Superanalysis
Mathematics and Its Applications
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Superanalysis
Mathematics and Its Applications
Managing Editor.
M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 470
Superanalysis by
Andrei Khrennikov Department of Mathematics, Statistics and Computer Sciences, University of Vdxjo, Vdxjo, Sweden
and Department of Mathematics, Moscow State University of Electronic Engineering, Zelenograd, Moscow, Russia
KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON /LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 0792356071
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid free paper
This is a completely updated and revised translation of the original Russian work of the same title. Nauka, Moscow 01997
All Rights Reserved 01999 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
Printed in the Netherlands.
This book is dedicated to Professor Vasilii Vladimirov.
Table of Contents
Introduction I
1
Analysis on a Superspace over Banach Superalgebras 1.
Differential Calculus
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7 7
2. CauchyRiemann Conditions and the Condition of A. . Linearity of Derivatives . . . . 3. Integral Calculus 4. Integration of Differential Forms of Commuting Variables . . . 5. Review of the Development of Superanalysis 6. Unsolved Problems and Possible Generalizations . .
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21
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II
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Generalized Functions on a Superspace Locally Convex Superalgebras and Supermodules 2. Analytic Generalized Functions on the VladimirovVo. . . . lovich Superspace . 3. Fourier Transformation of Superanalytic Generalized . . . Functions . 4. Superanalog of the Theory of Schwartz Distributions 5. Theorem of Existence of a Fundamental Solution . . 6. Unsolved Problems and Possible Generalizations . . .
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51
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43
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60 63 74 92 100 106
Table of Contents
viii
III Distribution Theory on an InfiniteDimensional Superspace 1.
2. 3. 4. 5.
109
Polylinear Algebra over Commutative Supermodules Banach Supermodules . . Hilbert Supermodules . Duality of Topological Supermodules . Differential Calculus on a Superspace over Topological Supermodules . Analytic Distributions on a Superspace over Topological Supermodules Gaussian and Feynman Distributions . Unsolved Problems and Possible Generalizations .
6. 7. 8.
IV
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Pseudo differential Operators in Superanalysis
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158 166 180
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183 197
205 221
Fundamentals of the Probability Theory on a Superspace 227 . Limit Theorems on a Superspace . 2. Random Processes on a Superspace 3. Axiomatics of the Probability Theory over Superalgebras . . . 4. Unsolved Problems and Possible Generalizations .
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VI
144
183
1. Pseudo differential Operators Calculus . 2. The Correspondence Principle . 3. The FeynmanKac Formula for the Symbol of the Evo. . . lution Operator . 4. Unsolved Problems and Possible Generalizations . .
110 116 130 141
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NonArchimedean Superanalysis .
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258 264 267 269
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270
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244 254
257
Differentiable and Analytic Functions . 2. Generalized Functions . 3. Laplace Transformation . 4. Gaussian Distributions . . 5. Duhamel nonArchimedean Integral. Chronological Exponent . . . . . . 1.
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227 240
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ix
Table of Contents
Cauchy Problem for Partial Differential Equations with . . 273 . . . Variable Coefficients 7. NonArchimedean Supersymmetrical Quantum Mechan6.
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ics ............................ Trotter Formula for nonArchimedean Banach Alge
. . . . bras 9. Volkenborn Distribution on a nonArchimedean Super. . space . 10. InfiniteDimensional nonArchimedean Superanalysis 11. Unsolved Problems and Possible Generalizations . .
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VII Noncommutative Analysis 1.
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VIII Applications in Physics 1.
2.
278 279 283 289
293
Differential Calculus on a Superspace over a Noncom. . mutative Banach Algebra Differential Calculus on Noncommutative Banach Al. gebras and Modules Generalized Functions of Noncommuting Variables .
276
294 298 309
313
. . . Quantization in Hilbert Supermodules Transition Amplitudes and Distributions on the Space
314
of Schwinger Sources
315
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References
329
Index
345
Introduction
The foundations of the theory of functions of commuting and anticommuting variables were laid in the wellknown work A Note to the Quantum Dynamical Principle by J. Schwinger [172] published in 1953.
Schwinger presented the analysis for commuting and anticommuting variables on the physical level of strictness. He assumed that there existed a set of points (which was later called a superspace) on which commuting and anticommuting coordinates were given and a differential calculus was constructed. This set was similar in many respects to Newton's differential calculus. However, the superspace was not
defined on the mathematical level of strictness (although the work [172] contained a remark concerning the construction of a superspace,
namely, it was proposed to define a superspace as a subset of the algebra of quantum field operators). A problem arose of constructing a mathematical formalism adequate to Schwinger's theory. Investigations in this direction were stimulated by applications in physics in which functions dependent on commuting and anticommuting variables would play an increasingly important part. The first mathematical formalism that made it possible to operate with commuting and anticommuting coordinates was Martin's algebraic formalism proposed in 1959 [114, 115]. Martin did not follow the way paved by Schwinger, neither did he try to give a mathematical definition of superanalysis  a set of superpoints with commuting and anticommuting coordinates. Instead, he developed a purely algebraic theory in which the "functions" of anticommuting variables were
2
Introduction
defined as elements of Grassmann algebra (an algebra with anticommuting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used.
Later, during the next twenty years, the algebraic apparatus developed by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G I. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticommuting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanalysis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" maps of sets and not as elements of Grassmann algebras. In 1974,
Salam and Strathdee proposed a very apt name for a set of superpoints. They called this set a superspace. Psychologically, physicists associated the introduction of the term "superspace" with the fact that this set was defined on the mathematical level of strictness, and, after the works by Salam and Strathdee [123] and by Wess and Zumino [184] were published, the superspace became a foundation for the most important physical theories. A paradoxical situation took shape by the end of the 1970s, namely, mathematicians continued an active development of an algebraic superanalysis whereas physicists used a different formalism which was considerably simpler and visual (there was no need to use here, as it was done by mathematicians, the language of algebraic geometry and the theory of bundles). It was clear that the use of such words as a
Introduction
3
ringed space and a structural bundle could not elucidate anything in physical theory and only made a very simple intuitive formalism more complicated. A serious problem arose in algebraic superanalysis in connection
with the construction of supersymmetric theories (D. Yu. Gel'fand, E. S. Likhtman, 1971; V. P. Akulov, D.V. Volkov, 1974; J. Wess, B. Zumino, 1974). The transformation of supersymmetry (SUSY) includes a SUSY parameter e which is a constant for a fixed transformation of SUSY, and not an ordinary constant but a constant anticommuting with the other anticommuting coordinates. However, there are no anticommuting constants in algebraic superanalysis. Here the concept of a constant is mixed up with that of a function since a "function" is a constant, an element of Grassmann algebra. However, from physical considerations, it was necessary to distinguish in SUSY between the constant e and anticommuting variables. For the first time,
this problem was subjected to a detailed discussion by J. Dell and 1. Smolin in 1979 [28]. This was, apparently, the first work in which the authors pointed out the difficulties that arose in superanalysis in connection with the attempts to use it for the description of SUSY. Moreover, a purely mathematical problem arose in algebraic superanalysis which was very disturbing. It was a problem of a change of variables in the Berezin integral. The simplest changes of variables (such, for instance, as those encountered in Rudakov's example, see [42]) led to senseless answers. Because of all these problems (the nonagreement of formalisms used by mathematicians and physicists, anticommuting constants in SUSY, a change of variables in Berezin's integral), some mathemati
cians and physicists tried to realize, on the mathematical level of strictness, Schwinger's idea concerning a set of superpoints. Several mathematical models of superspace were proposed. The first model was constructed by Batchelor in 1979 [82]. However, this model, constructed as a point realization corresponding to the graded Kostant manifold, did not answer the idea that physicists had of a superspace, and, despite the beautiful mathematical theory, was discarded.
Introduction
4
Practically at the same time as Batchelor published his article (and, perhaps, a little earlier), De Witt wrote his book Supermanifolds. Although this book was published only in 1984 [27], many mathematicians and physicists got acquainted with it in 1979. References to this book can be found, for instance, in [119]. De Witt proposed his model of a set of superpoints based on Grassmann's infinitedimensional algebra. De Witt constructed a well developed theory (differential and integral calculus, differential geometry, generalized functions). However, De Witt's model of a superspace had one drawback, namely, the topology that he proposed was not a segregated topology.
Models of superspaces endowed with an ordinary topology of a Banach space were proposed by Rogers [119, 120] and Vladimirov and Volovich [19, 20]. Roger's models of a superspace were based on Grassmann algebras endowed with normed topology. Vladimirov and Volovich constructed a superanalysis over an arbitrary (supercommutative) Banach superalgebra. The VladimirovVolovich superanalysis is invariant with respect to the choice of Banach superalgebra. Moreover, they not only achieved the greatest generality of mathematical constructions but also realized the following principle (the VladimirovVolovich principle of superinvariance of physical theories): all physical formalisms must be invariant with respect to the choice of a supercommutative Banach algebra that serves as the basis for the superspace.
I have analyzed practically all applications of superanalysis in Any formalism can be realized over an arbitrary Banach
physics.
superalgebra. It is natural to call the analysis developed in the works of De Witt, Rogers, Vladimirov, and Volovich a functional superanalysis [65]. It
is an analysis of "real functions" of commuting and anticommuting variables  maps of a set of superpoints called a superspace. The functional superanalysis is a mathematical realization of Schwinger's formalism of 1953 whereas the models of a superspace of De Witt, Rogers, Vladimirov, and Volovich are different mathematical models of what Salam and Strathdee called a superspace. The first chapter of this book is devoted to the VladimirovVolovich superanalysis. In this chapter we consider differential and integral cal
Introduction
5
culus on a superspace over a Banach superalgebra. The other chapters constitute the exposition of my D. Sc. dissertation, 1990 [68]. My contribution to superanalysis consists of (1) the theory of generalized functions, (2) the theory of pseudo differential operators, (3) an infinitedimensional superanalysis, (4) the theory of generalized functions on infinitedimensional superspaces and its applications to functional integration, (5) probability theory on a superspace. The book also includes a number of applications of functional superanalysis to the quantum theory of a field and a string. These models are considered only schematically. However, I hope that hav
ing read this book, any specialist in the quantum theory of a field and a string and of gravitation will be able to use easily functional superanalysis in his research. It should be emphasized once again that one of the main advantages of functional superanalysis is its simplicity and visuality. As for mathematicians, functional analysis constitutes for them a whole field of new unsolved problems. Although the fundamentals of superanalysis are similar to those of ordinary mathematical analysis, new nontrivial mathematical constructions arise in its further development. We can formulate here a number of general problems whose solution would lead to the creation of new mathematical theories such as, for instance, the construction of a spectral theory of selfadjoint operators in Hilbert supermodules. Nothing has been done yet in this direction. A large number of mathematicians and physicists took part in the discussion of the results exposed in this book. I want to use the opportunity to express my deepest gratitude to all of them. I feel myself especially indebted to V. S. Vladimirov, I. V. Volovich, B. S. De Witt, C. De WittMorette, O. G. Smolyanov, A. A. Slavnov, Yu. V. Egorov, Yu. A. Dubinskii, V. I. Ogievetskii, R. Cianci, T. Hida. I am also very grateful to my wife, Olga Shustova, for her constant support.
Chapter I
Analysis on a Superspace over Banach Superalgebras
Here we follow the works by Vladimirov and Volovich [19, 20].
1.
Differential Calculus 1.1. Superspace over a commutative Banach superalgebra.
Recall that a linear space L is Z2graded if it is represented as a direct
sum of two subspaces L = L° ® L1. The elements of the spaces L° and L1 are homogeneous. The parity is defined in the graded space L = L° ® L1 if it is said, in addition, that the elements of one of these spaces are even and those of the other space are odd. We assume, in what follows, that L° is a subspace consisting of even elements and L1 is a subspace consisting of odd elements. For the element f E L = L° ®L1 we denote by f ° and f 1 its even and odd components. The symbols 7r° and ir1 denote the projectors onto L° and L1. A parity function is introduced on the Z2graded space, namely,
Jal =0ifaEL°and Jal =1 ifaEL1. In the Z2graded space L = L° ® L1 we introduce a parity automorphism a: L + L by setting a (f) = (1) I f I f for homogeneous elements. Note that a2 = 1 and a(f) = f if and only if f E L°. A superalgebra is a Z2graded space A = A° ® Al on which a structure is introduced of an associative algebra with a unit e and
Chapter I. Analysis on a Superspace
8
even multiplication operation (i.e., the product of two even and two odd elements is an even element and the product of an even element by
an odd one is an odd element: jabl = lal + JbI (mod 2). In particular, the subspace Ao is a subalgebra of the algebra A. Everywhere in this book, linear spaces are considered over a field
K = R or C. The nonArchimedean superanalysis is presented in Chap. VI. The supercommutator [a, b} of the homogeneous elements a and b from the superalgebra A is defined by the relation (1.1)
[a, b} = ab  (1)161 Iblba
The supercommutator is extended to nonhomogeneous elements by linearity.
The superalgebra A = AO ® Al is said to be (super) commutative if, for the arbitrary homogeneous elements a, b E A, we have [a, b} = 0.
(1.2)
We introduce an annihilator of the set of odd elements (A1annihila
tor) by setting 'A1 = {\ E A: \A1 = 0}. In the sequel, the concept of an A1annihilator will play an important role when we construct the theory of generalized functions and harmonic analysis on a superspace. In this book, we use the abbreviation CSA for (super) commutative superalgebra.
Example 1.1. A finitedimensional Grassmann algebra (an exterior algebra) is CSA Gn = Gn(g1, ..., qn) whose elements have the form [[nom
f = fo +
E fi1...ikQi1 ...qik L k=1 il 0 and 77 = 77(t', 77'), and let G' x Am + Am be a Sdifferentiable function being invertible from the class S"+1 with respect to if, with det and the quantity M = [X E RA'm; X = cp o Q(t', 77'), (t', r7') E G' x Ai ], where the mapping Q(t', 77') = (t(t'), 77(t', 77')); G' x Am > G x Am.
Performing the indicated change of variables with respect to 77 for every fixed t E G on the righthand side of (3.12) and using relation (3.8) for a change of variables in the integral with respect to odd variables and then performing a change of variables t = t(t') (this is an ordinary change of variables in R"), we obtain fn
'1"
f [IGI
r7')))
sdet J(cp)(o,(t', rj )) det1
877
det
a17
at dt', drj at
1
fAm
[f f(
po o,(t', rl )) sdet J(cp o o) (t', T7') dtl drj J
where we used the relations
sdet J(u) = det1 sdet
J(W) (o (t', 77'))
det &,
sdet J(a) (t', 77') = sdet
J(cp o o) (t', 77") .
Suppose that a change of variables X = F(Y), Y = (y, ) is defined and let M = M((p, G) be a singular supersurface; M1 = M1(cp1, G) is
3. Integral Calculus
35
a singular supersurface induced by the mapping F from M: M, = F1(M), cp1 = F1 o cp. Then the formula fM f (X) dX = fMi f (F(Y)) sdet aY dY
(3.13)
for a change of variables is valid.
The sufficient condition for the existence of integrals and their equality is that m
f (x, 0) = E ff
(x)0',
ICI0
where fE E Sm+1IEI.
In order to prove relation (3.13), it suffices to use expression (3.12)
for writing out the integrals over the singular supersurfaces M and M, and employ the theorems of the derivative of a composite function and an inverse function.
Example 3.1. Let G = (a, b) C R, cp(t) = te, M = M(cp, G) (a, b)e C A0; F: AO + A0, F(y) = y  a, a E A0, and then M, = F1(M) = (ae + a, be + a) _ {y E Ao: y = a + se, s E (a, b) }. Relation (3.13) yields be+a
be
fea
x dx =
fae+a
(y  a) dy.
Example 3.2. Let G = (a, b) C R, cp(t, 01, 02) _ (te, 01 02), M = (ae, be) x A2 c RA 2; F: RA2 4 RA2, F(y,.1, e2) _ (y + U2, e1, e2), and then sdet ar = 1 and relation (3.13) yields i
be
fA2
x dx dO, d02 = j f i
befae
e(if2
(y +
dy d 1 dC2 = 0.
Chapter I. Analysis on a Superspace
36
Example 3.3. Let G = (a, b) C R, co(t, 0) = (te, 0), M = F(y, (y  i ), 77 = const E Al, and (ae, be) x Al C ,
then sdet ay = 1 and relation (3.13) yields fAl
f
6e
ae
fl
x dx dO =
be+i
J ae+nf
(y 
dy d = 0.
Note that when an algebraic approach to superanalysis is used (Martin [114], Berezin [4]), these changes of variables serve as counterexamples (see Sec. 5.10). The integral calculus on singular supersurfaces can be directly generalized to singular supermanifolds: a singular supermanifold is locally homeomorphic to a domain in the space RI x All.
3.4. NewtonLeibniz formula. The NewtonLeibniz formula LB f (x) dx = F(B)  F(A),
(3.14)
where F(x) is an antiderivative of the function f (x), holds for the continuous function f (x) of a real variable. Similarly, let f (z) be an analytic function of a complex variable and let ryAB be a contour connecting the points A and B in C. Then the integral fYAB f (z) dz fA f (z) dz does not depend on the choice of a contour and formula (3.14) holds true. The NewtonLeibniz formula can be generalized to the contour integrals in A0. Let the algebra AO be representable in the form AO = Re ® CO, where Co is a subalgebra (say, A = G'). We use the symbol 1(C0) to denote nilradical (a set of nilpotent elements) of the algebra Co. We can extend every function f E C°° (R, A) to the algebra UO = Re (D N(Co) with the use of Taylor's formula
AX) = E f (n!
t) a",
(3.15)
n=0
where x = to + a, t E R, a E N(CO). If we provide the algebra UO with a suitable pseudotopology (see Chap. II), then the function f (x) will be infinitely Sdifferentiable.
3. Integral Calculus
37
Let F(t) be an antiderivative of the function f (t), t E R. Then
f (n I) (t)an
F(x) = F(t) +
(3.16)
'
n=1
In what follows, we shall only consider contours which lie entirely in the algebra UO.
Theorem 3.2 (De Witt). Let f E C°° (R, A). Then the integral fA f (x) dx does not depend on the contour that connects the points A, B E U0, and the NewtonLeibniz formula (3.14) holds true.
Proof. Let A = a + a, B= b + /3, a, b c R, a,# E N(CO); 'AB (t) = ((p (t), V) (t)) E uO = Re ® N(CO). Then B
fA f (x) dx = f of f (w (t) +V) (t)){'(t) + '(t)} dt,
dt =
J ' f ((p(t) +
E
(n1) (b)on]
f (n1) ((p(t)) d n(t)
n=1
00
f
00 n=1
f' 1
 Li _ f (n1) (a)an] n=1
00

n!
dt
jol
(we have used formula (3.15)). Next we have
f =
f
tj
f (cp(t) + 0(t))0(t) dt
tj
00
f(i(t)) dp(t) +
,
dt.
f of
n=1
The final result is LB
+
00
f (x)
nl f(n1)
dx = f O1 f (V (t)) dcp(t) 00
(b)on]
I
1
f (n1) (a)an]
Chapter I. Analysis on a Superspace
38
= F(B)  F(A) (we have used relation (3.16)).
Example 3.4. Let us calculate the Gauss integral (a is a nilpotent +00
element): I = f exp{2 (1 + a)zxz} dx. In this integral, we make a
change of variables y = (1 + a)x; I = (1 + a)' fy exp{y2/2} dy, where the contour y: v = te, u = at, t E R, v E Re, u E N(C0). Let us consider the integral along the closed contour rN = rN U rN U FN;
rN: v = te, u = at; rN: v = Ne, u = at; r
:
v = te, u = 0;
0 < t < N (clockwise). Then we have frN exp{yz/2} dy = 0, and, consequently, z jexp{
2
} dy=f
z
z
+00
eXp{2
} dy2NmofNexp
2
dy.
However,
f
z
exp 
N
Nz
21 J
dy = exp
2 f
N
PN (t) dt
where pN(t) is a polynomial of degree k = k(a), i.e.,
limo fNexp and I =
2 dy = 0
27x/(1 + a).
3.5. Calculating Gauss integrals. Let the matrix M have a block structure: A C M= C* B
where the matrices A = (n x n) and B = (2k x 2k) consist of even elements, the matrix C = (n x 2k) consists of odd elements, the matrix A is symmetric, and the matrix B is skewsymmetric: A = A',
B = B*. The choice of parities of the elements of the matrix M ensures the evenvaluedness of the bilinear form corresponding to the
3. Integral Calculus
39
matrix M; this is also equivalent to the fact that the linear operator corresponding to the matrix M maps the superspace R',2k into itself. Let the elements of the matrices A and B belong to the algebra UO, i.e., A = a + a, B = b + /3, where a and b are numerical matrices and a and Q are matrices with nilpotent elements. We shall assume, in addition, that the matrices A and B are invertible (since a and 0 are nilpotent, this is equivalent to the invertibility of the matrices a and b) and to the fact that the matrix a is positive definite, a > 0. As the first step in the calculation of the Gauss integral
I=
f
exp {  2 (X, MX) } dX J
R^exAik
we make a change of variables x = x+A'CO, 0 = 0; the Jacobi matrix II
of this change has the form J l =
0
A
IC
,
and, consequently,
sdet Jl = 1. In the new coordinates, the Gauss integral has the form
I = f exp {_(x,MX)} dX, sl
where S1 is a singular surface in U x Aik obtained from R"e x Aik as a result of the change and M is a diagonal block matrix
M=
A
0
0 B + C"A1C
Since the matrix a is real, symmetric, and positive, there exists an orthogonal (numerical) matrix 01, det 01 = 1, such that O1aOi = diag()1i..., an), )j > 0. Similarly, for the matrix b there exists an orthogonal matrix 02, det 02 = 1, such that O2bO2 = diag I
0
_µl
p1 0
0
_µk
Pk 0
We make a new change of variables X = LX, where
L = diag(Ol (I + a'a)1/2,
O2[I + b1(Q + C'A1C)]112
Chapter I. Analysis on a Superspace
40
(the square roots are defined by their binomial expansions). For this change
sdet J2 = det (I + a1a)1/2det [I + b1(Q + C'A1C)]1/2
and the Gauss integral reduces to the form
I=f exp{2[A12i + ... +.\n2n] [µ1e102 + ... + µke2k102k]} sdet J2 1 dX.
The integral with respect to anticommuting variables can be immediately calculated. To calculate an integral with respect to commuting variables, we must use the multidimensional analog of the reasoning given in Example 3.4. Thus, we obtain I = (27r)n/2(det a)1/2(det b)1/2(sdet J2) 1
_
(27r)n/2(det A)1/2[det (B + C'A1C)]1/2
= (2ir)/2(sdet M)1/2 For the sequel, we shall need one more property of a bilinear form corresponding to the matrix M. The bilinear form is symmetric, i.e., (Y, MX) == (X, MY) for X, Y E RA,2k. Indeed, using the symmetries and the parities of the matrices A, B, C, we obtain (Y, MX) =
yiAijxj + E yiCijOj + E ej (C?j)xi + E (IBijej O Bijej = (X, MY).
