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\a\ := 0\U...Ucr^ are measurable r^ —> T with respect to the respective cr-algebras ( r ^ ) and TM- Finally a measure F^ is defined on (Fg, TM) by
where /J?{{d}) = 1 and for k ^ 1, jj,k is the product measure on A4k- A repeatedly used consequence of nonatomicity is that, for each d ^ 2, the complement of the set r ( d ) := {a G Td : cr* H <x,- = 0 for all i + j} is (r /1 ) d -null. The resulting measure space F^ := ( F s ^ ^ T ^ , ) is called the symmet ric measure space of X, and was first described systematically by A. Guichardet. Integration with respect to the symmetric measure will be written simply J ■ ■ ■ do. Remarks. The u-algebra F » is generated by the family {T]A ■ A e M}, where TJA : F -> C is the function a i—> #{onA). The measure T^ has 0 as an atom of measure unity, and is nonatomic on IJfc^i ^V If N is a null set of M then {o 6 T : a D N ^ 0} is a null set of F ^ . Exercise H . Let cp, ip : S —> C be measurable. Show that (a) -K^ is measurable. (b) If ip = ip fi- a -e-, then n^ = 7ty F^- a-e. (c) If
i Sa, g e ic => f *w g e g. Proposition 2.3. (IC, *w) is an abelian associative unital algebra, with identity 1 and vx 1 + ... + v~l < 1. Then g := fi *w ■ ■ ■ *w fn € G(v-i), where v = (wf1 + . . . + v~^) , and n
(2-6)
Ni(.-i) fn)
£ G(vi-1)
X . . . X G(vn-\)
'
> / l *W ■ ■ ■ *W fn £
G(v-1)
which is implicit in (2.6); and the density of £ in each Gc- To prove (b) one must delve a little (but not much!) deeper into the representation of the Clifford algebra. We shall touch on this later. As a consequence of the theorem we have the following result.
Integral-sum kernel operators 11 Corollary 2.6. (a) The duality transform f i—> Jrf(o)dWa restricts to an al gebra homomorphism from (/C, *ty) into np Jr f (a) dCa restricts to an algebra homomor phism from (K, *c) into f|p = i (at> ~ av) > a n d cf = ( / £ + /) • Exercise O. Check that: (a) Each of the above operators leaves the subspace K. invariant. (b) The following hold on K: Kk\\2
< \Wf f du{l + #u)
IMf
< h\\2JdLo#uj
{av)* *P
n a
x *v
= a'v, "X.
—
\k{u)\\
\k(w)\2,
(nx)* = n^,
(i^)* = £?,
Tt-yOiu) ~T~ ^x^p
= at>nx + a*x>P
a* A; = <pofc, (a* + n x ) A; = x • A, /*fc = >po_k, q^k =
(a* + n x + oy) k = cvk =
ip*w k,
k, ip*ck.
X*P
The operators a',0,, and n\ are the usual Boson Fock space creation, annihila tion and number operators, viewed on Guichardet space, and the operators /* and fv are the usual Fermion Fock space creation and annihilation operators, viewed on Guichardet space; both courtesy of Proposition 1.5. From the point of view of
Integral-sum kernel operators 13 quantum probability the last three relations (with what has already been said), fully justify the terminology quantum Wiener integrals, and the notations
/ \ p d A \ I' ipdN, I' cpdA, j ipdT, I\>dF*, j C is measurable, and
///'
dadf3dj
\h (aU P) x' (a, f3,y) k (P U -y)\ < 00,
then the function a 1—> h (a) (Xk) (a) is integrable, and j dah{a)(Xk){a)
= j77dadpd-y
h{aU P)x' (a,/3,7) fc(/? U 7 ) .
Exercise Q. (a) x = 0 a.e. if and only if x' = 0 a.e. (b) If x = 0 a.e. or k = 0 a.e. then Xk is defined and zero a.e.
Integral-sum kernel operators 15 We therefore have a partially defined bilinear map
L°
(r 3 ) x L° (r) —» L° (r),
(x', k) >—+ xk,
and trilinear functional L°(T) x L°(r 3 ) x L°(T) —> C,
(ft, x', fc) >—» f
h(Xk).
