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a=0 n=1 1
l,,. (A) _ {x
We consider the norm II the space co(A).
:
Ilxll... = E sup ll7r,,xnll < oo}. ck=0
IIP
n
in the space 1P(A) and the norm II III in
2.3. Conjugate supermodule. For the Banach CSM M and N we use the sumbols G1,,. (M, N) and Ll,i (M, N) to denote the submodules of the CSM L1,r (M, N) and L1,1(M, N), respectively, which consist of continuous mappings. If M = N, then these CSM will be denoted by L1,, (M) and L1,1(M). For the space of Klinear continuous operators, we use the standard notation £(M, N) (G(M)).
The space L(M, N) is a Banach space with respect to the ordinary uniform norm IIILIII = sup{IIL(m)II: IImll < 1}. This norm induces the structure of the Banach spaces on the CSM Ll,r (M, N) and
117
2. Banach Supermodules
L1,1 (M, N). The norm III' I I is consistent with the Z2graduation since I
7r°(L) = 7r°L7ro+7r1L7r1, 7r1(L) = 7rIL7r°+7r°L7r1i i.e.,
We introduce a norm I I L I I= I I I11° (L) I I I+ I
I
IIIraIII < 2IIILIII.
I II1(L) I I I This norm is
Z2graded.
Note that the restriction of the canonical isomorphism I to the CSM £1,,.(M, A) is an isomorphism of the CSM £1,,.(M, A) and G1,1(M, A)
Theorem 2.1. The isomorphism I is an isometry of the Banach CSM L1,r(M, A) and L1,i(M, A).
Proof. Let the operator L E LI,r (M, A). Then I (L) = L° + L1 o a, and, consequently, III (L) II = IIL°II + IIL1 o all, sup
IIL10 all =
IILI(x°  xl)II
11x011+I1x1ll
sup
IIL1(x°+x1)11 = IILIII.
11x011+Ilxlll<1
By virtue of this theorem, we can identify the Banach CSM Gl,,.(M, A) and G1,1(M, A): M' = MO' ® M1' = GI,r(M, A) G1,1(M, A), where
M.' = Gi,r (M, A) '" Gi i (M, A), a = 0, 1.
The CSM M' = MO' ® M1' is a conjugate of the Banach CSM M = M° ® M1. By analogy with the ordinary theory of Banach spaces, the elements of the CSM M' will be called functionals. Here is an example of a Banach CSM M such that M' = {0}. Example 2.1. We set A = C[O, 1] (a space of continuous functions on the interval [0,1]), M = LI[0,1] (a space of summable functions on the interval [0, 1]). Then M is a Banach commutative module for the commutative Banach algebra A.
Let the functional L belong to M'. Note that A C M, and for g E A we have L(g) = L(1)g, where L(1) E A. However, the set A is everywhere dense in M, and therefore, for any g E M, there exists a sequence g,, E A, gn 4 g in M. However, the sequence L(1)gn converges to L(1)g in M, and, consequently, L(1)g = L(g) E A, but for any continuous function L(1) A 0 we can choose a summable function g such that L(1)g V A.
Chapter III. Distribution Theory
118
Here is an example of a Banach CSM M such that the even part in M is trivial (there are no even elements except for the zero element) and in the conjugate CSM M' the odd part is trivial (there are no odd elements except for the zero element).
Example 2.2. Let A = G1 and M = G1,1 = {x E G1: x = uq, u E K}. There are no odd functionals (except for the zero one). All functionals are even and have the form L(x) = Ax, A E K, M' = K.
Example 2.3. Let M = A. Then M' is isometric to A, and we have a relation
IIILIII = sup II(L,x)II = IIIL°III + IIILIIII = IILII
(2.1)
Ilxll<1
Example 2.4. Let A = G2 and Mo = {x E G2: x = tQ1g2i t E K},
M1={xEG2: x=Bg2iBEK}. Then MM=K, and Ml={xEG2: x = tQ1, t E K}. In this case we have relation (2.1) and the supremum is attained on the odd part of a unit ball.
Proposition 2.1. We have an inclusion lq(A) C 1,(A), 1 < p < oo, 1/p+ 1/q = 1; 11(A) C co(A).
Proof. Every vector u E lq(A) is associated with a Alinear functional L E 1,(A) defined by the relation (L, x) = > unxn, and 1
IILIIlp(A)
 E sup 11E unxn
IIuIIq(EIIxnIIP)'
IIUIIgIIxIIP.
a=0 11416<1
The case u c 11(A) can be considered by analogy.
Open questions 1. Are the inclusions lq(A) c l' (A), 1 < p < oo, 1/p + 1/q = 1; 11(A) C c'0(A) isometric?
2. Are the spaces lq(A) and 1'(A), 1 < p < oo, 1/p + 1/q = 1, and 11(A) and c' (A) isomorphic?
For what classes of Banach CSM does the equality of norms (2.1) hold true? Let us consider the norms II  IIP, 1 < p < oo on the CSM M = An. Then the CSM M' is algebraically isomorphic to the CSM M for any 3.
2. Banach Supermodules
119
one of these equivalent norms. It is not known, however, whether the Banach spaces (A', II . II ,)' and (An, II ' II9), 1 < p < oo, 1/p + 1/q = 1 are isometric.
Proposition 2.2. The Banach CSM 1', (A) and l,,,, (A) are isometrically isomorphic.
Proof. The series x = i enxn, where en = (0, ...,1, 0, ...) is a 00 n=1
canonical basis in 11(A), converges in the Banach space 11(A). Consequently, we have (L, x) = E Lnxn for the functional L E 1'(A). Let ILO = > sup I I Ln I Then VE > 0 3N, a = 0,1, such that a
n
I
sup I I Ln < I I LN I I+ E It remains to note that n
IILIIl1(A) >
11 (L", eNQ)II > 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
Chapter III. Distribution Theory
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<1
> 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..
For any E > 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<1 j=1
Open questions 1. Are finitedimensional Grassmann algebras Ealgebras? 2. Are there finitedimensional CSA which are Ealgebras?
2.5. Duality of Banach supermodules over Esuperalgebras. Theorem 2.4. Let A be a Ealgebra. Then (1) the CSM l,(A) is isometrically isomorphic to the CSM lq(A),
1
Proof. Let the functional L E l,(A). Note that the series x = 00
n
Therefore (L,x) _
> xnen converges in the Banach space lP(A).
n=1
Lnxn, Ln = (L, en). We introduce vectors 1
U.  (n=1IILn
Ii°)1P
L
en0enjILli°1'
n
n=1
where i = 0, 1; a = (al, ..., aN), IIa1I < 1, and a is a homogeneous vector. Then IIuN.IIP =
(E IILnII9)VP(E n=1
n=1
IILnIIP(91))1/P'
Chapter III. Distribution Theory
122
note that p(q  1) = q, and, consequently, Ilur,,,IIP < 1. Thus the vectors uN. lie in a unit ball of the Banach CSM lp(A). Furthermore, IILII = IIL°II + IILIII =
>
i=o
sup II(Lx)II I1x16<_1
sup II (L uNJII i=o IIaHI.<1
E sup i=0 11 IIoo_
1
IE Lnan
IILnIIqll (r
n=1 1
IILn!Iq)1/P
n=1
N
_ E(Y
IILnII9)1iq
= IILIiq
i=0 n=1
It remains to use Proposition 2.1.
Corollary 2.1. Let A be a Ealgebra. Then the Banach CSM lp(A), 1 < p < oo are reflexive. Corollary 2.2. Let A be a Ealgebra. Then the CSM l2(A) is isometrically isomorphic to the CSM 12(A).
Corollary 2.3. Let A be a Ealgebra. Then the CSM (An, II IIP), and (An, II l 1q), 1 < p < oo, 1/p+1/q = 1 are isometrically isomorphic. 2.6. Topological bases in Banach supermodules. A basis (topological basis) in a CSM M is a system of vectors {ap} such that any vector x E M can be uniquely represented as the sum x = > xpap,
xp E A.
We assume, as usual, that no more than a countable set of coefficients in the sum is nonzero.
Example 2.5. M = A', en = (0, ...,1, 0, ..., 0). Example 2.6. M = lp(A), 1 < p < oo, e = ( 0 , Example 2.7. M = P(Ak, A), e,3 (0) = OP.
...,
1,
0,
. . .)
.
2. Banach Supermodules
123
In these examples, the bases are homogeneous; in Examples 2.5, 2.6 the bases are even and in Example 2.7 lepl = 1,81 (mod 2). There exist Banach CSM without bases, these CSM may even be finitedimensional Klinear spaces. Example 2.8. Let p be a fixed element of the CSA A. We denote by M the CSM pA, i.e., Mo = pAo, M1 = pA1. If p2 = 0, then there is no basis in the CSM M. Indeed, assume that these is a basis {ap} in M. Then x = E xpap, where xp E A. However, ap = pbp, by E A, p
and, consequently, x =
ypap, where yp = xp + p.
Theorem 2.5. Suppose that there exists a body projector in the CSA A. Then the number of vectors in the basis of the CSM M over A is an invariant relative to the CSM M.
Proof. Let {aa} and {ap} be two bases in the CSM M. Then a. _ >p &pap, ap = E. Kp,,a,,,. The matrices K = {K,,,p} and K = {&p} are associated with the operators K, K: M + M, and KK = 1 and KK = 1. It follows from these relations that KbKb = 1 and KbKb = 1. The statement of the theorem immediately follows from these equalities.
Open question Is the dimension of the CSM over a CSA in which there is no body projector an invariant?
Theorem 2.6. In every finitedimensional CSM M over a CSA A with a quasinilpotent soul there exists a basis consisting of homogeneous elements.
The proof of this theorem (which is long enough) repeats verbatim the proof of a similar statement for a CSA with a nilpotent soul which can be found in the book by De Witt [27, pp. 2123]). Of the most interest for applications are modules over the super
algebra G. Open question Is there a homogeneous basis in the finitedimensional CSM M over a CSA A whose soul is not quasinilpotent?
Chapter III. Distribution Theory
124
In his book, De Witt advanced a conjecture that infinitedimensional CSM with a basis may not contain a homogeneous basis even if the soul of the CSA A is nilpotent. This conjecture has not yet been proved.
2.7. Operator matrix. Let M and N be CSM with bases {aa} and {ba} and let L: M 4 N be a left Alinear operator. Then
Lx = > xaLaa = > xa > Lapbp a
a
p
xaLap, bp, p
a
i.e., the operator L is realized by the matrix L = {Lap}. In this case, when the operator acts on the vector from the CSM, the vector row is postmultiplied by the operator matrix. Right Alinear operators can be considered by analogy.
Let now M be a CSM with a homogeneous basis {ice}, where a = 0, 1 and liael = a. A lefthanded coordinate system is defined for every vector x E M: x = E > xaiiae = xaiiae (we use the stipulation a i
concerning the summation over the repeating indices). We set eai = iae and introduce a righthanded coordinate system:
x = E Ei eaiiax = eaitax. Q
Note that xai = Let the operator U E £1,1(M). Then, in the lefthanded coordinate system, y = Ux = xaiu(iae) = xaiiaU13'13e, i.e., yp7 = xaiiaUp7. Consequently, the operator matrix U has a block structure ora(iax).
u=
(ou°
luo
ou1
lul l
If the operator U is even, then the matrices 0U° and 1u1 consist of even elements and the matrices 0W and 1U° consist of odd elements, i.e., JQU1J = (1)a+0+IuI; if the operator U is odd, then the matrices oU° and 1u1 consist of odd elements and the matrices 0U' and 1U° consist of even elements, i.e., IaU1l = (1)a+p+IUI again.
2. Banach Supermodules
125
Suppose now that U E £1,,(M). In this case, it is convenient to use a righthanded coordinate system, y = Ux = eQfl Uaji°x, j13y = j#U,,i"x. The operator matrix again has a block structure
u=
°u1
lu01u1)
Ou°
the relationship between the parities of matrix elements and the parity of the operator is the same as for a left Alinear operator.
If the operator U is even, then it is simultaneously right and left Alinear, with the right and left matrices related as j16uai = (_1)a(Cf+#)i.upj.
2.8. Continuous operators in Banach supermodules 1P(A). The continuous right Alinear operator L: lP (A) 4 1P (A), 1 < p < oo, is defined by its matrix 00
(Lx)i = E Lijxj,
i = 1, 2, ...
(2.2)
j=1
The space l... (A) does not have a topological basis, and therefore not every continuous operator is defined by a matrix. In this case, however,
we shall again restrict the consideration to an operator of the form (2.2).
We shall prove, as usual, that for the continuity of the operator L in the Banach space 1P(A), 1 < p < oo, it is sufficient that
IIL11 = E (I(E
II7II9)P/4)1/P < 00;
cx=0 i=1 j=1
in the space l.. )(A) it is sufficient that 1
IILijII < oo;
sup
IILII00 Q=0
j=1
and in the space 11 (A) it is sufficient that 1
00
a<=0
j=1
oo.
IILIII=
Chapter III. Distribution Theory
126
Theorem 2.7. Let a CSA A be a Ealgebra. Then the norms IILII and IILII coincide for the operator L E G1,,.(l... (A)).
Proof. For any e > 0 there exists an iE such that 00
Lij11 < E+ j iiL°jii.
sup =
j=1
j=1
N
R
Let us consider vectors upN = > ej/j, where the vector Q = (01i ..., NN) 7=1
is homogeneous and I1 Q 11 OO < 1. Then n
IILII > sup IIL'uaN11. = sup jjELj'Ojjj i,II0II00<1 j=1 II011051 N
> Consequently,
I
I L" 11 >
00
I
I L° j 1 > I I La 11 I
7=1
Laicjaj
 E.
We have a similar result for left Alinear operators.
2.9. Adjoint operator. Let the operator L E £1,1(M, N). A adjoint operator L' is defined by the relation (Lx, y) = (x, L'y),
y E N'.
x E M,
Proposition 2.3. (LR). Let the operator L E £1,1(M, N). Then the adjoint operator L' E £1,,.(N', M'), with IILII < IIL11
Proof.
The functional uy(x) _ (Lx, y) is left Alinear and continuous for any y E N', i.e., the operator L': N' + M' is defined. 2. The operator L' is right Alinear 1.