E xiAijyj + E Oj (Cij)yi + >
Let us calculate the Fourier transformation of the Gauss "measure"
f =
exp { l
f exp l
2 (X, MX) + i(X, Y) }
(X
J
dX (27r)nsdet M1
 iM1Y, M(X  iM1Y)) 
(M1Y, y) J
2
41
3. Integral Calculus
x
dX
= exp
(27r)nsdet M1
l
2
1(M1Y, Y) } J
.
Here we have used the symmetry of bilinear form
(M1Y, MX) _ (X, MM1Y) = (X, Y).
Remark 3.4. The even number of odd variables in the Gauss integral is caused by the requirement of the nondegeneracy of the matrix B. Remark 3.5. With respect to odd variables, the Gauss "measure" is a generalized function (see Chap. II). It stands to reason that it can be extended as a Alinear continuous functional to the space C(Aik) (see Remark 3.3). However, such an extension is not unique since the set of polynomials of anticommuting variables is not dense in the space of continuous functions. It should also be pointed out that all calculations become considerably more difficult if the quadratic form is not evenvalued. Let us consider a simple example.
Example 3.5. Let 'y = yo + r', 0 = 0° + 01 E A, the elements 'y° and /3° being invertible. We calculate the Gauss integral with respect to anticommuting variables
f exp{91ry92  83/394 + i 1: 4
I=
d91...d94.
j1
It is more convenient to calculate the Gauss integral
I = f exp{0102'y93940+i
}d91...d94. vi i1
(which easily reduces to the preceding one). J = f (1 + 2191)...(1 + 2494) 1
X
(1o1o2o3o4/3+o1o2o3o4P)
d91...d94,
Chapter I. Analysis on a Superspace
42
p = y/3 + /37. Furthermore, we have
J=
2
+ f(1O12O2e3O3e4O4 + 60160203040 +e303e40401027) d91 d02 d03 d04
_ [1 + e1C2e3e42P1 7P1/3
Lemma 3.1. /3p17 +
Proof
e1e22/3P1

e3e427P1]2
= 1.
p = (2/3° y° + /31.y° + 71/30);
p1
= 21(,0yo)1(1 + (/jo7o)1(170 + 71/30))1
= 21(0o7o)1(1  (/3170 + y10o)(0o yo)1) Furthermore, /31)(/3°7°)1(1
/32P17 = (/30 +
 (/317° + 71/3°)(/3°7°) 1)(7° + 7')
+/31(/3°)1(7°)1)(1 /3'(/3°)'
=1_
 71(7°)')(7° +7')
/31(0)171(70)1
Similarly, 72p1/3 = 1  71(70)1/31(/30)1 Thus we have /32p17 + 72p1/3
= 2 + 71/31(/3°7°)
1

71/31(/3°7°)1 = 2.
Using this lemma, we obtain 3S427P'} = 1  e1e22/3P
+7P'/3P1] =
1

U427P1
2,Jp1
Setting /3 = a(/3), 7 = a(7), p = a(p), where or is an automorphism of parity, we find that
I = 
2u(7P')e354}.
4. Integration of Differential forms
43
If the elements y and Q are even, then p = 20y and
I = ayexp {
2
_
exp
M=diag1( 0
{(M'e
0)'(
11
1
)}
det M,
Q 00)).
0
The question concerning the calculation of the Fourier transform of the quadratic exponent in the case where the quadratic form assumes noncommuting values remains open.
It should be pointed out that when the number of odd variables is odd, the Fourier transform of the quadratic exponent may be not a quadratic exponent.
Example 3.6. fexp{_[0102 + 0283 + 0103] + i[Bi6 + 02e2 + 03e3]} d91d02d03 = i(6 C2 3  e1 + e2  6) 
4.
Integration of Differential Forms of Commuting Variables 4.1. Definition of Sforms. Let A be a commutative Banach
algebra with identity e. The theory of Alinear differential superforms (Sforms) in algebra A exposed below is constructed by analogy with the theory of ordinary differential forms in a Banach space exposed, for instance, by Cartan in his book [35]. The main difference is that instead of the Rlinearity for ordinary differential forms we require the fulfilment of the condition of Alinearity for Sforms. Recall that ordinary differential forms of degree p introduced in the domain 0 of the Banach space E with values in the Banach space F are introduces as mappings of 0 into the set of Rpolylinear skewsymmetric mappings of the space E9 = E x ... x E in F. We shall denote the value of the form w of degree p at the point x E 0 on the vectors y2 E E, j = 1, p, by w(x yl yP).
Chapter I. Analysis on a Superspace
44
For Sforms of in commuting variables the role of E will be played
by the space Am and the role of F by the arbitrary Banach CSA A=AoED A1 with A0 j A. We denote by GP = Gp(Am, A) a set of Apolylinear skewsymmetric
mappings of the space Am x ... x Am = Amp into A, i.e., f c Gp is equivalent to (1) f (yl, ..., ayj + Qz4, ..., yp) = of (yl, ..., y4, ..., yp) + of (yl) ..., Z47 yp), y4, z4 E Am, a, Q E A, y°(P) = E(a) f (yl, ..., yP), where E(or) is the signature (2) f
of the permutation a: (1, ...,p) + (a(1), ..., a(p)). The mapping w: 0 * Gp is an Sform of degree p defined on an open set 0 C Am with values in A. We denote by w(x; y', ..., yP) the
value of the Sform w at the point x E 0 on the vectors yJ E Am, j = 1, .... The Sform w of degree p belongs to the class S'(0) if the function w: 0 + Gp is k times continuously Sdifferentiable. By c(pk) (0; A) we denote the set of all Sforms of degree p in 0 with values in A of the class Sk(O). Example 4.1. An Sform of degree 1 is a mapping 0 + L(Am, A), where L(Am, A) is a space of Alinear mappings of Am into A.
It follows from the definition that every Sform is, at the same time, an ordinary differential form. We shall widely use the following criterion which connects ordinary differential forms with Sforms.
Criterion 4.1. For the differential form w to be an Sform, it is necessary and sufficient for it to be Alinear with respect to y', i.e., that it satisfy the condition w(x; hy1, ..., yp) = hw(x; y1, ..., yp),
h E A.
As usual, we introduce differential dxj: dx,(y) = yj, y = (yi, , ym) E Am as well as an outer product of differentials dx1 A dx; (yl, y2) _ dx; (yl) dx, (y2) dxi (y2) dxi (y') = y: y2  y2 1; similarly,

dx,1 A ... A dxip(yl,..., yP) = det IIy pIIQ,A=1.
4. Integration of Differential forms
45
Theorem 4.1. Every Sform w E
q(k) (O; A) can be uniquely
represented in the canonical form
w=
E
wi1...i9(x)dxi1 A ... A dxip,
(4.1)
1 1, p > 1 and dw = 0 in 0, where 0 is a star domain with respect to the
47
4. Integration of Differential forms
origin. Then there exists an Sform a E SZpkll (0; A) such that da = w, i.e., every locally closed Sform is exact. Proof. Let the Sform w be defined by expression (4.1). Then the Sform fi
dt
[(_1)k_1xikdxj, X
n ... A dxipJ
a= P
11
(4.2)
k=1
is a form of the class
A) and da = w.
4.3. The preimage of an Sform. Let A' be a commutative Banach algebra and 0' be a domain in (A')m'. Suppose that we are given a function x = W(y) of the class Sk+1(0') which maps 0' into O C Am. For every Sform w E S2pk)(O; A) defined by (4.1) we shall determine the Sform*w E q(k) (O'; A) from the formula W*w
=
wi, ...ip (W (y)) dWi1(y) A ... A dWip (y) it 1; thus the generalization of the CauchyPoincare theorem holds for any finitedimensional real algebra A of dimension higher than or equal to 2 (i.e., except for A = R).
It is interesting to compare, for m = 1, the results obtained with those from Sec. 3.4. It follows from the CauchyPoincare theorem that the NewtonLeibniz formula holds for any function from class
S' and there is no restrictions of the type of nilpotency on the path of integration. However, De Witt theorem does not follow from this. In De Witt theorem we can consider the extension of any function f E C°° to the algebra UO. However, generally speaking, functions of this kind cannot be extended to the whole Banach algebra A. Therefore a necessity arises of constructing an analysis on the VladimirovVolovich superspace not only over Banach CSA but also over arbitrary topological CSA (in particular, over CSA in which the even part has the form Uo, see Chap. II). Since all entire analytic functions can be extended to entire Sanalytic functions on Ao, the CauchyPoincare theorem on the VladimirovVolovich superspace makes it possible to extend essentially the class of Gaussian integrals over a superspace that we are considering (cf. Secs. 3.4, 3.5). In the commutative Banach algebra A, for any a c A there exists a limit lim jja"jjl/n = p(a) called a spectral radius [13]. The element a E A with p(a) = 0 is said to be quasinilpotent [13]. For instance, for the CSA Gam, G0,0 = Re ® Co all elements of the subalgebra Co are quasinilpotent (see [119]).
5. Review of Superanalysis
51
Numerous examples of the CSA A = AO ® Al in which the subalgebra Co consists of quasinilpotent elements were constructed in [99].
Example 4.4. Let us calculate the Gaussian integral from Example 3.4 in the case where a is a quasinilpotent element. We shall show, in the first place, that this integral converges. Note that for the quasinilpotent element Q for any E > 0 we have Ile°x211 < CEef22,
CE > 0,
x E R.
It follows from this estimate that the integral converges and that the integral along the path 1,N in Example 3.4 tends to zero. Consequently, for the quasinilpotent a we get the same answer as in Example 3.4.
Repeating the calculations from Sec. 3.5, we calculate the Gaussian integral for block matrices M in which A = a + a and B = b + Q, and a and ,Q are matrices with quasinilpotent elements.
Remark 4.2. If x is a quasinilpotent element, then the function (1 + x)' is correctly defined by its Taylor series. The same refers to the function ln(1 + x). For elements which are not nilpotent Gaussian integrals may diverge.
5.
Review of the Development of Superanalysis
5.1. Model of De Witt superspace [27]. In this model, the CSA A = G,,(K), n = 1, ..., oo. The superspace Kin is endowed with the (not Hausdorff) topology induced from K' by the canonical projector E: Kin + K' (a body projector, see Chap. II).
5.2. Model of Rogers superspace [119]. In this model, the CSA A = Gn or G. The superspace K'" is endowed with a Banach topology (see Sec. 1.1).
5.3. Model of Batchelor supermanifold [83]. The model space for the Batchelor supermanifold is the De Witt superspace. However, the definition of the Ssmoothness for transition functions is different from Definition 1.2 used in the works by De Witt, Vladimirov,
Chapter I. Analysis on a Superspace
52
Volovich, Rogers. In the work [83] a mapping is said to be Ssmooth if it is approximated by polynomials with coefficients from the field K (and not from the CSA A).
5.4. Model of Jadzyk and Pilch superspace [98]. The works by Jadzyk and Pilch were based on Roger's article [119]. Instead of Grassmann algebras, the authors of [98] considered BanachGrassmann
algebras. The BanachGrassmann algebra is the CSA A = Ao ® Al satisfying the following conditions. 1. For any continuous Aolinear mapping f : A,. 4 A, (r, s = 0, 1) there exists a unique element u E A,+, such that I I u I I= I I f I I and f(A)=u.1,AEA,.
2. Ao = K®A', 1ju+vjj = 1jull + jjvjj, u E K, v E A', where Ao is a Banach subalgebra of the algebra Ao generated by even products of elements from Al. Condition 2 is technical in nature and Condition 1 is the basis of the Jadzyk and Pilch differential calculus. By virtue of this condition, the Aolinearity implies that the operator f is a multiplication operator by the element A. However, the space KA'm is a Aomodule, and therefore the Sdifferentiability on the superspace over the BanachGrassmann algebras can be defined as follows: the mapping F: KAn'm 4 A is Frechet differentiable, F'(x): KAn'm > A is a Aolinear operator (cf. Theorem 2.3). Note that GI is a BanachGrassmann algebra.
5.5. Model of Boyer and Gitler superspace [79]. The article [79] continues the investigations of Rogers. Boyer and Gitler proposed that in the definition of the Sdifferentiability, not only the Z2grading
must be taken into account in the Grassmann algebra, but also the grading corresponding to the degree of the monomials e; = g11...gi,, e{ k: Gn = GI + ... + Gn; G° = K (in this case, Gn o = GnP GnP+l).
Gnj =
n
P
P
The ideals IP = GP + ... + Gn and the quotient algebras G$') _ n Gn/IP are introduced. The Sdifferential calculus is developed for the mappings f : KG ` > GnP). In the work [79], the authors obtained
5. Review of Superanalysis
53
the CauchyRiemann conditions in superanalysis which were similar to those proposed by Vladimirov and Volovich.
5.6. Model of Kobayashi and Nagamashi superspace [101, 102]. The authors of [101] introduced a superspace over asymmetric
Grassmann algebras, where a: r x r + K is a sign function on the finite Abelian group r = ro ® F1, where Fo = {a E F: cr(a, a) = 1},
F1 = {a E F: 01(a, a) = 1}.
5.7. MartinBerezin algebraic superanalysis [114, 115, 3, 4]. This approach to superanalysis was proposed in 1959 by Martin and, independently, by Berezin early in the 1960s. The MartinBerezin theory is of a purely algebraic character. The elements of the Grassmann algebra Gn are considered to be functions of anticommuting variables in the algebraic theory of Martin and Berezin. For the element E = (1.3) use is made of the symbolic notation e = E(ql,..., qn). The derivatives of the element e of the Grassmann algebra Gn with respect to the generators of this algebra are defined according to algebraic rules. A very good exposition of algebraic superanalysis can be found in Berezin's monograph [4].
5.8. Algebraic supermanifolds. The theory of algebraic supermanifolds (graded manifolds) was developed by many eminent mathematicians (see, e.g., [114, 4, 40, 42]). In classical geometry, there exist two approaches to the study of the geometry of manifolds. Traditionally, differential geometry regarded
a space as a primary object, but, at the same time, there existed a point of view of algebraic geometry according to which the geometry of space was studied via the algebraic structure of its bundle of functions. The theory of algebraic supermanifolds is based on the formalism of algebraic geometry. For a smooth real manifold M, the bundle of functions consists of algebras C°° (U), where U is a system of open subsets of the manifold
M. In order to obtain an algebraic supermanifold, we must extend the algebras C°°(U) to the algebras A(U) containing anticommuting
Chapter I. Analysis on a Superspace
54
elements. Thus, an algebraic supermanifold (a graded manifold) is an ndimensional smooth manifold M with a bundle of CSA A(U). It is customary to consider CSA of the form A(U) = C°°(U) ®G(U), where G(U) are Grassmann algebras.
5.9. Local realization of an algebraic supermanifold. "Since a supermanifold does not consist of points" according to Berezin (see [4, p. 20]), the use of algebraic supermanifolds led to a loss of geometrical visuality of physical theories. Batchelor was the first to give a local construction for an algebraic supermanifold (see [82]). She suggested to consider, for a smooth manifold M, the exterior vector
bundle Ext (M), namely, a bundle with fiber AIR. The following statement was proved in [82]. Let I'(Ext (M)) be a pencil of sections of a vector bundle Ext (M).
Then any graded manifold (Kostant [40]) over M is isomorphic for r(Ext (M)) of a certain exterior algebra AIR.
5.10. Nonequivalence of the two approaches to superanalysis. Here are the main differences between the algebraic superanalysis (it is more natural to call it a superalgebra) and the analysis on the superspace K A" presented in this chapter. 1. In algebraic superanalysis all "functions" of anicommuting variables (elements of Grassmann algebra) have a polynomial form (1.3). There are no nonpolynomial continuous or differentiable "functions." 2. In algebraic superanalysis there are no anticommuting constants. Any odd Grassmannian element E under an algebraic approach is a "function" E = E(ql, ..., qn). In superdifferential geometry, this leads to the existence of supermanifolds over KA'm with bundles of functions which are more extensive than the bundle of graded manifold considered in Sec. 5.8. Graded manifolds coincide [83] with supermanifolds over the model of De Witt superspace (5.1) with transition functions from Sec. 5.3. The topology of graded manifolds in the anticommuting sector is trivial. 3. In the commutative sector the MartinBerezin analysis is an analysis on Rn and not on By virtue of item 2, it is impossible to realize the SUSY transformation [95, 75, 84, 124] in the algebraic approach to superanalysis A0.
5. Review of Superanalysis
55
since the SUSY parameters E;' are the same "variables" as the coordinates P. However, the interpretation of SUSY as the transformation of variables 0'`, E'` is meaningless (a detailed discussion of the reason why SUSY cannot be realized in the algebraic approach can be found in the article [82]).
Item 3 implies that Berezin's integral with respect to commuting variables is an integral over R" and not over Ao A. An integral of this kind possesses a number of "pathological" properties which make its application in physics impossible. For instance, the theorem of a change of variables in Berezin's integral is valid only for functions with compact supports on R". One of the first counterexamples for functions with noncompact supports was constructed by Rudakov (see [42]). b
Counterexample 1. In the integral I1 = f x dx = a (b2  a2) we a make a change of variables x' = x + a, where a is a fixed nilpotent element. In accordance with the laws of algebraic superanalysis [4], the integration limits do not change because the real part of the variable does not change (since a is nilpotent):
f x dx = f (x  a) dx = b
x: I1 =
b
a
a
1
2
(b2
 a2)  a(b  a). b
Counterexample 2. In the integral 12 = f f f x d91d92dx = 0 we a make a change of variables x' = x + 0102, 0 = 03 Then .
b
12
(a  b). = f af f (x  0102)d91d92dx = b
Counterexample 3. In the integral 13 = f f x d0dx = 0 we make a a change of variables x' = x + 770, 0' = 0, where 77 is a fixed odd element. Then b
I3 = f f(x_i70)dodx=zi(a_b). a
constructed by Vladimirov, Volovich, De Witt, In the integral and Rogers these changes of variables are correct.
Chapter I. Analysis on a Superspace
56
6.
Unsolved Problems and Possible Generalizations
1. The sufficiency for the CauchyRiemann conditions for CSA which are not finitedimensional Grassmann algebras. The following special cases are of a considerable interest. 1.1. The sufficiency of the CauchyRiemann conditions for finitedimensional CSA which are not Grassmannian. 1.2.
The sufficiency of the CauchyRiemann conditions for the
infinitedimensional Grassmann algebra of Rogers.
2. Construction on a supermanifold of an integral calculus similar to the integral calculus with respect to singular supermanifolds. Remarks Any finitedimensional Grassmann algebra G,ti can be realized as a matrix algebra. Therefore, for finitedimensional Grassmann algebras superanalysis is included into the general theory of functions of matrices. In this connection, we must point out the works by LappoDanilevskii [41] in which the general theory of analytic functions of several matrices was constructed.
Chapter II
Generalized Functions on a Superspace
Generalized functions were introduced by Dirac [29] in his quantomechanical research in which the famous 6function was systematically
used. The fundamentals of the mathematical theory of generalized functions were laid by Sobolev ([131 (1936)]) and used to solve the Cauchy problem for hyperbolic equations. Schwartz ([69 (1950)] and [51]) gave a systematic exposition of the theory of generalized functions and indicated a number of its important applications. Generalized functions appeared in connection with quantum physics problems. This relationship served and continues to serve as the basis for the further development of the theory of generalized functions. Practically all new divisions of the theory of generalized functions emerged from physical problems. In particular, the works by N. N. Bogolyubov exerted a considerable influence on the development of new divisions of the theory of generalized functions. Bogolyubov was the first to show the fundamental role played by generalized functions in the description of the local interactions of elementary particles (see [10]). Later, generalized functions were widely used for constructing the axiomatic quantized field theory [7, 8].
In connection with the applications to the quantized field theory, we must point out the works by Vladimirov [15, 17], Vladimirov, Drozhzhinov, Zavyalov [18], Streater and Wightman [57], and Jost [34].
Chapter II. Generalized Functions
58
The further development of the quantum field theory (superfield theory, superstring theory, supergravitation [25, 27, 28, 40, 76, 123]) led to the theory of generalized functions on a superspace. Just as in the ordinary theory of generalized functions, physicists were the first here. For the first time generalized functions on a superspace (on the physical level of exposition) were considered in De Witt's monograph [27] (1984).
The first mathematical theory of generalized functions on a superspace was proposed in [141, 144] (198687). De Witt's generalized functions are analogs of Schwartz distribu
tions (although De Witt did not deal with problems of functional analysis on a superspace and did not introduce spaces of test and generalized functions, it is clear conceptually that the mathematical realization [27] leads to the supertheory of L. Schwartz). In my works [65, 68, 146148] I constructed the theory of analytic generalized functions. Another essential distinction between De Witt's theory of generalized functions and my theory is the class of CSA over which generalized functions are considered. I have introduced generalized functions on the VladimirovVolovich superspace over an arbitrary Banach CSA with a trivial annihilator of the odd part. In the monograph [27] generalized functions are introduced over a nonBanach CSA in which all even elements with zero numerical part are nilpotent. Note that one of the fundamental problems of the theory of generalized functions is the problem of the choice of CSA. For certain CSA the theory of generalized functions is essentially simplified and for other CSA practically insurmountable difficulties arise. In particular, restricting our consideration to the CSA A = AO G Al with a trivial A1annihilator [65, 68, 146148], we obtain a simple theory of generalized functions on a superspace whose exposition differs but little from the standard theory of generalized functions. The choice of a CSA is closely connected with the class of problems being studied. The type of a CSA is a new parameter of the theory of generalized functions; when solving a specific problem, we choose a CSA which is adequate to the problem at hand.
In the standard theory of generalized functions the part of this
59
parameter is played by the number field over which the theory of generalized functions is constructed. The theory of generalized functions
over the field of real functions is adequate to one type of problems and that over the field of complex numbers is adequate to some other type of problems. Theories of the generalized functions over the field of padic numbers arise in applications to mathematical physics (see [21, 72, 66]). It stands to reason that we can consider CSA not only over fields of real and complex numbers. The analysis on a superspace over an arbitrary locally compact number field was developed in [19]. The theory of generalized functions on a superspace over an arbitrary nonArchimedean number field (not necessarily locally compact) was proposed in [163].
Nagamashi and Kobayashi [103] constructed a theory of generalized functions on CSA which is a strict inductive limit of finitedimensional Grassmann algebras (this algebra is nonBanach).
In [152, 153] I proposed a theory of generalized functions on a pseudotopological CSA which is an inductive limit of nilpotent sets. The theory of generalized functions over Banach CSA with a trivial annihilator of the odd part is most adequate to Caushy's problem for linear differential equations on a superspace, in particular, in the framework of this theory the existence of a fundamental solution of Cauchy's problem on a superspace was proved for an arbitrary linear differential equation with constant coefficients [146]. The theory of generalized functions over CSA [152, 153] is more adequate to the problem of the fundamental solution of a linear differential operator on a superspace. In particular, the theorem of existence of fundamental solution of a linear differential operator with constant coefficients on a superspace was proved in the framework of this theory [153]. In contrast to the real case, in a supercase there exist differential operators which do not have a fundamental solution.
60
Chapter II. Generalized Functions
1.