Our key tool in moving towards operators is the family of norms indexed by three positive numbers a, b and c: \\x'\\a,b,c : = { /
da
, , . s a h r (a,/3,7) =
/ ^ess.sup^K^Ja,/?^)!
H
a:'(a,/?,7) =,
T h e o r e m 4 . 1 . Le* p > a > 0, g > c > 0, Ze* ft, A; € L ° ( r ) and /ei x € Then
L°{T3).
With the techniques developed so far, it is not hard to verify this inequality. We may write the basic estimate as an operator norm inequality:
\\xk\\(a+bir^\\xX^2jk\\{c+b2) for a, b\, 62, c > 0. In particular, for any e > 0,
ll^ll(e) o\ ,
(4.3)
leave K, invariant; in particular they are densely defined on Q. Insisting that there is b > 0 such that ||z'|| a ), c < 00 for all a,c > 0 ensures that the adjoint operator is also densely defined on Q and leaves K. invariant too, since its integral-sum kernel is (p, a, T) 1—> x (r, a,p)*. Two further special cases of interest are: ll^'lladc x
\\ '\\
wriere
a < 1 =>
n\—TT,—r < °°> where a, c < 1 = >
X is densely defined on Q, X is bounded on Q.
4.2. Uniqueness of the kernel. There are various senses in which an integralsum kernel operator has a unique kernel. Here is one. Let « be the linear span of {1E : E C I, compact}. Proposition 4.2. Let x G L° (T3) be locally integrable in its third argument, in the sense that for E C I compact,
L
doj \x(a,/3, u))\ < 00 for a.a.
(a,/3).
IT(B)
Then Xk is defined for k G £K, and Xk = 0 for all k e £K = > x = 0 a.e.
16 J. Martin Lindsay Alternative forms of determining sets for integral-sum kernel operators arise from restricting re, or from replacing SK by finite particle spaces. 4.3. Reconstruction of kernel from operator. There is a systematic way of as sociating a decreasing sequence {Fu} of finite measure sets of T (I) to each element = {jj w of T such that p| n >i ^ - The association depends on a Vitali system for the Lebesgue measure space I. Theorem 4.3. Let X be an integral-sum kernel operator, and suppose that its kernel is locally integrable. Then x> (p, a, r) = Jim {r L e b . (F U r ) } " ' j
for a.a. (p,a,r)
1^,
(xi^)
,
£ T3.
Example. Let I = R+ and let X be the shift operator on L° (F), given by Xk(a)
= k{a + 1),
where { s i , . . . , sn} -f t := {si + t,..., sn + t}, and 0 + t := 0. Then X leaves each Qa invariant and is a nice unitary operator on each of these Hilbert spaces, however the sequence f lP(n) (Xl„(„) ) is eventually zero, so the shift cannot be an integral-sum kernel operator. 4.4. Algebras of integral-sum kernel operators. If X and Y are operators with integral-sum kernels x and y respectively, then how about the operator product XY1 Exercise R. By systematic application of the quantum Ito relations, and the J"-^ Lemma, give a heuristic derivation of the following formula for the kernel x*y of XY at
{P,(T,T):
Y^
/ dw x
(Q 2 ,
A U /?2 U a 3 ,
7I
U W U 73) y (QI U W U a 3 , 71 U f32 U #,, 72)
(4.4)
where the sum is over all partitions of p, a and r into a, /3 and 7 respectively. The special case in which x and y vanish except when their middle argument is the empty set is important. In this case x* y is of the same form, and the formula simplifies to x*y{p,T)=Y\
Y\a
[dux(atpUu>)y(uUa,P).
*■—'act* ^—'PCT
J
(4.5)
'
Note also how the action of operator X on vector k is contained in (4.4): V (P, cr,T) = k (p)
< 00 for all a,c < 00 j
(c.f. (4.3) ), then we may summarise as follows.