(x, L'(yA)) = (Lx, y\) = (Lx, y) A = (x, L'(y) A).
3. Let the operator L be homogeneous. Then 1
1
IIL'yll = E sup II (L'yQ,x)11 < IILII > 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<1,uEco(A)
sup
11(u, zcl) II,
lulloo<1,uE1.(A)
a = 0, 1. Thus we have IILII = IILII, where L' E ,C1,,(l.(A)), for the operator L. It remains to note that I I L' 11 = I I L' I I . = 11 L1 I I As a consequence, we get a similar result for the Banach CSM A. .
Suppose now that the operator L E C,,,(M, N). The adjoint operator L' is defined by the relation (L'y, x) = (y, Lx),
y E N',
x E M.
Proposition 2.3. (RL). Let the operator L E G1,r(M, N). Then the adjoint operator L' E G1,i(N') M'), with IILII < IILII Propositions 2.4, 2.5, and the analog of Proposition 2.6 are valid for right Alinear operators.
2.10. Banach supermodules of type lp with arbitrary graded bases. Banach CSM of Asequences considered above were defined as
Chapter III. Distribution Theory
128
Aspans of the even canonical basis {en}. This construction admits an obvious generalization to the case where the canonical basis is not even.
Suppose that A and B are finite or countable sets of indices, {noe}nEA are even vectors and {nle}nEB are odd vectors.
We introduce Banach CSM ip,A,B (A)
_
{x = xaiiae. Jjxjjp
1
= a=0 2 (E iEA
11(xoi)aM1p
+
II(x1i)10jjP)h'P < oo}, iEB
1
Proposition 2.7. The inclusions lq,A,B(A) C lp,A B (A), 1 < p < oo, 1/p + 1/q = 1 are valid.
Lemma 2.1. Let the elements al, ..., ak, bl,..., bi E G00' (T). Then k
I
EIlanll+EIlbnll n=1
n=1 k
sup
I E anon + E bn/3
UaHIoo,IIAlloo<1 n=1
I,
n=1
where aj, /33 E A, Iaj I = 0, 1/3, I = 1.
The proof of this lemma is similar to the proof of Theorem 2.3 (the elements of the algebra G , (T) can be "moved apart" not only with the aid of generators but also with the aid of pairwise products of generators).
Theorem 2.8. Let the CSA A = G' (T). Then the CSM lq,A,B (A) is isometrically isomorphic to the CSM lq,A,B(A), 1 < p < oo, 1/p + 1/q = 1.
2. Banach Supermodules
129
Proof. Consider the case of countable sets of indices. We introduce vectors N
n
U N = M II(ioL)°`III+ i=1
i=1
N
N
x (EQiioe
II(ioL)aIII1 +
EQiile
II(i1L)1allq1),
i=1
i=1
where a = 0 ,
1/P
II(i1L)1°Ilq)
IQl J = 0, I/ 1= 1; i.L = (iae, L). < 1 (by virtue of the choice of coefficients 0 and Q, the vectors Ll. are even). Furthermore, 1;
IiBMMOO, IIQII. < 1 ,
Then 11 U.O ll P
1
IILII
II(U
sup
,L°)II
a=o IIC1II<1,11I311oo<1
N
1
_E
sup
IE/3 0(L°`)
1I(ioL)'jIq1
a=o 1101151,11,611<_1 i=1
II (i1
L)1allq1([
i=1
E II(i1L)1'IIq)1/PII, i=1
i=1
a = 0, 1. with io(L) = (ioL)", i1(L") = It remains to use Lemma 2.1. Corollary 2.4. The Banach CSM IP,A,B(A), 1 < p < oo, A = (i1L)1a,
G , (T) are reflexive.
Corollary 2.5. The Banach CSM 12,A,B(A), 1 < p < oo, A = G ,(T) is isometrically isomorphic to 12,A,B(A).
We can see from the proof of Theorem 2.8 that it remains valid for any Banach CSA A which possesses property (2.3). Requirement (2.3) is stronger than that in the definition of the Ealgebra, where we considered homogeneous vectors. We do not yet know anything concerning CSA which possess property (2.3) and are different from G100 (T).
Open question Is Theorem 2.8 valid for an arbitrary Ealgebra?
Chapter III. Distribution Theory
130
3.
Hilbert Supermodules
3.1. Hilbert supermodule with an even basis. Let the CSM M be Banach and let CSM M and M' be dual. Then the form of A separates the points of the modules M duality ( , ): M x M' and M', is continuous, and possesses property (1.5) (and, in particular, is even).
Definition 3.1. The continuous bilinear form ( , ): M x M 4 A, where M is a Banach CSM, which separates the points of the CSM M and possesses property (1.5), is known as the scalar product on M.
Definition 3.2. The Banach CSM 12(A) provided with the scalar product (x, y) _ >00xnyn is called a coordinate Hilbert CSM (with an n=1
even basis).
Proposition 3.1. For the coordinate Hilbert CSM 12(A) we have the CauchyBunyakovskii inequality II(x,y)I1
(3.1)
11X11 IIyII
Definition 3.3. The triple (M, I I . , (. , )), where M is a Banach CSM with norm I I I I and scalar product ( , ) is known as a Hilbert II
CSM (with an even basis) if there exists an isomorphism S: M * l2 (A) which preserves (1) the scalar product (Sm1i Sm2) = (m1i m2), (2) the norm IISmMI = IImII
Note that in the definition of a Hilbert CSM it suffices to require the existence of a Rlinear isomorphism of linear spaces M and 12(A) which possesses properties 1 and 2.
Note, in the first place, that if the operator L is left and right Alinear, then it is even. Indeed, L(x)Q = L(x/) = L(crHHH (,3)x) = a1'1(Q)L(x),
3 E A.
Next, we assume that there exists an Risomorphism S: M + 12(A) which possesses property 1. Then (Am1i m2) = (S(Am1), Sm2),
3. Hilbert Supermodules
131
but A(m1, m2) = (.\Sml, Sm2), i.e., (S(.\ml)  .1Sm1i Sm2) = 0. It remains to note that S(M) = l2(A). The right Alinearity can be proved by analogy.
It follows from the definition of Hilbert CSM that the CauchyBunyakovskii inequality (3.1) holds true, the scalar product on M possesses not only property (1.5) but also property (1.4), and there exists an even topological basis in the CSM M.
3.2. Riesz theorem for Hilbert supermodules. Corollary 2.2 for a coordinate Hilbert supermodule immediately implies a superanalog of the Riesz theorem.
Theorem 3.1. Let M be a Hilbert CSM over a Ealgebra A. For every continuous Alinear functional L on M there exists a unique element m E M such that L(x) = (m, x),
x E M,
(3.2)
Conversely, if m E M, then relation (3.2) defines a continuous Alinear functional L on M such that IILII = IImII Thus, the CSM M and M' are isometrically isomorphic.
with IILII
=
IImII.
Open question Is there a superanalog of Riesz theorem for Hilbert CSM over an arbitrary Banach CSA?
3.3. Hilbert supermodule with an arbitrarily graded basis. The Banach CSM 12,A,B(A) provided with a scalar product (x) y) = x0:or a(Y'),
(3.3)
is called coordinate Hilbert CSM with a basis of parity (A, B). The CaychyBunyakovskii inequality (3.1) also holds for the scalar product (3.3). The scalar product (3.3) no loger possesses property (1.4) (if B j4 0). Property (1.4) is valid only for vectors of the form x = x0ijoe, and for the vectors x = x1';le we have a relation (x, ±) = (_1)1X141+1(±' x).
Chapter III. Distribution Theory
132
(
)) of parity (A, B) is defined with the aid of the isomorphism of the coordinate Hilbert supermodule The Hilbert CSM (M, 1 1
1
1
,
,
12,A,B(A) which preserves the scalar product and the norm. The definition of the Hilbert CSM of parity (A, B) and Corollary 2.5 yield
Theorem 3.2. Let M be a Hilbert CSM of parity (A, B) over the CSA A = G ,(T). Then the CSM M' is isometrically isomorphic to the CSM M.
The definition also implies the CauchyBunyakovskii inequality (3.1).
Definition 3.4. The homogeneous basis {iae} in a Hilbert CSM is said to be orthogonal (ONB) if (jae, ipe) = kap.
Proposition 3.2. A scalar product in coordinate with respect to ONB has the form (3.3).
3.4. Orthogonal operators. The operator U E £1,1(E, F), where E and F are Hilbert CSM, is orthogonal if U maps the CSM
E onto the CSM F (Im U = F) and preserves the scalar product ((Ux,Uy) = (x,y)). Proposition 3.3. Every orthogonal operator is even, and therefore an orthogonal operator is also right Alinear. Proposition 3.4. Every orthogonal operator is invertible, and the inverse operator is also orthogonal. Proof. By virtue of Banach inverse operator theorem, it suffices to show that KerU = {0}. Indeed, let us assume that x E KerU. Then (x, y) = 0, y E E, but the scalar product separates the points of the CSM E and, consequently, x = 0 (note that the relation (x, x) = 0 does not, in general, imply x = 0). Corollary 3.1. Orthogonal operators in a Hilbert CSM M form a group.
3. Hilbert Supermodules
133
A group of orthogonal operators is denoted by Os(M).
Proposition 3.5. The operator U E L01', (E, F), ImU = F is orthogonal if and only if it maps an ONB into an ONB.
Proof. 1. Let {iae} be an ONB and U be an orthogonal operator. We set iaa = uiae. Since ImU = F, any vector y E F can be represented in the form y = xaiiaa, i.e., {iaa} is a topological basis in F; it remains to use the fact that an orthogonal operator preserves a scalar product. 2. Let the operator U: E + F be even, continuous, and Im U = F, Then with {iaa = uiae} being an ONB for the ONB {ice}.
(ux,uy) = xai(iaa, ypjjpa) = xaiaa(yai) = (x, Y) 
We can see from the proof that it is sufficient that at least one of the ONB be mapped into an ONB.
Proposition 3.6. The operator U E C0,1 (E, F), ImU = F is orthogonal if and only if its matrix elements for the ONB satisfy the relations iau7kQ7(9pu7k) = 6i,jap.
Proof.
1.
(3.4)
Let the operator U be orthogonal. Then the vectors
{iaa = uiae} are ONB for any ONB {iae}, i.e., 8ijap = (iauryke,jpuµssµe) = iau7k(k7e,aµe)oµ(Apuµ9).
2. Conversely, it follows from condition (3.4) that the operator U preserves the scalar product. Example 3.1. (the orthogonal group 0.(12,{1},{1}(A)). Conditions (3.4) for the matrix 1
u = (OUoo °ul
IOU° I
= Bull = 0,
Iou11 =IiU°I=1,
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134
have the form (ou°)2
 (°ul)2 = 1,
ou°lu° + oullul = 0,
(lu°)2 + (lul)2 = 1, 1U°0u°
 luloul = 0,
i.e., the orthogonal group Os consists of matrices u =
0 C
b
I
,
where
a2 = b2 = 1, with a and b being even.
3.5. L2supermodules. For applications we need superanalogs of the Hilbert spaces L2.
We denote by cpa(x) Hermite (normed) functions on R. The Hilbert CSM L2 (R", dx) of functions, cp: R" + A, which are square summable with respect to the Lebesgue measure on Rn, is introduced as a completion of the tensor product A ® L2 (R", dx); here L2 (R", dx) is a space of realvalued square summable functions 00
L2 (Rn, dx) = If (x) = E facoa(x): f,, E A, Q=o 1
00
IIf112 = i(i IIfQ7112)1'` <
00
y=o a=O
This is a Hilbert CSM with even basis {cp,,(x) } which is isomorphic to 12(A). The scalar product on the CSM L2 (R", dx) has the form
(g, f) =
JR^
9(x)f (x) dx = E faga.
If the CSA A is a Ealgebra, then, by virtue of the superanalog of the Riesz theorem, any Alinear continuous functional L on the Hilbert CSM L2 (R", dx) is defined by the function
9(x) E L2 (R", dx): L(f) =
f g(x)f(x) dx.
Suppose now that (Q, a) is a measurable space and p is a orfinite countably additive measure on (Q, Q). The Hilbert CSM L2 (R", dµ)
3. Hilbert Supermodules
135
can also be introduced as a completion of the tensor product A 0 L2 (1,dµ). Under natural constraints imposed on the measure p, this CSM is also isomorphic to 12(A).
3.6. Nested Hilbert supermodule. Suppose that M is a Hilbert CSM, E is a locally convex CSM which is topologically em
bedded into a CSM M. We assume that the Eannihilator in M' is trivial:
1E = Ann (E, M) = {L E M': LIE = 0} = {0}. For this annihilator to be trivial, it is not necessary to require that the CSM E be densely embedded in the CSM M.
Example 3.2. Suppose that the set of indices A consists of one element and the set of indices B is empty. Then M = 12,A,B (A) = A (the basis in M is a unit element e from the CSA A). Let us consider the case A = G. Then E = is a (Banach) CSM embedded into M, with E _ {0} although E is not dense in M. If lE = {0}, then the CSM M' is embedded into the CSM E' and we get a quadruple of embedded CSM
ECMCM'CE'.
(3.5)
Quadruple (3.5) is known as a nested Hilbert CSM. If the CSA A is a Ealgebra, then M M', and quadruple (3.5) is replaced (just as in ordinary functional analysis) by a triple of embedded CSM
ECMCE'.
(3.6)
The framing of the CSM L2 (R", dx) is carried out, as usual, with the aid of the CSM of test and generalized Avalued functions on R. For instance, G(R", A) C L2 (R", dx) C (LA(R", dx))' C G'(R", A) or
D(R", A) C L2 (R", dx) C (L2 (R", dx))' C D'(R", A).
Chapter III. Distribution Theory
136
Now if the CSA A is a Ealgebra, then
g(R", A) C L2 (R", dx) C g'(Rn, A), D(Rn, A) C L2(Rn, dx) C D'(Rn, A).
Here the triviality of the annihilator follows from the density of embedding of the CSM of test functions into the CSM L2 (Rn, dx).
3.7. Superalgebras with involution. Recall that an algebra D over a field C is called an algebra with involution if the operation f 4 f * possessing the properties (1) (f *) * = f , (2) (Af + µg)* = A f * + µg*, (3) (f g) * = g* f *; f , g E D, A, p E C is defined in D. In a topological algebra D it is assumed that the involution is continuous. If f * = f , then the element f is real and if f * _  f *, then the element f is imaginary. Any element h is representable in the form h = x + iy, where x, y are real elements, x = 2 (h + h*),
y = 2i (h  h*).