Locally Convex Superalgebras and Supermodules
1.1. Locally convex superalgebras. A topological linear space is a linear space E over a field K (K = R, C) endowed with topology relative to which the algebraic operations ((x, y) H x+y, E x E + E; (A, x) H Ax, K x E + E) are continuous. The topological linear space E is locally convex if there exists, in this space, a basis of the neighborhoods of zero {U} consisting of convex sets (neighborhoods of the
form {a + U} form the basis of convex neighborhoods of the point a c E). Just as in a normed space, the topology can be defined with the aid of a system of prenorms (a prenorm is a map x H 11x11 satisfying the same axioms as a norm, except for the fact that, generally speaking, IIxII = 0 does not imply x = 0). We shall use the symbol FE to denote the system of prenorms { } which defines the topology in the locally convex space E. A topological algebra is the algebra A endowed with a topology relative to which the algebraic operations are continuous. A topological algebra is said to be locally convex if its topology is locally convex. It follows from the continuity of the multiplication operation that for any prenorm II II E IFA there exists a prenorm ' E rA such that llxyll < CIIxII'llyll' (the constant C depends on
Let the CSA A = AO ® Al be a topological algebra, where the direct sum AO ® Al is topological (i.e., the projectors 7r,,: A 4 A,, are continuous). A = Ao ® Al is called a topological CSA. If the topology in A is locally convex, then A is called a locally convex CSA. A locally convex space whose topology is metrizable is known as
a Frechet space. This is equivalent to the fact that the system I'E is countable. Frechet algebras and Frechet CSA can be defined by analogy.
1.2. Locally convex supermodules. A supermodule is a Z2graded space M = Mo®M1 on which a structure of a twosided module over the CSA A = Ao ® Al is defined with an even multiplication operation by the elements of the CSA A = Ao®A1 (Iabl = lal Ibl, mod 2, for homogeneous elements from the algebra and the module). The su
1. Locally Convex Superalgebras
61
percommutator [a, b} of homogeneous elements from the algebra and the module is defined by relation (1.1), Chap. I. The supermodule M = Mo ® M1 is said to be (super) commutative if, for arbitrary homogeneous elements [a, b} = 0 (i.e., the even elements of the algebra
commute with all elements of the module and the odd elements of the algebra anticommute with the odd elements of the module). Instead of the term a (super) commutative supermodule we shall use its abbreviation CSM. A CSM is said to be unital if ex = x for all x E M (e is an identity in CSA). In this book we consider only unital CSM. The concept of a CSM is equivalent to the concept of the representation of a CSA A = A0®A1 in a Z2graded linear space M = Mo®M1. In this case we consider representations consistent with the structure of parity. Topological CSM over topological CSA are defined in a natural
way, as well as locally convex CSM over locally convex CSA and Frechet CSM over Frechet CSA.
We shall not give here any examples of CSM, they will appear later, in the theory of generalized functions on a superspace.
1.3. Conjugate supermodule. The successive exposition of the duality theory for CSM is given in Chap. III. Here we introduce only the main concepts which are necessary for constructing the theory of generalized functions. Let M = Mo ® Ml be a CSM. The linear functional 1R: M + A
is said to be right Alinear if IR(mA) = lR(m)A for any m E M, A E A. Left Alinear functionals 1L can be introduced by analogy. On the space of functionals which are right Alinear we introduce the structure of a module setting (AIR)(m) = AIR(m), (IRA)(m) = lR(Am), A E A, m E M. By analogy we introduce the structure of a module on the space of functionals which are left Alinear. A functional l is even if 1 1 (m) I= 0 for Im l = 0 and 11(m) I= 1 for Im l = 1. A functional l is odd if 1l(m)I = 0 for Iml = 1 and 1l(m)I = 1 for Iml = 0.
A Z2grading and a parity function are introduced on the space of functionals which are right (or left) Alinear. These spaces become CSM over the CSA A = Ao ® A1.
Chapter II. Generalized Functions
62
A is a right Alinear It can be noted that if 1R = 1R ® AR: M functional, then the functional IL = IL ® 1i defined by 10
= 10
IL' (m') = lR(m°);
1L(m') = lk(ml).
(1.1)
where Im° _ laI = Ili = a, a = 0, 1, is a left Alinear. Similarly, every functional which is left Alinear defines a right Alinear functional. Identifying the spaces of left and right Alinear functionals, we obtain a CSM M' = MO* ® M1 which is an algebraic conjugate of the CSM M = MO ® M1. Considering continuous Alinear functionals on the topological CSM M = MO ® M1, we get a CSM M' = MO '(D M1' which is a topological conjugate of the CSM M = MO ® M1. 1.4. Differential calculus for mappings from a subspace
into a supermodule. Let KA'm' = Ao x Am be a superspace over a topological CSA A = AO ® Al and let M = MO ® M1 be a CSM over the CSA A = Ao ® A1. The following definition of Sdifferentiability is a trivial generalization of Definition 1.2 from Chap. I. This generalization proceeds in two directions. First, the functions assume
values not in a CSA but in a CSM, and, consequently, the derivatives are multiplication operators by elements from the CSM. Second, generally speaking, there is no norm on a superspace, and therefore we must replace Frechet differentiability by some differentiability in a topological linear space [54]. By way of example, we shall consider differentiability with respect to a system of bounded subsets [54].
Definition 1.1. The mapping f : U + M, where U is an open subset of KA'm, is Sdifferentiable at a point x E U if representation (1.5), Chap. I, is valid, where the partial derivatives are elements of the CSM M and the remainder o(h) satisfies the condition m o(th)/t = 0 uniformly on any bounded subset B of the superspace Kn'm (t E K). Recall that the set B in the topological vector space E is bounded if, for any neighborhood V if zero in E, there exists a A > 0 such that B C AV. If E is normed, then the bounded subsets coincide with subsets bounded in norm and the differentiability with respect to the system of bounded subsets coincides with the Frechet differentiability.
2. Analytic Generalized Functions
2.
63
Analytic Generalized Functions on the VladimirovVolovich Superspace 2.1. Superalgebras with a trivial annihilator of the odd
part. As was pointed out in the introduction to this chapter, CSA with this property serve as the basis for the theory of generalized functions on the VladimirovVolovich superspace.
Theorem 1.1. Let A = AO ® Al be a CSA with a trivial A1annihilator, and there exist odd elements whose product is nonzero. Then the subspace Al is infinitedimensional.
Proof. Assume that Al is finitedimensional and let a1, ..., am be the basis in the linear space A1. We shall prove by induction that
aiaj =0foralli,j: 1. Since a,...am E' Al always, it follows that a,...am = 0. 2. Assume that for any collection j1 < ... < jk: aj1...aik = 0. 3. Let us show that it follows from item 2 that for any collection 71 < ... < jk1: a,j,...ajk_, = 0. Assume that there exists a,1...ajk_, 0. Let A = c1a1 +... + cmam m be an arbitrary element of A1. Then a,j,...a7k_1A = > cjaj,...ajk_laj. 3=1
If aj = a 1, then a ... ajk_, = 0 since a = 0.
If a3
ail, then
aj,...ajk_, = 0 by virtue of item 2. Everywhere in what follows in this chapter we assume that in the Banach CSA A = AO ® AO the A1annihilator is trivial or the CSA A is a commutative Banach algebra (i.e., A = A0). The main model example of a CSA with a trivial A1annihilator is the Banach exterior algebra Gl(B). Let us prove this simple assertion. Let
=
[
f  k=OJjl Ra. x°`90 converges in the 1.1=0101=0
to the function fR(x, 9). The continuity of the functional u implies that the series space
00
m
(1)1 R' u.#
(fR, u) = E
(2.7)
1.1=0101=0
converges. The convergence of series (2.7) for any R yields (2.6). The system of norms {II IIR} defines the Frechet topology in the CSM E'(Cnm). We introduce spaces of generalized functions A' (C m) = {u E A'(Cn'm): I U I IR < oo}; they are Banach CSM. The space of generalized functions is endowed with a topology of inductive limit: A'(Cn'm) = lim ind A' (Cn'm); A'(C m) is a complete locally convex I
CSM.
Propositions 2.6 and 2.7 can be used to prove
Proposition 2.8. The spaces A"(Cpm) and E"(Cn'm) which are conjugates of the spaces of generalized functions A'(Cn'm) and E'(Cn'm) coincide with the spaces of the test functions and E(Cn'm).
Complete proofs of Propositions 2.62.8 can be found in [146]. Furthermore, we shall use the symbol of an integral
f w(x, 9) p(dxdO) = (W, p)
(f p(dxdO) cp(x, 0)
cp))
to denote the action of a generalized function on a test function.
2. Analytic Generalized Functions
69
Theorem 2.1. The mapping z H µZ from the superspace CA 'm into the space of generalized functions A'(Cn'm) (or E'(Cp'm)) is Scomplete for any generalized function p E A'(CA'm) (or p E
Proof. Using the fact that for any test function f its Taylor series converges in the space of test functions, we obtain (µ=, f) _ (µ, f=)
i.. V
(1)101+101
a!
00
xaea
as
aL
ax° aemm
ao p1
f
0
µ' axa ao#' ...a9p1 f)xaO'(1)IkI+If1101
_
3
a.
axea9aR
mm
...
m
n Jal
_ E ail j=1
aOx"90' f ) 1
1131
E oi. i=1
It remains to verify that the power series µy,9) =
1
,a
x° 9o
cfo
converges absolutely in the space of generalized functions (or E'(CAm)). By virtue of Remark 2.1, here again everything reduces to a commutative case.
2.4. A direct product and convolution of generalized functions. The direct product of the generalized functions 1L1 and µ2 from 'm2 is the spaces A i(CAn"mi ) and A '(CAn2 'mz ) or E '(CAn"mi ) and E ' (CAnz) correctly defined by the relation
f µ1 ®p2(dx dy d9 a
(1.5)
for any A, Q, a E A and m E M, m* E M* (i.e., LO,, Proof. Suppose, for instance, that Im* _ m = 1. Then (m*, m) _
Ir(m*)(m) _ Ir(m*)(a(m)) _ I(Ir(m*))(m) _ Ii(m)(m*) _ (m*, m), Ir(.Am*,Q)(ma) = .\Ir(m*/3)(m)a = AIr(m*)(Qm)a.
Chapter III. Distribution Theory
116
2.
Banach Supermodules
2.1. Z2graded norms. A CSM M over a CSA A is called a Banach CSM if M is a Banach module over the Banach algebra A (i.e., IlAmll < IILII Ilmll, A E A, m E M) and the direct sum M = Mo ® M1 is topological (i.e., the projectors 7r,,, are continuous). The norm on the CSM M is said to be Z2graded if Ilxll = Ilx0 II + IIx1lI. In this case, Il7r II = Ilir°II +
Ilirlll for x E M.
We shall only consider Z2graded norms on Banach CSM. In this section, the symbols M = MO ® Ml and N = No ® N1 are used to denote Banach CSM over the Banach CSA A = AO ® A1.
2.2. Supermodules of Asequences. The CSM introduced below constitute a natural generalization of Banach spaces of sequences of real (or complex) numbers 1P and co. We introduce Banach CSM consisting of Asequences x = (x1) ... xn, ...), xj E A,
co(A) = {x :m xn = 01; 00
1
1 (A)={x:
I17raxnllP)1'P IILII00  2E.
We find, as a consequence, that the Banach spaces (An, II II1)' and (An, 11.11 I) are isometric.
The situation realized in Example 2.1 is pathological. In order to exclude cases of this kind from the consideration, we give the following definition.
Definition 2.1. The CSM M and M' are dual if M' separates the points of M. If M and M' are dual, then M is embedded into M": x '* lx E M",
1x(y) = y(x) for y E M'. It suffices to verify that the operator is M + M", x H lx belongs to the class LO,, (M, M") = C',, (M, M"). Indeed, lax(y) = (lax, y) = (\x, y) = )(x, y) = Aix(y), i.e., lax = )lx and i E G1,i(M,M"). The parity of the operator can be immediately verified.
Definition 2.2. A Banach CSM M is semireflexive if is M 4 M" is an algebraic isomorphism, i(M) = M'. Definition 2.3. A Banach CSM M is reflexive if M is semireflexive and the canonical isomorphism i is an isometry. In the theory of linear Banach spaces, the semireflexivity implies reflexivity since the canonical inclusion of E into E' is an isometry (the
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120
concepts of reflexivity and semireflexivity differ only for topological linear spaces) .
Theorem 2.2. Suppose that for any vector x E M there exists a vector y E M' such that IIyII < 1 and IIxII = IIy(x)II. Then the canonical inclusion of M into M" is an isometry.
Proof. Let x E M. Then there exist ya E M', a = 0, 1 such that II(xa,ya)II
= IIxa!I and IIyall < 1. Consequently, IIl2II = sup 11W"011 Ilvll IIxalI, i.e., 111 "11 = IIxII and 111.11 = IIxII
Open question Is the Banach CSM (An, II II1) reflexive?
It has been shown that (An, II II1)' is isometric to (An, II . II00). It is not known, however, whether the CSM (An, II  II,)' is isometric to (An, II 111) for the arbitrary Banach CSA A.
2.4. Esuperalgebras. The following definition play an important part in duality theory for Banach CSM. Definition 2.4. The Banach CSA A is called a Ealgebra if, for any elements a1, ..., an E B, we have n
n
> IIajlI = sup 11Eajajll,
j=1
I10,1151 i=1
where a = (a1, ..., an) are homogeneous vectors from An. The algebra K is the simplest example of the Ealgebra.
Theorem 2.3. The CSA G1 00 (T), where T is an arbitrary commutative Banach algebra, is the Ealgebra.
Proof. Let 00
a7 =
E 1:
aj,q,, ...gryn
a77 E T,
j = 1, ..., m.
n=0 71 G.. 0 there exists a finite set of indices r such that Ilai air II < E
m,
where air =
yEr
2. Banach Supermodules
121
We set I3 = qK3, j = 1, ..., m, where qK, # q.y, for all indices 'y E IF and kj j4 k; when j i. Then m
m
IIE Qjajrl = > IlajrM. j=1
j=1
Furthermore, m
M
m
IEajajl <EIlajli <E+EIlajrll
sup
j=1
I1c11o051 j=1 M
j=1
m
=E + 11 E Ojaj
E+
j=1
m
Qj (air  aj) I + 11 E Qjaj j=1 j=1 M
< 2E + sup
11E
ajaj
.
Ila11oo
i=o
sup II(Lx)II I1x16 11y'II = IILII IlyM1. Q=0 11451
0=0
2. Banach Supermodules
127
It remains to use the following statement.
Proposition 2.4. The relation (L')' = (L°)', a = 0, 1 is valid. It is easy to construct an example of an operator for which IIL'1I < IILII.
Proposition 2.5. Let the CSM M and M' be dual and let the canonical inclusion of M into M" be an isometry. Then IILII = IILII.
Proof. Using the fact that the inclusion into the second conjugate module is isometric, we have 1
1
IILII = E sup 11(y,Lx°)11 < IIL'II E IIx'11= IILII IIxII. a=O
a=O I1y1151
Proposition 2.6. Let the CSA A be a Ealgebra. Then the relation I I L I = IILII1 I
holds for any operator L E G1,r(l1(A)).
Proof. Note that co(A) = l1(A) and li(A) = l.. )(A). It follows from the first relation that sup
IIzC'111=
II (u, z°) II
I1ulloo
Under a new approach to a superanalysis, a boson field assumes values in Ao and not in Km. Example 5.4 (superspace of a boson string with FaddeevPopov ghosts). We denote by M a CSM consisting of paths q: [0, 7r] + AD,
D = 26, satisfying the boundary conditions q'(0) = q'(7r) = 0 and by N a CSM consisting of paths z = (c, c): [0, 7r] + A2 satisfying the boundary conditions c'(0) = c'(ir) = 6(0) = 6(7r) = 0. We can impose different conditions of smoothness on the paths and topologize the CSM M and N respectively. The superspace X = MO ® N1 = {W:
cp = (q, z)} is a coordinate space of a boson string with FaddeevPopov ghosts, Mo is the string part of the superspace, N1 is the ghost part of the superspace [25]. Under the new approach to a superspace, a boson string assumes
values in Ao and not in R'. The topological basis (Schauder basis) in the superspace X = Mo® N1 is the topological basis (Schauder basis) in the covering CSM Lx =
M ®N. Let {ej}jEJ and {ai}iEJ be topological bases in the CSM M and N respectively. Then {ej; ai}jEJiEI is a topological basis in the space X:
x = E x°ej + E x1ai, jEJ
1x; l = lej 1,
lxi 1 = 1  jail.
iEI
Example 5.5 (Hilbert superspace). Let M, N 12(A) be Hilbert CSM and let {ej} and fail be canonical bases in M and N respectively. The superspace H = Mo®N1 is known as a Hilbert superspace, {ej, ail is Schauder basis in the Hilbert space H, 00
00
j=1
i=1
The scalar product 00 (x)
y) = (x°, y°) + (x1, y1)
j=1
00
X59 + > xi yi i=1
Chapter III. Distribution Theory
146
assumes values in A0.
Let M and N be Hilbert CSM with involutions `. The involutions in the CSM M and N induce an involution in the Hilbert superspace H (since I' I = 0); the *scalar product has the form 00
00
(x, y) = (x°, y°) + (x1, yl) _
x°(y9)i + E xi (yi )'. j=1
i=1
By analogy, we can consider a superspace over the Hilbert CSM M, N 12,A,B (A)
5.2. Superdifferentiability. Let us recall, for the beginning, the definition of a differentiable mapping of topological Klinear spaces El and E2. In differential calculus, there are several dozens of definitions of differentiability for topological Klinear spaces. In order to define differentiability, we must, first, fix a certain class of Klinear operators 11(E1, E2) to which the derivatives will belong and, second, fix a type
of smallness of the remainder, i.e., define in what sense o(h) E E2,
hEE1. The mapping f : U + E2, where U is an open subset of the space
El, is said to be differentiable at the point x E U if, for all h E El such that x + h E U, we have
f(x + h)  f(x) = A(h) + o(h),
(5.1)
where the operator of the derivative A = f'(x) belongs to the class 7 l (El, E2).
The differentiability with respect to systems of bounded and compact subsets is especially widely used. For the differentiability with respect to a system of bounded subsets we take £(E1, E2) as 3l(E1, E2) and define o(h) as o(th)/t 4 0,
t + 0,
t E K,
uniformly on any bounded subset of the space El. For the differen
tiability with respect to a system of compact subsets f(E1, E2) = K(E1, E2), and in the definition of o(h) we use a uniform convergence
5. Differential Calculus
147
on compact subsets. The Gateauz differentiability is also of importance. For this differentiability we also have 3l(El, E2) = G(E1, E2) and o(th)/t 4 0, t 4 0 for any vector h in El. For Banach spaces, the differentiability with respect to a system of bounded subsets coincides with the wellknown Frechet differentiability (see Sec. 1.2, Chap. I). Using a pair, namely, a superspace and a covering supermodule, we can extend the definition of differentiability for topological Klinear
spaces to a supercase. We fix a certain class of Alinear operators Hi,r(Lx, Ly) (7i1,1(Lx, Ly)) (it is customary to assume that this class is a CSM) and a certain definition of o(h).
Definition 5.3. The mapping f : U a Y, where Y is a superspace and U is an open subset of the superspace X, is said to be right (left) Sdiferentiable at a point x E U if, for all h E X such that x+h E U, we have relation (5.1), where the operator of the Sderivative A = 8f (x) belongs to the class 3ll,r(Lx, Ly) (f1,1(Lx, Ly))
When necessary, the righthand Sderivative will be denoted by aR f (x) and the lefthand derivative by aL f (x). By a complete analogy with the case of topological Klinear spaces, we can define Sdifferentiabilities with respect to systems of bounded and compact subsets and the Gateauz Sdifferentiability. The definitions of o(h) can be extended without changes; for the Sdifferentiabi
lity with respect to a system of bounded subsets and the Gateauz differentiability we take the CSM Gl,r(Lx, Ly) (G1,1 (Lx, Ly)) as the class Ni,r(Lx, Ly) (711,1(Lx, Ly)); for the Sdifferentiability with respect to a system of compact subsets I,r(Lx, Ly) = IC,,, (Lx, Ly) (' 11,1(Lx, Ly) = K1,1 (Lx, Ly))
Definition 5.3 can be reformulated as follows: the mapping f is differentiable as the mapping of Klinear topological space X and Y, the derivative belonging to the class of Alinear operators R1,r (f1,1)
Remark 5.1. We cannot restrict the consideration to some type of linearity on a superspace and not on a covering supermodule. A superspace is only a A0module, but the A0linearity of the derivative is insufficient for Sdifferentiability (see Example 2.4, Chap. I).
Chapter III. Distribution Theory
148
Let M=M°®M1, N=No ®N1, R=RoED R1, S=So ®Sl be topological CSM over a topological CSA A, X = M°®N1 i Y = Ro®S1.
The function f : X 4 Y, f = (f °, f 1), f ° E Ra, f 1 E S1, is right Sdifferentiable. Then
of
of
of
= 19x°, axl
of
aa axe
=
,
aE
where the operators ax E 7{° o l,r (M, R),axr E 1ll ,r(N R) , 3li r(M, S), a E Vi,r(N, S). If the CSM R and S coincide, Y = R° ® R1 = R, then we can regard the Sderivative a f as an element of the space W1,r (Lx, R): e
of=of°ED af'. Definition 5.4. The mapping f : U 4 N, where U is an open subset of the CSM M, is said to be right (left) Sdifferentiable at a point x E U if, for all h: x + h E U, we have relation (5.1), where af(x) E Hi,r(M, N) (af (x) E 71,1(M, N)) In contrast to an analysis on a superspace, an analysis on a CSM is very meagre and is of no particular interest.
Example 5.6. Let us consider the mapping f : A 4 A, f (x) _ ax,3, where a, fl E A1. We shall regard the domain of definition of A as a CSM = CSA and not as a superspace KA1 = A° x A1. The mapping f is not (right or left) Sdifferentiable. The same mapping regarded as a mapping of superspaces is both right and left Sdifferentiable. By virtue of Remark 5.1 and Example 5.6, it is obvious that an analysis on a pair (a Klinear superspace, an Alinear covering CSM) is a "golden mean" between an analysis on a Klinear space and an analysis on an Amodule.
Proposition 5.1. If the operator CSM fl,r = Ll,r and 711,1 = L1,1 or 7ll,r = 1Cl,r, 711,1 = 101,1, then the right and left Sdifferentiabilities
are equivalent.
This proposition is a direct corollary of Proposition 1.4.
5. Differential Calculus
149
Theorem 5.1 (Leibniz formula). Let M, N, S be Banach CSM; suppose that the functions f : X = MO ® Nl * S and g: X + S' are Frechet S differentiable at a point a E X, and g(a) is a homogeneous element of the CSM S'. Then the function W(x) = (f (x), g(x)) is Frechet S differentiable and we have a formula
l
ax° J f (a)
axe (a)
+(1)°I9(a)I
{o.19a)I
axf
(a)l g(a),
a = 0, 1.