Integral-sum kernel operators 17 Theorem 4.4. (/C'3',*) is a unital associative algebra with involution; the corre sponding algebra of integral-sum kernel operators leaves K, invariant. Moreover, in a natural way (/C'2', *) may be viewed as a unital subalgebra of (/C'3', *) Formula (4.5) and a version of Theorem 4.4 were found by H. Maassen, its extension (4.4) was given by P.-A. Meyer. 4.5. Four argument integral-sum kernels. There are several reasons for extend ing our mixed quantum Wiener integrals to include time, so that we are considering operators of the form:
X = jf (I x {a, 0,7, S) dA'a dN0 dA^ dT5. We shall demonstrate just one of these. It should be noted at once that uniqueness is lost in the extension (cf. the next subsection). However this is offset by the fact that the convolution of such integral-sum kernels becomes purely combinatoral. Theorem 4.5. Let X and Y be four-argument integral-sum kernel operators, with sufficiently regular kernels x and y then XY has integral-sum kernel x*y where the convolution is given by x*y{p,
a, T, u) = Y^x
(a2,ftUftUa3,
7i U (52 U 73, 5{)y («i U 0, we have | / | < 1 almost everywhere with respect to P. So / G L°°(fl, E,P"). Finally, since the operators X and Mf are both bounded and coincide on the dense subspace L°° of TL, they are equal. □
26 Hans Maassen Step 3 . We now drop the commutativity requirement to arrive at the following definition. Definition. By a noncommutative probability space we mean a pair (A, E(S) := l*-i(s) to an (injective) *-homomorphism L ° ° ( 0 ' , E ' , P x ) -» L°°(ft,E,P). Still the projection E(S) stands for the classical event X~1(S) that the random variable X takes a value in 5 e E'. In Step 3 this is now generalised to the following notion. Definition. By a generalised random variable on a noncommutative probability space (.4, tp) we mean a *-homomorphism from some other von Neumann algebra B into A mapping l g to 1^. The probability distribution of j is the state ip := tp o j on B. We denote this state of affairs briefly by
Quantum probability applied to the damped harmonic oscillator 27 If B is commutative, say (B, ip) = L°°(Q.',E',¥'), to take values in fl', and j can be written
then the random variable j is said
j(f) = f f(X)E(dX) , where E denotes the projection-valued measure given by E(S) := j(ls),
(SeZ').
In the particular case that fl' = K, j determines a unique self-adjoint operator on the representation space 7i of A: Theorem 1.2 (Spectral Theorem, von Neumann). There is a one-to-one correspon dence between self-adjoint operators A on a Hilbert space H and projection-valued measures E : E(K) —> B(7i) such that A = f XE(dX) . JR
When E(S) € A for all S in the Borel cr-algebra E(R), then A is said to be affiliated to A- Moreover: Theorem 1.3 (Stone's Theorem). There is a one-to-one correspondence between strongly continuous unitary representations 11—> Ut of the abelian group R into A and self-adjoint operators A affiliated to A such that Ut = eitA . Here the right hand side is to be read as the strongly convergent integral eit.A
._ j eME(dX)
_
JR
where E is given by the spectral theorem. If we put e t : 1 - t C : i H eztx, then the connection with j can be written 3{eL) = eUA . So altogether we can characterise a real-valued random variable or observable in any one of four ways: (1) by a self-adjoint operator A affiliated to A; (2) by a projection-valued measure E in A; (3) by a normal injective *-homomorphism j : L°°(R, E(M),P) —> A; and (4) by a one-parameter unitary group (£/i)*eR in A. Interpretation of quantum probability. It is a surprising fact that nature — at least on small scales — appears to be governed by noncommutative probability. Quantum probability describes manipulations performed on physical systems by certain mappings between generalised probability spaces called operations. (The same has been said about classical probability see, e.g. [Kam].) These mappings will be treated in some detail in Section 3. The generalised random variable which we just saw is such a mapping. It represents the operation of restricting attention to a subsystem. Another such mapping is the conditional expectation, describing the
28 Hans Maassen immersion of a physical system into a larger one. Yet other operations are the time evolution and the transition operator: they represent the act of waiting for some time while the system evolves on its own, or in interaction with something else respectively. At the end of a chain of operations we land in some probability space (A, A, followed by the observation of the compatible events {1} x {0,1} and {0,1} x {1}. The quantum coin toss: 'spin'. The simplest noncommutative von Neumann algebra is M 2 , the algebra of all 2 x 2 matrices with complex entries. And the simplest noncommutative probability space is (M 2 , | t r ) , the 'fair quantum coin toss'. The events in this probability space are the orthogonal projections in M 2 : the complex 2 x 2 matrices E satisfying E2 = E = E* . Let us see what these projections look like. Since E is self-adjoint, it must have two real eigenvalues, and since E2 = E these must both be 0 or 1. So we have three possibilities. (0) Both are 0; that is E = 0, the impossible event. (1) One of them is 0 and the other is 1. (2) Both are 1; that is E = 1, the sure event. In Case (1), E is a one-dimensional projection satisfying tr E = 0 + 1 = 1 and det E = 0 • 1 = 0 .
