k-th copies of A and f3 there exist in Xo(A) and Xo(f3) large enough projections which act as units on the given copies of A and f3, respectively, and all ‘earlier’ copies. Step 3. Finally, we construct the ’closures’ of Xo(A) and X o ( f 3 )w.r.t. the sequences (am) and (4;) and denote them by %(A)and X ( D ) , respectively. Namely, %(A) is the unital *-algebra of equivalence classes
[X,, em], where Xnem = Xme, and X;e, = X A e , for m < n where (em) = (f&(m)) is a subsequence of (qm) such that ( k ( m ) )t co (we construct “(a) in a similar way). We can say that the closures are taken with respect (4,) and (&), respectively. Moreover, sequences (X,) and (Y,) come from increasing sequences of *-subalgebras
X m E U k < m A ( k ) * c[41,4 2 , 4 3 7 . * . ] / I
These equivalence classes are called monotone closed operators (for details of this construction, modelled on the algebra of closed operators affiliated with *-algebras, see Ref. 5). What is more important, this concept of closure allows us to consider infinite series on the level of tensor products, except that the usual tensor product of *-algebras has to be replaced by the monotone tensor product. We refer the reader to Ref. 5 for details. Let us only mention here that the definition of 3 is similar to the von Neumann algebra tensor product and the closure is taken w.r.t. the sequence (qm 8 4;). Thus, we can define a unital *-homomorphism j :A * B
+ X(A)SX(B)
by the formulas
where X ( k ) ’ s and Y ( k ) ’ sdenote copies of X E A and Y E f3 in A(k) and B ( k ) , respectively and (&), 03:) are sequences of orthogonal projections given by pk = qk - Q k - 1 , pi = 4; - qLV1 with Qo = 4; = 0.
330
Theorem 4.1. For given states 4 and $ on A and 23, respectively, there exist states and on %(A) and %(B) and a unital *-homomorphism
4
3
j : d * 23 + %(d)B%(B)
--- --
such that (4@$) o j agrees with the free product of states 4 * $. Proof. The homomorphism j is given by (4.3). The states be defined by the following conditions: w(qnX(n)qn
- 4(X)qn)v,
3 and 4can
w - Q l W l E ker3,
and
for every with
together
for every m E N. That such states exist can be shown by lifting tensor products of boolean extensions of 4 and $ to %(A)and %(B),respectively, of the type used in the proof of Theorem 3.2. For details, see Ref. 5. This result can be generalized to an arbitrary family of unital *-algebras.
Theorem 4.2. For given states $1 on unital *-algebras dl there exist states & on %(Al) and a unital *-homomorphism j : *ladl + %%(Ai) 1€L
such that the state
3o j
agrees with the free product of states
*zE~c$i.
Proof. For notational simplicty we assume that L = N and that dl = A for every 2 E L. In that case we can use permutations 7r : XBm + given by m
k=l
M
k=l
and their extensions to the monotone closures. Denote by ~ 1 the, trans~ position which exchanges 1 and n (if n = 1, we get the identity). Then the unital *-homomorphism j can be written as
331
where X E A, and
_ _
Pk
=qk@qk@.
..- qk-l@'qk-l%.
.
I
with q(0) = 0. Roughly speaking, this formula replaces Pk = Qk - q k - 1 and p i = q6 from Eq.(4.3) - now the 'free action' of a variable from a given algebra is extended on all algebras (by abuse of notation we use (qm) 0 at all tensor sites). A detailed proof can be found in Ref. 5 .
Remark. It is worth noting that in this formulation it is very natural to associate with each algebra a sequence of states. Since states are obtained by lifting the tensor product states Tyw from to 31(dl), one can lift the tensor products of boolean extensions of different states @ k > l & , k , where $l,k'S are for each E L posssibly different states on dl. In particular, if we take
xy"
&
z
$1
if k = 1
we obtain the so-called conditional (or, $-) independence of Ref. 8. The case of general sequences ( $ l , k ) k > l is a special case of the model of freeness for seqeuneces of states, see Ref-17. 5. Mixed types of independence
This section is motivated by the model of A-freeness of Mlotkowski, see Ref. 6, which is a mixture of tensor independence and freeness. In particular, we show that A-freeness can be reduced to tensor independence. In an analogous fashion we introduce mixtures of tensor independence and boolean independence which we call A- boolean independence, and a mixture of tensor independence and monotone independence - A-monotone independence. In view of the fact that the corresponding product functionals are restrictions of the tensor products states, we immediately conclude that these functionals are states. The idea of producing such mixtures is based on placing projections onto the cyclic vectors at tensor sites corresponding to appropriate copies of the algebra d and keeping the identity at the remaining sites. For simplicity assume that L = N and let A C N x N. We again form free products of copies of each algebra, except that we extend them by sequences of projections related to A. Let
A = U k c W d l ( k ) * C[Qi Q 2 , Q 3 , - - .]/JA 7
332 where (Qn) is a sequence of commuting projections and JA is the two-sided ideal generated by commutation relations
Q n X ( k ) = X(k)Qn for { k , n } E A where X ( k ) is the k-th copy of X E di. We can now construct a state on the free product of An’s, which may be called the A-boolean product of states since the algebras An and A, commute if {n,m } E A, and otherwise they are boolean independent. The associated notion of independence may be called ‘A-boolean independence’.