The involution * in the CSA A = A° ® Al is consistent with the Z2graduation if *: A° ® Al + A° ® Al is an even operator. In this book we only consider involutions consistent with the Z2graduation. Example 3.3. (involution in G1(C)). It suffices to determine the involution on the generators qj; qj* = qj (i.e., the generators are real
elements). Then, by virtue of property 3 of the involution, we have the monomial qjl...qj is real if 2n(n1) is even (q and imaginary if 2n(n  1) is odd. By analogy, we can define the involution in the CSA G00' (B), where
B is a commutative algebra with involution. Let A be a banach CSA with involution *. We can introduce in A an equivalent norm I I I I I satisfying the condition I aI I I= I I Ia* I I (a star norm), for which purpose we must set I I a I I I= max(I I a I I, I I a* I I) Setting IIaII' = IIIa°III + IIIa'III, we obtain an equivalent norm satisfying the star condition I I a I I' = I I a* I I' and a Z2graduated a I I' = I I a° I I' + I I a' I I'. Everywhere in the sequel, we only consider star Z2graduated norms on CSA with involution. I
I
I
.
I
3. Hilbert Supermodules
137
Note that the norm on G'00(B) is a star norm if the norm on B is a star one.
3.8. Hilbert supermodule with involution. A module W over the algebra D with involution * is called a module with involution if the operation (denoted by the same symbol ') f H f' possessing the properties (1) (f*)* = f, (2) (.*f) = f*)*, (3) (f + g)' = f ` + g1,
f, g E
W, A E D is defined on W. Just as we did in Sec. 3.7, we define real and imaginary elements and the involutions in the CSM consistent with the Z2graduation. A Banach CSM 12(A) over a CSA A with involution * endowed with *scalar product 00
(x, y) = L xnyn n=1
is called a coordinate Hilbert CSM (with an even basis) with involution.
Here are the main properties of a *scalar product. 1. Hermiticity: (x, y)` = (y, x). 2. `Alinearity: (Ax, pya) = \(xa*, y)µ'. 3. The CauchyBunyakovskii inequality: (x,y)
x112y2 Similarly, coordinate Hilbert CSM with involution with bases of an arbitrary graduation; *scalar product in 12,A,B (A) over a CSA A with involution is defined by the relation (x, y) = x' (y°')' In contrast to the scalar product (. , ), the properties of a *scalar product ( , ) do not depend on the graduation of the basis.
Definition 3.1. (*). The bilinear form ( , ): M x M + A which separates the points of the Banach CSM M and possesses properties 12 is called a *scalar product on the CSM M. Definition 3.2. The triple (M, 1 1 , ( , )), where M is a Banach CSM with norm I I II and *scalar product ( , ) is called a Hilbert CSM with involution if there exists an isomorphism S: M + 12,A,B (A) which preserves the *scalar product and the norm. 11
If a CSA A is a Ealgebra, then a superanalog M' ^_' M of the Riesz theorem is valid for Hilbert CSM with involution.
Chapter III. Distribution Theory
138
3.9. Selfadjoint operators in Hilbert supermodules with involution. Let M be a Hilbert CSM with involution. The operator at which is an (Hilbert) adjoint of the operator a E £1,1(M, M) is defined by the relation (ax, y) = (x, a*y). Just as we did for the opeator a', we can verify that a*: M' + M' is continuous. However, in contrast to the operator a', the type of Alinearity for the operator a* is the same as for a, at E £1,1(M', M'). Indeed, (x, a* (Ay)) _ (ax, Ay) = (ax, y) A* _ (x, Aa*y)
Let the CSA A be a Ealgebra. Then M' ^_' M and the operator a*: M + M. The operator a is selfadjoint if a* = a. Unbounded selfadjoint operators are introduced in the same way as in the ordinary functional analysis. As usual, if (1Qa0j) is a matrix in the ONB {i,,e} of the selfadjoint operator a in a Hilbert CSM with involution, then the matrix elements satisfy the condition iaaAi =
(jfa")*
Conversely, any Alinear continuous operator satisfying condition (3.7) is selfadjoint.
3.10. Unitary operators in Hilbert supermodules with involution. An operator U E £1,1(E, F) (G1,,.(E, F)), where E and F are Hilbert CSM with involution, are said to be unitary if U maps the CSM E onto the CSM F (ImU = F) and preserves the *scalar product ((Ux, Uy) = (x, y)). As distinct from an orthogonal operator (Proposition 3.3), a unitary operator is not necessarily even.
Example 3.4. Let M = lz,{1},{1}(A) _ {x = xooe + x11e: xa E A},
where oe is an even base vector, le is odd.
Consider an operator
U: M * M, Uoe = 1e, 111e = oe. Then (Ux,Uy) = (x,y), U E L'1'1 (M, M) Proposition 3.4. (*). Every unitary operator is invertible, and the inverse operator is also unitary.
3. Hilbert Supermodules
Corollary 3.1.
139
Unitary left (right) operators in a Hilbert
(*).
space CSM form a group. Groups of unitary left and right operators are denoted by U,,, (M) and U,,,. (M) respectively.
Proposition 3.5. (*). An operator U E G1,,(E, F) (U E G1,r(E, F)), Imu = F, is unitary if and only if its matrix elements for ONB satisfy the relations u7k
u7k *
= joij.
Example 3.5. Let the CSM M be the same as in Example 3.3 (one even vector and one odd vector). Then the group of unitary operators U,,,(M) consists of matrices U=
°U0 0ul ) lu°
lul /
whose elements satisfy the conditions OUOOUO* + OU1OU1*
= 1,
lu°lu°* + lullul* =
1,
= 1. If the CSA A = G ,, then, for instance, any matrix of the form OU01UO* + 0U11U'*
U=
1+gtg3
qj
qj 1
+ gtg3
satisfies these conditions.
Proposition 3.7. Let the operator a E £1,,(M, M) be selfadjoint. Then the operator U = e" is unitary. The proof repeats the standard one.
3.11. L2supermodules with involution. Let A be a real CSA and A' = A ® iA be its complexificatoin. We introduce a superspace
Chapter III. Distribution Theory
140
T ,m = Ao x (A')' in which even coordinates are real and odd coordinates are complex. Let us consider a system of functions go :
V,' ) Ac, cp0p (x, 0) = cp,,(x)0P, where {cp,,(x) } are Hermite func
tions on R"; the parity of the functions W,,# is 101 (mod 2). We introduce a Hilbert CSM with basis {cpa }:
(
L2`(Tn'm, dxdO) = If (x, 0) ffQ E AC, 11f
112 =
_
(x, 0): aQ
E
IIf.10112+
161=0
(mod 2)
191=1
(mod 2)
+(E IIffp112 + > 191=0
IIf.10112)1,2
IIff0112)1/2
< oo}.
161=1
(mod 2)
(mod 2)
This Hilbert CSM is known as a CSM of A`valued functions which are square summable with respect to the distribution dxd0 on a superspace
TA''. The involution in this CSM can be introduced coordinatewise: f ` _ {f}. pThis involution is consistent with the Z2graduation: the *scalar product (f , g) _ ffpg.*,6 can be represented as an integral over a superspace: (f,g) = fSfl,m f (x, 0) (g (x, 0))`dx dB` d9. A
In order to frame the Hilbert CSM L2 ` (TA'm, dxdO), we must do the following. Instead of the superspace TA'm we must consider a narrower superspace Tu'm, where U is a pseudotopological CSA with
nilpotent soul constructed with the use of the CSA A. The framing of the Hilbert CSM L2`(Tn'm, dxd0) is carried out with the aid of the spaces of test and generalized functions on the superspace Tu'm. In the second chapter, we introduced spaces of types G and D on the superspace Ru'm = Uo x Ltl `; by a complete analogy, we can introduce
spaces of types g and V on the superspace Tu'm = U x (Ul )"` (here the anticommuting variables become complex, but the functions with respect to these variables are polynomials). In the case where A is a Ealgebra, we have (TT'm, A`) C L2 ` (TA'm, dxd0) c
(Tu'm, A`)
4. Duality of Supermodules
141
or
D(TT'm, A`) C L2`(TA'm, dxdO) C D'(TT,m, Ac).
4.
Duality of Topological Supermodules Everywhere in the sequel we use the symbols M and N to de
note topological CSM over the topological CSA A. By Ln,r and we denote the submodules Ln,,. and Ln,i of the CSM that consist of continuous mappings; Ln (L  L,) is a space of continuous nlinear mappings over the field K. In addition to continuous polylinear mappings, we shall be interested in mappings which are continuous on compact sets. The spaces of these mappings are denoted by JKn,r and lc,,,, respectively. If the topology on a CSM is metrizable, then the continuity on compact sets implies continuity, i.e., in this case we have Ln,r = Kn,r, Ln,1 = Kn,l.
The analogs of Propositions 1.11.8 are valid for the CSM Ln,r, Ln,l, Kn,r, Kn,l
4.1. Topologies of a bounded and compact convergence on a conjugate supermodule. Just as in a Banach case, by identifying the CSM Ln,r(M, A) and Ln,l(M, A) we obtain a CSM M' which is topological conjugate of M. The forms of duality ( , ): M x M' * A and ( , ): M' x M + A possess properties (1.4) and (1.5). As in the theory of topological Klinear spaces, in the theory of topological CSM an important role is played by the topologies of uniform convergence on the conjugate CSM M'. Let us consider the case of locally convex CSM M and CSA A. Every bounded subset B C M and every prenorm II II from the defining system of prenorms IPA on the CSA A are associated with a prenorm on the conjugate CSM M': IIILIIIB = sup
IIL(x)II.
(4.1)
xEB
If {B} is a system of bounded subsets in the CSM M, then the topology of bounded convergence (topology of uniform convergence on bounded subsets) is defined by the system of prenorms (4.1). In
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142
the same way we can introduce the topology of compact convergence on M' (the topology of uniform convergence on compact subsets). In a standard way, from prenorms (4.1) we can obtain equivalent Z2graded prenorms JIL11B = UUIL°I11B+1HIL'IIIB. If M is a Banach CSM over Ba
nach CSA A, then the topology of bounded convergence coincides with the normalized topology on M'.
4.2. Topological bases and Schauder bases. As in the Banach case, we can define the topological basis {ep} in the topological CSM M.
Example 4.1. Consider functional CSM A(Cm) and The monomials ec,p = x°9p form a topological basis of these CSM.
Example 4.2. Consider CSM of generalized functions and E'(C"m). The generalized derivatives e'Q0 = ;i axae A
of the
Jfunction form a topological basis in these CSM. It follows from the definition of a topological basis that there exists
a family of functionals ep E M* such that x = E (x, eQ)ep, and the families {ep} and {eQ} are biorthogonal. Just as in ordinary functional analysis, we shall call the topological basis {ep} a Schauder basis if the functionals ep are continuous, ep ep E M'. For the Schauder basis we have
x=
(x, eQ)ep.
(4.2)
p
For a wide range of locally convex Klinear space, the pointwise convergence of a sequence of linear continuous operators implies a uniform convergence on compact subsets. Locally convex spaces of this kind are known as barreled space [71]. In particular, metrizable and Banach spaces are barreled spaces. For Banach spaces this is well
known BanachSteinhaus theorem. Moreover, every inductive limit of barreled locally convex spaces is barreled. Consequently, inductive limits of Banach and metrizable spaces and, in particular, practically all spaces of test functions that are encountered in analysis are barreled. We can use all these results in superanalysis since modules are
4. Duality of Supermodules
143
Klinear spaces and Alinear operators are, in particular, Klinear. In this case, series (4.2) converges uniformly on compact subsets of these CSM M.
Example 4.3. A CSM A(C"m) is a Frechet space, and, consequently, the series f (x, 0) = i f,,px'013 converges uniformly on any Qp
compact subset of a functional CSM.
Example 4.4. A CSM E(Cn'm) is an inductive limit of Banach spaces, and, consequently, the statement of the preceding example is valid here as well.
4.3. Dual topological supermodules. An essential difference from the theory of locally convex Klinear spaces is that the duality forms do not, in general, separate the points of the CSM M and M' (even in a Banach case, Sec. 2). In order to exclude these pathological cases from consideration, we distinguish a class of "good" CSM with the aid of the following definition.
Definition 4.1. Topological CSM M and N are said to be dual if there exists a bilinear form ( , ): M x N + A (form of duality) continuous on compact sets which separates the points of the CSM M and N and satisfies condition (1.5).
In particular, a duality form is even. The duality form ( , ): N x M  A is defined by the relation (1.4). Dual bases are the topological bases {ep} in M and {e'} in N, biorthogonal relative to the duality form, with respect to which the series converge uniformly on compact subsets of the CSM M and N respectively; a duality form in coordinates is written as (x, y) = > xpyp
Example 4.5. Let the CSM M = A(C" m) and the CSM N = A'(Cnm). Then they are dual CSM with dual bases
and {e' p} (it should be pointed out that the space of generalized functions A'(CA'm) is an inductive limit of Banach spaces, and therefore series (4.2) converges uniformly on compact subsets).
Adjoint operator. For topological CSM, an adjoint operator is defined in the same way as for Banach CSM.
Chapter III. Distribution Theory
144
Proposition 4.1. Suppose that the conjugate CSM M' and N' are endowed with a topology of compact or bounded convergence; the operator a E G1,,(M, N). Then the adjoint operator a' E C,,r(M', N').
The continuity of the operator a' follows from the fact that for any compact (bounded) subset B C M the set a(B) is compact (bounded) in N.
Example 4.6. Let the CSM M =
(Run", Ac) or D(Ru", AC), Then the operators e E £1,1(M), j = 1, ..., n, and Am), E(Cnm) A(C M' =g'(Rum, Ac), D'(Ru'm, A`), the adjoint operators (B A'(Cn'm), E'(Cn'm) respectively. .
5.
Differential Calculus on a Superspace over Topological Supermodules 5.1. Superspace over topological supermodules.
Definition 5.1 [144]. A Klinear topological space X = MO ® N1, where M = Mo ® M1 and N = No ® N1 are topological CSM, is called a superspace over a pair of CSM M and N.