Quantum probability applied to the damped harmonic oscillator 29 As E* = E and tr E = 1 we may write E
%y 3 = \2 \x l+ V . iy ^~ \ — z\ - with(x,y,2)eIR v i»i y
Then det i? = 0 implies that i((l-z2)-(x2 + /)) = 0
==>
a'2 + ? / 2 + z 2 = l .
So the one-dimensional projections in M2 are parametrised by the unit sphere S2. Notation. For a = (ai, a2, 0,3) G R 3 let us write ff fl
(
:=
n 4- in
= aiCJl
„
+ a2ff2 + a3(73 '
where o-!, 02 and CT3 are the Pauli matrices "0 ll ai
'-
[1 o\ '
f0 -i] a2
'-[i
\1
oj ■
ff3:=
0 "
[0 -i_ •
3
We note that for all a, b € K we have a(a)a(b) = (a,b)-l+ia(axb)
.
(1.1)
Let us write £ ( « ) : = A(l + HK be given by VK(x):=5x+M. Then for all x,y £ S: (VK(x), VK{y)) = (Sx +M,Sy+ M)C/N = (4,8 y )c = K(x, y) . Now let V : S —> fi be a second minimal Kolmogorov dilation of K. Then we define a map U0:£~*H:
AK^A(I)1/(I).
xes This map vanishes on N: for A £ M we have 2
|t/oAf =
^A(x)K(z) 165
= J2Y,Mx)K(x,y)X(y)
= (A,A) £ = 0 .
165 !/€5
So f/o may be considered as a map C/M —> W. By the same calculation we find that UQ is isometric. Since \j V(S) is dense in H and \j VK{S) is dense in Tin, U0 extends to a unitary map U : HK —> H mapping VK{X) to V(x). O Examples 1.6. (a) Let S be any set and let K(x,y) := SXiV. Then Tt = l2(S) and V maps the elements of <S to the standard orthonormal basis of H.
Quantum probability applied to the damped harmonic oscillator 31 (b) Let S := Hi x Tt2, the Cartesian product of two Hilbert spaces H\ and 7i2Let K((ipi,f2),(Xi,X2)) ■= {ipi,Xi) ■ ii>2,X2) ■ Then H = Hi®H2, the tensor product of Hi and H2, and V(tpi,ip2) = i>i®ip2(c) Let <S be a Hilbert space; call it AC for the occasion. Let K(4>,x) '■= (i>,x)2Then H is the symmetric tensor product AT )X) : = e w ' x > . Then the Kolmogorov dilation is the Fock space T{K.) over /C, defined as T{K)
: = C © AC © i (AC ® s AC) e g (AC ® s AC ® s £ ) ©
and V(^>) is the so-called exponential vector or coherent vector
Exp (VO : = 1 © V1 © (V> ® VO ® W ® ^ ® V') © ' ' • (e) Let S = E and let if : R x R -> C be given by K{s, t) := e-v\s-t\+Ms-t)
(V>0,UJER).
t
The Kolmogorov dilation of this kernel can be cast in the form H = L2{R, 2r]dx) ;
V : t >-► vt G L 2 (R) :
vt{x) :=
efo-iw)(x-0
0
if a;
t
2. S O M E QUANTUM MECHANICS
Quantum mechanics is a physical theory that fits in the framework of noncommutative probability, but which has much more structure. It deals with particles and fields, using observables like position, momentum, angular momentum, energy, charge, spin, isospin, etc. All these observables develop in time according to a certain dynamical rule, namely the Schrodinger equation. In this section we shall pick out a few elements of this theory that are of partic ular interest to our main example: the damped harmonic oscillator considered as a quantum Markov chain. P o s i t i o n a n d m o m e n t u m . Let us start with a simple example: a particle on a line. This particle must have a position observable, a projection valued measure on the Borel tr-algebra £(R) of the real line R: E : £(R) -► B(H) . The easiest choice (valid when the particle is alone in the world and has no further degrees of freedom) is H := L 2 (R) ; E(S) : $ -» Is • i> ■ In this example the Hilbert space H naturally carries a second real-valued random variable in the form of the group (Tt)teR of spatial translations: (Ttil>){x) := 1>(x - ht) ,
(2.1)
32 Hans Maassen according to the remark following Stone's theorem (Theorem 1.3). This second ob servable is called the momentum of the particle. The constant h~ is determined by the units of length and of momentum which we choose to apply. The associated self-adjoint operators are Q and P given by (Qip)(x) = xip(x) ; (Pil>)(x) = -ih-^Mx)
.