(5.2)
Proof. It suffices to note that P = ` axe (a) V, g(a)) =
(a),
I9(a)I (_(a)h'\"
by virtue of Proposition 1.7 (relation 1.4). If jg(a) I = 0, then P=
((a)f (a)ho /
if jg(a) I = 1, a = 0, then P
f
= Kg(a)' a ('9x° (a)) ho)
if jg(a)j = 1, a = 1, then P
(g(a), a
((a)) hl)
These three equalities yield the righthand side of relation (5.2). The ordinary theorems of differential calculus (on an inverse function, ...) are valid in superanalysis. They are obtained in Sec. 1, Chap VII in a more general case of analysis on a superspace over an arbitrary noncommutative algebra.
Chapter III. Distribution Theory
150
We introduce a right annihilator of the superspace X = M° ® N1 by setting 1X  Ann (X; ll,r(Lx, R)) = {a E 'Hj,!(Lx, R): Ker a D X}, where R is a CSM. By analogy, we introduce the left annihilator.
Proposition 5.2. If the condition 1A1  Ann(A1; R) = 0
(5.3)
is satisfied, then Ann (X; Ll,r(Lx, R)) = 0 (and Ann(X; L1,i(Lx, R))= 0)
Proof.
1.
Suppose that the operator a E Ll,r(Lx, R) and the
restriction of a to M is zero. Then we have a(m10) = a(m')O = 0, i.e., a(ml) E 1A1, for all 0 E Al and ml E M1. Consequently, aim = 0. 2. Let a E L1,r(Lx, R) and ajN1 = 0. Then we have a(n°9) _ a(n°)9 = 0, i.e., a(n°) E 'A1i for all 0 E Al and n° E N°. Conse
quently, a iN = 0.
Thus, by virtue of Proposition 5.2, the triviality of the annihilator of the superspace for any class of Sderivatives Hi,r(Lx, R) follows from the triviality of the A1annihilator for the module R (see (5.3)). As was pointed out in Chap. I, an Sderivative is, in general, not uniquely defined.
Let the function f : X 4 Y be Sdifferentiable (with a space of Sderivatives ?'ll,r(Lx, Ly)). By factoring the space of Sderivatives with respect to the annihilator of the superspace, we get a onetoone mapping aR f : X +'Hl,r(Lx, Ly)/Ann(X, Lv). Consequently, we can define the second derivative aRf (x) E Ni,r(Lx, W i,r(Lx, Ly)/Ann(X, Ly)) and higherorder derivatives. 5.3. Supersymmetries of higherorder superderivatives. Let us restrict the consideration to the case of singlevalued Sderivatives: 1X = 0. By virtue of Proposition 5.2, it suffices to require that 1A1 = 0.
5. Differential Calculus
151
Thus, suppose that 'Al = 0. Proposition 5.3. For the Frechet Sdifferentiability on a Banach superspace, the restriction of higherorder derivatives to a superspace is symmetric.
Proposition 5.3 is a direct corollary of Frechet differentiability in Klinear Banach spaces [37].
Remark 5.2. When formulating the results concerning Sdifferentiability, we consider Banach superspaces only for the sake of simplification of the exposition. We can also consider Sdifferential calculi over topological and pseudotopological superspaces (cf. the Klinear case [38, 541).
Lemma 5.1. Let X = MO ® N1, b E Lp,r (LX, R) and let the restriction of b to XP be symmetric. Then we have the symmetry b(yl, ...) yk, ..., y;, ..., yP) (5.4) ak+j(yj), ..., Qk+j(yk), ..., yp) _ onYP, Y = M1oNo. Proof. Let 03 E Al for j = 1, ..., p. Using relations (1.1) and (1.2) b(yl,...,
and the symmetry on XP, we obtain b(yl01, ..., yk8k, ..., yjOj, ..., yp9P) = b(y101,...,y3O,,...) ykOk,...,yyOP)
=
j1 y;, ... , QP1 yp)e1...Bk...B;...9p yk, ..., Q j1 P1 k1 yp Bl...e;...ek...ep. Q Or yke ..., Or
b (yl, Oyz, ..., 0'
 b (yl
k1
y2r Let us set yj = Qilyt Then
b(yi, ..., yk, ..., y', ..., y'P)el...ek...e;...ep
_
b(yl, ..., or
k+j yk,
I I ..., a k+j y;, ..., yP)el...ek...B;...ep
In order to prove relation (5.4), it remains to use the triviality of the Alannihilator.
Chapter III. Distribution Theory
152
Lemma 5.1 yields
Proposition 5.4. Let X = Mo ® N1, b E LP,,.(LX, R) and let the restriction of b to XP be symmetric. Then the restriction of b to M? and No is antisymmetric.
Lemma 5.2. Suppose that the conditions of Lemma 5.1 are fulfilled. Then b x1Q1, ..., xkQk, ..., xQi , ... , xPQP) i1
QkQ;+(Qk+Q;)
Q;
b(x1..., xp', ...)xkk, ..., xQP),
(5.5)
where/3,=0,1,xAEMo for/3=0 and XP ENo for /3=1. Proof. Let 03 E A1, j = 1, ..., p. Then b(x#1
0Q1
, ..., x1 k akk , ..., X13' 01' , ..., xpp 9pp )
xA'9p, ..., xkk ekk, ..., xpp epp)
= b(xp1
=
b(xQ1 1,
9Q1...9Qk ..OOP ..., xak k, ..., xQi k ...B~i 7 P j , ..., XOP) P
Xk = b(xQ11, ..., xQi 7, ..., k, ..., xQ9)0Q1...Bp'...BQk...BQP 7 k P P 1
i1
X(1)
Qk1;+(Qk+Q;)+ > Qi :=k+1
It remains to use the triviality of the A1annihilator. Lemmas 5.1 and 5.2 show that the symmetry of an A1linear form on a superspace entails many (rather unexpected) supersymmetries on a covering CSM.
Lemma 5.3. Any form b c Lp,,.(LX, R) is uniquely defined by its restriction to the superspace XP.
Proof. Let a, Q = 0, 1. We regard the sum a + Q mod 2. Let zC1Q j = 1 , ..., p belong to M, , for / 3 = 0 and to N, , for 0 = 1 For 0 E Al .
we obtain 0Q1+Q1 Bop+QP b(za1 , ..., z°P 1Q1 1 PQP P )
5. Differential Calculus
153
(1)6b(zQ' lfl,
,
zOP °P )BQ1+Q1 1
0Qp+Op .. p
where 5 = b(a, Q). Note that the vector z = (z1A1B11+A1,
,
P+Rp)
belongs to XP. Therefore, if blxp = 0, then P
> (Q; +Ai )
b(z) E L (Aj ' l
Theorem 5.2 (on the properties of higherorder Sderivatives). Let the function f : X + Y, where X and Y are Banach superspaces, be n times Frechet right Sdiferentiable at a point x E X. Then its Sderivative of order n is uniquely defined and belongs to the CSM Gn,r(LX, Ly); the restriction of the Sderivative to the superspace is symmetric and supersymmetries (5.4), (5.5) hold on the covering CSM.
Theorem 5.2 is also valid for other types of Sderivative on locally convex superspaces, for instance, for differentiability with respect to a system of compact sets.
Example 5.7. Let M = A2, N = A2, X = RA2, Lx = A4 and let the function f (x, 0) = ax1x2 + /30102i where a, /i E A, x = (x1, x2) E A2, and 0 = (01 i 92) E A2. Then 8Rf (x, 0) (h, h') = a(v2u1 + v1u2) + Q(e1712e2711), h = (v, e), h' = (u, 71) E A4. The restriction of 8Rf(x, 9)
to the superspace RA2 is symmetric and the restriction to o ® A2 and to A2 ® o is antisymmetric. Note that 8Rf (x, 0) has many Klinear extensions to the covering CSM A4. For instance, (h, h') + v2au1 +vi au2 + bft  e20711. These extensions are not right Alinear if a, /3 V Ao. A higherorder Frechet Sdifferentiability is defined by the classes
of operators 'Hn,r = Gn,r (Wn,i = Gn,l) to which the Sderivatives belong and by the Frechet differentiability in Klinear Banach spaces. Similarly, every differentiability in topological Klinear spaces and any sequence of classes of forms Wn,r C Ln,r (fn,l C L, ,1) are associated with a higherorder Sdifferentiability.
5.4. Taylor formula. It follows from Theorem 5.2 that if Taylor formula holds for the differentiability in Klinear spaces [38, 54], then,
Chapter III. Distribution Theory
154
for the corresponding Sdifferentiability, we have Taylor's formula on a superspace
f (x) _
aRf xo) Ti.
n=0
(X l
 x0i ..., x  xo) + Tm(x  x0),
where Tm is the remainder, 8R f (xo) E fn,r, and the restriction of oR f (xo) to Xn is symmetric.
In particular, the Taylor formula holds for an Sdiffrentiability with respect to a system of bounded (compact) subsets: rm(th)/tm 4
0, t + 0, uniformly on bounded (compact) subsets of the superspace X; in Banach superspaces, this is equivalent to the fact that IITm(h)II/IIhII + 0, h + 0.
5.5. Superanalyticity. The Taylor formula on a superspace leads to the following definition of Sanalyticity. Definition 5.5. The mapping f : U t + Y, where U is the neigh
borhood of the point x0 E X, is right (left) Sanalytic at a point xo if, in a certain neighborhood of the point x0i the mapping can be expanded in power series 00
f(x)=Ebn(xxo,...,xx0),
(5.6)
n=0
where Anlinear forms of bn belong to the supermodules (L'r, Ly) (Ln , Ly)) and the restriction of these forms to X n is symmetric. We obtain various definitions of Sanalyticity corresponding to dif
ferent classes Anlinear forms ?in,,, fn,t and to different types of convergence of the power series (5.6).
Definition 5.6. The mapping f : U
Y, where Y is a superspace over a locally convex CSM, is said to be compact (bounded) Sanalytic at a point xo E U if Kn,r(Ln,r), in,t = 1Cn,i(Ln,i) and if there exists a neighborhood V = V(xo) of the point xo in the covering CSM
LX such that for any compact (bounded) subset B C V and any prenorm II II E FL,, we have 00
Ilf IIB = E Sup Ilbn(x1  x0, ..., xn  xo)II < 00. n=0 xi E B
(5.7)
5. Differential Calculus
155
It follows from Proposition 1.4 that the left and right compact (bounded) Sanalyticities coincide.
It follows from Definition 5.6 that every compact (bounded) Sanalytic function f : U(xo) * Y can be extended to the neighborhood of the point x0 in the covering CSM L. From estimate (5.7) it follows that every compact (bounded) Sanalytic function is infinitely Sdifferentiable with respect to a system of compact (bounded) subsets.
If the function f is compact (bounded) Sanalytic on the whole superspace X, then it is said to be compact (bounded) Sentire. An Sentire function can be extended to the covering CSM Lx. Consider compact and bounded Sentire functions on a superspace KA'm over a locally convex CSA A with a trivial Alannihilator.
Proposition 5.5. The spaces of compact and bounded Sentire functions f : KA'm + A coincide.
Proof. Let f be a compact Sentire function:
f (y) = 001: 1:
bk(ejl, ...,
ejk)yjl...yjk,
k0 jl...j
is a canonical basis in KA'm, and for any compact set where B C An+m and prenorm II . II E I A we have 00 IIfI
bk(ejl, ..., ejk)zljl...zkjk 11 < 00.
I B = E sup
k=0 zyCB ji...jk
We set BR = {Rel, ..., Ren+m}, and then IIfMIBR =
E00 sup IIbk(ejl, ...,
ejk)IIRk
< 00.
k=0 it ...7k
Consequently, 00
IIfIIR=
Ilbk(ejl,...,ejk)IIRk < IIfMIBr < 00, k=0 ji...jk
(5.8)
Chapter III. Distribution Theory
156
r = R(n + m). Suppose now that If M R < oo for all R > 0. Then, for any bounded set B C An+m: IIf JIB < If IIR, where R = sup up JJxJJ.
Corollary 5.1. Let A = A be a Banach algebra. Then the space of compact (=bounded) Sentire functions f : KAn'm + A coincides with a space of Sentire functions in the sense of the VladimirovVolovich definition (Chap. I). The corollary follows from estimate (5.8).
Remark 5.3. It should be emphasized that convergence (5.6) in Definition 5.6 holds on compact (bounded) subsets of the covering CSM Lx and not only of the superspace X.
The article [118] is a brilliant illustration to this remark. The author of this work constructed a "counterexample" by virtue of which (in his opinion) "Khrennikov's theory of superanalyticity (1988) is not wellgrounded." The author of [118] considers a superspace KA1, where
A = lim ind Gn is a NagamashiKobayasi topological CSA [103]. It is easy to show that the series E Q1...gkxkB converges uniformly on every compact k=0
subset of the superspace KA1 and defines the function f (x, 0) on KA'1 (it follows from the properties of inductive topology [71] that every
compact subset in A is contained in one of the finitedimensional Grassmann algebras Gn). This function possesses a number of "pathological" properties. In quantum theory, the fields of the function cp(x, 0) are fields on a superspace. Of a considerable importance is the expansion of a field in the powers of 0: cp(x, 0) _ cop(x)0', (5.9) where the coefficients cpp (x) are physical tensor fields. Expansion (5.9) does not hold for the function f (x, 0).
5. Differential Calculus
157
However, the function f (x, 0) is not Sanalytic in the sense of Definition 5.6. In the covering CSM A2 we take a compact set KE consisting of one point {Ee, Ee}, where E > 0 is any arbitrarily small number. Assume that 00
Ilf IIK, =
L
IIg1...gkEk+lII < oo
(5.10)
k=0
for any prenorm I II E I'A. Then the series E Q1...gkek+1 converges to k=0
A, and, consequently, the sequence of partial sums of this series must be contained in one of the finitedimensional Grassmann algebras G,,. Therefore (5.10) does not hold true.
Corollary 5.2. For every compact (bounded) Sentire function go: KA'm + A expansion (5.9) holds true; the coefficients Wp(x) are compact Sentire functions.
This corollary can be immediately obtained with the aid of estimate (5.8). It follows from the triviality of A1annihilator that the coefficients are uniquely defined.
If the superspace X is of an infinite dimension (over A), then the Sanalytic function f of anticommuting variables can be a nonpolynomial.
Example 5.8. Let X = N1, where N = 12(A); the function f (0) = exp{ w 8;93+1} is bounded Sentire, but is not a polynomial. 71
In an infinitedimensional case, the compact Sanalyticity does not imply a bounded Sanalyticity. We denote the space of compact Sentire functions f : X + A by A(X) (by virtue of Corollary 5.1, for the superspace X = KK'm this
notation is consistent with that used in Chap. II); the topology in the space A(X) is defined by a system of norms (5.7), where II I is a norm on A, x0 = 0.
Chapter III. Distribution Theory
158
6.
Analytic Distributions on a Superspace over Topological Supermodules
The theory of distributions on a superspace over CSM was constructed in [141, 144, 65]. It was pointed out in Chap. II that this theory was based on two ideas, namely, Alinear functionals are considered instead of Klinear functionals and the theory of distributions is developed over a CSA with a trivial A1annihilator. Only over a CSA A with a trivial A1annihilator can we consider functions of an infinite number of variables (infinitedimensional superspaces over CSM is of the main interest for physical applications).
6.1. Dual superspaces. A superspace X' = Mo ® N1', where M' = MO 'E) M1' and N' = NO ED N1' are CSM conjugate to M = M° ®M1
and N = No ® N1, is a conjugate of the superspace X = M° ® N1. The form of duality between the superspaces X and X' is defined by the relation (m° ® nl, u° ® v') = (m°, u°) + (n', v'),
(6.1)
where ( , ) are forms of duality between conjugate CSM. In contrast to the form of duality between CSM, the form of duality between superspaces assumes values in A0.
Generally speaking, the form of duality does not separate the points of the superspaces X and X' even if the forms of duality separate the points of the conjugate CSM M and M', N and N'.
Example 6.1. Let M = Am, N = An. Then M' = Am, and
N'=An,X=KA'm=X'; m
(m° ®nl, u° (D vl)
n
 Ej=1m°uj + j=1 E nj
m °,u° EAo,
n 1,v l E A. Ai
Let the CSA A be a Grassmann algebra Gk, k < oo. If the number of generators k is odd, then the form of duality does not separate the points of the superspaces X = KA'n , X' = KA'n although the form of
6. Analytic Distributions
duality (a, b) _
d
159
a3bj separates the points of the conjugate CSM Ad
and Ad.
Definition 6.1. The superspaces X = M° ® N1 and Y = RO ® S1 over the pairs of dual CSM M and R, N and S are said to be dual. The form of duality is defined by relation (6.1) with the aid of the forms of duality between the CSM.
Definition 6.2. The dual bases in the dual superspaces X and Y are dual bases in the covering CSM LX and Ly. Recall that we have accepted the notation MI (M')'C, a = 0, 1. In precisely this way must we interpret all corresponding symbols in the subsequent examples.
Example 6.2. Let X = M° ® N1 be a Hilbert superspace and Y = X; {e3; a canonical basis of X. The orthogonality of the basis means that it is dual to itself. The basis lies in M° ® N°.
Example 6.3. Let M = A(CA'm), N = A(C; °), X = M° Ni ®= Ao(CA'm) ® A1(CA9), Y = X' = A°(CiA'm) ® Al(Cr'A9). The basis in the superspace X is formed by the monomials onomials e p (x, 0)
= X419, x c Ao, 9 E Am, e7P(z, ) =
z E Ap, S E M;
f(x,z,0, ) = f°(x,0) +fl(z,O E ffpe,p(x, 0) + E f7Pe7P(z, S), ap
where
7P
I f7Pl = 1  e7Pl'
The basis in the superspace Y is formed by the derivatives if the 5function a
a
ax' ae) 
1 a°+p6(x, 0) a! axaaop
1 a7+P6(z,e) e7P
a
a
a
P (ax, az' 190'
a 111 9C
7!
az7 a °
a
,
a
a
 P (ax' aB) +P
1
a
( az' al;)
Chapter III. Distribution Theory
160
°
,
a
a
1
P°,6e.,c ax(' ae ) +
P7Pe,P
7P
where 1 P.1# 1 = l eQp I , I PP J = 1  l e7P 1.
Ca
al
az a
The form of duality between
the superspaces X and Y is written as (f, P) = (f °, P°) + (f 1, P1)
_ E fpPap + E f7PP7P at3
(E Ao).
7P
Example 6.4. X = E0 (Cnm)
(D El (COQ),
X = E°(Cnm) ®
y = X' = Eo (C m) ®Ei Y = X' = EE(CAm)
Example 6.5. X = 9(RR'm, Ac) ®G(Ru4, Ai),
Y = X'= c'(Ru'm, A') ®g'(R f, A1); X = D(Rum, Ac) e D(Ru°, Ai), Y = X' = D'(Ru'm, Ao) ®D'(Ruq, Ai)
Proposition 6.1. Let X = M° ® N1 and Y = R° ® S1 be dual superspaces over CSM over a CSA A with a trivial A1annihilator. If there exist dual bases in the superspaces X and Y, then the form of duality (6.1) separates the points of X and Y.
Proof. Let {ej, ak} and {e'., ak} be dual bases: ej E M, ak E N, e E R, ak E S (by definition, the bases lie not in superspaces but in covering modules). Then we have x = (x°, ej)ej + E(x', ak)ak j k for x = x® ® x1 E X. Assume that (x, y) = 0 for ally E Y. Then (x°, O c a'.) = 0 for all le = 0 and (xl, ak) = 0 for all jakj = 1. Let A1, and then (x°, e')0 = 0 for all lej1 = 1 and (x', ak)0 = 0 for all IakI = 0. It remains to use the triviality of the A1annihilator.
6. Analytic Distributions
161
6.2. Fourier transform. Let V = Mo ® Ni and W = R® ® S1 be dual superspaces over pairs of dual CSM (M, R) and (N, S) over the Banach CSA A.
For every vector w E W we introduce on V a function f,,(v) _ e`(","). We denote by 'Y(V) a certain CSM consisting of Ssmooth (or Sanalytic) functions f : V + A and containing all functions f,,, w E W. We choose T (V) as a space of test functions on the superspace V ; W' (V) is a space of distributions on the superspace V.
Definition 6.3. The Fourier transform of the distribution of L E 'k'(V) on the superspace V is the function .F(V) on the dual superspace W defined by the relation F(L)(w) = f L(dv) f,,(v). We denote by D(W) the Fourierimage of the space of distributions W'(V). If Ker F = {0}, then we can define the space of distributions on the dual superspace W:
M(W)=III eV(W):IL =µg, 9E`y(V), fco(y)(dy) = f .F1(w)(dx)9(x),
co E
ID(W)j.
Thus, every function g from the space of test functions on V is associated with a distribution p9 on W. The funcion g is called a Fourier transform of the distribution µ9 and is denoted by .i'(µ9). A harmonic analysis arises on a pair of dual superspaces V and W :
`y'(V) 4''(W), 11(V)  M(W). By definition, we have a Parseval equality
f (L)(w)µ(dw) = fL(dv)Y()(v). Then we use the notation of cp for
(6.2)
cp E' (W).
Theorem 6.1 (on the kernel of a Fourier transform on a superspace). Let the covering CSM LV be complete and locally convex.
Chapter III. Distribution Theory
162
Suppose that in the superspaces V and M there exist dual topological bases; A = A is a Banach CSA with a trivial A1annihilator, the space of test functions W(V) = A(V) is a space of compact Sentire functions. Then the kernel of the operator of the Fourier transform .F is zero.
Proof. Let a E Ker.F. Then OO in
> (a, (. no n1
(a,
,
w)n) = 0
for all w E W.
It follows that (a, ( , w)n) = 0 for all w E W, n = 0, 1, ... Let {ej; a;} and {ej; ai} be dual bases in the superspaces V and W. Let ej' I = 0 or I aiJ = 0. Then, setting w = e' or w = ai, we obtain (n = 1): (a( , e'j)) = 0 or (a( , e;)) = 0. Let le'1 = 1 or Jail = 0. Then, setting w = or w = ai0 for any value 0 E A1, we obtain (a(. , e'))9 = 0 or (a(. , ai)) = 0. It remains to use the triviality of the .
A1annihilator.
By analogy, we obtain (n = 2,...): (a,
0
for all jl... jk, il...im If the superspaces V and W are finitedimensional, then the theorem is proved. Suppose that the dual bases are countable. We introduce projectors 00
7rmkv = E (v°, j =M
00
E(v1, a')aj. j =k
Uniformly on compact sets of the covering CSM Lv, 7rmk + 0,
m,k+oo.