Theorem 5.1. Let j : UnENdn 3 2 be the unital *-homomorphism given by
-
. j ( X ) = Qn 8... @ Qn @ X ( n )@ Qn8.. n-1 times
where X E An. Then, there exist states &n o n Xn such that X E A, and Y E A, commute w.r.t. $ o j if { m , n } E A, and otherwise they are boolean independent. Proof. The states & are obtained by lifting tensor products of boolean extensions. We let
where
q ( X ( n ) ) = I@(~-’) @ x @ 1mrn q(Qn) 1
QD
pj @
QD
li
i$A(n)
jEA(n)
for any X E dl and A(n) = { m : { n , m } $ A}, where Pj = P coincide with the projection by which we extend the algebra An to get its boolean extension. Let
-
j ( X ) = Qn @ . .. @ Q n @ X ( n )@ Qn 8.. . n-1 times
j ( Y )= Qm @ .. . @
Qm
@Y(m) Q m @ . .
m-1 times
where X E A, and Y E Am and m # n. By definition, if { n , m } E A, then j ( X ) and j ( Y ) commute. Otherwise, j ( X ) and j ( Y ) are boolean independent w.r.t. the state since
8
QnY(m)Qn- +m(y)Qn E ker&, if m E A(n)
333
QmX(n)Qm - 4n(X)Qm E ker4n if n E A(m)
+.
which follows directly from the definition of Thus, if {n,m} $ A, they are boolean independent w.r.t. the state This completes the proof. 0
4.
Example 5.1. Let L = { 1,2,3} and suppose A consist of one pair only: { 1,3}. Instead of infinite tensor product it is enough to take = Al €4 2 2 € 4 & a n d & = & @ ~ & @ &L~e t X ~ d l , Y ~ d 2 a n d Z ~ Note d3. that if we take
A
j ( x ) = x(1)8 QA 8 QA j ( y ) = QB 8 y(2) 8 qB j ( Z ) = QC 8 QC €4 2(3)
where A = {2}, B = {1,3}, C = {2}, then we get qAz(3) = z(3)QA QCX(1) = X(1)QC which implies that j(X) and j ( 2 ) commute. Moreover, we can identify QBX(1)QB= 4 1 W Q B qAx(2)qA = 42(x)qA QBZ(3)QB= 4 3 ( a n 3 QcX(2)qc = 42(X)QC in the weak sense, which implies that the pairs {j(X),j(Y)} and 0 {j(Y),j ( Z ) } are boolean independent w.r.t. the state
4.
In an analogous fashion one can construct a mixture of tensor independence and monotone independence. The state obtained in the theorem below may be called the A-monotone product of states since it has the following properties: if { n , m } E A, then d, and d,,, commute and otherwise they are monotone independent with respect to the product state. The associated notion of independence may be called A-monotone independence.
-
Theorem 5.2. Let j : U n E N d n
+ A be the *-homomorphism given by
j ( X ) = Qn € 4 . . . € 4
Qn
@X(n)8 Qn 8 . ..
n-1 times
where X E An. Then, there ezkt states & on An such that X E An and Y 6 dmcommute u.r.t. 0 j if { m ,n} E A, and otherwise, they are monotone independent.
4
334
Proof. The proof is similar to that of Theorem 5.1. The only change is that in the definition of q(Qn) we replace A(n) by A+(n) = (m > n : b m ) $! A). 0. Example 5.2. Consider the same setting as in Example 5.1, the only difference being that we require now j ( X ) and j ( 2 ) to commute, whereas the pairs ( j ( X ) , j ( Z ) )and ( j ( Y )j,( 2 ) )to be monotone independent (order is important). Note that if we take
j ( x )= x(1)€9 QA €9 QA j ( y )= QB €9 y(2)8 QB j ( Z ) = QC €9 qc c 32 ( 3 ) where A = { 2 ) , B = ( 3 ) and C = 0, then q c X ( 1 ) = X(1)qc qAz(3)
Z(3)QA
since 1 $! C and 3 $! A , hence j ( X ) and j ( 2 ) commute, and we can identify qAY(2)qA = 4 2 ( y ) q A
q B X ( 3 ) q B = 43(z)qB
in the weak sense, which implies that the pairs ( j ( X ) , j ( Y ) )and ( j ( Y ) , j ( Z )are ) monotone independent with respect to the state &. Let us finally reproduce the A-free product of states. For simplicity we assume that L = N. For a given subset A of the set of pairs from N,we can define
XA(dI) = UrnEMdl(m) *C[Qm;mE M]/IA where M = N x
N and IA is the two-sided ideal generated by relations Q n x ( m ) = x ( m ) Q n if { m l , n l ) E A
X ( m ) = Q n X ( m ) if { m l , n l } 4 A and m2
< 122
where n = (721,712) and m = (ml,m2). Then we can form the monotone closure of % A ( d , ) as in the free case and we denote it % A ( d n ) . We thus let i l = % A ( d , ) in ( 1 . 2 ) .