(2.2)
Just as we have Tt = e~ltP, it is natural to introduce Ss := els® whose action on His Ssi>{x) := eisxij(x) . (2.3) The operators P and Q satisfy Heisenberg's canonical commutation relation (CCR) [P, Q] = -ih-l.
(2.4)
A pair of self-adjoint operators (P,Q) satisfying (2.4) is called a canonical pair. Representations of the canonical commutation relations. What kinds of canonical pairs are there? Before this question can be answered, it has to be reformulated. Relation (2.4) is not satisfactory as a definition of a canonical pair since the domains on the left and on the right are not the same. Worse than that, quite pathological examples can be constructed, even if (2.4) is postulated to hold on a dense stable domain, with the property that P and Q only admit unique self-adjoint extensions ([ReS]). In order to circumvent such domain complications, Weyl proposed to replace (2.4) by a relation between the associated unitary groups (Tt) and (Ss), namely: TtSs = e~ihstSsTt ,
(s,teR).
(2.5)
It was von Neumann's idea to combine the two into a two-parameter family W(t)S):=e%siTtSs,
(2.6)
2
forming a 'twisted' representation of K , as expressed by the Weyl relation: for all s,t,u,v £ M, W{t, s)W(u, v) = e " f ('"-»")w(t + u,s + v) . (2.7) This relation captures the group property of Tt and Ss together with the relation (2.5). Formally,
W(t,s)=eiW-tP)
.
We shall call the representation on L 2 (R) of the CCR given by (2.1), (2.2), (2.3) and (2.6) the standard representation of the CCR. Here and in the rest of the text we shall follow the quantum probabilist's convention ([Mey]), namely that k=2. Theorem 2.1 (von Neumann's Uniqueness Theorem). Let (Wit, s)) t R be a strongly continuous family of unitary operators on some Hilbert space H satisfying the Weyl relation (2.7). Then H is unitarily equivalent with L 2 (R) ®K, such that W(t,s) corresponds to Ws(t,s) ® 1, where W$ is the standard representation of the CCR.
Quantum probability applied to the damped harmonic oscillator 33 Proof. Let W : R2 -> U{U) satisfy the Weyl relation (2.7). For each integrable function / : R2 —» C with J f \f(t, s)\dt ds < oo, define a bounded operator A(f) on H by the strong sense integral /*O0
OO
/
/
f(t,s)W(t,s)dt ds .
oo J —oo
We find the following calculating rules for such operators A(f) and their kernels f: A(f) + A(g)
=
A(/+S);
^(/)*
=
Mf),
i4(/)A(5)
=
^(/*ff).
_
where/(t,s):=/(-i,-s);
Here the 'twisted convolution product' * is defined by /-DO
CO
e- i ( t "- s u ) /(t-w>s-t;)ff(u,t;)dud?; .
/
/
oo J — oo
Moreover we claim that an operator (on a nontrivial Hilbert space) can have at most one kernel: A(f) = 0
=»
W = {0}or/ = 0.
(2,8)
Indeed, if A{f) = 0 then we have for all a, b G R, OO
0 = W(a,b)*A{f)W(a,b)=
/
/"OO
/
e 2 i ( a s - M ) /(i,s)W(4,s)dids
Applying the linear functional A — i > (ip,Aip) with tp,ip & H to both sides of this equation, we find that for all ip,ip £H the (integrable) function (t,S)^/(i,S)(^W(t,S^> has Fourier transform 0. By the separability of H, either W(t, s) = 0 for some (t, s), (that is H = {0}), or f(t, s) = 0 for almost all (i, s). The key to the proof of uniqueness is the operator /-OO
OO
E:=l
/
/
e-^t2+s')W{t,s)dtds
■oo J ~oo
It has the remarkable property that for all a,b G R, £W(t,s)-E is a scalar multiple of E: EW(a,b)E
= e~i(ai+b2)E.