Definition 5.2. A CSM Lx = M ® N is said to be a covering of the superspace X = MO ® N1.
Example 5.1 (VladimirovVolovich superspace). Let M = Am, N = A". Then X = MO ® N1 = KA ", LX = Am+" In particular, if A = G"(K), G', then X is a Rogers superspace. De Witt superspace is the simplest example of a nonBanach space (Chap. I, Sec. 4).
Example 5.2 (superspace of Fermi fields). We denote by M a CSM consisting of some class of functions (perhaps, generalized) cp: Rk + Am. Then X = M1 = {cp E M, W: Rk + Am} is a superspace of Fermi fields.
Example 5.3 (superspace of a system of intersecting Bose and Fermi fields). We set M = {cp: Rk  Am}, N = {co: Rk + A"} (in
5. Differential Calculus
145
specific models, conditions of smoothness and boundary conditions are imposed on functions from the classes M and N); X = MO ®N1 = {cp: KK'"}. Rk >
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 <_ CL Sup
Ilbn(v1,...,vn)II
viE By
Thus,
Sup Sup IIABv < CL E 1 n=0 n! vj EBo Wj EBW
II(v1,w1)...(vn,wn)II
It remains to use the continuity of the form of duality on compact sets.
Proposition 6.3. Suppose that the superspace W = V' is endowed with a topology of uniform convergence on compact sets. Then the functions from the space (D(W) are of the first order of growth. 6.2.
The proof of this proposition is contained in that of Proposition Indeed, the estimate I f (w) I < C exp{ I I Iw I I }, where I Iw I I I= I
I
I
I
sup II (v, w) II is a continuous prenorm on W, is valid for functions from
vEBv
the space 4)(W).
Proposition 6.4. Suppose that the sequence of vectors {rj},'?_1 is contained in a compact subset B covering the CSM Lv, and let
6. Analytic Distributions
165
{Ljl...;n} be a sequence of elements of the CSA A satisfying the condition 00 00 Rn
CL =
E 1141 ... in I I< o0
jl...in
n=O
for a certain R > 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 <
1 sup n! zjE R
Ilanf(0)(zl,...,zn)II
where BR = RB. We have not obtained an explicit description of the space of test functions D(W).
Remark 6.2. The problem of describing Fourier transforms of the spaces of measures and distributions on infinitedimensional spaces was posed in the wellknown work by Fomin [62]. However, no essential advances (even in the Klinear theory) have been obtained.
Remark 6.3. The operators .F and F of the Fourier transformation belong to the spaces LO,, (A',,D) = L0,1 (A', c) and LO,, (M, A) _ L°,i(M, A).
The direct product and the convolution of distributions on the superspaces V and W are defined as usual (see Chap. II). The operations of direct multiplication and convolution in the spaces A' and M are
Chapter III. Distribution Theory
166
correct. The spaces of distributions A' and M are convolution CSA. The operators F and F' of the Fourier transformation transform a direct product and a convolution of distributions into a product of Fourier transforms. Remark 6.4. For the superspace CA'm the space of generalized functions M(CA'm) = E'(CAm) is a topological conjugate of the space of test functions (Cn'm) = E(Cnm). In the general case, the question concerning the topology in the space of test functions (D(W), in which M (W) = c' (W) , remains open.
7.
Gaussian and Feynman Distributions
Out of numerous of definitions of Feynman integrals over infinitedimensional space (see, e.g., [1, 3, 5, 7, 24, 26, 45, 59]) we choose a definition based on the Parseval equality. This definition (for Klinear spaces corresponding to boson theories) was used by many authors. We can point out the articles by De Witt and Morette [97], Albeverio and HoeghKrohn [1], Slavnov [126], Uglanov [133], Smolyanov [127], and Khrennikov [138141, 145, 67].
This definition is based on a simple but very fruitful idea. Note that (in the sense of the theory of generalized functions) the Fourier transform of a quadratic exponent 2n , x E R, is also a quadratic 71 et22
exponent Let the function cp(x) be a Fourier transform of the function cp(s) (generalized, in general): cp(x) = f d. Then ein(2/2.
eix2/26
J co(x)
27rib
eix2/26
dx

ix2/26
_
0,
F ( e 27rib)) =
Jd(e)e2/2.
(7.1)
It is an infinitedimensional analog of the Parseval equality (7.1) that is used for defining the Feynman integral. The notation det B 1 w 1
(x)e`(Blx,x)/2
00 ( dx;
H =1
27ri
7. Gaussian and Feynman Distributions
167
where B is a linear operator in an infinitedimensional space, is set, by definition, equal to
f
ei(Bt,t)/2(P(<)
= ((P,
e=(B(,()/2),
(7.2)
where (p is a distribution on an infinitedimensional space, (p(x) = (the space of test functions of an infinitedimensional f argument must contain quadratic exponents). A detailed analysis of this definition, in particular, the choice of spaces of test functions, can be found in [128]. Note that the analog of (7.1) holds for Gaussian measures on R, and in an infinitedimensional case definitin (7.2) turns into an equality:
f (P(x)vB(dx) =
fe_'2(de),
where vB is a Gaussian measure with a covariant operator B. As was pointed out in Chap. II, Gaussian integrals even over a finitedimensional superspace are integrals with respect to generalized func
tions and not with respect to measures. Therefore, in a supercase, it is natural to use the Parseval equality for defining both Feynman integrals and Gaussian integrals.
7.1.
QuasiGaussian, Gaussian, and Feynman distribu
tions. Definition 7.1. A quasiGaussian distribution on a superspace W is the distribution rya,B E M (W) with the Fourier transform Jr'(7a,B) (v) = exp{ 2 B(v, v) + i(v, a) },
(7.3)
where (mean value) a E W, B (the lefthand covariance functional) belongs to the space K2,1(L2 , A), the restriction of B to the superspace V2 is symmetric. By virtue of Parseval equality (6.2), we have fw
(P(w)'Ya,B(dw) = fv (p(dv) exp{2B(v, v) + i(v, a)}.
Chapter III. Distribution Theory
168
If there exist projectors b of the body and (c = 1  b) of the soul in the CSA A, then we can define Feynman and Gaussian distributions on a superspace. Definition 7.2. A Gaussian distribution is a distribution 7a,B for which Re b(B) > 0, Im b(B) = 0.
Definition 7.3. A Feynman distribution is a distribution rya,B for which Re b(B) = 0.
7.2. Feynman distributions in the quantum field theory. Countably additive Gaussian measures on Rlinear spaces are Gaussian distributions. These distributions can be extended from a space 4> of test functions to a space of continuous exponentially growing functions.
Feynman "measures" for boson systems (see [7, 26, 50, 53]) are Feynman distributions. Of especial interest are the following examples of Feynman distributions with fermion degrees of freedom.
Example 7.1. (Feynman distributions for a spinor field). We set V = q(R4, A8) and W = 9'(R4, A8). It is convenient to write the elements of the superspaces V and W as v = (r)j, 77j)'o and w _ ('0j, ,0j);.0. Then the form of duality 3
(v, w) = = j=0
,/,
,/1
(x) j (x) + ; (x) l'j (x)) dx.
R
14
We denote by ryS a Feynman distribution on the superspace W with zero mean and covariance functional
BS(v, v) = 2i I l(x)Sc(x
 y)77(y) d4x d4y,
where
S`(x  y) =
1
(27r)4
+m I P2 Pm2 + io
ei(P,xy)d4P
is Green's causal function of the Dirac equation
(i8  m)Sc(x) = 6(x)
7. Gaussian and Feynman Distributions
169
3
3
µ=0
µ=0
(we use the notation j3 = E yµpµ, a = E 'y, where yµ are Dirac µ
matrices). The Feynman integral f U(,0, z,)'ys(d dzb) is defined for the func
tions U from the class 4>(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 < CPI
2(BQXBp)
By analogy with the PDO calculus on Rn (see, e.g., [5, 32, 44, 73, 74]), we can rewrite formula (1.8), which connects various types of symbols and the composition formulas (1.9), (1.10), (1.11), with the aid of infiniteorder differential operators. We define a differential operator with constant coefficients as a PDO with a polynomial symbol (of class A(P); it should be emphasized that in an infinitedimensional case there exist polynomials which are not compactly Sentire) which depend only on momenta:
R(iaL)(co)(q) =
f
R(o,(p))e=(n,a)c(dp)
(1.13)
Proposition 1.1. Let the form b E 1Cn,,.(Lp A) and suppose that the restriction of b to the superspace Pn is symmetric. Then we have a relation b(i8L)(co)(q)
Chapter IV. Pseudodifferential Operators
192 n
k=0
"n
b(e71, ..., ejk, ail, jl...jkEJ il...ikEI 1 0 1 xpjl0 ...pjkpLil ...pLin_k
(1.14)
Proof. It follows from formula (1.13) that n
b(iaL)(co)(q) _
(1)nkCn
k=0
x f b((p0)k,
(p1)nk)ei(p°,q°)+i(p1,g1)gdpodpl)
n
_ E(1)nkCC k =O
E b(e71, ..., elk, ail, .., ain_k jl...jk il...in_k
x f p1
(1.14).
Proposition 1.2. Let a be a PDO with a symbol depending only on momenta, a E A(P). Then the operator ea: 1>(Q) * 1>(Q) is defined which is a PDO with symbol e°().
The proof is a direct consequence of (1.12) for PDO with an Sentire symbol. In particular, an exponent is defined for any differential operator with constant coefficients.
Theorem 1.8. If at, a, E 1)(Q x P) are t and ssymbols of the PDO a, then aL aL l j  s)Caq' ap)}a,(q, p). at(q,p) = exp{i(i
(1.15)
Proof. Using formula (1.8), we have at (q, p) = f a,(dp dq') exp{i(p', q) + i(q', p) + i(s  t) (p', q')}.
1. Pseudodifferential Operators Calculus
193
We set X = P x Q, Y = Q x P, and then a,(dx)ei(x,Y)ei(sc)b(x)
at(y) = f
where b(x) = (p', q'). Note that JbI = 0 and b(a(x)) = b(x). Hence, we obtain at(y) = f ei(,t)b(a(x))ei(x,y)a,(dx) = (1.15). Theorem 1.9. Let a, a1, a2 E D(Q x P) be 7symbols of the PDO a, a1i dz and a = al o a2. Then aaL
aL
a(q,p) = exp1zr( g1, apt) + z(T  1)
aL 41L
aqz apl (1.16)
xal(g1,p1)a2(g2,p2)I91=q2=q P1=P2=P
Proof. Using formulas (1.9)(1.11), we obtain
a(q, p) = f a1
®a2(dp'dq'dp"dq")e(17)i(p q')Ti(P',q")
x ei(P',q)+i(P",q)+i(q',P)+i(q",n)
We set X = Pz X Qz, Y = Q2 X Pz, and then
a(q,p) = f a1 ®
a2(dx)e(1T)ib1(x)rib2(x)ei(x,v)
Y=(q,P,q,P)
where bi (x) = (p", q'), b2 (x) = (p', q"). Note that I b1 I _ I bz = 0 and bi(o(x)) = bi(x), bz(o(x)) = b2(x). Therefore
a(q,p) = f
e(1T)ib1(a(x))irb2(a(x))
x ei(x,+J)a1 0 az (dx) l y=(q,r,q,P) = (1.16).
Let the function cp belong to the class A(P). Then it follows from formula (1.14) that
_ P(21L) _
n,aRW(0)((iaL)n).
Chapter IV. Pseudodifferential Operators
194
Theorem 1.10. Let a, al, a2 E 'ID(Q x P) be 7symbols of the PDO a, al, a2 and a = a1 o a2. Then
a(q,p) = al(q+iT
aL
api
,p+i(T  1)
x a2 (q + ql, p + a (pl))
'9L
)
9ql (1.17)
P1=0.
91=0
Proof. Using formula (1.16), we obtain a(q,p) = f (al o'yr) ® a2(dpi dgldp2dg2) x et(P2,g1)+1(p1,g2)+i(P2,q)+i(g2,P)
where ryr is a Alinear continuous operator, 'yr(f) (pl, ql) = ei(P',q)+i(g1,P) f (Tpl, (1  T)ql),
yr: A(P x Q)
A(P x Q).
Furthermore,
(a1 o'yr)(dpi dgl)a2(q+ql,p+a(p1))
a(q,p)
Let us calculate the Fourier transform of the distribution a1 o ryr
.P(a1 o'yr)(q',p) =
f al
07r(dpldgl)et(p1,q')+i(g1,P')
= al (q  Tq', p + (1  T)p') In conclusion, we shall use formula (1.4) for a PDO with a symbol depending only on momenta. We set X = P x Q and Y = Q x P, and then a(q,p) = bgp(fgp) (0) _ (1.17), where bqp is a PDO with a al(qTq',p+(1T)p'), fqp (x) = a2(q+q',p+a(p')), symbol bgp(x) = x = (p', q'). Let a be a PDO with a qpsymbol a(q, p). Then, by virtue of Theorem 1.3, we can symbolically write this operator as 2
2
1
1
0 1 0 1 a = a(40,43,P3,P31),
1 1
4 = 4L,
1 1
P = PL
1. Pseudodifferential Operators Calculus
195
Here we use Maslov's notation [44]. The digits over the operators denote the order in which the operators of the coordinate and momentum act. For instance, for the pqsymbol a(q,p) we have 1
2
1
2
0 0 1) = a(Qj, 4j,pj,jj 1
The PDO calculus that we constructed above constitutes rules for dealing with ordered resolutions in terms of operators of (left) coordinates and (left) momenta satisfying the commutation relations pk }
= [qj
qk }
l0j , qk } _
= 0,
(1.18)
In many applications there appear resolutions of operators a in terms of the operators u and v that satisfy commutation relations different from (1.18). Let C: P 4 P be a fixed even Alinear operator which is diagonal in the basis {ej; ai}; Cej = c°ej, Cai = c;a;, Ick = 0. To every
operator C there corresponds its own PDO calculus with Sanalytic symbols. The PDO ac can be determined from the relation (1.19)
ac(cp)(u) = f a(u,
where the symbol a(u, v) E A(QxP). The PDO ac can be symbolically written as 2
2
1
1
ac = a(u°, u;, v°, where the operators ucl ,
[v
,
vjcl
, a = 0, 1, satisfy the commutation relations
vk} _ [zt , uk} = 0,
[v uk } = ihCk J.pbjk. ,
(1.20)
The calculus for PDO (1.19) can be constructed by a complete analogy with the PDO calculus with qpsumbols. If C = I is an identity operator, then formula (1.19) coincides with formula (1.12) and the commutation relations (1.20) coincide with (1.18). The most important part in applications to the quantum field theory is played by Wick (normal) symbols. We can obtain these symbols
Chapter IV. Pseudodifferential Operators
196
if we choose the operator of multiplication by i in P as the operator C. In this case, the operators f i9 denoted by a, and the operators In the field theory, the operators a°' are the operators of vjc' by a,". creation of bosons, the operators a'* are the operators of creation of fermions, the operators a° are operators of annihilation of bosons, and aj are operators of annihilation of fermions. These operators satisfy the canonical commutation relations
[a
[aJ ak } = 0,
[ai , ak
ak.