Theorem 5.3. For given states 4% on unital *-algebras An there exist on X(d1) and a unital *-homomorphism states
&,
j : *nENdn -+
&A(dn) nEN
335
such that the state
3=
3 0j , agrees with the A-free product of states 41 where
BnCNJn.
Proof. We define the states Jn
Jn
by
=(@
&,m)
ov
mEM
where
&,m
= & for every m and
The homomorphism j
Where
is defined by where
for
P(n,k)
= Q ( n , k ) G Q ( n , k ) G ** *
- Q(n,k-l)@Q(n,k-l)@.
*.
Note that this is a dilation of the representation of free random variables. Thus j agrees with the A-free product of states. The details are rather technical and are omitted.
30
It is worth pointing out that in our approach, positivity of the product states associated with A-boolean independence, A-monotone independence and A-freeness is an immediate consequence of the positivity of 4. A
References 1. A. B. Ghorbal, M. Schurmann, “Non-commutativenotions of stochastic independence” Math. Proc. Camb. Phil. SOC.133 (ZOOZ), 531-561. 2. N. Muraki, ‘‘The five independences as natural products”, Inf. Dim. Anal. Quant. Probab. Rel. Topics 6 (2003), 337-371. 3. R. Lenczewski, “Unification of independence in quantum probability”, Inf. Dim. Anal. Quant. Probab. Rel. Topics 1 (1998), 383-405. 4. U. fianz, “Unification of boolean, monotone, anti-monotone and tensor independence and LBvy processes”, Mat. Zeit. 243 (2003), 779-816. 5. R. Lenczewski, “Reduction of free independence to tensor independence”, Inf. Dim. Anal. Quant. Probab. Rel. Topics, to appear.
6. W. Mlotkowski, “A-free probability” , Inf. Dim. Anal. Quant. Probab. Rel. Topics 7 (2004), 1-15. 7. L. Accardi, M. Boiejko, “Interacting Fock spaces and Gaussianization of probability measures” , Centro Vito Volterra preprint No. 321 (1998). 8. M. Boiejko, R. Speicher, “$-independent and symmetrized white noises”, in Quantum Probability and Related Topics VI, Ed. L. Accardi, World Scientific, Singapore, 1991, 170-186. 9. M. Boiejko, “Uniformly bounded representations of free groups”, J. Reine Angew. Math. 377 (1987), 170-186. 10. N. Muraki, “Monotonic independence, monotonic central limit theorem and monotonic law of large numbers”,Inf. Dam. Anal. Quant. Probab. Rel. Topics 4 (2001), 39-58. 11. Y . G. Lu, “On the interacting Fock space and the deformed W i p e r law”, Nagoya Math. J . 145 (1997), 1-28. 12. R. Lenczewski, “Filtered random variables, bialgebras and convolutions”, J. Math. Phys. 42 (2001), 5876-5903. 13. D. Avitzour, “Free products of C*- algebras”, Trans. Amer. Math. SOC.271 (1982), 423-465. 14. D. Voiculescu, “Symmetries of some reduced free product C*-algebras”, in Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132, Springer, Berlin, 1985, 556-588. 15. U. Franz, R. Lenczewski, “Limit theorems for the hierarchy of freeness”, Prob. Math. Stat. 19 (1999),23-41. 16. U. Franz, R. Lenczewski, M. Schiirmann, “The GNS construction for the hierarchy of freeness” , Preprint No. 9/98, Wroclaw University of Technology, 1998. 17. T. Cabanal-Duvillard, “Variation quantique sur l’independence: la a-independence” , preprint, 1993.
LEVY PROCESSES AND JACOB1 FIELDS
E. LYTVYNOV Department of Mathematics University of Wales Swansea Singleton Park Swansea SA2 8PP U.K . E-mail: e.lytvynov0swansea.ac.uk
We review the recent results on the Jacobi field of a (real-valued) LBvy process defined on a Riemannian manifold. In the case where the LQvy process is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an extended Fock space. We also give a unitary equivalent representation of the Jacobi field in a usual Fock s p x e . This representation is inspired by a result by Accardi, Franz, and Skeide’ .