Indeed, £ has kernel g(t,s) := Je"5*-* + s ), and the product W(a,b)E h{u,v) := i e - * < — H ■ e-h{(«-u?Hb-v?)
(2.9) has kernel
34 Hans Maassen So EW(a, b)E has kernel (g*h){t,s)
I'e-i(tv-su)g{t-u,s-v)h{u,v)dudv
=
I
=
— /" /' e -i("'-™) e -K( t - u ) 2 + ( s -'') 2 )e- i ( a ''- H e-K( a ~ , ') 2 +( i , - l ') 2 )du(i'y
= J_ e -K a2+(,2 ) e -5( t2 + s2 ) / f =
-Ie-5(o2+2)e-§(t2+s2) /" f
=
I e - I ( a 2 + " 2 ) . e-H H : ((*, s), tp) >-» e^t2+s2)W(t,
s)y
is a Kolmogorov dilation (cf. Section 1) of the positive definite kernel K : (R2 x K) x (R 2 x K) -» C,
({t, s), e{t-isKu+iv){y,i>)
.
(2.10)
By explicit calculation you will find that E$ is the orthogonal projection onto the one-dimensional subspace spanned by the unit vector Q(x) := ^ 7 ( 1 ) , where 7(2;) := —7=.e 2
.
So in the standard case the dilation is Vs : R 2 -► L 2 (R) : (t,s) >-► e i ( ' 2 + s V 5 V C T fi(s; - 2i) . By Kolmogorov's Dilation Theorem, there exists a unitary equivalence U : L2(R) ® K. —> H such that for all a, b e R and ip £ !C: U(Ws(a, b)Q ®i>)= W(a, b)ip , and therefore for all a, b 6 R: W(a,fo) = f / ( H / S ( a , 6 ) ® l ) t / - 1 ,
Quantum probability applied to the damped harmonic oscillator 35 provided that the range of V is dense in H. Let C denote the orthogonal complement of this range. Then C is invariant for the Weyl operators; let W0(t, s) be the restriction of W(t, s) to £. Construct E0 := A0(g) in terms of W0 in the same way as E was constructed from W. Then clearly EQ < E, but also E0 J_ E. So E0 = A0(g) = 0 and by (2.8) we have C = {0}. □ Exercise. Calculate the minimal projection Es in the standard representation. Energy and time evolution. The evolution in time of a closed quantum system is given by a pointwise strongly continuous one-parameter group (at)tew. of ^auto morphisms of the observable algebra A. Like in the case of a particle on a line, for a finite number n of (distinguishable) particles in d-dimensional space we take A = B(H) with 7i = L2(Rnd). Since all automorphisms of this algebra are implemented by unitary transformations of Ti, the group (at) is of the form at{A) = UtAUf1 . It is possible to choose the unitaries so that t H-+ Ut is a strongly continuous unitary representation R —> U(H). We denote its Stone generator by H/h: Ut = eltHlh . The self-adjoint operator H corresponds to an observable of the system of parti cles, called its energy. The operator H itself is known as the Hamilton operator or Hamiltonian of the system. As the Hamiltonian commutes with the time evolution operators, energy is a conserved quantity: at(H) = UtHUf1 = H . The nature of a physical system is characterised by its dynamical law (a term of Hermann Weyl, see [Wey]). This is an equation which expresses the Hamiltonian in terms of other observables. For n interacting particles in Kd in the absence of magnetic fields the dynamical law takes the form nd
# = V-
..
Pf +
V(Q1,Q2,...,Qnd)
for some function V : (M.d)n —> R, called the potential. The positive constants m*, k = 1, • ■ • , n are the masses of the particles. (Incidentally we put k(j) := 1 + \(j — l)/d], where [ ] denotes integer part, in order to attach the same mass to the coordinates of the same particle.) Free particles. If V = 0, then Ut factorises into a tensor product of nd one-dimensional evolution operators, all of the form Ut = itH'h = e i 5 ^ p 2 . Since the Hamiltonian H = P2/2m now commutes with P, momentum is conserved: at(P) = P .