} = hbapbjk.
,
(1.21)
The PDO f with the Wick symbol f (a', a) E A(Q x P) can be symbolically written as 2
2
1
1
Here the operators of annihilation are the first to act, and then their action is followed by the action of the creation operators. The relativistic PDO calculus also plays an important part in the quantum field theory. Let T and R be dual superspaces P = TD, Q = RD, p = (p")D o , q = (q")D o , and suppose that {gµ"} is Minkowski metric (goo = 911 = = 9D1D1 = 1). Let the operator C act according to the rule Cp" = g""p", v = 0,..., D  1. Then the PDO a defined by relation (1.19) can be symbolically written as 2
0 v,
2
1 (qk)",(pk)
1 (Pk)"),
a = a((qk
where the operators of the coordinate and momentum satisfy the relativistic commutation relations [(pj )", (Pk)µ} =
[(qj)", (qk)µ} = 0,
[(P7 Y, (qk )µ} = 2h6.#6jk9µ".
This formalism can be generalized in many directions, for instance, we can choose an operator C which is nondiagonalizable or odd. The generalization of the calculus of pqsymbols can be constructed by
197
2. The Correspondence Principle
analogy. Of special interest here are antiWick symbols. PDO with antiWick symbols can be symbolically written as 1
f=
1
f(a;.,a,
z
z
In contrast to Wick PDO, here the operators of the creation of bosons and fermions are the first to act, and their action is followed by that of the annihilation of bosons and fermions. The PDO calculus developed in this section is the calculus of operators which are right Alinear. In particular, for 7symbols the operators can be resolved in terms of left operators of the supercoordinate and supermomentum. By analogy, we construct the calculus of PDO which are left Alinear. PDO with the Tsymbol can be resolved in terms of the right operators of supercoordinate and supermomentum. Mixed calculi are also possible in which we use both left and right operators of supercoordinate and supermomentum. These PDO are also very important for applications and a successive exposition of the theory of leftright PDO would be of a considerable interest. We can
obtain another interesting generalization of the PDO calculus on a superspace by considering the Tsymbols for the parameter T E A.
2.
The Correspondence Principle With an operator approach, the procedure of quantization of a
physical system which has boson and fermion degrees of freedom can
be realized as follows. The functions a(q,p) on a phase superspace (finitedimensional or infinitedimensional) are put into a correspondence with a PDO (see formula (1.2)): et (W) (q)
= f a((1
 10q + rq', p)co(q') eXp{ h (q  q', p') }d9 dP
,
where h is a Planck constant. In this case, the operation of pointwise multiplication of functions becomes an operation of quantum multiplication * (see composition formula (1.16)): al (q, p) * ai (q, p)
Chapter IV. Pseudodifferential Operators
198
= exp{ihr( aL , aL) + 2h(,  1)( aL , aL )} 9q2 api aqi ape x al (qi, pi )a2 (g2, p2) I q1=92=q D1=P2=P
[al, a2}. = al * a2 
1
1
)IaiIIa2l a*2a1.
The principle of correspondence between classical and quantum
theories consists in the fact that in the approximation h = 0 the quantum theory turns into the classical theory: lim ai (g, p) * a2 (g, p) = ai (g, p) a2 (q, p),
o
(2.1)
=: {al, a2}(q,p),
u o h[ai, a2}.(q,p) where I. , .1 are Poisson brackets on a phase superspace.
Remark 2.1. The Poisson brackets on a superspace were introduced by Martin in his work [114].
We introduce Poisson brackets on the phase space Q x P setting aL agz
If, 9}
aL
api)

x f (qi, pi)9(g2, p2)
aL
aL
ape C age
)]
91=92=q
D1=p2=P
Using the formula that connects the righthand and lefthand derivatives (Chap. I, formula (1.6)), we obtain 09f
{f,9}= [Ca p°'a4 )\ 19g
af
ag
,apo)]
+[/aRf 9L9) + /aRf aL9 \ 9ql , apl \ '9P1 aq l ,
where Ip°l = jq' I = a, a = 0, 1.
2. The Correspondence Principle
199
Definition 2.1. The Lie superalgebra is a Z2graded linear space G = Go ® G1 in which the bilinear operation [ , ] (called a commutator) is defined, and the identities I[x,y]I = 1x( + IyI; (_1)lxl lyl+1[y,
[x, y] = [x, [y,
z]](1)1x1IZI + [z,
x];
[x, y]](1)IZI IYI + [y, [z, x]](1)IYI I=I = 0
are valid for homogeneous elements. The last identity is a superanalog of Jacobi identity (the second and
the third term in it result from a cyclic permutation of the elements x, y, z in the first term).
Definition 2.2. The Lie superalgebra G = Go ® Gl which is a CSM over a CSA A and in which the commutator is Abilinear, [)xa, y/3] = \[x, ay]Q,
A, a, Q E A,
x,yEG,
is a Lie superalgebra over the CSA A.
Proposition 2.1. A CSM
x P) with a commutator equal to a Poisson bracket is a Lie superalgebra over a CSA A, and
if, g} =  f x ((l /, q')  (',
f
®g(dp'dq'dp'dq")
q"))e'(P'+p",q)+i(q'+q",p)
Proof. Since there are dual topological bases in the superspaces Q abd P, we can restrict the consideration to the case p E C"1, q E We set u(e, 77) = e'(f,q)+'(+r,p), where q and p are fixed. Then f f ® 9(dpi dg1dp2dg2) x(paq + pagi  gzpi + g2pi)u(pl + P2, q1 + q2)
= f f (dpldgl) (q° f 9(dp2dg2)pzu(p2, q2)
Chapter IV. Pseudo differential Operators
200
P'l f 9(dp2dg2)gzu(p2, q2) + f 9(dP2dg2)pau(P2, g2)gi
+ f 9(dp2dg2)g2'u(p2, g2)p1)u(pi, qi) =
f
f (dpldgl)4°u(p1, q1) f(dp2dq2)pu(p2,q2)
 f f (dpi dg1)p°u(p1, qi) f
.
(dp2dg2)gzu(p2, q2)
+(1)1§1+1[f f (dpldgi)giu(pi, q1) f.(dp2dq2)pu(p2,q2)
+ f f (dpldgl)piu(pl, q1) f 9(dpd2)g2 (p2, q2)]
of ag
of ag
 [ap0 aq°  aq° app
+ (_1)Isl+1
aRf aRg 0R! aRg apt aql + aql ap1) _
1f,
g}.
The commutator [ , }, on the space of symbols 1 (Q x P) is known as a quantum commutator. The results of Sec. 1 yield [
,
Proposition 2.2. A CSM D(Q x P) with a quantum commutator }, is a Lie superalgebra over the CSA A.
Theorem 2.1. (correspondence principle). Relations (2.1) and (2.2) are valid for any functions al, a2 E c(Q x P).
Proof. Formula (1.16) immediately yields (2.1). From the same formula we get
l o h [al, a2 }. (q, p) =[TCaL aL )+(T1)\aL
aL aq2' apl
aqi ape
x a,(g1,pl)a2(g2,p2)
91=92=9
Pl =P2=P
+(1)1alI1a21 [T(
aL
aL ,
age
19P1
+ (T  1) \
aL
aL
aql 'ape
J
2. The Correspondence Principle
201
xa2(g2,p2)a1(g1,pi)I q1=q2=q = {al, a2}. P1=P2=D
Thus, the Lie quantum superalgebra over the CSA A, namely, (41(Q x P), [ , },), is a continuous deformation (cf. [46]) of the clasx P), { , }). sical Lie superalgebra over the CSA A, i.e., Apparently, the concept of Lie superalgebra is not adequate to the correspondence principle in the quantum field theory. In a finitedimensional case (supersymmetric quantum mechanics), the Lie su
peralgebra 1(Q x P) (a space of entire Sanalytic functions of the first order of growth) contains symbols of all real Hamiltonians. In an infinitedimensional case, the Lie superalgebra D(Q x P) only contains symbols of model Hamiltonians. Indeed, the Lie superalgebra D(Q x P) contains only polynomials which satisfy constraints of the type of nuclearity (see Proposition 4.4, Chap. III). For instance, the Lie superalgebra (D(Q x P) contains the Hamiltonian function H(q, p) = (Ap, p) + (Bq, q),
where A and B are nuclear operators (boson system), whereas real quantumfield Hamiltonian functions have the form (for free theories)
H(q,p) =
1(p,
p) +
(B'q,
q),
where B is a nuclear operator.
Definition 2.3. A Lie supermodule over the Lie superalgebra G = Go ® Gl is a Z2graded linear space M = MO ® M1 in which the bilinear commutators [ , ]: G x M * M, [ , ]: M x G + M are defined, and the identities I[9,m]I=191+Iml; [g, m] = (_ 1)19111+1 [m, 9] [91, [92,
+ [m, [91,
m]](_1)'91I I1 + [92, 92]](_1)1L 1921
= 0,
are valid for homogeneous elements.
[m,
9, 91, 92 E G,
91]](_1)19211911
mEM
Chapter IV. Pseudodifferential Operators
202
If M and G are CSM over the CSA A and the commutator is Abilinear, [,\ga, m/3] = ,\[g, am]/3,
a, /3, .1 E A,
g E G, m E M,
then M = Mo ® Ml is a Lie supermodule (over the Lie algebra G = Go ® Gl) over the CSA A.
Let f be a PDO with a qpsymbol f E A(Q x P) and let g be a PDO with a qpsymbol g E 1)(Q x P). Then g maps the space 4) into 0 and f maps the space
fog: D(Q) 4 A(Q) is defined.
Furthermore, relation (1.4) makes it possible to extend the domain of definition of the PDO g from the space c to A. Thus, the operator
g o f : (Q)
A(Q)
is defined.
We denote by 00 and OA the spaces of PDO with symbols from the classes 1 and A respectively. We introduce a commutator
[f g} = fog ,
(1)111 M§01.
Proposition 2.3. The space OA is a Lie supermodule over the Lie superalgebra O, (over the CSA A). Proof. It suffices to prove that the space OA is a twosided module over the algebra O0 relative to the operation of composition of PDO. We set u = f o g. When proving Theorem 1.1, we established that g(W) _ .P((g ®,p) o S), where S is an even operator. Using (1.12), we obtain
u(co)(g) = f f (q, ha(p))e",9)g((P)(dp).
(2.4)
Note that the direct product g ®cp is a left direct product, i.e., first the functional cp acts from the left and then its action is followed by
2. The Correspondence Principle
203
that of functional g. Therefore we must transfer the functional g(cp) in relation (2.4) from right to left: '
(co)(q) = (1)I9(v)I Ifl f 9(()(dp)f (9,
ha(p))e'(P,q).
Next, using the definition of the operator S, we have u(W)(q) _ (1)I9(W)I
X f (q, hu(P +
_
III
f 9(dP dq% (dp) p))e'(p+P',q)+i(P,q')
(_1)I9(W)I IfI+Igl(lvl+Ifl)
f(
x (f f (9, ha(p'
(dp)e'(P,q)
+p))e'(P'q)+i(P,q)§(dp
dql )
= f u(9, ha(p))e'(P,q)(P(dp), u(q, p) = Jf(g,hcrQi) +
p)e'(P',q)+'(q',P)9(dp'dq')
E A(Q x P).
We set v = g o j. From relations (1.4) and (1.12) we have
v(W)(q) = f
9(dp'dgl)e'(P'q)
x f f (q + hq', hu(p))e=(P,q+hq)(Pldp)
= f v(q, ha(p))e'(v,q)c(dp), v(q, p) = f 9(dP dq')f (q + hq',
p)e'(P',q)+=(q',P)
The commutator [. , } on the operator Lie supermodule A(Q x P) induces a commutator on the space of symbols:
[f, 9}. = f f (9, ha(p) + p)e=(P'q)+i(q'P)9(dP d9 ) (1)III I9l r 9(dP dq')f (q + hq',
p)e'(q',P)+i(P',q>
(2.5)
Chapter IV. Pseudodifferential Operators
204
The Lie supermodule A(Q x P) over the Lie superalgebra (D(Q x P) with supercommutator (2.5) is known as a quantum Lie supermodule. Commutator (2.5) is an extension of the commutator which can be defined on the space 4D(Q x P) with the aid of the * operation.
Proposition 2.4. The space of Sentire symbols A(Q x P) with a Poisson bracket as a commutator is a Lie superalgebra (over the CSA A)
In particular, the space of symbols A(Q x P) is a Lie supermodule over the Lie superalgebra (D(Q x P). This Lie supermodule is said to be classical.
Theorem 2.2. The classical Lie supermodule A(Q x P) is a limit as h  0 of the quantum Lie supermodule A(Q x P), i.e., (2.2) holds
for any f Proof. Transforming relation (2.5) for a quantum commutator, we have
r (f (q, hu(p') + p)  f (q + hq', p)) xe'(q',p)+i(p',q)g(dp dq')
=
i(aa f (q, p), f o,
i(af a (q, p), f aRf aLg ,
,9p
aq)
(p')e`(q'p)+=(q,a(p'))9(dp
q'e'(q',P)+i(p',q)9(dp dq'))
aRf (aq
d4 ))
+ o(h)
aL (ap)g) + o(h)
= { f, g} + o(h).