1. Introduction
This paper is devoted to study of the Jacobi field of a real-valued LCvy process defined on a Riemannian manifold X. We recall that a LCvy process in this case is defined as a generalized stochastic process with independent values in the space V’-the dual of the space V of all smooth, compactly supported functions on X (cf. Ref. 16, see also Ref. 31). The notion of a Jacobi field in the Fock space first appeared in the works by Berezansky and Koshrnanenko7$*,devoted to the axiomatic quantum field theory, and then was further developed by Bruning (e.g. Ref. 14). These works, however, did not contain any relations with probability measures. A detailed study of general commutative Jacobi fields in the Fock space was carried out in a serious of works by Berezansky, see e.g. Refs. 3, 4 and the references therein. We start with recalling, in Section 2, the classical results on the Jacobi matrix and its spectral measure, which is a probability measure on R As examples, we discuss the Jacobi matrices which correspond to the orthogonal polynomials of Meixner’s type27. In Section 3, we discuss the chaotic decomposition for Gaussian and
337
338
Poisson process and corresponding Jacobi fields, which act in the Fock space. We recall that the construction of the unitary isomorphism between the Gaussian, respectively Poisson space and the Fock space through the multiple stochastic integrals is essentially due to It618i19(one also has to add the names of Wiener and Segal in the Gaussian case). The Jacobi field of the Gaussian measure is the classical free-field in the quantum field theory, and the Jacobi field of the Poisson measure was independently discovered by Hudson and Parthasarathyl’ and Surgailis30, though these authors did not use the term Jacobi field. In this paper, we present a generalization of the results of Refs. 9, 10, 21, by introducing a parameter X E R connecting the Gaussian case (A = 1) and the Poisson case X = 1 (see also Refs. 25, 26). In Section 4, we review the recent results on the Jacobi field of a general LCvy process on X, Refs. 23, 11. Now, the corresponding Jacobi field acts in the so-called extended Fock space, which indeed extends the usual Fock space in a natural way. In Section 5, we study the special case of LCvy processes of Meixmer’s type, i.e., gamma, Pascal, and Meixner processes, Refs. 22, 20, see also Ref. 5. We characterize these process as those LCvy process which respect the set of finite, smooth vectors in the extended Fock space. We also show that the Jacobi field of such a process has a much simpler form than in the general case. Finally, in Section 6, we again consider the general case of a LCvy process. We first recall the unitary isomorphism between the L2-space of the L6vy process on X and the L2-space of a Poisson random process on R x X. Using this isomorphism, we construct a unitary equivalent representation of the Jacobi field of a LCvy process in the usual Fock space over L2(R x X,v 8 o),where Y is the LCvy measure of the process and o is its intensity. This representation is inspired by a result by Accardi, F’ranz, and Skeidel . The corresponding creation, neutral, and annihilation operators now have a much simpler representation than in the extended Fock space. However, a drawback of this representation is that the n-particle subspaces of the usual Fock space do not correspond to the orthogonal chaoses of the LBvy process.
339 2. Jacobi Matrix and its Spectral Measure
Let us consider the Hilbert space with e,=(O
,...,0,
l2
spanned by the orthonormal basis 1
v
,&(I...).
n-th place
(Jn,m)cm=O
An infinite matrix J = is called a Jacobi matrix if Jn,n=:an E R for n E Z+, Jn,n-l = Jn.-l,n=:bn > 0 for n E N, and = 0 for In - ml > 1. Thus, the matrix J is symmetric and has non-zero elements only on the three central diagonals. We denote by l 2 , o the dense subset of l2 consisting of all finite vectors, i.e.,
Z+such that
3N
t!2,0:={(f(n))r=0 : E
f(") = 0 for all n
2N}.
Each Jacobi matrix J determines a linear symmetric operator in domain e2,O by the following formula:
J e , = bn+len+l
+ anen + bnenPl,
L2
n E Z+, ePl:=O.
with (2.1)
We denote by J the closure of J , which evidently exists since the operator J is symmetric. Under some appropriate condition on the behavior of the coefficients a,, b, at infinity, the operator J can be shown to be self-adjoint (see e.g. Ref. 2 for details). We have2:
Theorem 2.1. Assume the operator is self-adjoint. Then, there edsts a unique probability measure p on (R,B(R)) (B(R) denoting the Bore1 B algebra on R) and a unique unitary operator I : e,
+L2($p)
such that Ieo = 1 and, under I , the operator 3 goes over into the operator of multiplication by the variable, i.e.,
(~J~-lf)= ( zxf(x), )
f
E I(Dom(J)) =
{g
E
L 2 ( R , p ) : L x 2 g ( x ) 2p(&)
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