36 Hans Maassen On a formal level the time development of the operator Q is found by solving the differential equation
i«(Q) = &M1
= ^FMQ)),
(2.ii)
a solution of which is at{Q)=Q
+ —P. m According to the Uniqueness Theorem the canonical pairs (P, Q) and (P, Q + —P) are indeed unitarily equivalent. So we expect the evolution of the Weyl operators to be the following: at(W(x,y))
= at (e-ixP+i^)
= e-^+W+^J
= e-i(x-£v)P+iyQ
= W(x_±y
\
y
m
Proposition 2.2. Let P := —ih-j^. denote the momentum operator on H := L2 and let W : R 2 -► U{H) be given by (2.7). Let Ut := e'5^Rp2 . Then UtW(x,y)Ut-1
=
wfx--^y,y
Proof. From the definitions of Tt and EQ it follows that for all measurable sets B c K and all t e R: TtEQ{B)Trl = EQ{B + ht). By the uniqueness theorem irreducible representations of the CCR have the symmetry Q —» P, P —> —Q. So we also have the exchanged imprimitivity relation SVEP{B)S-1
VSe2(R)Vy6R :
= EP(B + hy) .
Hence for all y, t G E, r
l x2 e- ^ Ep{d\)\
S~l
— CO
/
CO
/
e'^^-W
EP{d\)
-co
= EL, ■ T_XvV ■ .
Its
e-^v
Multiplying by Ut on the left and by Sy on the right we find UtW{Q)V)U-1
= USyUr1
= e-i^T.^Sy
= W ( - l y , y) .
As Tx commutes with Ut we may freely add (2;, 0) to the argument of W, and the proposition is proved. D
Quantum probability applied to the damped harmonic oscillator 37 By imposing some state ip on A = B(L2(R)), all stochastic information on the model (.4, f,at) can be obtained from the evolution equation at(Q) = Q + ^P- For example, at large times t the random variable \at{Q) approaches ^P in distribution, provided that tp does not favour large Q values too much. So a position measurement at a late time can serve as a measurement of momentum at time 0. This puts into perspective the well-known uncertainty principle for position and momentum at equal times. The Schrodinger picture and the Schrodinger equation. The type of description of a system given so far, namely with random variables moving in time, and the state ip given once and for all, is called the Heisenberg picture of quantum mechanics. In probability theory this is common usage, and we shall adopt it also in quantum probability. However, quantum mechanics is often thought of in a different way, where one lets the state move, and keeps the operators fixed. This is close to Schrodinger's 'wave mechanics', and is therefore called the Schrodinger picture: If we take for
,A1>), ( t f € W , ||tf|| = l), then we can express all probabilities at later times t in terms of the wave function T/>(zi,...,x nd ;i) := (Uf1ip)(xu...,xnd)
.
This wave function satisfies the Schrodinger equation, a partial differential equation reflecting the dynamical law: ._ d ,, -ih-Q-ip{x1,...,xnd;t) nd
= ^
. „2
-7)
« ^~2 VKzi> • • ■ > x„d; t) + V(xu ..., xnd)ip(xu
..., rnd; t) .
If E is an orthogonal projection in H, then the probability of the associated event can be calculated in the Schrodinger picture by 0) .
(2.16)
(We apologise for a clash of notation: Tt is not related to translations.) This spiralling motion in the plane compresses areas by a factor e~2r,t, so that for t > 0 the operators Tt(Q) and Tt(P) disobey the canonical commutation relation, and Tt cannot be extended to an automorphism of B(H). Yet this damped oscillatory behaviour occurs in nature, for instance when an atom is loosing its energy to its surroundings by emission of light. So it would be worth while to make sense of it. There are two basic questions related to this model. Question 1. How should Tt be extended to B(H)7 Question 2. Can (Tt)t>o be explained as part of a larger whole that evolves by *-automorphisms of the form at(A) = UtAU^1, where Uf1 satisfies a Schrodinger equation? Spirals and jumps. In Heisenberg's matrix mechanics atoms were supposed to move in a mixture of two ways. Most of the time they were thought to rotate according to the evolution Uf* as described above, but occasionally they made random jumps down the ladder of eigenvalues of the energy operator H. Each time an atom made such a jump, it emitted a quantum of light whose (angular) frequency u> was related to the size E of the jump by E = tku . The probability per unit of time for the atom to jump was given by Fermi's 'Golden Rule', formulated in terms of the coupling between the atom and its surroundings, and it is proportional to the damping rate rj. In the following sections we shall describe this behaviour as a quantum Markov process. Both jumps and spirals will be visible in the extension of our Tt to the atom's full observable algebra. This will be our answer to Question 1, for which we shall need the notion of completely positive operators. Our answer to Question 2 will be a reconstruction of the atom's surroundings: a dilation. There we shall see how the atom can absorb and emit quanta. 3. CONDITIONAL EXPECTATIONS AND OPERATIONS
We shall now give a sketch of the operational approach to quantum probability which was pioneered by Davies, Lewis and Evans ([Dav], [EvL]). Conditional expectations in finite dimension. In this section we choose for definiteness: A ■= Mn, the algebra, of all complex n x n matrices, and f : A -> C :
A H-> tr (pA) ,
where p is a symmetric n x n matrix with strictly positive eigenvalues and trace 1, so that cp is faithful.