Let B c Lv be a compact set. For every absolutely convex neighborhood U in the CSM Lv we can construct a finite Unet of the set 00 00 U U 7rmk (B) = C. Indeed, for any neighborhood U in the CSM Lv m=1 k=1 there exist mo, ko such that 7rmk(B) c U for all m > mo, k > ko. However, the sets 7rmk(B), m = 1, ..., mo, k = 1, ..., k° are compact,
6. Analytic Distributions
163
and therefore there exists a finite Unet for them. Since the space Lv is complete, the existence of a finite Unet for any absolutely convex neighborhood U implies that the set C is compact. Let the form b E IC,a,,(L ', A). Then, for any c > 0 there exists a neighborhood U of zero in L such that sup jIb(v1,..., vn)jj < E. V2 E Cf U
Therefore, for any e > 0 there exist mo, ko such that sup Ilb(7rmkv1, ..., 7f,nkvn)II < E v, EB
lim b(®(1  7rmk)) in the space for m > mo, k > ko. Thus b = m,k+oo 1
A(V). It remains to note that for any function f E A(V) there exists m a sequence of forms {bn}, bn E Kn,r(LV, A), f = lim E bn in the m_+oo n=0 space A(V). In particular, Theorem 6.1 is valid for all superspaces considered in Examples 6.26.5.
Remark 6.1. One must distinguish between two causes for the noninjectivity of a Fourier transform on an infinitedimensional superspace. The first cause for the noninjectivity is not connected with infinitedimensionality. It is the nontriviality of the A1annihilator (see Chap. II). The second cause is not connected with the superstructure and is due to the infinitedimensionality of the space. The conditions of injectivity of a Fourier transform on Klinear infinitedimensional spaces were discussed in the articles [137141, 67]. It was shown there that for a Fourier transform to be injective, a weaker condition was sufficient, namely, the fulfilment of the approximation property [71]. The approximation property for superspaces can be defined as for Klinear spaces; an identity operator can be approximated uniformly on compact sets by finitedimensional Alinear operators. Everywhere in what follows, we consider the theory of generalized functions in which the space of test functions %P(V) = A(V); V and W are complete locally convex superspaces with dual bases, and the A1annihilator is trivial.
Chapter III. Distribution Theory
164
Proposition 6.2. The functions from the space Z(W) are compact Sentire. Proof. Every function f E 1>(W) has the form 00
in
f (w) _n=E ?n1 (L, (,
)n),
where L E A'(V). Let B,, be a compact subset of the CSM Lw. Then 11f
fIIBW =
00
1
SUP II(L, (.,wl))...(.,wn)II.
n=0 n' w, EBW
However, since L is continuous, there exists a compact set B in the CSM Ly such that for any form b E 1Cn r (LV, A) we have II(L,bn)II 0. Then the function 00
00
E E Lil...in(ril
1
Wl)...(rin, Wn)
n=0 n. 71...in
belongs to the space (D(W).
Proof. We introduce a functional 00
E n=0
00 1 ni
anb(y)
,
jl...in
a;
is a generalized derivative of the sfunction in the direction of rj. Let us prove that L E A' (V ) where
II(L,.f)II (W). This integral is a strict mathematical definition of a Feynman path integral for a spinor field [7, 50, 53]:
fU(b,ii)exp{ifR
(x)(za  m) V) (x) d4x} II dV) xER4
(in this symbolical notation the determinant of the operator (ia  m) is included in the normalization of the functional differential
II d'(x)di/i(x)) xER4
Example 7.2 (Feynman distribution for a chiral superfield [123]). In the superspace Ru4 we consider a SUSY transformation (cf. Chap. I (1.8)):
9'=0,+EQ.
xµ=xµ+2EyµO,
The generators of the SUSY transformation a i3 S.=89°+2
3
a
(ryµ9)ax
µ=0
µ
where e = C10, C is a charge conjugation matrix, satisfy the commutation relations IS., Sa} = i ('y C).a aiµ . The operators of the µ=0
covariant differentiation on the superspace D°
3
a
a
E (fµe)° ax = ae  2 µ=0 µ
are invariant relative to the SUSY transformation; they satisfy the commutation relations 3
[D., Da}
i E (1'µC).0 8 µ=0
µ
Chapter III. Distribution Theory
170
We !introduce spaces of chiral scalar neutral superfields A): (1 + 75)1)(p± (x, 9) = Q.
9(R4,4
Gt(RA4, A) = {cPf E
,
We set
V=
9+(R4,4
,
w=
A) ®G_
(R4,4
,
A),
A) ®G'
A).
The form of duality between the superspaces V and W is defined by the relation
(v, w) = f ( DD) 2
(v+w+ + vw_) d4x.
We denote by 'yss the Feynman distribution on the superspace W which has zero mean and covariance functional
f[1(v)2(v + a 2 + M2 + io v) 1
Bss (v, v)
(DD) V+
1/2M 1/2M d4x' + v_ v)] v a2+M2+io a2+M2+io +
M>0,
a=(9XA a )3 µ=o
For any function Z E O(W), the Feynman integral f Z(w+, w)'yss(dw+dw) is defined which is a strict mathematical definition of a Feynman path integral for a chiral neutral superfield [123]:
f Z(w+, w) exp{i f[D)2(w+w)
1(DD) (DD) (2M(w+ +)]}
II
dw+(x, e)dw(x, O).
(z,O)ER '4
Example 7.3 (interaction of a boson and a spinor field). We set V = G(R4, RA8), W = G'(R4, RA8) and denote by rySB the Feynman
7. Gaussian and Feynman Distributions
171
distribution on the superspace W with a covariance functional BSB = Bs + BB, where BS is a covariance functional for a spinor field,
BB(V, V) = i fR8 cp(x)D`(x  y)co(y) d4xd4y
is a covariance functional for a scalar boson field, where D`(x) is Green's causal function for the KleinGordon equation D`(p)=p2_m2+io,
f u((P,.,, ) x exp{i fR4
f `
fi(x)
x 11
(
2
2m
)W(x)
+ (x)(ie 
f u((P,'),0)'YSB(dcpdV)
xER4
U E 1)(W).
Remark 7.1. It should be emphasized that with our approach to the superanalysis, a boson field assumes values not in the field of real numbers R but in the even part of the algebra A. The simplest example of the Gaussian distribution on a superspace is a distribution on R`n,,m (see Chap. II). Its infinitedimensional generalization is given by the following example.
Example 7.4 (Gaussian distribution on a Hilbert superspace). Suppose that M and N are Hilbert CSM, V = Mo ® Ni , W = V. We denote by y the Gaussian distribution on the superspace W with a covariance functional B(y ® ®t, y ®C ®) _ (y, y) + 2(e, e). Then, by definition, we have
f U(x,9,6)exp{2(x,x)  (9, 6)}dxdOd9 f U(x, B, 6)'y(dx dO d6).
Chapter III. Distribution Theory
172
Let, for instance, V = 12(A0) ®12(A1) ®12(A1). Then 00
ry(dxdOdB)
=exp{1 Exj 29=1
00
E0363} 11 00 dxjdOjd#j. 7=1
9=1
If V = L2 ° (Rn, dx) ® L21(Rn, dx) ® L21 (R n, dx), then
ry(dcp dib dpi)
= eXp{
2
1R^ 1P 2(x)dx  fR' i(x)0(x)dx}
x II dcp(x) dz/'(x) di'(x). xER^
7.3. Formulas for integration by parts and for an infinitesimal variation of a covariation. In the theory of Gaussian measures on infinitedimensional spaces an important part is played by formulas for integration by parts (an extended stochastic integral [26, 6], Malliavin calculus [113, 6]). The simplest formula for integration by parts for the Gaussian measure 'YO,B with a covariance operator B is f (cp'(x), Ba)ryo,B(dx) = f (p(x)(a, x)'YO,B(dx).
(7.4)
For Feynman integrals, formula (7.4) was obtained in [53, 126, 145].
The formula for integration by parts on a superspace differs essentially from formula (7.4). When integrating by parts, in a quasiGaussian integral another integral appears which is not quasiGaussian in addition to the standard quasiGaussian term (see [151]). Let T be an associative Banach algebra with identity e. For t, s E T we set
exp{t; s} = e +
tksnk 1)i
n=1 (n + k=0 This function is twoparameter generalization of the exponent:
exp{t; t} = exp{t}.
7. Gaussian and Feynman Distributions
173
For the forms c, b E 1C2,1(L2 , A), we denote by x(c; b) the distribu
tion belonging to the space M (W) and having a Fourier transform .F'(ic(c; b)) (v) = exp{2c(v, v); 2b(v, v)}.
For the bilinear form b E 1C2,1(L2 , A), whose restriction to the superspace V2 is symmetric, we introduce a bilinear form b_ (v, v') = b° (m e
n, m'en') bl(men,m'en'), where v = m®n, v' = m'® n' E Lv. For the restriction of the bilinear form b_ to the superspace V2 we have a relation b_ (v, v) = a(b(v, v)),
v c V.
For instance, suppose that we are given a superspace CA,m and a form
m
m n
b(v, v') = i,j=1
mimjaij + E Dminj + njmi)Qij i=1 i=1 n
+ I (ninj  njni)7ij.
(7.5)
i,j=1
Then the form b_ has coefficients a(aij), a(Qij), a('Yij) We also introduce a diagonal and an antidiagonal form a+(b)(v, v') = b(m, m') + b(n, n'),
a(b)(v, v') = b(m, n') + b(n, m'). For the bilinear form (7.5) we have a+(b) (v, v')
m
n
= > mimjaij + > (ninj  njni)'Yij; ij=1
i,j=1
m rn
a (b) (v, v') =
njmi) i=1 j=1
In what follows, we shall use the notation ryb to denote a quasiGaussian distribution with zero mean and a covariance functional b and the notation Kb to denote the distribution ic(b_, b).
Chapter III. Distribution Theory
174
Theorem 7.1 (formula for integration by parts). Let the function cp belong to the class 1(W) and let the vector a E Lv. Then f co(y) (a, y)'yb(dy)
f [a+(b°)(a, aR) + a (b1)(a, aR)] (co) (y)yb(dy)
+ f [a (b°)(a, aR) + a+(b1)(a,
(7.6)
Proof. Note, in the first place, that for arbitrary m E M and n c N the functional p = Ann: A(V) + A, P(f) = i
8x°
(0) (m) + 8x1 (0) (n)
is continuous.
Furthermore, (a, y) = F(p)(y), a = m ® n. Thus, we have fw co(y) (a, y)'yb(dy)
 fv
fv
= fc*p(dv)exp{_b(v,v)}
p(dvl) exp{
2b(v + vi, v + vi)}
We set gv(vl) = [b(v + v1i v + v1)]n, and then, for m° E M°, we have 8x°v (0)(m°) =
> 2bk(v, v)b(m°, k=0
v)bnk1(v,
v);
l
similarly, for n1 E N1 we have aLgv
8x1
n1
(0)(n1) = E 2bk(v, v)b(nl,
v)bnk1(v, v).
k=0
Furthermore, b(m°, vo ® v1) = A° ®A1 E A° ® A1, where .1° _ b°(m°, v°) + bl(m°, v'), Al = b°(m°, v') + bl(m°, v°); similarly, b(v° ®
7. Gaussian and Feynman Distributions
175
v1, v° ® v1) = a° ®a1 E Ao ® A1i where a° = b°(v°, 0°) + b°(v1, v1) + 2b1(v°, v1) and a' = bl(v°, v°) + bl(vl, v1) + 2b°(v°, v'). Therefore we
have b(v, v)b(m°, v) = (a° ® a')(\° ® A1) = )°b(v, v) + Alb (v, v). Similarly, for the vector n1 E N, we have b(v, v)b(n', v) = µ°b(v, v) + p'b_(v, v), where µ° = b°(n1, v1) + bl(n1, v°) E Ao, µ' = bl(nl, v1) + b°(n', v°) E A,. Employing these formulas, we obtain q9a 8x°v (0) (mo) = 2n\°[b(v, V)] n1 n1
+2)1 >
[b_
(v, v)]k [b(v,
v)]nk1 ,
k=0 19
2nµ°[b(v, 8 v (O)(n1) = n1 +2µl E [b_ (v, v)]k[b(v,
v)]n1
v)]nk1
k=0
Let us calculate the integral with respect to the distribution P= Pmonl E A'(V):
f p(dv,)exp{2b(v+vl,v+vl)} 1 2
2n 1)n! (n[b(v,
v)]n1(A° + µ0)
n1
+(A' + µ1) L. [b(v, v)]k[b(v,
v)]nk1)
k=0
= i(A° + µ°) exp{2b(v, v)}
+01 + µl) exp{2b_(v, v); 2b(v, v)} Finally, for the vector a = m° ® n' E V we have
f
w (y) (a, y) yb(dy)
Chapter III. Distribution Theory
176
= if cp(dv)[b°(,m°, v°) + b'(rn°, v') + b°(nl, vl) +b1(nl,v°)]eXp{Zb(v,v)}+if cp(dv)[b°(m°, v1) + bl (m°, v°) + bl (n', v') + b° (nl, v°)] x
exp{2b(v, v); 2b(v, v)}.
Extending relation (7.6) in the left Alinearity from the superspace V to the covering CSM Lv, we get formula (7.6) for the vectors a c Lv.
Corollary 7.1 (cf. (7.4)). Suppose that the function cp belongs to the class '(W), the vector a E Lv, and the restriction of the covariance functional b to the superspace V2 is evenvalued. Then f cp(y) (a, y)'yb(dy)
= f b(a, aR) (cP) (y)'yb(dy)
Example 7.5. Consider the Gaussian distribution on the superspace Ai: 'yb(d9) = exp{01027  0304/3} dO 2p 1,
where ry = 'yo +'y', /3 = 00 + 01, /3,, y, E Aj, j = 0,1; there exist /30 1, 7o 1; p = 'y/3 + /3ry. Then (see Example 3.5 Chap. I)
('Yb)( ) =
f
exp{66A 
e)},
where A = 2/3p1, B = 2ryp1. Consequently,
exp{2b(e,
_b(,)}
= e  4 [b (e, e) + b(e, e)] + +b_ (e,
4!
[b2
b(e, ) + b2 (e, )] = e  [e11;2Ao + 3&4Bo]
7. Gaussian and Feynman Distributions
+1
177
r(B)Q(A) + a(B)A + cr(A)B
+AB + BA] = e  66A0  U4B0 + exp{S1C2Ao
 U4Bo}.
Thus, in the example that we are considering 'b is a Gaussian distribution kb(d9) = exp{9192Ao 1  0304Bo 1}d9AoBo.
Note that
A=(7o1010o1'Y17o2)y1'Yo2=Ao+A1, B=(0011'1'Yo1Q1Qo2)01002=Bo+B1, AO 1
= ('Yo
1
 0100
1'Y1'Yo 2)
1 = 'Yo + Q1 Q0 17'1,
Bo 1 = (00 1  'Y1'Yo 1Q10o 2) 1 = 00 +'Y1'Ya 101
Let us calculate the integrals I1 = Jb0(,aR)()(o)yb(do); 12
_ I1 =
f
= f b1(C,a.)((P)(0)r1b(de),
_ ae2) a ae 2)
aR
We1
+ f Lf aRae4 A3) _
09Rae3 4)
AO exp{0102ry  93940}d9 2p Bo exp{9192ry  93940}d9 2p1
= Ill + 112.
Using the formula for integration by parts in the integral with respect to anticommuting variables (Chap. II, formula (3.27)), we obtain 111 = f (o(9) [e1Ao aL a91
19L
exp{0102'y  9394Q}
a02
exp{0102Y  93940}] d9 2p1
Chapter III. Distribution Theory
178
= fw(O)[eioi +e292]Ao('Y 
0304P)d02p1.
By a complete analogy, we get 112 = JP(0)[e303 + e404]Bo (N  0102p/2)de 2p
1.
In the same way we can calculate the integral 12: 112 = f (P(e)[e181 + e2e2]A1(Ao 1  0304Ao 1Bo 1) dOAoBo,
122 = fw(9)[3O3 + e484]B1(BO 1  0102Ao 1BO 1) dO AoBo.
Consequently, Il + 12 = f W(e)[e1e1 + e202]
X [(Ao7 +
A1Bop)
+ f cP(0)[6 03 + G O4] [(Bo1 +
2p1
 9304(Ao + A1) 2] dO B12 Ao
 0102(Bo + B1) 2]
d92p1
Furthermore, Ao'y + Al Bop 2
11"Yo 2/30
= ('Yo 1
 010o 1'Yl'Yo 2) ('Yi +
1(1 _ 'Yl'Yo 1Ql/3o 1)(/3o'Yo + 01'Yo +'Y1Qo) = 1; AP 2
Similarly, Bo/3 + B12 ° = 1,
=
2QP1P = Q. 2
= y. Thus we have
I = f(9)[ei9i + e202](1  00384)dO +
YO)
f
W(9)[6O3+5484](1'YB182)dO
2p1
2p1
f (P(e) [ 01 + e282] (1  01027  0304Q + 101e2B3e4P) dO
2p1
7. Gaussian and Feynman Distributions
179
+ f cP(9)[683 +e404](1  0102'Y  0304/ + 1 01 02
0304P)d02p1
= fco(0)(, 0)yb(d9).
It is easy to give an example showing that for the distribution 'yb(dO) considered above the standard formula for integration by parts (7.7) is not valid.
Example 7.6. Let the function cp(0) = 029394. Then
f w (0) (e, 0)yb(d9) = f b(e, aR) ((P) (0)yb(d9)
f ei01 02 93 04 7b(d9)
= 2j1P1i
= f e1A91029394y d9 2P 1 =
(7.8) (7.9)
If (7.8) = (7.9) for all 1 E A1i then Ay = 1 since the A1annihilator
is trivial. Let y = 1 + yl, Q = 1 +,31. Then Ay = 1  01y1 j4 1 if Q1y1 j4 0.
For countably additive Gaussian measures on Rlinear spaces, the following formula of infinitesimal variation of covariance is valid (see, e.g., [24]): dt f co(y)'Yb(t) (dy) =
2
Jb(t)(a,o)(2)(Y)Yb()(dY),
(7.10)
where yb(t) is a family of Gaussian measures with covariance function
als b(t), t is a parameter, b(t)  dtt Just as the formula for integration by parts, formula (7.10) for an infinitesimal variation of covariance cannot be directly extended to a supercase. The following theorem can be proved by analogy with Theorem 7.1 (see [151]).
Theorem 7.2 (infinitesimal variation of a covariance). Suppose that the superspaces V and W are Banach and b: AO + £2,1(L,, A) is a continuously Sdiferentiable function. Then, for any function cp from the class '(W), we have a fcQ(Y)7b(t)(dY) =
2
[f (bo (t) (acx, a.0)
180
Chapter III. Distribution Theory +b° (t) (a', aR) + 2b' (t) (aR, aR))co(y)7b(t) (dy)
+ f(bl (t) (aR, a°R) + bl (a', aR) + 2b' (t) (aRl, aR))'(y)nb(t) (dy)]
.
Corollary 7.2. Suppose that the conditions of Theorem 5.2 are satisfied and the restriction of the covariance functionals b(t) to the superspace V2 is evenvalued. Then formula (7.10) holds true.
8.
Unsolved Problems and Possible Generalizations
In this chapter we outlined the main directions of development of analysis on a superspace over CSM. We hope that this analysis will be successfully developed. In general, an infinitedimensional analy
sis has much in future. We think that with the aid of an infinitedimensional analysis and, in particular, the theory of distributions on infinitedimensional spaces, we shall be able to expose, on the mathematical level of strictness, the quantum theory of a field and a string outside of the framework of the perturbation theory; probably, some other infinitedimensional objects will appear in physics. We observe a standard situation where, along with infinitedimensional bosons commuting coordinates (boson fields, strings, string fields, membranes) there are also infinitedimensional fermion coordinates. The infinitedimensional superspace X = M°®Nl over a pair of CSM M = M°®M1 and N = No ® Nl arises in practically all quantum models. The ordinary infinitedimensional analysis developed during the last hundred years, beginning from the works by Volterra, Frechet, Danielle, Wiener, Levy, Gateaux, Hadamard (see [22]) and following to the works by Gross, Fomin, Smolyanov, Berezanskii, Daletskii, Hida, Uglanov, Khrennikov, Shavgulidze, Bogachev [2, 6, 26, 5455, 62, 64, 6568, 96, 127130, 133, 134]. It is natural that in the framework of this book we cannot propose as well developed infinitedimensional superanalysis, the more so that a considerable part of the book is devoted to finitedimensional super
8. Unsolved Problems
181
analysis. Therefore a wide range of problems remain unsolved (many problems may give rise to whole theories). Topological supermodules. 1. Theorems of the type of HahnBanach and KreinMil'man theorems.
2. Topologies on conjugate CSM. 3. Superanalog of Mackey topology. 4. Reflexivity theory for locally convex CSM. 5. Weak topology Q(M, M'). Weak compactness. 6. Topological properties of spaces of test and generalized functions on a superspace.
7. Unbounded operators in Hilbert CSM: selfadjoint operators, unitary groups, Stone theorem. 8. Theory of semigroups of operators in Banach and locally convex CSM.
9. Operators of trace class and HilbertSchmidt operators in Hilbert CSM. 10.
Nuclear locally convex CSM. Superanalog of Grothendieck
theory. Sdifferential calculus. 1. Successive exposition of differential calculus on topological and pseudotopological superspaces.
Distribution theory. 1. Theories of nonanalytical superdistributions. 2. Existence theorem of a fundamental solution for a linear differential operator with constant coefficients on an infinitedimensional superspace. 3. Cauchy problem for linear differential equations with variable coefficients on an infinitedimensional superspace. QuasiGaussian distributions. 1. Extension of the class of integrable functions of an infinitedimensional superargument. 2. Formulas for integration by parts and an extended stochastic integral.
182
Chapter III. Distribution Theory
Remarks Sec. 2. These results were published in [65, 166]. Sec. 3. These results were published in [166]. Hilbert modules over C`algebras were introduced by Paschke [116]; in connection with the applications to the theory of pseudodifferential operators they were studied by Mishchenko [48]. The main differences between the theory of C'modules
and the theory of supermodules are generated by the differences in the properties of Banach algebras over which these modules are considered. All proofs of the theory of C'modules are based on positive linear functionals, in superanalysis these methods are inapplicable. Sec. 5. A superspace over a pair of CSM was introduced in article [144]. The Sdifferential calculus on these superspaces was developed in [65, 68, 148].
Sec. 6. Here wide use was made of the methods of infinitedimensional analysis. Actually, the results of the works [136141, 145, 67] were extended
to the supercase. In turn, these works were based on the investigations of Fomin, Smolyanov, and Uglanov concerning the theory of distributions on infinitedimensional spaces.
Sec. 7. Uglanov was the first to define Feynman's "measure" as a distribution on an infinitedimensional space. I have done this for a supercase.
Chapter IV
Pseudodifferential Operators in Superanalysis In this chapter we expose the theory of PDO on a superspace over topological CSM. These superspaces can have a finite as well as infinite number of supercoordinates. Thus, the proposed PDO calculus serves as a mathematical basis for the quantization of physical supersystems with a finite as well as infinite number degrees of freedom. In a finite
dimensional case, we obtain quantum mechanics on the superspace and in an infinitedimensional case we obtain a quantum theory of a superfield, in particular, that of a superstring and superstring field, and fermion theories and boson theories with anticommuting FaddeevPopov ghosts. Only the first steps have been made in the PDO theory on a superspace. In [65, 68, 153] I constructed a PDO calculus (composition formulas,...), proved the correspondence principle, investigated evolutionary pseudodifferential equations. No considerable results concerning a superspace (even for a finitedimensional one) are available in many important branches of the PDO theory (such as, for instance, a parametrix, spectral properties).