Note that commutator (2.5) can be written as
[f, g}* = exp{ ih(
, aL
aL
aq2 aP1
) } (f (qi, pi)g(g2, p2)
(1)1h1 i9ig(q,,Pl)f(g2,P2))I q1=q2=q P1 =P2 =P
3. FeynmanKac Formula
205
Relation (2.6) allows us to extend the domain of definition of the commutator [ , }. with A(Q x P) x ''(Q x P) to wider classes of symbols. If the function [1 , g}.(q, p), which can be calculated from formula (2.6), belongs to the class A(Q x P), then, by definition, we
set [f,.} = [f,g}. (i.e., the PDO [f,.} is defined as a PDO with an Sanalytic symbol [f, g}.). Note that this PDO can even be defined in the cases where both operators f o g and g o f are not defined. This fact is of a special interest in an infinitedimensional case (even for ordinary numerical coordinates). All results obtained in this section can be immediately generalized to the PDO a defined by formula (1.19). In particular, the correspondence principle for the quantum Lie supermodule A(Q x P) also holds for Wick symbols.
It should be emphasized that qpsymbols (as well as all generalizations of these symbols (1.19)) play a special part in an infinitedimensional case. For other symbols (including pqsymbols and Weyl symbols), we cannot construct anything similar to the quantum supermodule A(Q x P). For these symbols, we can only consider the quantum Lie superalgebra 4)(Q x P) which, as has been noted, is insufficient for applications to the quantum field theory.
3.
The FeynmanKac Formula for the Symbol of the Evolution Operator
The new functional approach to superanalysis makes it possible to consider continual integrals over spaces of paths in a superspace. We
interpret a path as a real mapping of the range of variation of the evolution parameter into a superspace. When we used an algebraic approach to superanalysis, we used the symbol of an integral along paths but no paths (maps) really existed, and the continual integral with respect to anticommuting variables constituted a purely algebraic
construction and was rather an object of algebra than that of the functional analysis. In this section, we shall consider a Feynman path integral over a space of paths in a phase suprespace which appears in the Feynman
Chapter IV. Pseudodifferential Operators
206
Kac formula for the symbol of the evolution operator of a PDO. Use is made of the PDO calculus on a superspace developed in the preceding sections. We restrict our consideration to a finitedimensional (dimension over A) case.
3.1. Composition formulas on CA''. For the superspace coincides with the space the functional space 4)(CAm) = of entire Sanalytic functions of the first order of growth E(CA'm). The PDO a with a qpsymbol a(q, p) E E(C" '2m) is extended, with the use The of formula (1.4), to a continuous operator in the space
PDO a with a qpsymbol a(q,p) E A(Cl '2m) is defined by relation into A(Cn'm). This operator is (1.12) and maps the space also continuous.
In what follows, when deriving the FeynmanKac formula on a phase superspace, we shall use the composition formula for the PDO qpsymbols. Formula (1.10) can be rewritten in terms of Fourier transforms a and b of the symbols a, b E E(CA '2m), a * b(q, p) = f a(dp'dq')b(dp'dq'")
x exp{i(p", q) + i(p' + p", q) + i(q' + q", p)}.
(3.1)
Now if a is a PDO with a qpsymbol of the class E(C2 '2m) and b is a PDO with a qpsymbol of the class A(C2 '2m), then we shall use the composition formula a * b(q, p) = f a(dpdq')b(q + q', p) exp{i(p', q) + i(q', p)}.
(3.2)
3.2. Representing the qpsymbol of the evolution operator as a quantum chronological exponent. In the space of Sentire symbols of A(C2 '2m) we consider the Cauchy problem 8a(t, q, p) = h (q, p) of
*
a( t , q, p) ,
a(0, q, p) = 1.
(3 3) .
(3.4)
3. FeynmanKac Formula
207
This Cauchy problem is equivalent to the Cauchy problem for the operatorvalued functions
dta(t) = ha(t),
a(0) = I.
Thus, the solution of problem (3.3), (3.4) is the symbol of the evolution
operator for the PDO h. In the following theorem, the evolution parameter t belongs to the commutative Banach algebra A0.
Theorem 3.1. Let the PDO symbol h have the form h(gx,g9,px,po) _
h.Q(gx,px)geqe) as
q = (qx, qo),
p = (px, po)
E Cp m, h,Q(dpdq')e'(q'e,P=)+i(P'e,q)
hcip(gx,px) = fR2,.
where hc,p are Avalued measures with compact supports on R2i. Then there exists a unique, Sanalytic with respect to t E A0, solution of the
Cauchy problem (3.3), (3.4), a: Ao 4 A(C2 '2m). The solution is defined by the relation tk
a(t, q, p) _ (*) exp{th(q, p) } 
h*
.k.
*h(q, p)
k=0 oo tk
=
k=0 k!
f
h(dpidgi)...
k
fii(dpdq) k
x exp{i > (pj, qi) + i(y:pj, 4) + i(E qj, p)}. 1<1<j
j=1
(3.5)
j=1
Proof. The symbol h belongs to the class E(C2 '2m) and, consequently, the expression h * ... * h has sense and the expression h * a has sense for any function a of the class It suffices to prove the convergence of series (3.5). Since all functions are polynomials with respect to Qei pei it suffices to prove the
Chapter IV. Pseudodifferential Operators
208
theorem for an even case. We restrict the consideration to the case
n=1,m=0. In the first place, we have
Ila(t)IIR <00:
IIk
IIh * ... * hIIR.
k=0
We denote by IhI the variation of the measure h and by var h the total
variation, varh = IhI(R), where K is the support of the measure h, and by K = sup{Ikj: k E K}. Using the composition formula (3.1), we obtain
00
00 Rn+m
IIh*...*hIIR<w
n=0 m=0 n!m! k
xJ 00
k
Qjm
KIhl(dpidq')...fxlhi(dpkdgk)IEPiInI, 7=1
7=1
00 Rn+m
<EE
n=0 m=0 n !m!
(var h)krn+mkn+"` = (var h)ke2Rk
Consequently,
IIa(t)IIR < exp{IItII varhe21}.
We denote by M(K) the space of Avalued measures whose supports are in the compact set K. The space M(K) with norm var p is a Banach space. We set
M(Rn) = lim ind M(K), K
In the following theorem the evolution parameter t is real.
Theorem 3.2. Let h(t) be a family of PDO which depends on the parameter t E R, 0 < t < a, with symbols of the form h(t, 4x, q0, px, pe)
hap (t, Qx, px)4B pe
1
crp
h,,,6 (t, 4x, px)
= f.2n
h,,p(t, d4 dp)e`(ge,p:)+i(P'e,9s)
3. FeynmanKac Formula
209
[0, a] + M(R2n) are continuous functions. Then there
where
exists a solution of the Cauchy problem for the symbol of the evolution operator 8a(t, q, p) = h(t, q, p) * a(t, q, p), at a(0, q, p) = 1.
The solution belongs to the class C1((0, a), A(C2 '2m)) and is defined by the quantum chronological exponent: t
a(t, q,p) = T(*) exp{I h(s, q,p) ds} 00
k=O 00
=
k
t
f 0dtlh(tl, q, p) * ... * fo
tk1
dtkh(tk, q, p)
t
O
f dtl... ft'' dtk f h(tl, dpi, dqi)... f h(tk, dpk, dqk 0
a
k
k
x exp{i E (pj, qj) + i(E pj, q)+ i(E qj, p)}. 1
)
j=1
(3.6)
j=1
Proof. We can again restrict the consideration to the case n = 1, m = 0. We shall show that series (3.6) converges uniformly on the interval [0, a]. We set A = max var h(t), r. = sup lc(t), lc(t) tc [O,a]
tc [O,a]
sup{IrI: r E supp h(t)}, and then we have 00
00
00 Rn+m
IIa(t)IIR << k=0 n=0 m=0 X
f
c,
dt1... f
tk
1 dtk f I h(tl)I (dpidgi) In
X J I h(tk)I (dpkdgk)
00 (ya)k 00 00 Rn+m
1:
k!
f
n=O
m=0
n m'
k
pj j=1
k=O
n!m!
j=1
qjlm
Chapter IV. Pseudodifferential Operators
210
By analogy, we can prove the convergence of series (3.6) which we have differentiated with respect to t. It is natural to call the first part of (3.6) a quantum chronological exponent.
Everywhere in this book we use an additive representation of a chronological exponent.
Suppose that R is an arbitrary topological algebra, h(t) is a function acting from the interval [a, b] into the algebra R (satisfying a number of constraints that depend on the algebra). Then the additive representation of the chronological exponent T exp{ fa h(t)dt} is the series
00
r
k=oJa
6
dtlh(tl)
r
Ja
tk
tl
dt2h(t2)... f dtkh(tk) a
(for whose convergence we impose the constraints on the function h(t)). If the algebra R is Banach, then the continuity of the function h: [a, b] * R is sufficient for this series to converge. A chronological exponent defines the solution of the Cauchy problem for the evolu
tionalry equation u(t) = h(t)u(t) in the algebra R. If R is a matrix algebra, then the additive representation for a chronological exponent is called a matrizant (see Gantmacher [23, p. 405], series (40)). A more complicated situation is considered in Theorem 3.2, namely, the operation * of the composition of PDO is chosen as a multiplication operation (therefore this chronological exponent is called a quantum exponent). The function h(t) maps the interval into the algebra E(C2 '2m). An additive representation series can be written for this function. However, this series does not converge in the topological algebra E(C2 '2i`). The series converges only in a wider space A(Cl '2m) which is not an algebra relative to the operation *. Along with an additive representation of a chronological exponent, use is often made of a multiplicative representation (a multiplicative chronological integral). However, we do not use this representation
in this book. A detailed exposition of the theory of multiplicative integrals for operatorvalued functions can be found in [26] (see [23, p. 407] for the description of a multiplicative integral for matrixvalued functions).
211
3. FeynmanKac Formula
3.3. Feynman path integral in the framework of perturbation theory. The class of functions of an infinitedimensional argument for which the Feynman distribution integral is defined (see Chap. III) is too narrow to be applied to the FeynmanKac formula. This class can be extended considerably with the use of perturbation theory on an infinitedimensional superspace. Let Y be an infinitedimensional superspace and let the function f (y) belong to the class A(Y) with its Sderivatives belonging to the class 1(Y). If the series 00
1
E n1 f f(n) (O) (y, ..., Y)'Ya,B(dy) converges, then the function f (y) is said to be summable in the framework of the perturbation theory with respect to the Feynman distri
bution 'Ya,B and the sum of series (3.7) is denoted by the integral fy .f (Y)'Ya,B (dy)
3.4. Representation of the symbol of an evolution operator as a Feynman path integral over a space of paths in a phase superspace. We denote by B([O, t], Cam) the Banach space of Borel bounded paths q: [0, t] + Cn'm (i.e., the mapping q of the interval [0, t]
into the Banach space C m is measurable relative to the aalgebras of Borel subsets and IIgII00
= sup
Ijq(s) jI < oo).
s E [O,t]
We set Q([0, t], CA'm) _ {q E B([0, t], CAm): q(t) = 01; P([0, t], Cn'm) = {p E B([0, t], Cn'm): p(0) = 0};
P([O, t], CA') = Q'([O, t], CA'm);
Q([0, t], Cn'm) =
Pl([O,
t], CA'm);
Y([0, t], Cn '2m) = Q([0, t], Cn'm) X P([0, t], Cn'm). X ([O, t], Cn'2m) = P([0, t], CAm) X Q([0, t], CA'm);
Chapter IV. Pseudodifferential Operators
212
On an infinitedimensional superspace X([0, t], C2 '2'") we consider a quadratic form t
b(x, x) = f f 0(s  T) (p(dT), q(ds)) 0 0 x = p ®q,
(P
where B(s) is an ordinary Heaviside function, and 0(0) = 0 (the last condition is very important and is actualy equivalent to the choice of a qpsymbol for a PDO). On the dual superspace Y([0, t], C2,,2,) we introduce a Feynman distribution v with zero mean and correlative functional (2ib).
Theorem 3.3. Suppose that the conditions of Theorem 3.2 are fulfilled. Then the symbol of the evolution operator can be represented as a Feynman path integral (in the framework of perturbation theory) over the space of paths in a phase superspace t
a(t, q, p) = fY([0,t],C2m` T exp I h(T, q + q(T ),
p+P(T)) dT} v(dq(.)dp(.))
(3.8)
Proof. Note, first of all, that functions on the paths in a phase superspace
uk(q(.), P(.)) = f0t dt1...
ftk1
dtk h(tl, q + q(tl), p + p(tl))
...h(tk, q + q(tk), P + p(tk)) = F(Lq,p,k) (q(.), p(.)) E ID(Y),
where the distributions Lq,p,k E A' (X) are defined by the relation ft
(Lq,p,k, f) =
1
dt1... f tk dtk f h(t1, dpi, dqi)... f h(tk, dpi) dqk) k
k
x exp{i(E pj, q) + i(E qq, p)} j=1
i=1
3. FeynmanKac Formula
213
k
x f\
k
pj6(T  tj), j=11
q,6 (r  ti)) j=1
E A(X), where b(T  tj) are Dirac for the test functions f measures concentrated at the points tj. Using relation (5.3) from Chap. III, we find that
a(t,q,p) = >00f F(Lq,,,,k)(q(),p(.))v(dq(.)dp(.)) Y k=0
00
a fx Lq,p,k(dp(.)dg(.)) exp{i f t f t 0(s  T)(p(dr), q(ds))} dtk f h(tl, dpi, dqi)... f h(tk, dpk, dqk) f0 dt1... f 00
tk
1
0
k=0
k
k
exp{i(>Pj, q) + i(> qq,p)}
x
j=1
x exp{i
j=1
Jot 1 0(s  T)J(T  tj)
(p,, q j=1 1=1
00
 ti)drds} _ E
xb(s
k=0
ft
dt1...
f 41 dtk 0
0
x f h(t1, dpi, dqi)... fil(tk,dp,dq) k
k
k
x exp{i(E p q) + i(E qq, p) + i ,
j=1
j=1
00
j=1 1=1
tk
k=0 fo
t
1 dtk f h(t1, dpi, dqi)... f h(tk, dpk, dqk)
dt1... fo
k
k
x
k
E(pj, q)0(tt  tj) }
exp{i(E p q) + i(E qq, p)+ i E ,
j=1
j=1
(pj,qi)}.