42 Hans Maassen Let A be a symmetric n x n matrix with the spectral decomposition
A=
]T
aEa.
aesp(A)
The orthogonal projections Ea, a € sp(yl), form a partition of unity. Measuring the observable A means asking all the compatible questions Ea at the same time. Precisely one of the answers will be 'yes'> as stipulated in the interpretation rules. If the answer to Ea is 'yes', then A is said to take the value a. This happens with probability \ Y2 assp(^l)
aEa
yaesp(^l)
) = fW ■ J
So the state
i =
a|B
= /31 =
a&p(A)
2(^)i ^
/3j
Note that this is a function, / say, of 0. Seen as a quantum random variable this conditional expectation is described by the matrix f(B): E(A\B):=f(B)=
£
f(m=
£
^ S )
Note that
C : A — i > tr (pA) with p strictly positive and tr (p) = 1. Then the following are equivalent. (a) There exists a linear map P : Mn —> B such that V^ e W „V F 6 £ ( e ) :
cp(FAF) = y{FP(A)F)
.
(3.3)
(b) There exists a linear map P from Mn onto B such that (i) P maps positive definite matrices to positive definite matrices.
(ii) P(l) = 1; (iii) ip o P = ip; (iv) P2 = P. (c) Bp = pB. If these equivalent conditions hold, then the linear maps P mentioned in (a) and (b) are the same. It is called the conditional expectation onto B compatible with ip.
44 Hans Maassen Proof, (a) = > (b): suppose P : Mn —> B is such that (3.3) holds. Let A > 0 and decompose P(^4) as J2pesp(B) apF& w i t n ^a G £(^)- Then apip(Fp) = ip(Ff)P{A)) = ip{F0P{A)Fp) = tp(FpAF0) > 0. So ap > 0 for all /? and P{A) > 0. Putting ,4 = 1 in (3.3) we find that for all 0 6 sp(P): (c): Make a Hilbert space out of A = Mn by endowing it with the inner product (X,Y)v:= 0 by /3Ssp(B)
/3Ssp(B)
Then from the positivity property (b)(i) it follows that VA6C:
P((\-1-A)*(\-1-A))>0.
This implies that for all (5 6 sp(B) and all A € C, \\\2-(\a0
+
Xad+bp>Q,
from which it follows that M
2
< b/3 ,
that is
P(.4)*P(yl) < P(A*A) .
Applying tp to the last inequality and using (iii) yields the statement (3.4). So P is an orthogonal projection Mn —> B, that is for all A € Mn, A - P{A) ± „ B ■ This means that for all A e M„: p{P{A)B) and ^ ( S A ) = tp(BP(A)) . But then, since B is commutative, ^ ( i M ) = (P(A)fl) = • (a): Suppose that Bp = pB. Then for all F £ £(B) and all A G M n , ^ ( P T I P ) = tr {pFAF) = tr (PpP.4) = tr {pF2A) = tr {pFA) = M 2 M2 maps 0 1
0 0 0 0
0 0 0 0
l" 0 to 0 1
1 0 0 0
0 0 0 0 10 10 0 0 0 1
i.e. it maps a one-dimensional projection to a matrix with eigenvalues 1 and — 1.
46 Hans Maassen Operations on quantum probability spaces. A quantum probability space {A,