1.
Pseudo differential Operators Calculus Let us begin with considering PDO on a space R". Recall (see,
e.g., [73, p. 178]) that a PDO a in a space of functions on R' with
Chapter IV. Pseudodifferential Operators
184
the Tsymbol a(q,p) is an integral operator
a*')(q) = f a((1  T)q
+Tq',p)(P(q')e`(qq',P')
(2)
For T = 0, 1, 1/2 we obtain qp, pq, and Weyl symbol respectively. Let us now consider the case of an infinitedimensional Klinear space (Hilbert, Banach, locally convex). In an infinitedimensional
case, the Lebesgue measure dq dp is absent, and therefore PDO are introduced either as limits of finitedimensional PDO [87] or by proceeding from polynomial operators [3], or, else, with the use of the distribution theory on infinitedimensional spaces [74, 127, 129, 136, 139, 141] (on the mathematical level of strictness, the first variant of PDO calculus in the framework of the distribution theory on infinitedimensional spaces was proposed in [127]). In my works [139, 141] I introduced a Feynman integral on a phase space and defined an infinitedimensional PDO by the relation
a(W)(q) = f a((1  T)q +Tq',P)W
(q')e'(qq'P'1dq'dp',
(1.2)
where the symbol ei(qq'P')dq'dp' was used to denote Feynman distribution on a phase space (the same formulas as (1.1), but the normalizing factor 1/(27r)°° was "driven in" the Feynman distribution).
I have constructed spaces of functions of an infinitedimensional argument which possess a remarkable property, namely, every formula of the PDO theory in R" is also valid for infinitedimensional
PDO in these spaces with a replacement of the complex measure e`(9q',P'P)dq'dp',
a E C, by a Feynman distribution on a phase space. In [144, 146] I introduced a Feynman integral over a phase super
space and defined the PDO a with the Tsymbol T E Ao, a(q,p) E O(Q X P) by relation (1.2). This definition of the PDO is used in the sequel.
Let P and Q be dual superspaces satisfying the constraints imposed
in the process of construction of the distribution theory (see Sec. 4, Chap. III). The superspace Q x P is known as a phase superspace. The superspaces Q x P and P x Q are dual.
1. Pseudodifferential Operators Calculus
185
We set b.(p®q,C(D 77) = 2Z [(p°,77°)+(q°,C°)]
aEAo,p,eEP,q,r7EQ. The form b,, is Alinear both on the right and on the left, continuous on compact sets, and symmetric. We shall denote by i
f co(q',p) exp {(q'  q, p'  p)} dq'dp an integral with respect to Feynman distribution on a phase superspace with mean a = q ®p and covariance functional (2b,,). It is this symbol that is used in definition (1.2) of PDO. Theorem 1.1. Every PDO a with a 7symbol a E (D(Q x P) is a right Alinear operator in the CSM (D(Q).
Proof. For every q E Q, we introduce a Alinear continuous operator Sq: A(P x Q)  A(P2 x Q) by setting Sq(f)(P1,P2, q') =
e'(1'r) (Pi,q)f
(7pi + p2, q').
We can present the integrand Vq(q', p') = a(7q' + (1  7)q, p')cp(q') in (1.2) (q plays the part of a parameter) in the form z/)q(q', p') = T ((d 0 () o Sq)(q', p'). Consequently, this function belongs to the space 4)(Q x P). Thus, the operator a is defined on the whole space
(Q)

We shall show now that a(,p) E ID(Q) for any cp E O(Q). Using the
Parseval equality (5.3) from Chap. III, we obtain
a(co) (q) = I a 0
cp(dpdq'dp")
x exp{iT(p', q') + i(p', q) + i(p", q' + q)}.
(1.3)
We introduce a Alinear continuous operator S: A(P) + A(P2 X Q) setting S(f) (p', q', p") = eir(P',q')+i(P",q) f(p' + p"). Then we can represent the function &(W) (q) as
a(W)(q) = F((a 0 ) o S)(q) E (D (Q)
Chapter IV. Pseudodifferential Operators
186
Theorem 1.2. Let a be a PDO with a 7symbol a E 1>(Q x P). Then we have a representation dgi)e'(Pl,a+'rq1 )W(q
f a(dpi
+
qi).
(1.4)
Proof. Using relation (1.3), we obtain et ((p) (q)
(P(dp2)e''2,q+qi))
= f a(dpi dgi) (f xe=(pl,q)+iT(pi,gi) = (1.4).
Let {ej, ai}jEJ,iEI and {e'j, a=}jEJ,iEI be dual bases in the superspaces P = Po ® Pl and Q = Qo ® Q1 consisting of even elements
p = Ep°ej +>piai, jEJ
q=
jEJ
iEI
q°ej +Eq'ai iEI
We introduce the (left) operators of the coordinate and momentum corresponding to the resolution with respect to the bases P°, 4°, j E J; jii, 4ii, i E I. These operators satisfy the canonical commutation relations on a superspace (which coincides with (1.7), Chap. I in a finitedimensional case).
Theorem 1.3. Suppose that a is a PDO with a rsymbol a E c(Q x P) and the superspaces P and Q have even dual bases. Then, for any function cp E 1(Q) we have relations a(cp) (q) = fa(duciv)
x exp{i >(Tu°v° + u°4°) + i (TUkv1 + uk4Lk) } jEJ
kEI
x exp{i > v°p° + i jEJ
vk7Lk}w(q); kEI
a(cp)(q) = fa(dudv)
(1.5)
1. Pseudodifferential Operators Calculus
187
x exp{i E((7  1)v°u° + vjp°) + i E((T  1)ukvk + vkpLk) } jEJ
kEl
x exp{i E(u°4° + i E(ukgLk}W (q); jEJ
(1.6)
kEI
&(W) (q) = fa(dudv)
x exp{i E((T  1/2)u°vjo + v°pj + u°4jo) jEJ
+i E((T  1/2)2Gkvk + ukgLk +
vk11
PLk)}co(q)
(1.7)
kEI
Theorem 1.3 is a direct corollary of representation (1.4). Formula (1.5) is considerably simpler for the qpsymbol, formula (1.6) is simpler for the pqsymbol, and formula (1.7) is simpler for Weyl's symbol.
Example 1.1. Let P = Q = Al, and then 4i(cp)(q) = qw(q), pL(c')(q) = i5LCo(q). Let the function a(q,p) = qp, and then a = 4p for the qpsymbol, a = p4 for the pqsymbol, and a = z (4p  p4) for Weyl's symbol (i.e., Weyl's symbol with respect to anticommuting variables leads to antisymmetrization). Theorem 1.4 (on the relationship between the symbols for PDO). Let at and a, be, respectively, t and ssymbols of the class D(Q x P) of PDO a and let t, s E Ao. Then
at(P,q)= f a,(q',p)eXp{t
(q'q,P P)}dq'dp.
Proof. Using formula (1.4), we obtain h(cp)(q) = f a,(dp'dq') exp{i(s  t) (p', q')} x exp{i(P', q) + it(p', q')}W(q + q'), i.e., at (q, p)
= fas(d7idq')
(1.8)
Chapter IV. Pseudodifferential Operators
188
x exp{i(s  t)(p', q') + i(p', q) + i(q', p)} _ (1.8).
Theorem 1.5 (composition formula). If a, al, a2 are PDO with Tsymbols (T E Ao E {0,1}), a, al, a2 E (D(Q x P), and a = al o a2, then a(q,p) = fai(q',p')a2(qh',p") x exp{
1
T
(q  q", p  p) + T (q  q', p  p') }dq'dp"dq"dp'. (1.9)
Proof. Using formula (1.4), we obtain et (W) (q)
=
f a1 ®a2 (dp dq'dp"dq")
x exp{i(p',q+Tq') +i(p",q+q'+Tq'")}cp(q+q'+q") =
f(ai ® a2) o B(dpdq')e=(P'q+Tq')W(q + q'),
where B is a Alinear continuous operator defined by the relation
B(f)(p,q,p",q") = exp{i(1 r) (p",q) xf(p'+p',q'+q"),
 iT(p',q")}
B:A(PxQ)+A(P2xQ2).
Using formula (1.4) once again, we have a(q,p) = f (al ® a2) o B(dpdq')e'(P',q)+i(q,P)
= f al 0
a2(dp'dq'dp"dq")
x exp{i(1  T)(p", q')  iT(p , q") + i(q' + q",p) + i(p' + p", q)} _ (1.9).
Passing to the limit in relation (1.9) as T 3 0 (T
1), we obtain
composition formulas for the qp (pq) symbols:
a(q, p) = f ai (q, p)a2(q',
p)e'(q'q,P'P)dq
dp,
(1.10)
1. Pseudodifferential Operators Calculus
189 p)e'(q'q,P'P)d9
a(q,p) = f a 1 (q', p) a2 (q,
(1.11)
dp'.
Theorem 1.6. Let a be a PDO with a qpsymbol a E 4)(Q x P). Then, for any function cp E 4)(Q) we have (dp).
a(W)(q) = f a(q,
(1.12)
Proof. We set bq(p) = a(q, p) (where q plays the part of a parameter). Then a(W)(q) = f bq(P)ip(9
)_i('
f bq(P)(f ei(P",q')c(dp"))ei(q'q,P')dgdp
ff
=
bq(p')ei(q'q,P')+i(p",q')dq'dp'(
(dp")
(we have used the supercommutativity of the operation of direct multiplication of distributions and the fact that a Feynman distribution on a phase superspace is even). Consequently, et (W) (q) = f (f bq ®(SPii (dq'dp') x e'(P'q')+i(P',q)) gdP 1)
= f (f bq(dq')e'(P",q'))ei(P",q)( (dp")
=
f (f
bq(d4)e'(q',0r(P")))ei(P",q)c(dp
)
From methodological point of view, it is useful to consider formula (1.12) by way of a simple example.
Example 1.2. Let P = Q = A1i a(p) = p, W(q) = aq, a E A. Then a(cp) (q) = a(a). Recall that every functional u E A' is associated with two functionals ur = Ir (u) E Gl,r (A, A) and ui = Ii (u) E G1,j(A, A) and that ui = I (ur) = u° (Dur o a. For the functional gyp, the right Alinear realization has the form cpr = 228W Indeed, .
esPl)
= (a Z
app)' eiP9)
= a((S(p), aa g) _ W(q)
Chapter IV. Pseudodifferential Operators
190
Note that the generalized function
al OR6(p)
0
Pr Therefore
2
ap
i
I
cpt = (APT) =
app
is odd. Consequently,
ao M(P)
1
cp, = i
,
ap
al ORb(P) ® ao 19R6(P) o Q.
ap
i
ap
i
Thus we have
f 1) =
al i
aRJ(P) C
ap
f) +
ao 'M(P) (
ap
a0 d (o) + iaO ap
,
a(f))
(o)
i ap
Using the formula for the transformation from the lefthand derivatives to the righthand ones (Chap. I, formula (1.6)), we have
(f7 (A) =  (al
a ap(f)
ao
(0) +
a f (0)) _ 
(o) te(a)
Consequently, aLb(P) cr(a) ap
i
,
in this case,
W(4) = (&
a (P) o (a) ). ,
P
Furthermore, f a(o(p))e'P9c (dp) =
(pe`P4,
a a(P) a(a)) _ _(a) P
Theorem 1.7. Let the function a E A(Q x P). Then the PDO a defined by relation (1.12) maps the space (D(Q) into A(Q).
Proof. Consider an arbitrary function b E A(Q x P) and a p c A'(P). Let us estimate the norm of the function
functional
1. Pseudodifferential Operators Calculus
191
g(q) = f b(q, p)p(dp) in the space A(Q). For the arbitrary compact subset BQ in the CSM LQ we have 11
ro(0, p) (hl, ..., hn)P(dp)II
sup
I19I1BQ = 00
f
n=0 n. h, EBQ
q
1
00
00
GCPLn n=0
m=0 7n !
'
an+'nbb
x sup sup II f L agnapm hj EBQ u, EBp
0) (hl, ..., hn, ul, ..., um) I,
where Bp is a compact subset in the CSM Lp which exists by virtue of the continuity of the functional J. Using this estimate, we have k
00
I1911BQ
CP E k=0
x
sup V3 EBQxBp
IIaL9(0)(v1,
kl
lE Cn n=0
vk)II a.
The space W(A0) is isomorphic to the space of
infiniteorder differential operators 00
{P(a) &
=
E 0)
n=0
Pn6(n)(t): IIPIIP
= E IlPnlln!pn < C)0' p > o}. n=0
3. FeynmanKac Formula
217
Theorem 3.4. Let the symbol h(q, p) belong to the class E(CA ,2m). Then there exists a unique generalized solution of the Cauchy problem (3.3), (3.4), the solution of the problem
f a(t, q, p)cp(t) dt + h(q, p) *
a(t, q, p)cp(t) dt = 0,
J
fa(t,q,p)o(t)dt= 1. The solution is defined by relation (3.5). Proof. It suffices to show that series (3.5) converges in the space of generalized functions W'(A0, A(C2 '2m)) if the symbol of the PDO belongs to the class E(C2 '2m). Since the topology in the space A(C2 ,2m) is a topology of coefficientwise convergence for an expansion in terms of anticommuting variables, we can restrict the consideration to a purely
commutative case. We can assume, without loss of generality, that n = 1. Since the functional h E A'(A2), there exists R > 0 such that 11
f h(dpdq)f (p, q) I < Ch max
Iz1I,Iz21qj+i 1 M11(Sk) P'(Sk1)II k=1
n
= Sup IE CYk(Pi(Sk)  P(sk1)) II0kII!S1 k=1 n
sup I\I ak(q',  /y Y
IkakII!5l
k=1
n
< 1011 sup I ak(gk  k_1)1100 11001 Cn,,m" we introduce a mapping j,:Un,f,m, U(W), co y wo7r. The CSM U(W) is endowed with a topology of an inductive limit of the family j.r}, 7r E 11 (W). By j,., 7r E II (W), we denote an operator from U'(W) into Un',r ,n,r which is an adjoint of 3a.
Definition 2.1. A cylindrical distribution on a superspace W is a Alinear continuous functional on the CSM of cylindrical functions U(W) (i.e., the element µ E U'(W)).
2. Random Processes on a Superspace
241
Definition 2.2. Finitedimensional distributions of a cylindrical distribution p is a collection of distributions {A r = jrp}, 7r E II(W) on finitedimensional superspaces.
Theorem 2.1 (the condition of consistency of finitedimensional distributions). Let µ be a cylindrical distribution on a superspace W, the projector 7r E II(W), and let A be a Alinear operator from. An+minto A'+' which maps the superspace Cnw,m r
into C. Then, for any
function cp E U1,3, we have
f
cp o A(w)p.,,(dw) = f cp(w)pAO,(dw).
Cnir.mx
(2.1)
CA A
A
Conversely, let the family 7r E II(W), p E Un,,,nw satisfy the condition of consistency (2.1). Then {µ,} is a family of finitedimensional distributions of the cylindrical distribution p defined by the relation
ff(w)p()= f co(w)µff(dw),
f= W o rr.
Cner.m, A
Remark 2.1. If the algebra A = R, then condition (2.1) coincides with the condition of consistency of finitedimensional disributions for cylindrical measures on locally convex spaces. 2.2. Cylindrical random processes with continuous tra
jectory. We set SZ = {x E C([O,1], Cp,°): x(O) = 0}. This is a Banach superspace with a norm JJxJJ. = max 11x(t)II. We set o 0 is called a Wiener process on the superspace C" m. If the soul of the CSA A is quasinilpotent, then the results of Secs. 3.5 and 4.5 from Chap. I are applicable and we can write out finitedimensional distributions of a Wiener random process.
Suppose that a finitedimensional projector 7r on the superspace Q is defined by points, i.e., 0 = to < t1 < ... < ti, 7r = 7rtl...t,, 7rx = (x(t1), ..., x(t1)) E CA'21k. Then we have 11
µa(dx) = [((27r)n sdet M)1 fl (tj+1  tj) n2k F
12
J
j=0
x exp{9
(xj+1  xi,
M1(xj+1
 xj))/(tj+1  tj)}dx.
j=0
Note that if n = 2k, then the time disappears from the normalizing factor.
Chapter V. Probability Theory
244
2.6.
Representing the solution of the heat conduction
equation on a superspace as a probability mean. We consider a Cauchy problem 09U
at n
+2
2k
1
(t, x, B)
=2
a2
n
2
[il Ai' ax ax; a2
2k
i=1=1 c=' axiae;
i,7=1
+Bi'
aeiae
u t, x, 0)
+v(x) 0)u(t, x, 0),
u(0, x, 0) = cp(x, 0),
t E R+.
(2.2) (2.3)
Theorem 2.2 (FeynmanKac formula). Let the potential v(x, 0) = > ga(x)0', where ga are Fourier transforms of Avalued measures with a compact supports on Rn and let the initial condition cp(x, 0) belong to the class Then the probability mean t
u(t, x, 0) = MT exp{J v(x + w(T), 0 + e(T)) d7} xcp(x + w(t), 0 + l;(t)),
where w(t) = (w(t),e(t)) is a quasiWiener random process, defines the solution of the Cauchy problem (2.2), (2.3). In order to prove this theorem, we must pass to Fourier transformations for the potential and the initial condition, write the integral with respect to the quasiWiener distribution also in terms of the Fourier transformation (on an infinitedimensional superspace).
3.
Axiomatics of the Probability Theory over Superalgebras
We propose here the generalization of Kolmogorov's axiomatics to the case of probabilities with values in a CSA A and, in general, in an arbitrary Banach algebra.
3. Axiomatics of the Probability Theory
245
3.1. Measures with values in a Banach space. Let (SZ, a) be a measurable space and E be a Banach space. The mapping p of the aalgebra a into the Banach space E is called a vector measure if, for any linear continuous functional l on E, the set functions is a bounded (signed, in general) measure (aadditive) on (SZ, a) (i.e., a charge). The aadditivity of compositions of vector measure with all linear
continuous functionals on E (a weak aadditivity) entails a strong oadditivity, namely, p(U Aj) _ E µ(A3) in the sense of a normed i=1
i=1
topology on E for any contable family of pairwise nonintersecting sets.
3.2. Generalization of Kolmogorov's axiomatics to probabilities with values in a Banach algebra. Suppose now that E is an arbitrary Banach algebra over R with a unit element e. We want to obtain a generalization of Kolmogorov's axiomatics [38] to the case of probabilities with values in E. In accordance with Kolmogorov's axiomatics, probabilities are (aadditive) measures on a measurable space (1, a) which assume values in the interval [0, 1]; P(Q) = 1. It is obvious that the generalization of the concept of measure is a vector measure with values in the Banach algebra E. It remains to find out what serves here as an analog of the concept of a probability measure, i.e., a measure assuming values in the interval [0, 1]; P(11) = 1.
We propose the following approach to this problem.
Definition 3.1. A probability measure with values in the Banach algebra E is a vector measure p satisfying the following conditions. 1. For any set A E or the spectrum of the element µ(A) lies in the interval [0, 1], the spectrum being nonempty. 2. p(Q) = e, where e is a unit element of the algebra. Everywhere in what follows, we denote by Spec (A) the spectrum of the element A E E. Definition 3.2. The collection of objects (1, or, P), where (a) 0 is the set of points w;
Chapter V. Probability Theory
246
(b) a is the Qalgebra of the subsets of S2; (c) P is a probability measure with values in E, is called a Eprobability model. We have thus proposed a generalization of a standard probability model. If E = R, then Definition 3.2 coincides with Kolmogorov's definition. In the Eprobability model, the probability of the event A E a is an element of the Banach algebra E.
Example 3.1. We consider as E the algebra of continuous functions on the compact set T: E = C(T). Then Spec (p(A)) consists of the set of values of the function µ(A) (t), t E T, and the probability measures are vector measures p such that for all t E T we have the following: (1) 0 < µ(A)(t) < 1 for any t E T, (2) µ(S2)  1. As usual, discrete measures are the simplest examples of probability measures. The general scheme for constructing C[a, b]valued probabilities is as follows.
Let us consider an ordinary Rprobability model (S2, a, P) and a random process x(t, w), a < t < b, with values in R and with continuous trajectories satisfying the following conditions: (1) for almost all w E S2 we have 0 < x(t, w) < 1 for all t E [a, b], (2) Mx (t, w)  1. Then the vector measure PE(A) fA x( , w)P(dw) is a probability measure.
This example can be immediately generalized to the case of the algebra C(T) of continuous functions on the compact set T, the C(T)probabilities are constructed with the use of continuous random functions x(t, w), t E T.
In particular, if the set T is finite, then we obtain probabilities with values in R" corresponding to random vectors.
Remark 3.1. Example 3.1 will be important in the frequency interpretation of Eprobability models (see Sec. 3.8).
Example 3.2. Let us consider an algebra Mats (n x n) of complex matrices n x n as the algebra E. Then the probability measures are the vector measures p for which the eigenvalues A (A) of the matrices µ(A), A E a, lie on the interval [0, 1] and µ(S2) is an identity matrix.
3. Axiomatics of the Probability Theory
247
We can again construct numerous examples of discrete Matc (n x n)probabilities; the general construction is based on random matrices. Let a(w) be an ordinary random matrix, Ma(w) = e. Then, under certain constraints on the random matrix a(w), the vector measure PMatc(nxn)(A) = fA a(w)P(dw) is a probability. Into this example we can also include probabilities with values in finitedimensional Grassmann algebras. Consider, for instance, an algebra G2 and its regular representa
tion in Matc(4 x 4). Let ej(w), j = 0, ..., 3, be ordinary random variables satisfying the following conditions: (1) 0 < eo(w) < 1 a.e., (2) Meo(w) = 1, MCj(w) = 0, j i4 0. Then
Pcz (A) _
MAeO
0
0
0
MA1
MAeO
0
0
MAC2
0
MAeO
0
MAS3 MAC2 MAe1 MAeO where MAej (W) = fA j (w)P(dw) is a G2valued probability. The following example is an infinitedimensional generalization of Example 3.2.
Example 3.3. Let X be a Banach space. Consider an algebra G(X) as the algebra E. The probability measures are those G(X)valued measures p for which the spectrum of the operators µ(A), A E a, lies on the interval [0, 1] and p(Q) is an identity operator. We can use discrete measures in order to construct these probabilities; the general construction is based on random linear operators. Into this example we can include probabilities with values in the CSA G1 00 by considering the regular representation of this algebra.
3.3. A spectrum of en event and the multivalued probability theory. The set Spec (P(A)) constitutes the real probabilities of an event A E a. We actually deal with the multivalued probability theory. Every event A E a is associated with a whole set of probabilities Spec (P(A)). If the spectrum of the element P(A) consists of one point, then the set of real probabilities reduces to one number, namely, the probability
248
Chapter V. Probability Theory
of the event A. Conversely, we can regard every classical probability as an element of a Banach algebra. It is not necessary to take the field of
real numbers as this algebra, we can, for instance, regard this probability as a diagonal matrix and, in general, as any element of a Banach algebra which has a onepoint spectrum equal to this probability.