1<1<j
3.5. Generalized solutions. Recall that any entire Sanalytic function f (x, 0) on a superspace C"' is a polynomial with respect to
Chapter IV. Pseudodifferential Operators
214
anticommuting variables 0 and is therefore representable in the form
f (x, 0) = i fp(x)A, where fg(x) are entire Sanalytic functions of commuting variables x E Ao A. We shall use the fact that the topology in the functional space A(C" m) is equivalent to the topology defined by the system of norms Ilf I1x = max
maxEC
IIb(zle,
..., z,le)
The Cauchy problem (3.3), (3.4) in the functional superspace A(Cl '2m) is of a very general nature and includes, in particular, all partial differential equations with entire coefficients of the first order of growth if we consider the symbols h E E(C2 '2m) as well as systems of equations of this kind. The existence of a solution of problem (3.3), (3.4) which would be classical for t is very problematic even in the absence of superstructures. The coefficients from the superalgebra make the matter even more complicated. In what follows, we construct a solution of this problem that is generalized with respect to t. We can write Cauchy problem (3.3), (3.4) as a Cauchy problem for an ordinary differential equation in the Frechet CSM A(CA ,2m),
a(t) = h.a(t),
a(0) = 1,
t c A0,
(3.9)
where h.: A(C2 '2m) A(C2 '2m), a H h * a. As has been pointed out, this operator is continuous if the symbol h(q, p) belongs to the class E(CA '2m).
Let us consider, in an arbitrary locally convex CSM M, the Cauchy problem (3.10) r(t) = Ax(t), x(0) = xo, t E A0,
where A is a Alinear operator in M. The formal solution (3.10) of this problem can be written as an exponential series 00 tnAnx0 x(t) = 
ni
(3.11)
n=0
If series (3.11) converges in the CSM M (at least for sufficiently small
t), then it defines the mapping from the range of variation of the
3. FeynmanKac Formula
215
evolution parameter t (a ball in the Banach space Ao) into the CSM M which is a classical solution of the Cauchy problem (3.11). However, the qualitative estimates of series (3.11) for the general symbols from the class E(C2 '2m) show that the series diverges for arbitrarily small
time moments t (i.e., the solution is "chucked out" from the space of analytic functions in an arbitrarily small time moment). In this situation we can consider a generalized solution with respect to t. For the linear differential equation from (3.10), the generalized solution with respect to t can be defined as a solution of the equation
fx(t)c(t)dt+Afx(t)co(t)dt = 0,
(3.12)
where cp(t) belongs to some space of test functions U(Ao) and the generalized solution x(t) belongs to the space of (vectorvalued) generalized functions U'(A0, M), namely, the space of Alinear continuous operators from the space of test functions It (Ao) in the CSM M.
Remark 3.1. It stands to reason that the classical solution of the Cauchy problem (3.10) may exist even if series (3.11) diverges. For instance, for an unbounded selfadjoint operator in a Hilbert space a continuous solution of the Cauchy problem ±(t) = iAx(t), x(0) = xo exists for all initial conditions xo from this space although series (3.11) may diverge (series (3.11) converges for analytic vectors). Since the solution a(t) is "chucked out" from the functional space A in an arbitrarily small time moment, it is natural to choose as test functions the functions W(t) which affect the solution only dur
ing infinitely small time intervals, i.e., functions with a support at zero. However, these are generalized functions. Thus, we propose a new conception concerning a generalized solution, namely, spaces U containing generalized functions can be chosen as spaces of test functions.
Remark 3.2. The ordinary classical solution fits in this scheme. The classical solution is a generalized solution with a space of test functions U(R) that contains generalized functions (measures) W, (t) =
b(t  s). If the space U(R) = D(R) or G(R), then we obtain an ordinary definition of a generalized solution.
Chapter IV. Pseudo differential Operators
216
Remark 3.3. Spaces of o additive differentiable measures on infinitedimensional spaces are widely used as test spaces in the theory of infinitedimensional distributions. Let x(t) be a classical solution of the Cauchy problem (3.10) from the class A(Ao, M). This solution can be regarded as a linear continuous functional on the space of test generalized functions A'(Ao):
f
x(n (0)
dt =
n!
n
(t , P(t)),
n=0
and the value of the classical solution at a point is defined by the relation x(s) = (x(t), 5(t  s)). Therefore it is natural to choose as a space of test functions U(Ao) a certain subspace of the space of generalized functions A'(Ao). We set
W(A0) = { E A'(Ao): IIWil,,
= E n! II(tn,
,)Ile,n2
< oo
n=O
va > 0}.
The topology in the space W (Ao) is defined by a system of norms Il,,}, a > 0, which is a countably normed space (a Frechet space). Lemma 3.1. If W(t) E W(Ao), then the generalized derivatives of all orders cp(k) (t) E W (Ao) .
Proof. Indeed, =
i(p
W(k))Ileanz
ll(tn' nE n!
00
= n=k E 1,n II(tn, Lemma 3.2.
co)Ile'(n+k)2 <
CII(PIIa
a' > 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,Iz21
11f (zle, zee) jI, zj E C.
Consequently,
,1721 Irilma
I f h(dpldgl)... f h(dpkdgk)
k
k
j=1
j=1
xexp{iTl>pj+iT2>qj+i 1<1<j
pjgllll
k
max
max
IriI,IT2I
exp{ZT1 E zlj j=1
k
+2T21: z2j+i E j=1
zljz21 }I
1<1<j
< Ch exp{2RR1k + R2k(k  1)/2}.
Chapter IV. Pseudodifferential Operators
218
Let cp c W(A0), and then 11f a(t, q, p)W(t) dt IRl
< 000k (tk, W) 11 exp{2RRlk + R2k(k  1)/2} k=0
< Bh,R 11WIlp,
p > R2/2.
Representing a generalized solution as a Feynman path integral over a space of paths in a phase superspace. 3.6.
Our aim is to obtain formula (3.8) for a generalized solution of the Cauchy problem (3.3), (3.4). We shall consider here symbols which do not depend on the time t but belong to the class E(C2 '2m). We shall consider the time t from the commutative Banach algebra A0, i.e., the Feynman path integral will be an integral along the paths from A0 into a phase superspace. The Feynman path integral in the FeynmanKac formula (3.8) was considered in the framework of perturbation theory. We shall also use here the definition in the framework of perturbation theory, but the series of perturbation theory converges in the space of generalized functions W'(A0, A(CA '2m)) A closed interval in A0 is the set
[0, t] = {s E A0: s = at, a E [0, 1] C R}. Recall (see Chap. I) that an integral over a closed interval
f f (s)ds = t f 0t
I
f (at) da.
We denote by B([0,t],C"m) a space of bounded Borel paths q: [0, t] + CAm. This is a Banach superspace with norm s
Just as we did in Sec. 3.4, we introduce superspaces X([0, t], Cn'Zm), 1'([0, t], c'2) and define the Feynman distribution v on the infinitedimensional phase superspace Y([0, t], C2 '2m). Theorem 3.4 and calculations similar to those used in the proof of Theorem 3.3 give a FeynmanKac formula in a phase superspace for generalized solutions.
3. FeynmanKac Formula
219
Theorem 3.5. Let the symbol h(q, p) belong to the class E(CA '2m). Then the symbol of the evolution operator a(t, q, p) can be represented as a Feynman path integral over a phase superspace
f a(t, q, p)cp(t) dt f
=/ J
r
[
J
t
T exp{f h(q + q(T),
Y([O,t],cn '2m)
p + p(T))
cp(t) dt,
where cp E W(Ao).
3.7. Paths in a phase superspace on which the Feynman distribution is concentrated. Here we discuss a problem concerning the choice of phase superspace Q([0, t], Cnm)P([0, t], CA 'm) the continual integration over which gives the symbol of the evolution operator. By virtue of relation (5.3), Chap. III, we have fy
V(q('),p(.))v(dq(.)dp(.))
= fx dcP(P(.)q(.)) eXp{i f Jot 9(s  T) (p(dT), q(ds)) }, where cp is a linear continuous functional on the space of entire Sanalytic functions on a dual phase superspace X. Consequently, as a space of the trajectories of coordinates Q([0, t], CA") and those of momenta P([0, t], Cn'm) we can choose any superspaces of paths for which the covariance functional of the Feynman distribution is continuous. We shall call an infinitedimensional phase superspace which possesses this property a superspace of paths of the Cauchy problem (3.3), (3.4).
It stands to reason that a phase superspace constructed on the basis of a superspace of bounded Borel paths is a superspace of paths of the Cauchy problem (3.3), (3.4), but this superspace is too large and it is interesting to find narrower superspaces of paths.
Chapter IV. Pseudodifferential Operators
220
We shall restrict the consideration to a real evolution parameter t. We denote by VQ0, t], CA ") the superspace of functions of the bounded variation v: [0, t] + CA "`; j jv M v = var v l o which is the total variation on the interval [0, t]. We denote by R([0, t], Cn'm) the superspace of functions without discontinuities of the second kind r: [0, t] + CA'"`, Ilrlloo _ sup 1jr(T)II. TE[o,t}
Theorem 3.6. Let Q([0, t], Cam) = {q E R([0, t], Cn'm): the coordinate q(T) is right continuous, q(T + 0) = q(T), and q(t) = 01, P([0, t], Cn'm) = {p E V([0, t], Cn'm): p(0) = 0}. If the CSA A is a Ealgebra, then the infinitedimensional phase superspace Q([0, t], CA'"`) X
P([0, t], CA'"') is a superspace of paths of the Cauchy problem (3.3), (3.4).
Proof. Let us consider a family of functions q,(T) = 9(s  T),
0 < s < t.
These functions have no discontinuities of the second kind, are right continuous, and q, (t) = 0, 0 < s < t. Consequently, and the function q, E
1(s) =
fq(r)(dr) == (gs(T),P(r)), t
1 < j < n + m,
is well defined. We shall show that this function has a bounded variation (cf. the proof of Riesz theorem [37]). Let 0 < so < sl < ... < sn = t be partitioning of the interval [0, t]. Then we have nn
,,.,7
> 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<1 k=1
221
4. Unsolved Problems <11911,
P E P([0, t], Cn'm) = Q'([0, t], Cn m).
Thus, IIpMIv < IIPII We note that qo(T) = 0, and therefore p(O)
_
(qo,P) = 0. Thus, p(t) E P([0, t], Cam), and the quadratic form
x = P ® q H (p, q)
is well defined, with II (p, q) II < Consequently, the covariance functional of the IIpIIvMIgMI < IIPII I. Feynman distribution v is continuous.
Theorem 3.7. Let Q([0, t], Cn'm) _ {q E V ([O' t], CA'): q(t) = 0},
P([0, t], CA'm) _ {p E R([0, t], CA 'm): momentum p(s) be left continu
ous, As  0) = p(s) and p(O) = 0}. If the CSA A is a Ealgebra, then the superspace Q([0, t], C m) x P([0, t], CA m) is the space of paths of the Cauchy problem (3.3), (3.4).
Proof. Let us consider the family of functions pr(s) = 9(s  T), 0 < T < t. These functions have no discontinuities of the second kind, are left continuous, and pT (0) = 0, 0 < T < t. Consequently, pT E P([0, t], Cnm) and the function q3 (T)
qi) is well defined.
By analogy with the proof of the preceding theorem, we find that varglo < IIgII Then we have pt(s)  0 and, consequently, q(t) _ (pt, q) = 0. The next part of the proof repeats that of the preceding theorem.
Theorems 3.6 and 3.7 reflect the uncertainty principle. If the trajectories of coordinates have a bounded variation, then we can only require that the moments have no discontinuities of the second kind, and, conversely, if the trajectories of momenta have a bounded variation, then we can only require that the coordinates have no discontinuities of the second kind.
4.
Unsolved Problems and Possible Generalizations
As was pointed out in the introduction to this chapter, only the first steps have been made in the PDO theory on a superspace. It
Chapter IV. Pseudodifferential Operators
222
should be emphasized that the PDO calculus is constructed on an infinitedimensional superspace. Thus two problems have been simultaneously solved here. The first problem is connected with an infinite dimensionality of a phase space and the second problem is connected with a superstructure. It has turned out that these (seemingly quite different) problems have much in common, namely, both in an infinitedimensional case and in a case of a superspace we have to solve the same problem, the problem of construction of a PDO theory on a
phase space on which the dx measure which is shift invariant (the Lebesgue measure, the Haar measure) will be absent. The reasons for the absence of dx in an infinitedimensional case and in a supercase are different but the problem is the same. We have first overcome this difficulty in an infinitedimensional case and then applied the methods employed in an infinitedimensional case to superanalysis. Therefore we have practically not used the techniques of the standard PDO the
ory on R. We think that in further investigations it is expedient to split the PDO theory developed in Sec. 1 into three separate theories, namely, infinitedimensional PDO over the field of real numbers, PDO on a finitedimensional superspace (dim over A), and PDO on an infinitedimensional superspace. The third theory is the most general and the most difficult. Below we formulate certain problems which can be solved by the
methods developed in this book. We have only to overcome some technical difficulties (though, perhaps, rather essential).
4.1. PDO calculus on a finitedimensional superspace. A superanalog of the ordinary PDO theory on R" is the PDO theory on the superspace R"' over the pseudotopological CSA B in which the Biannihilator is trivial and all even souls are nilpotent. The definition of the PDO a of order rn can be directly extended to this case (see, e.g., [32, p. 57]). Here use is made of the theory of generalized functions (D(RBm), D'(RBm)). For these PDO, we can try to construct a systematic theory which would include (1) PDO on supermanifolds, (2) a parametrix for superelliptic differential operators,
4. Unsolved Problems
223
(3) canonical transformations, (4) Maslov's canonical operator, (5) Fourier integral operators, (6) spectral PDO theory on supermanifolds, (7) boundary value problems, (8) wave fronts.