3.4. Splitting real probabilities. Another essential peculiarity of the new probability theory is that by extending the field R to the algebra E we can "split" the real probabilities of an event. We can demonstrate this in the most visual way by considering a zero probability. In the ordinary probability theory an event A of zero probability is not at all impossible, it is only "very rarely" realized. Thus, the zero probability comprises a large class of events which are "very rarely" realized. However, the frequency of realization of events of zero probability may differ rather considerably. It would be expedient to sort out somehow events of zero probability. We can do this by extending the number field R to the Banach algebra E. In contrast to the number field, a Banach algebra may contain arbitrarily many elements with a zero spectrum, i.e., the zero of the number field extends somehow to form a Banach algebra. We can now add various elements of the Banach algebra E which have a zero spectrum in the Eprobability model to the two events A and B which have zero probability in the Rprobability model. In the same way we can split any other real probability. Thus, from this point of view, the new probability theory makes it possible to study finer properties of a probability model which disappear under real approximation. Here we can draw an analogy with the quantum field theory. We begin with real probability (an analog of the free theory) and then consider the perturbation of this initial approximation by operator probabilities. This perturbation describes finer effects which were not taken into account under the first approximation.
3.5. The soul and the body of probability. We choose an algebra G' as a Banach CSA. Any element of this algebra can be represented as) = b) + c), where cA is a quasinilpotent element. Consequently, Spec) = {b)}.
3. Axiomatics of the Probability Theory
249
This means that all G.probability models are singlevalued. Any event A is associated with a unique real probability, namely, the body of the element P(A). Any G'probability assumes values in the set [0, 1]e x c(G ,). Any event A E or whose probability belongs to the soul c(G'00) has a zero real probability, i.e., an element of the number field R is extended to the soul c(G100 ). In general, G1probability models can be regarded as a splitting of ordinary Rprobability models with a layer c(G'). For any G1probability P we can distinguish a body of the probability Pb(A) = bP(A) and a soul of the probability PE(A) _ cP(A): P = Pb + PP. The body of the probability Pb is an ordinary probability and the soul Pc is the c(G100 )probability. The real probabilities for the soul PP are zero.
Thus, we can represent the G'probability model as follows. We have an ordinary probability model (S'l, or, Pb). Then we extend
the zero element of the field R to the algebra c(G100), i.e., we ascribe new values P(A) = Pc(A) to the events of zero probability, Pb(A) = 0; in the same way we extend the other probabilities. If the CSA A # G', then the spectrum of the elements of the soul may be nonzero and the real probabilities may be multivalued.
3.6. Conditional probabilities. Independent events. The majority of concepts and constructions for Rprobability models can be easily extended to Eprobability models.
Suppose that (SI, a, P) is a Eprobability model, A, B E a, and P(B) is an invertible element of the algebra E. The right (left) conditional probability of the event A relative to the event B is an element
P,.(A/B) = P1(B)P(A n B) (P,(A/B) = P(A n B)P1(B)). Note the formula that connects the right and left conditional probabilities, P,(A/B) = P1(B)P1(A/B)P(B).
Proposition 3.1. Let E be a subalgebra of the algebra of continuous functions C(T) or E = G. Then the conditional probabilities are Evalued probabilities.
Chapter V. Probability Theory
250
Proof. Consider the case E = G. We have Pr(A/B) = (Pb(B) + PC(B))1(Pb(A n B) + Pc,(A n P)), and, consequently,
Spec (P,.(A/B)) = {Pe 1(B)Pb(A n B)} E [0, 1].
An event A is said to be right (left) independent of an event B if
P(A n B) = P(B)P(A) (P(A n B) = P(A)P(B)). If the event A is right independent of the event B, then the event B is left independent of the event A and vice versa. If the event A is both right and left independent of the event B, then the events A and B are independent. The probabilities of independent events commute. If the probability of the event B is invertible, then the event A is right (left) independent of the event B if and only if
Pr(A/B) = P(A) (P1(A/B) = P(A)). We must also note the formula for total probability of Eprobability models. Let {A1, ..., An} be a complete group of incompatible events and P(Ai) > 0, i = 1, ..., n. Then n
P(B) _
P(Ai)Pr(B/Ai)
P(B n Ai) _ i=1
i1
n
Pt(B/Ai)P(Ai) i=1
As usual, this gives the Bayes theorem (we give the right version)
PP(Ai/B) = P1(B)P(A n B) n
_ [E P(Ai)Pr(B/Ai)]1P(Ai)Pr(B/Ai) i=1
Let us generalize the concept of independence to n events.
Let or = (jl, ..., jn) be a permutation of the subscripts (1, ..., n). The events A;, j = 1, ..., n are said to be orindependent if P(nA;) =
3. Axiomatics of the Probability Theory
251
The events A3, j = 1, ..., n which are aindependent for any permutation a are said to be independent.
3.7. Random variables. The spectrum of expectation. A random variable is any measurable map C: (S2, a) 4 (E, p), where Q is a aalgebra of Borel subsets of the algebra E. The expectation of the random variable C(w) is the integral
M = fC(w)P(dw).
(3.1)
We shall not discuss here the mathematical definition of integral (3.1). Note that this integral can be embedded into the general theory of a bilinear integral on locally convex spaces (a bilinear form is defined by the multiplication operation in algebra). The spectrum Spec (MC) constitutes the real expectations of the random variable C(w). Here again we encounter multivalued quantities. However, if E = G1, then the real expectation (just as the probability) is a singlevalued variable since Spec (MC) = bMC. In
this case, the random variable can be decomposed into the sum of the body Cb(w) = bC(w) (of an ordinary real random variable) and the soul C ,(w) = ca(w) (of c(G')valued random variable), and we have relations bMC = MbCb
=
f
Cb(w)Pb(dw),
cMC = fCc(41() + f Cb(w)Pc(dw) We can represent the random variable in G' as the result of a perturbation of the ordinary random variable Cb(w) by a certain random variable Cc(w) which is nonzero although the spectrum of this variable is zero, i.e., C ,(w) "assumes a zero real value."
3.8. Frequency interpretation. As is known (see Kolmogorov [38]), the axiomatic probability theory was preceded by the frequency probability theory. All Kolmogorov's axioms reflect some properties of relative frequencies. The most systematic exposition of the frequency probability theory was proposed by von Mises [47]. His frequency theory is based on the concept of a collective.
Chapter V. Probability Theory
252
Let S be an experiment with a set of outcomes H. For simplicity, we shall assume that the set II is finite, II = {irl, ..., 7rn}. If we repeat the experiment N times and record the outcome after each experiment, then we get a finite sample x = (X1, ..., xn) in which we can calculate the relative frequencies v' = n2 /N, where n1 is the number of realizations of the outcome 7rr in the first N trials. A collective is a mathematical abstraction of a finite sample, it is an infinite sequence x = (x1; x21 ...) xm, ...),
(3.2)
where xj E II, for which there exists a limit of the sequence of the relative frequencies
P = lim v3, Noo
v3 = n3 IN,
for each outcome irk (there occurs a statistical stabilization of relative frequencies). This limit is called a probability of the outcome 7rj. In the frequency probability theory, a collective is regarded as a
fundamental object and the whole frequency probability theory reduces to various operations performed on the collectives. Consider now an infinite sequence of outcomes (3.2) for which there is no statistical stabilization of relative frequencies for some characteristics Irk. Such a sequence is not a collection and, consequently, cannot be regarded as an object of the frequency probability theory. However, a sample of this kind also carries some information concerning the event which is investigated in the experiment S, and it would not be wise to reject all events in which there is no statistical stabilization. We propose the following formalism (which is, naturally, only the first step in this direction). For every characteristic Irk we denote by Spec (vk) the set of limit points of the sequence of relative frequencies {vk}. The probability of the characteristic 7rk is an element of a Banach algebra E whose spectrum coincides with Spec (vk). The algebra E and the rule according to which every characteristic is associated with an element of the algebra are defined in accordance with the properties of the probability model in question.
3. Axiomatics of the Probability Theory
253
This formalism does not cover the case of an empty set Spec (vk). However, we can include this case into the formalism considering elements with an empty spectrum in the definition of the Eprobability model, and then the characteristic Irk for which Spec (vk) = 0 is associated with an element Ak E E with an empty spectrum. In the following model we can also arrive at multivalued probabilities.
Suppose that we have a sequence of instances of time t E T and every instance is associated with a collective It = (xlt, ..., X t) ...). For each instance t we calculate relative frequencies vk (t) = nk (t) /N
and probabilities Pk (t) = Ntoo lira vk (t). We obtain a discrete vector probability on the set II of characteristics. If, for any one of the characteristics, the function Pk(t), t E T is continuous, then it is a C(T)valued probability (see Example 3.1). In Secs. 1 and 2 we considered more general probability models on a superspace. Distributions that we came across in these models
are not aadditive Avalued measures. Even if the algebra A = C, they are complexvalued distributions of, in general, an unbounded variation. These distributions and random processes are similar, in many respects, to constructions from the theory of quantum random processes [36].
With respect to the degree of complication of the mathematical formalism, the results obtained in this chapter should be arranged as follows: the frequency probability theory, the analog of Kolmogorov's axiomatics, the generalized probabilities from Sec. 1, and the generalized random processes from Sec. 2. We have exposed the results in the reverse order because the most interesting results have been obtained precisely for generalized Gaussiantype probabilities. It is also important that quasiGaussian distributions play a significant part in applications to the quantum field theory. No interesting models have been obtained as yet for oradditive Avalued probabilities. However, we can choose oradditive Avalued probabilities considered in this section as initial distributions p in the limit theorems from
Chapter V. Probability Theory
254
Sec. 1. The frequency interpretation for Gaussian distributions on a superspace can be realized in the same way as in the ordinary probability theory. Using the central limit theorem, we can represent a Gaussian distribution on a superspace as the distribution of a sum of infinite number of discrete Avalued random variables. The frequency interpretation is valid for these variables, and then the Gaussian distribution on a superspace is approximately considered to be equal to the approximating distribution with a sufficiently large number.
4.
Unsolved Problems and Possible Generalizations
We can see from the content of this chapter that only the first strokes with a paintbrush have been put in the probability superstructure. I can formulate some problems which we are of the most interest.
Limit theorems. 1. The superanalog of Lyapunov's theorem for the distributions µnk with noncommuting values. 2. Limit theorems, in which a limiting process takes place for the functions cp, which are continuous and bounded with respect to commuting variables and which are polynomials with respect to anticom
muting variables. The most interesting case here is that of a "purely Gaussian" distribution on a superspace. Nothing is known even in a finitedimensional case, even for the simplest Gaussian distribution 'YB,
0
0
0
0 1
0
1
0
1
B=
3. Limit theorems for dependent random variables. 4. Infinitely divisible distributions on a superspace.
4. Unsolved Problems
255
Random processes. 1. Stochastic differential equations and diffusion processes (at least on a finitedimensional superspace). This seems to be the most interesting problem. A Wiener process already exists. The stochastic differential equation can be written in the space of (D(1l)processes de(t) = a(e(t))dt + b(e(t))dw(t).
Now we have a problem of the existence and uniqueness of a solution in the space of 4) (1)processes. 2. The OrnsteinUhlenbeck process on a superspace and Malliavin calculus on a superspace connected with it. 3. Investigation of parabolic equations on a superspace with the aid of the theory of random processes. 4. Poisson superprocess. 5. Relationship between the theory of quantum random processes and the theory of random processes on a superspace. It seems to me that we can obtain here something like a correspondence principle.
Frequency interpretation. 1.
Of a considerable interest is a systematic exposition of the
frequency supertheory on a mathematical level of strictness (a superanalog of Mises theorem). 2. Construction of specific examples of the use of Eprobability model in natural sciences. Here the ideas can be realized of an extension of the number field R, in which the standard probabilities assume values, to a Banach (or even topological) algebra.
Remarks Sec. 1. The results of this section were published in [154, 161]. The central limit theorem for Feynman distributions on real locally convex spaces was formulated in the article by Smolyanov and the author [128]. A different version of the central limit theorem for Feynmann distributions was obtained by Ktitarev [108]. Krylov's articles [106] seem to be the first in which the central limit theorem was obtained for noncountably additive distributions.
256
Chapter V. Probability Theory
Sec. 2. The results of this section were published in [161]. The Brownian motion on an infinitedimensional superspace was introduced in [144], the FeynmanKac formula was obtained in the same article. Independently, the Brownian motion on a finitedimensional phase superspace was constructed by Rogers [122]. Sec. 3. Here the ideas were realized and used which I proposed when constructing a padic probability theory [169]. In general, the Mises frequency theory of probability (almost forgotten by now) is the most powerful method for the development of new probability formalisms. In the padic probability theory the reasoning was carried out according to the same scheme. A general principle of statistical stabilization of relative frequencies was advanced. By virtue of this principle, the convergence of sequences of relative frequencies can be considered not only in a real topology on the field of rational numbers (and all relative frequencies are rational) but also in any other topology. A random computer modeling was carried out in the padic probability theory as a result of which random samples were obtained for which statistical stabilization of relative frequencies does not exist in the field of real numbers but exists in the field of padic numbers. Here p is chosen in accordance with the properties of the probability model in question. The choice of algebra E plays a similar part in the formalism from Sec. 3. In the padic probability theory, the probability of an event may be a negative number, an imaginary unity, a natural number, exceeding unity, so that
the padic probability theory is, essentially, an intermediate step on the way from Rprobability models to Eprobability models.
Chapter VI
NonArchimedean Superanalysis
Traditionally, all constructions of mathematical physics were carried out over the field of real numbers R. However, a different point of view is also possible according to which on fantastically small distances (of order 1033) the spacetime has nonArchimedean structure and, consequently, cannot be described by real numbers. The philosophy and ideology of nonArchimedean physics were laid as a foundation by I. V. Volovich (1987), he also advanced an invariance principle (which got the name of the Volovich invariance principle). By virtue of this principle, rational numbers formed the experimental basis for any physical formalism, and physical formalism must be invariant with respect to the choice of the completion of the field of rational numbers. Thus, along with real physical theories, certain theories were worked out over other number fields, in particular, over fields of padic numbers (see [21]).
In order to describe nonArchimedean fermions and nonArchimedean superfields as well as nonArchimedean (and, in particular, padic) superstrings, we need a nonArchimedean generalization of superanalysis. The first work in nonArchimedean superanalysis was the article by Vladimirov and Volovich [19] in which the authors considered a superspace over an arbitrary locally compact field. In this chapter I expose my version of the theory of generalized functions, partial differential equations and Gaussian (continual inclusive) integrals on a nonArchimedean superspace (both finitedimen
258
Chapter VI. NonArchimedean Superanalysis
sional and infinitedimensional).
1.
Differentiable and Analytic Functions Recall that the absolute value (valuation) on a field K is said to be
nonArchimedean if, instead of the inequality Ix+ y J K< I x J K+ I y J K, a stronger inequality, namely, Ix + Y K < max(IxIK, I yI K), is satisfied. An absolute value is nonArchimedean if and only if I n I K n and lal > N. Suppose now that lil > 2N. Since Iil = Iyl + Ial, it follows that either IyI > N, or a > N and, consequently, IIcjII < e, i.e., cv E G°00 (B), with IIcIl < If ll00IIgMI00. It is obvious that the CSA Go (B) satisfies conditions 13 for any Banach algebra B for which rB = F. The monomials {gi1...gin } form a topological basis in the Klinear space Go00 (K). I
I
Remark 1.1. A series in a nonArchimedean normed space converges if and only if its general term tends to zero. This follows from the strong triangle inequality. Just as in Chap. I, we introduce a nonArchimedean Banach superspace KK,m, the norm IIull = max Ilnj II being nonArchimedean, 1<j bn(f, ..., f ), n=0
where the polylinear forms bn c Gn,,.(L jr, A), the restriction of bn to the superspace X being symmetrical and series (10.5) converging uniformly on the ball UR,P of the covering CSM LX (i.e., sup IIbn(f, ..., f) II f EURL,v
Note that pnlbnlR = sup
Ilbn(f,..., f)II
f EUR.v
In the space A(UR,P, A) we introduce norms
FR,=sup Sup n
f EUR.v
Ilbn(f,...,f)Il=sup pnlbnlR,
R, p E r.
n
We have inequalities IIFIIR,P/C< IFIR,P< IIFIIR,P'
(10.6)
Let us assume that AP (X, A) consists of functions which are Sanalytic
on balls of a fixed radius p with respect to all norms that define the topology in X. Note that UR2,P 3 UR1iP1 R2 > Rl and A(UR2,P, A) 3 A(UR1,P) A).
We set Ao(X, A) = lim ind Ap(X, A) (note that AP2 (X, A) 3 API (X, A), p2 < pi). The functional space Ao(X, A) is an infinitedimensional analog of the space of functions on KA''n which are Sanalytic at zero. Theorem 10.1 (on the approximation by cylindrical polynomials). A set of polynomials which depend on a finite number of variables is everywhere dense in the space .A0(X, A).
10. InfiniteDimensional Superanalysis
287
We introduce functional spaces BR,P = {F E Ao: IIFIIR,P < oo},
BP = lim ind BR,P.
By virtue of inequality (10.6), we have
Ao = l
o ind lim ind A(UR,P, A) = l m ind R m ind BR,P
We introduce spaces of sequences of elements of the CSA A:
IIs(A) = {7r = {7rn} 0: 7rn = {7r 3}, a = (al, ..., an), Q = (Ql) ..., On)) where 7r,,p E A, V R, p E r:
1
17fI
I R,P = Sllpn p n I I7rn I I R< o, I I7rn I I
R=
sup II7rc,0II RI1 < oo}. QQ
Theorem 10.2. The space A'(X, A) of generalized functions of an infinitedimensional nonArchimedean superargument is isomorphic to the space of Asequences IIs(A). Let us introduce a dual superspace Y = Mo ® N1' = A'(KK'q, Ao) ® A'(Kns, A1).
The covering CSM
Ly=M'®N'=A'(KAq, A)eA'(KKS,A). We set YR = MR,O ® NR,1, where
MR =A(URxA', A),
NR=A(URXAs, A).
Note that Y = lim ind YR.
The spaces of test and generalized functions on the dual superspace Y can be defined in the same way as on the space X. The only difference is of a topological character, namely, the superspace X is a projective limit of nonArchimedean Banach spaces whereas the superspace Y is inductive. The map F: YR 4 A is said to be Sentire if series (10.5) converges
uniformly on a ball of radius p E r in the covering CSM LyR (here
Chapter VI. NonArchimedean Superanalysis
288
the polylinear forms b. E Gf,r(Ly , A)). The space of Sentire maps on the superspace YR will be denoted by A(YR, A). The map F: Y > A is said to be Sentire if the restrictions of F to the spaces YR are Sentire maps for all R E F. The space of Sentire maps on the superspace Y will be denoted by A(Y, A). We denote a ball of radius p E F in the superspace YR by WR,P, WL R,p being the corresponding ball in the covering CSM LyR. The topology in the CSM A(YR, A) is defined by a system of norms IFIR,P = supra sup Ilbn(f, ..., f)II In the CSM A(Y, A) we introduce f EWR,,
a projective topology A(Y, A) = lim proj A(YR, A) which, in the CSM
A(Y, A), is equivalent to the topology defined by the system of norms I I F I I R,P = sup. pn I I bra I I R (by virtue of inequalities (10.6)). The space of Sentire functions on the superspace Y is a nonArchimedean Frechet CSM.
We introduce a space of Asequences IIs, (A) = {7r = {7rn}°Oo: 7rn = {7raf}, Trap E A, I R, p c F: II7r
IIR,P = SUPP nMI7rIIR < oo, n
II7rnIIR = Sup II70AII I Y. KR'a1 < oo}. aQ
Theorem 10.3. The space A'(Y, A) of generalized functions of an infinitedimensional nonArchimedean superargument is isomorphic to the space of Asequences Us1(A).
Now the Laplace calculus over a pair of dual superspaces X and Y is developed according to the usual scheme (see Chap. III). Theorems 10.110.3 give a nonArchimedean analog of a theorem of the PaleyWiener type for analytic generalized functions.
Theorem 10.4. The Laplace transformation L: A'(Y, A) p Ao(X, A)
is an isomorphism of a CSM.
11. Unsolved Problems
289
Now we introduce a Gaussian distribution on an infinitedimensional superspace Y = MO' ® N. This distribution serves as a basis for constructing a theory of continual Gaussian integrals in a nonArchimedean case.
In [167] we gave the proofs of the results contained in this section in the commutative scalar case A = K. The proofs for a supercase are similar.
11.
Unsolved Problems and Possible Generalizations
1. A substantive analysis on nonArchimedean supermanifolds. 2. Theory of differential equations on nonArchimedean supermanifolds.
3. Pseudodifferential operators on a nonArchimedean superspace. 4. Formulas of the type of FeynmanKac. 5. Gaussian and Feynman integrals over infinitedimensional nonArchimedean superspaces. 6. Theory of differential and pseudodifferential equations on infinitedimensional nonArchimedean superspaces. 7. Superconformal structures corresponding to Galois groups. 8. NonArchimedean Hilbert superspace. 9. NonArchimedean infinitedimensional superdiffusion. 10. Hida calculus on a nonArchimedean superspace. 11. Formulas for integration by parts for nonArchimedean Gaussian distributions and Malliavin calculi. 12. In their pioneer work, Vladimirov and Volovich discuss a num
ber of problems of superanalysis over an arbitrary locally compact field. The authors of [1] suppose that the majority of results which they obtained in this work for a field R can be generalized to a nonArchimedean case. However, this has not yet been done anywhere sufficiently accurately. 13.
Fundamental solutions for linear differential operators with
constant coefficients.
Chapter VI. NonArchimedean Superanalysis
290
14. We have only used spaces of Sanalytic generalized functions. It would be interesting to generalize to a supercase the theory of generalized functions over the spaces Ck which is widely used in nonArchimedean analysis (see, e.g., [72]).
Remarks The foundations of nonArchimedean physics were laid by I. V. Volovich
(1987); the padic quantum mechanics (with complexvalued wave functions) was constructed by Vladimirov, Volovich, Zelenov, Alacoque, Ruelle, Thiran, Verstegen, Weyers (see [21] and the bibliography therein). The padic quantum mechanics with padicvalued wave functions was constructed by Khrennikov [66, 156159]. Vladimirov and Volovich also considered padic quantum field theory with comlpexvalued fields. Models with padic fields were studied by the author [66, 162]. The monograph [21] contains practically everything that was done by now in the padic physics. For padic physics with padicvalued functions see (66].
The most simple, complete, and reasoned exposition of nonArchimedean analysis can be found in Schikhov's book [72]. More subtle problems of the number theory are exposed in the monograph by Borevich and Shafarevich [11].
Sec.1. The theory of Sdifferentiable maps on a superspace over an arbitrary locally compact field was given in [19]. In [153] the author considered a different version of a nonArchimedean superspace, namely, a nonstandard superspace.
Secs. 27. The results of these sections are given in [156159, 162164]. A padic Gaussian integral over a superspace is a natural generalization of a padic Gaussian integral proposed by the author in [157]. For the theory of padic Gaussian integration see also [66, 167]. M. Endo proved that a padic Gaussian integral cannot be extended to a linear continuous functional on a space of continuous functions of a padic argument, i.e., as distinct from a real case, a padic Gaussian distribution is not a measure. Sec. 8. Trotter's formula over nonArchimedean fields was obtained in [66], its complete proofs can be found in [160]. Of course, it is only the simplest version and wide generalizations of this formula are possible (cf. [26]).
Sec. 9. Volkenborn's integral plays a significant role in nonArchimedean
11. Unsolved Problems
291
analysis. It is possible that its superanalog can also be used for integral representations of special functions on a nonArchimedean superspace (cf. [72]).
Sec. 10. An infinitedimensional nonArchimedean analysis was presented in [66, 167]. Here we only outlined its supergeneralization. Sec. 11. 1. We mean investigations similar to the investigations carried out by De Witt, Volovich, Rogers, Buzzo, Cianci (see [27, 19, 20, 52, 8893, 80, 81, 111, 112]). 2. Compare, for instance, with the work by Cianci [112]. 3. The PDO theory on K" was proposed in [156]; the results of this article can apparently be combined with those from Chap. 3 from [156]. 4. It is interesting to try to generalize the results of this chapter to a nonArchimedean case. 5. Gaussian and Feynman integrals on infinitedimensional Klinear spaces were considered in [66, 167]. Of particular interest is the quantization of a nonArchimedean spinor field and of graded fields in the formalism of a continual integral. 6. For the secondary quantization over K" see [158]. 7. A number of models with a conformal structure over a Galois group were considered in [159] and [162]. 8. Nothing has been done here, but it is clear that the theory from
Chap. III must be generalized to a nonArchimedean case. 9, 10. The nonArchimedean white noise and nonArchimedean Hida calculus (nonArchimedean Brownian functionals) were introduced in the report that I made at the conference concerning Gaussian random fields, Nagoya, 1990. The padic theory of probabilities was proposed in connection with a probability interpretation of padic quantum mechanics with padicvalued wave functions (see also [169]). 11. Nothing has been done here. 12. Nothing has been done here either. 13. The reader should take the article by Vladimirov and Volovich [19] and Schikhov's book [72] and try to combine them. Although the first steps of this theory will be a trivial generalization, essential advances can be obtained in this direction.
Chapter VII
Noncommutative Analysis
When the main text of the book was ready, I got some ideas that allowed me to construct a noncommutative generalization of the supercommutative analysis exposed in the book. Here I again use the scheme which I used when passing from the Klinear ordinary mathematical analysis to Alinear superanalysis. In order to construct analysis over an arbitrary noncommutative algebra A (or on an Amodule), it is necessary to define, "in a natural way," the concept of Alinearity which will be used in the noncommutative differential calculus. In the first place, we can use here a further generalization of the methods from Chap. III and consider analysis on a pair (Asuperspace, Amodule) defining a superspace as a Klinear subspace of the Amodule. M. I constructed this theory as early as in my first works in superanalysis. It does not constitute an essential advance as compared to superanalysis (see Sec. 1). In this theory, the approximating Alinear maps are, as before, right Alinear or left Alinear maps on a covering Amodule. Precisely these maps are used as classes of Sderivatives in the analysis on a pair (Asuperspace, Amodule). A new essential progress in the development of noncommutative analysis can be obtained with the aid of a new class of Alinear maps (noncommutatively linear maps). New nontrivial algebraically topological constructions arise here such as an ordered projective tensor product of noncommutative Banach algebras (onedimensional noncommutative differential calculus), a projective tensor product of non
Chapter VII. Noncommutative Analysis
294
commutative Banach algebras which is ordered with respect to two indices (multidimensional noncommutative differential calculus) and similar constructions for Amodules (infinitedimensional noncommutative differential calculus). Apparently it will later be possible to generalize all main parts of this book to arbitrary noncommutative Banach (or topological) algebras and modules. It should be pointed out that the analysis on a pair (Asuperspace, Amodule) given in Sec. 1 is contained in the more general noncommutative analysis considered in Secs. 2 and 3. Choosing different Klinear subspaces in an Amodule and regarding them as Asuperspaces, we obtain Sderivatives (Sec. 1) as restrictions of the noncommutative derivatives from Sec. 2 to the Asuperspace.
As in the superanalysis, two equivalent approaches to the construction of noncommutative analysis are possible here, namely, an algebraic approach and a functional one. In this chapter we construct a noncommutative functional analysis, i.e., a theory of functions of noncommutative variables (of maps of sets with noncommuting coordinates). The Connes noncommutative geometry [105], the Wess and Zumino quantum differential calculus [85], the Soni quantum superanalysis [132] are versions of algebraic noncommutative analysis. It should be pointed out that the functional approach to the theory of quantum groups was used by Aref'eva and Volovich [78].
1.
Differential Calculus on a Superspace over a Noncommutative Banach Algebra
Everywhere in this section we denote by A an algebra over a field K which, in general, is nonassociative and noncommutative. All modules are modules over A.
Let Mk, k = 1, ..., n, and N be right modules. In the space Ln (fl Mk, N) we distinguish a subspace Ln,r: the right Alinearity. k=1
The map b c Ln,r if, for any xj E M3 and a E A, we have relations b(xl, ..., xka, xk+1, ..., xn) = b(xl,..., xk, axk+1, ...) xn);
1. Differential Calculus on a Superspace
295
b(xl, ..., xna) = b(xl, ..., xn)a.
By analogy we can introduce a space Ln,1 of maps which are left Alinear on the left modules Mk, N. As in Sec. 1 of Chap. III, we can introduce the structure of modules in the spaces Ln,, and Ln,1, but there is no canonical isomorphism between Ln,,, and Ln,1 for twosided modules in the general case.
For topological CSM of modules we denote by Ln,r(Kn,r) and Ln,1(Icn,i) the subspaces of the spaces Ln,r and Ln,1 consisting of continuous (continuous on compact sets) maps.
Definition 1.1. A Klinear topological subspace of a topological module M is a superspace over the module M. The module M is said to be covering for the superspace X.
Definition 1.2. The map f : X + Y, where X and Y are superspaces over the modules M and N, is said to be (right) Sdifferentiable
at a point x if f is differentiable (in a certain sense) as a map of topological Klinear spaces X and Y and there exists an operator aR f (x) E Ll,r (M, N) such that aR f (x) Ix = f (x).
As in the case of a superspace over a CSM, the Sdifferentiability
on a superspace over an arbitrary Amodule is defined by a class fll,r (M, N) of operators, which are right Alinear, to which the Sderivative belongs and by the convergence which defines the ordinary derivative in topological Klinear spaces. In the sequel we assume that the classes fl,r (M, N) are submodules of Ll,r (M, N). The left Sdifferentiability is defined by analogy. The derivatives are not uniquely defined.
The Sdifferentiabilities on a superspace X over a CSM and on a superspace coincident with the Amodule are special cases of Sdifferentiability. We shall formulate the fundamental theorems of Sdifferential cal
culus for Banach modules and for Frechet differentiability: fl,r = Li,r and f : X 4 Y is Frechet differentiable as a map of Banach spaces. The generalization of these theorems to the case of Sdifferential calculus in topological superspaces can be carried out by analogy with [54].
Chapter VII. Noncommutative Analysis
296
In the following theorems, we denote by U, V, 0 the neighborhoods of the points xo E X, yo c Y, and zo E Z, where X, Y, Z are superspaces over the Banach Amodules M, N, R.
Theorem 1.1 (chain rule). If the functions f : U + Y, g: V + Z are S differentiable at points xo and yo = f (xo), then the composite
function cp = g o f : U 4 Z is Sdiferentiable at a point xo and aRw(xo) = ORg(yo) ° ORf
(xo).
Theorem 1.2 (the differentiability of an implicit function). Let the function F: U x V + Z be continuous at a point (xo, yo) and let F(xo, yo) = 0. If there exist partial Sderivatives a and A , which are continuous at a point (xo, yo), and the operator 8y (xo, yo) has bounded inverse, then there exists an implicit function y = f (x) which is Sdiferentiable at a point yo = f (xo) and (aaF(xo,yo))1
ORf(YO) _
0 (aaF(xo'yo))
y
Theorem 1.3 (differentiability of an inverse function). Let the function f : U a Y be continuously Sdiferentiable and let the operator aR f (xo) have a bounded inverse. Then there exists an inverse function cp = f 1 which is Sdiferentiable at a point yo = f (xo) and 1RW(xo)
_
(ORf (xo))1
As in Sec. 5 of Chap. III, we introduce an annihilator 1X = Ann (X, £1,,.(M, N)).
If the map f : X + Y, where X and Y are superspaces over the modules M and N, is Sdifferentiable, then the derivative aR f is a single valued map from the superspace X into the right module £1,r(M, N)/Ann (X; £1,r(M, N)). Consequently, we can define the second derivative and higherorder derivatives: a2Rf: X > Gl,r(M, £1,r (M, N) /Ann (X; Gl,r(M, N)))
We shall restrict the further consideration to associative algebras A and superspaces with a trivial annihilator: 1X = 0.
1. Differential Calculus on a Superspace
297
Theorem 1.4. Let the function f : X + Y be ntimes Sdifferentiable in the neighborhood of the point xo. Then the nthorder Sderivative 8Rf (xo) E Gn,r (Mn, N) and the restriction of eR f (xo) to the superspace Xn is symmetric.
Proposition I.I. Let the polylinear form bn E Gn,r(Mn, N) and the restriction of bn to a superspace Xn is symmetric. Then the map f : X + N, f (x) = bn(x, ..., x), is n times Sdifferentiable and 8Rf = n!bn.
Theorem 1.5 (Taylor's formula). Let the function f : X > Y be n times S differentiable at a point xo E X. Then f (x)
E 8R n!f x0) (x  xo, ..., x  xo) + rn(x  x0),
M=0
where the Sderivatives 8Rf (xo) E Cn,r and their restriction to the superspaces Xn are symmetric;
0, h + 0.
The proof of theorems of Sdifferential calculus on superspaces over Banach modules repeats the proofs for the corresponding theorems of differential calculus in Banach spaces. We must only replace in these proofs Klinear operators by Alinear ones. Taylor's formula for Sdifferentiable maps on superspaces over Amodules leads to the following definition of Sanalyticity.
Definition 1.3. The function f : X + Y is right Sanalytic at a point xo c X if f can be expanded in a power series in some neighborhood of the point x0, i.e., W
f (x) = > bn(x  xo, ..., x  xo), n=0
where bn E Gn,r and the restriction of bn to the superspace X' is symmetric.
The type of Sanalyticity is defined by the choice of modules to which the coefficients of the power series belong and by the choice of the type of convergence of the power series.
Chapter VII. Noncommutative Analysis
298
The construction of a theory of distributions, pseudodifferential operators, and evolutionary differential equations on a superspace over a module over an arbitrary noncommutative algebra is an interesting unsolved problem.
Differential Calculus on Noncommutative Banach Algebras and Modules
2.
Everywhere in this section, we denote by A an associative Banach algebra over a field K of real or complex numbers in which there exists a topological basis (as in a Klinear Banach space) {en}n 1; {rynmk} 00 are structural constants of the algebra: enem = k Ynmkek k=1
We also assume that {en ® em} is a topological basis in the completion of a projective tensor product (we shall denote this completion by the symbol for an ordinary tensor product, i.e., A ® A). We introduce an operation of multiplication (a1 0 b1) x (a2 ® b2) = a1a2 0 b2b1 relative to which A 0 A is an associative Banach algebra. As before, we denote by G(A) the space of Klinear continuous operators U: A + A. We introduce a canonical map j: A®A 4 G(A) by setting j (a 0 b) (x) = axb, x c A.
Proposition 2.1. The map j is a continuous homomorphism of Banach algebras.
In order to prove this proposition, it suffices to use the representation j (> unmen ® em) (x) = E unmenxem and the implicit forms of n,m
n,m
norms in a projective tensor product (see, e.g., Schaefer [71]) and in G(A).
We denote the image of the tensor product A ® A under the homomorphism j: LA(A) = Im j ^_' A® A/Ker J by LA(A) and the Banach algebra A ® A/Ker j by 11(2) (A).
Proposition 2.2. The element u = E unmen®em of the projective n,m
tensor product A ® A belongs to the kernel of the homomorphism j if
2. Differential Calculus on Algebras
299
and only if we have a relation E unm E rynksYsmi = 0 n,m
(2.1)
s
for any k, i.
Proof. Let uh = 0 for any element h E A. Then we have Unmenhem = i unmhkenekem = E Unmhk i 7'nkj7'jmiei = 0, n,m
n,m,k
n,m,k
j,i
i.e., the relation E Unm"fnkjfjmiei = 0 n,m,j,i
holds for any k. Thus relation (2.1) is valid for any k and i.
Example 2.1. Let A = Gn be a Grassman algebra. Then all elements of the form u = g31...q,n® gil...gim, where j, = it for certain subscripts s and t, belong to Kerj.
Proposition 2.3. Let A = MatK(n x n) be an algebra of n x n matrices. Then Ker j = {0} and SZ(2) (A) = A ® A.
This proposition is a direct corollary of (2.1).
Definition 2.1. The map f : G + A, where G is an open subset of the algebra A, is noncommutatively differentiable (NCdifferentiable) at a point xo E G if f (xo + h) = f (xo) + V f (xo)h + o(h), where the operator V f (xo) E LA(A) ' Sl(2)(A) and
0,
h +0. Definition 2.1 can be reformulated as follows: the map f is NCdifferentiable if it is Frechet differentiable as a map of a Banach space
and the Frechet derivative f'(xo) E L(A) belongs to the operator algebra LA (A).
Example 2.2. Let A be a unital algebra and e be a unit element. The function f (x) = xn. Then f is NCdifferentiable and V f (x) _ e®xn1+x®xn1+...+xn1®e.
Chapter VII. Noncommutative Analysis
300
Example 2.3. Let f (x) = alxa2x...xan+l, where aj c A. Then f is NCdifferentiable and V f (x) = al®a2x...xan+l+alxa2®a3...xan+l+ alx...an ®an+1. If A is a commutative unital algebra, then V f (x) coincides with an ordinary derivative in the commutative Banach algebra. For instance,
for Examples 2.2 and 2.3 we obtain V f (x) = nxn1 and V f (x) _ nal Example 2.4. Let the operator U E G(A) \ICA(A). Then the map f (x) = U(x) is not NCdifferentiable although it is Frechet diferen...anxn1
tiable as a map of a Banach space.
The derivative of the NCdifferentiable map f : G + A, f (x) _ E fn (x) en, to be more precise, its representative in the algebra A ® A, n
can be represented in terms of base vectors {en ® em}: V f (x) =
JJ Vfnm(x)en ®em,
n,m
where V fnm(x) are numerically valued functions on the set G.
Theorem 2.1. Let f and g be NC differentiable maps on the set G. Then (1) the map cp = of /3 + Agp, a, /3, A, M E A, is NC differentiable, with
V (x) = aV f (x)/.3 + )Vg(x)p and
V psp(x) = E 1'knsrymrp(ak/rVfnm(x) + )1kµ'rVgnm(x)), n,m,k,r
(2) the map cp(x) = f (x)g(x) is NC differentiable; here the Leibniz formula
V (x) = V f (x)g(x) + f (x)og(x) holds true and Wnp(x) = >(7'mkpV fnm(x)gk(x) +'Ymknfk(x)V9kp(x)) m,k
2. Differential Calculus on Algebras
301
Theorem 2.2 (noncommutative chain rule). Let the maps f : G + A and g: W  A be NC differentiable at points x E G and y = f (x) E W respectively. Then the composite function cp = g o f : G 4 A is NC differentiable at a point x, with VV(x) = Vg(y) V f (x) and 7'nkp7'lmgVgnm(y)Vfkl(x)
V ppq(x) = n,m,k,l
Theorem 2.3 (on the noncommutative differentiability of an inverse function). Suppose that the function f : G + A is NCdifferentiable in a certain neighborhood 0(x) of a point x E G, the derivative V f :
0(x)
Q(2) (A) is continuous and maps the neighborhood
0(x) into a subgroup of inversible elements of the Banach algebra 1(2) (A). Then, in a certain neighborhood 0'(x) of the point y = f (x), there exists an NCdifferentiable inverse function g(y) = f 1(y), and Vg(y) = (VAX)),IX=g(Y)
We introduce higherorder noncommutative differentiability with
the use of Taylor's formula. As before, we denote by £(A, A) the space K of nlinear maps from An = A x ... x A into A. Next, we 10
e
n
shall need a new algebraic construction, namely, an ordered projective tensor product. Let or = (il, ,in) be a permutation of indices (1, 2, ..., n). Then we set
®A=A®...®A O tl
in
(everywhere we use the symbol of tensor product to denote a projective tensor product which is a completion of an algebraic tensor product in a projective norm), i.e., ®A is an ordinary projective tensor product A ®... ® A with ordered symbols of tensor products. Next, we introduce a direct sum of ordered tensor products with respect to all permutations or from the permutation group Sn: F(n+I) (A)
_ oESn ® (®A) o
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_
{x = E E a.1 ®a2 ®... ®a.n+1: IIxIIA®...®A Goo}. to it
oESn al...an+1
t2
Example 2.5. E(3) (A) =A®A®A®A®A®A, i.e., 2
1
2
1
x = Ea., ®aa2 ®a Q3 +Eb#1®bp2 p
2
1
of
2
®b031
1
where the elements a ® b ® c and a ® b ® c are not identified. 1
2
2
1
Let us consider the canonical map j: E(n+1) (A)  £, (An, A), j (a ®b ®c (9 ... ®d) (hi, ..., hn) = ahi1 bhi2 c...hind. i1
i2
i3
in
We set Gn,A(A', A) = Im j ' 1(n+1) (A) = E(n+1) (A)/Ker j (the map j is linear and continuous).
Definition 2.2. The map f : G + A is n times NCdifferentiable at a point xo E G if n
Vkf (xo) (h, ..., h) + o(hn),
f (xo + h) =
(2.2)
k=0
where the Kpolylinear forms Vk f (x0) belong to the classes Q(k+1) (A), Ilo(hn)II/IIhMI" + 0, h 4 0.
This definition can be reformulated as follows: the map f is n times
Frechet differentiable as a map of the Banach space A and Frechet derivatives belong to the classes cl(k+1) (A)
Example 2.6. Consider a map f (x) = axbxc, where a, b, c E A. Then
Vf(x)=a®bxc+axb®c, 1
1
V2f(x) = a®b®c+a®b®c. 1
2
2
1
Let us now consider a map f (x) = aixa2...xan+1. Then
Onf =
a1®... ®an+1 OESn
it
in
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Definition 2.2 of higherorder noncommutative differentiability leads to the following definition of noncommutative analyticity.
The map f : G + A is NCanalytic at a point x0 E G if, in a certain neighborhood 0(xo) of this point, f can be expanded in a power series
f(x) =
C'O
Ebn(xxo,...,xxo),
(2.3)
n=0
where the coefficients of bn E 11(n+1) (A) and the series converges in the sense 00
Ilf IIP = > PnllbnJJn(n+1)(A) < oo
(2.4)
n=0 for a certain p > 0.
We can reformulate this definition as follows: the map f is Frechet analytic as a map of the Banach space A and the coefficients of the Taylor series belong to the classes SZ(n+1)(A).
Proposition 2.4. Every NCanalytic map f is infinitely NCdifferentiable.
NCanalytic maps are series of the form
f (x) =
Ean,ai (x  xo)an,,Z...(x  xo)an,an+1 n=0 a
with coefficients that satisfy condition (2.4). In order to fulfil this condition, it is sufficient that 00
E Pn E
n=0
Ilan,a1II...Iian,an+1 II < 00.
a
Let us pass to further generalizations of the construction described above. Consider maps f : G + M, where G is an open subset of A and M is a Banach Amodule (twosided). A new algebraic construction arises here which is an ordered projective tensor product of n copies of the algebra A and one copy of the module M. An ordered tensor
Chapter VII. Noncommutative Analysis
304
product in which the module is at the (k + 1)th place is defined by the relation k+1
®(A, M = O
O M O ®...®A. i1
22
ik
in
ik+1
In particular, 1
®(A) M)=M®A®...®A, i2 in i1 or
n+1
ii in1 in Next, we define the direct sum of ordered tensor products with respect or
to all permutations of the indices a E Sn and the numbers of places occupied by the module M: k E(n+l)(A, M) _ ®1 ® ®(A, M)
k=1 oESn o
® (M®A®...®A®... (DA®A®...®M).
oESn
11
1
in
12
i2
in
Example 2.7.
E(3)(A,M) =M®A®A®A®M®A®A®A 1
2
1
2
1
®M ®M®A®A®A®M®A®A®A®M. 2 2
2
1
2
1
1
As before, we introduce a canonical map j: E(n+1) (A, M)
+ £ (A'1, M); q(n+1) (A, M) = E(n+1) (A, M) /Ker j.
Definition 2.3. The map f : G + M, where G is an open subset of the algebra A and M is a Banach Amodule, is said to be n times NCdifferentiable at a point xo E G if, in the module M, the relation (2.2) holds true, where Vk f (xo) E I (k+l)(A, M), and 11o(hn)JjM1jjhjIn _+ 0,
h+ 0. Example 2.8. Consider a map f (x) = axbxm, where a, b E A, m E M. Then
Vf(x) =a®bxm+axb®m;
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305
V2f(x) =a®b®m+a®b®m. 2
2
1
1
And now let us consider a map f (x) = mxaxb. Then
Vf(x) =m®axb+mxa®b;
V2f(x) =m®a®b+m®a®b, 2 2
1
1
and, finally, for the map f (x) = axmxb we have
Vf(x)=a®mxb+axm®b;
V2f(x)=a0m®b+a®m®b. 2 2 1
1
Let us now consider functions of several noncommuting variables. We shall use the same scheme as above. The map f : G + M, where G is an open subset of Am = A x ... x A, is said to be NCdifferentiable if it is Frechet differentiable as a map of the Banach space Am into M and the Frechet derivative (gradient) belongs to the class [SZ(2) (A, M)]m: m
V f (x) (hl, ..., h.) _ E Vj f (x) h;,
Vif (x) E
c(2) (A,
M).
Example 2.9. Consider a map f (x, y) = axbym, where a, b E A,
m E M. Then V f= a ®bym, V f= axb ®m. The higherorder differentiability for functions of several noncommuting variables will also be defined with the use of Taylor's formula. A new algebraic construction arises here which is a generalization of the ordered tensor products introduced earlier. In order not to com
plicate the consideration, we shall begin with studying the case of Avalued maps. We introduce a projective tensor product which is ordered with respect to two indices. In this product, every symbol of a tensor product has two indices, an upper index and a lower index.
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306
Suppose that we have a permutation or = (i1i ..., in) E S. and (with repetitions) from the set of indices a sample (1, ..., m).
The permutation or and the sample , are associated with a projective tensor product ordered with respect to two indices, namely, ®A=A®A®32
ar,K
_
it
32
31
{z =
...®A
12
in
7n
aQl ®aU2 ®... ®aan+1 Q1 ...Qn+1
12
11
IIZIIA®...®A