4.2. Infinitedimensional PDO. The most promising is the construction of a PDO theory on a locally convex space with a fixed Gaussian measure. Here it is easy to formulate a spectral problem on infinitedimensional PDO. It would be natural to consider, as the first step, PDO whose symbols satisfy constraints of the type of nuclearity. Also of interest is a PDO calculus on a locally convex space with a fixed diferetiable measure. This calculus can also be successively used to study the spectral properties of infinitedimensional PDO. In the framework of PDO calculus with analytical symbols, Sec. 1, we can obtain relations of the type of the FeynmanKac formula, similar to relations from Sec. 3. The construction of a PDO theory on infinitedimensional manifolds is a technically very complicated problem. Here we can evidently use the calculus discussed in Sec. 1.
4.3. PDO on an infinitedimensional superspace. We have to construct a superanalog of the space L2 with the use of Gaussian measure. This will be a Hilbert CSM. It is natural to develop this theory over Esuperalgebras. In this Hilbert CSM we can pose a problem concerning the spectral properties of PDO. However, this is a very complicated problem. At present, we do not have any results concerning the spectrum of the operators in Hilbert CSM. It is evidently easy enough to obtain FeynmanKac formulas, similar to the formulas from Sec. 3, for an infinitedimensional superspace. We can hardly say anything definite about PDO on infinitedimensional supermanifolds. We do not yet have a theory of infinitedimensional supermanifolds.
224
Chapter IV. Pseudodifferential Operators
Remarks The fundamentals of the PDO theory on a superspace were laid in the works [65, 68, 144, 153]. I managed to use here the methods of the theory of infinitedimensional PDO [67, 136, 139, 141]. For the first time, infinitedimensional PDO were considered in [127]. The main part in my constructions is played by the article [139] in which I introduced functional spaces that were used in all further considerations. Sec. 1. The results given in this section were published in [65, 68, 144, 153, 148]. For the infinitedimensional PDO theory over the field R see [127, 129, 136, 139]. For infinitedimensional PDO in spaces of functions square summable with respect to uadditive Gaussian measure on a locally convex space see [141, 67, 149, 150].
Sec. 2. The results considered in this section were published in [148, 153, 149]. Here the main difficulties are connected with infinite dimensionality of a phase space. The introduction of Lie supermodules is due precisely to infinite dimensionality. Sec. 3. The most systematic exposition of the FeynmanKac formulas for PDO symbols can be found in Berezin's review [5] (see also [3, 53, 26]). In these works, a sequential approach to the definition of the Feynman integral was used (the continual integral was defined as a limit of infinitedimensional integrals). We regard the Feynman integral as an integral with respect to the distribution on an infinitedimensional space. For the first time, this idea was realized on the mathematical level of strictness by Uglanov [133].
In the framework of algebraic superanalysis, the FeynmanKac formulas for PDO symbols (a sequential Feynman integral) were obtained in [3]. The FeynmanKac formula for PDO symbols on an infinitedimensional superspace in the framework of the new functional approach to superanalysis was obtained by Khrennikov [144, 68]. As has been pointed out, the main advantage of this functional integral is that it is actually a functional integral and not an integral with respect to the infinitedimensional Grassmann algebra (which is a purely algebraic object). This is more consistent with Feynman's concept [59] of continual integral as an integral along paths or over classical fields. The FeynmanKac formulas for the heat conduction equation and for the Schrodinger equation with a potential on a finitedimensional superspace were obtained in [146, 147]. In these works, the FeynmanKac for
4. Unsolved Problems
225
mulas connected with the Wiener process were also obtained. In [153] I considered the FeynmanKac formulas for the heat conduction equation and the Schrodinger equation with a potential on an infinitedimensional Hilbert superspace. Interesting formulas of the type of the FeynmanKac formulas on a phase superspace were obtained by Rogers and Ktitarev [122, 109, 110].
Generalized solutions (for t) of the Cauchy problem to the symbol of an evolutionalry PDO were studied in [68, 148] and generalized solutions (for t) of the Cauchy problem for the Liouville infinitedimensional equations were obtained in [170]. I was apparently the first to consider the Feynman path integral which converges in the sense of the theory of generalized functions (see [68]). Sec. 4. A boundedless sphere of action opens here indeed. It is
beyond doubt that a superextension of the results obtained by Egorov, Maslov, Treves, Hormander, Shubin [32, 4446, 58, 73] and by other authors
in the PDO theory on R" and on manifold over R" can be obtained on
the superspace RBm over the pseudotopological CSA B with a trivial Blannihilator and a nilpotent soul. In an infinitedimensional case, of especial interest is the spectral PDO theory on a locally convex space with a fixed Gaussian measure. For these PDO, we can try to get results similar to those from the theory of infinitedimensional differential operators on a space with a fixed Gaussian measure (see the systematic exposition of this theory in the monograph by Berezan
skii [2]). An important part can also be played by the Hida calculus of generalized Brownian functionals [64]. The theory of differentiable measures on infinitedimensional spaces was proposed by Fomin [62] in 1968, the modern exposition of this theory can be found in [6].
Chapter V
Fundamentals of the Probability Theory on a Superspace
In this chapter we consider probability models in which probabilities as well as expectations and variances are elements of Banach CSA and more general noncommutative algebras. Limit theorems of probability theory on a superspace are obtained, a Wiener superprocess is introduced, and a representation of a solution of heat conduction equation is obtained as a probability mean with respect to Wiener process. A spectral interpretation of Avalued probabilities is given and the connection with the multivalued probability theory is considered.
1.
Limit Theorems on a Superspace
Limit theorems play an exceptionally important part in the ordinary probability theory. We also begin the exposition of the fundamentals of the probability theory on a superspace from limit theorems, namely, the central limit theorem (CLT), the law of large numbers (LLN), and a supergeneralization of Lyapunov theorem. We begin with considering a general case where a superspace may be infinitedimensional. In the ordinary probability theory, limit theorems on infinitedimensional spaces are no less important than finitedimensional theorems.
Chapter V. Probability Theory
228
The methods of the theory of analytical distributions that we use
here allow us to obtain as limit distributions in the CLT not only Gaussian but also quasiGaussian distributions. In particular, we have obtained a CLT for Feynman distribution. This CLT is of interest by itself, without being connected with superanalysis.
1.1. Mean value and covariance functional. Definition 1.1. A momemt of order m of the distribution p E M(W) is the form bm(v) = f (v, w)mp(dw). The firstorder moment bl is called a mean value. The form a2 = b2  bl ® bl is the covariance functional of the distribution M.
Lemma 1.1. Let the distribution p E M(W). Then the relation F'(µ) (v) = f et("'W)p(dw)
(1.1)
holds for the Fourier transformation P. Proof. Assume that the kernel of the Fourier transformation .F': .M (W) 4 A(V) is a function g, c D(W ), v E V. Then
F'(µ) (v) = f
f.v(th.")F'(l2)(V'),
i.e., g is a Dirac 5measure concentrated at a point v E V. Therefore f(9v)(w) = e'(' ).
Lemma 1.2. Let the distribution µ E M(W). Then bm (v)
(_j)maT .F'(p)(0)(vm).
This lemma follows from relation (1.1). It follows from this lemma that Definition 1.1 and Definition 7.1 from Chap. III are consistent.
Definition 1.2. The distribution p E M (W) is normalized if
µ(W)= fp(dw)=1.
We use the symbol µc, where c E C, to denote the image of the distribution p under the homothety c and the symbol ('µ)" to denote the convolution of n copies of the distribution A.
1. Limit Theorems on a Superspace
229
1.2. Central limit theorem on a superspace. Theorem 1.1. Let µ E M (W) be a normalized distribution with zero mean and covariance functional b c £2,1(L ,, A). Then fco(w)'m(dw) = lim f W(w)(*µl/,/n)n(dw)
for any function W E c(W) (yb
(1.2)
rya,b)
Proof. It suffices to show that the Fourier transformations of the distributions (`µl/,/n)n converge to the Fourier transformation of the quasiGaussian distribution ryb in the functional space A(V). We set f (v) = F'(p)(v), and then we have
)=1
f (v/
2nb(v,
v)
00
+ni+1/2
k
k1n(k3)/2f (0)(v ) k=3 b(v,
v) )
+ 9(n, v)
2n
n1+1/2
We introduce functions fk,n(A, u), k = 0, ..., n, A, p E A by means of the relation n
(A + lpl p)n =
L. fk,n(A, 1 )IPk,
P E C.
k=O
These functions are homogeneous over the field C. With respect to the variable A the order of homogeneity is equal to (n  k) and with respect to the variable µ it is equal to k. It should also be pointed out that fo,n(A, µ) = An, and we have an estimate (1.3) Ilfk,n(A,/.L) < CnIJAIInkllllk. Using the function fn,k(A, µ), we can write [f (v//) ]n in the form [f (vl V'L)]n _ (1

b(v, )
)n
Chapter V. Probability Theory
230
+ kE
1
nk(1+1/2) fk,n
Furthermore,

( n
k=1
(1
 n
b(2)
((1
)
)n  exp{ b(2 y) }
b(y, y) k Cn
1
) (nk
2
 b(v,2nv) ), g(n, v ).
°O
Vii) +
( b(y,2 y) ) k
1
k=n+1
.
!
k
Having verified that (C  k,) + 0, n + oo, we find that (1
 b(v,
))n
exp{b(2
v) }
in the space A(V). Using inequality (1.3), we find that fn,k ((1  b(2n v) ), 9(n, v)) I B
< Cn(1+ II2n
)nk11g(n,v)Ila
for any compact set B from the covering CSM Lv. By virtue of this estimate, the functional sequence [f (v/V qn]n converges to exp{° zv } in the space A(V). The convergence in Theorem 1.1 differs from the weak convergence of probability measures even when there is no superstructure. Infinite dimensionality does not play a decisive part here either.
Let us consider Theorem 1.1 in a finitedimensional case for numerical coordinates. In this case, the space 0 coincides with the space E(Cn) of entire functions of the first order of growth. Thus, in the CLT we consider a weak convergence not over a space of continuous bounded functions on Rn but over a space of entire functions of the first order of growth. For ordinary Gaussian measures, on Rn, the standard CLT does not yield Theorem 1.1 and Theorem 1.1 does not yield the CLT.
1. Limit Theorems on a Superspace
231
What is new that Theorem 1.1 gives for Gaussian measures on R"? In this case, the main difference from the standard CLT is that we can choose as µ not only a probability distribution on R" but also a signed measure µ on Rn as well as a complexvalued measure or even a generalized function. The following example shows that the standard CLT is no longer valid for signed measures.
Example 1.1 (signed initial measure µ and a standard Gaussian distribution on R). Consider a discrete signed measure on the real line 3
1
2Jo2b1+2J3, where 8t is a Dirac bmeasure concentrated at a point t E R. We can immediately verify that this measure is normalized, its mean is +00 zero, and f t2p(dt) = 3 > 0. By virtue of Theorem 1.1, the limiting 00
process (1.2) takes place for any entire function of the first order of growth on C. Suppose now that co(t) is a continuous unbounded nonnegative function (not identically zero) whose support lies on the interval (oo, 0) (i.e., does not intersect the support of the measure µ). Then we have 0000
1
0
+00
for any n whereas f to(t)et2/6dt > 0. 00
This example shows that if we consider even the simplest probability supermodel with aadditive measures on the real line, which assume values in a Banach CSA, then there is no weak convergence on the space of continuous bounded functions on the real line with values in the Banach CSA A. In connection with example 1.1, we would like to discuss one more classical theorem of the probability theory. It is known that in the ordinary probability theory the convergence of characteristic functions implies a weak convergence of distributions. Example 1.1 shows that
Chapter V. Probability Theory
232
this is not true even for signed measures and all the more so for measures with values in a Banach CSA. A wider choice of initial distributions p leads to a wider class of limit laws. In an infinitedimensional case, the CLT for the Feynman distribution is especially important for applications. As the following example shows, in the absence of a superstructure, we can choose a countably additive complexvalued measure on an infinitedimensional space as an initial approximation of p for the Feynman distribution. Then the limiting process (1.2) can be regarded as a new method for calculating Feynman path integral. Under this approach, the function W(w) on an infinitedimensional space is said to be Feynman integrable if there exists a limit (1.2) which is called a Feynman path integral of the function co(w). In the framework of this approach, the class of functions 4) integrable (by virtue of Theorem 1.1) can be essentially extended.
Example 1.2. Suppose that H is a separable real Hilbert space, vl and v2 are Gaussian countably additive measures on the aalgebra of Borel subsets of H with zero mean and covariation operators B and 2B. It follows from the theory of measure on a Hilbert space that the operator B is a nuclear operator and the measures vl and v2 are mutually singular. Consider a countably additive complexvalued
measure on the aalgebra of Borel subsets v = (2v1  v2) + i(v2 v1). This measure is normalized, has zero mean, and its covariance functional is equal to i(Bv, v). By virtue of Theorem 1.1, the limiting process
1Hco(w)exp52(B1W,W)Jdw=L c(w)7iB(
= lim
r
corwi +... +Wn
Iv(dwl)...v(
Hn
)
n)
is valid for any function co from the class 4)(W).
1.3. The law of large numbers on a superspace. By analogy with Theorem 1.1 we can prove
1. Limit Theorems on a Superspace
233
Theorem 1.2. Suppose that the distribution µ E M (W) is normalized, and its mean value µ is zero. Then the limiting process
. f (P(w)(*A'/f)n(dw)
0(0) = n,lim
is valid for any function cP E c(W).
1.4. The central limit theorem for exponential distributions. QuasiGaussian distributions were introduced as distributions whose Fourier transform has the form of a quadratic exponent. A natural generalization is distributions whose Fourier transform has the form of an exponent of a polynomial on a superspace over a CSM. Definition 1.3. An exponential distribution on a superspace W is the distribution yp E .M (W) whose Fourier transform is .F'('yp) (v) _ eP(V), where P(v) = k,bk(vk) is a polynom on a superspace V. Here k=0
the forms bk E Xk,l(Lkv, A), and their restrictions to a superspace are symmetric. An exponential distribution is a fundamental solution of the Cauchy problem on a superspace of (t, w)
at
= P0) f (t w) , ,
f(0,w) = fo(w). Let b E 1Cn,i(Ln, A). We shall denote the exponential distribution with a Fourier transform F(ry) (v) = exp{ n, b(vn) } by ryb (if n = 2, then this is a quasiGaussian distribution with zero mean and covariance functional b). It follows from Lemma 1.2 that the moments of order 1, ..., n  1 of the distribution ryb are zero and the moment of order n is equal to b.
Theorem 1.3. Suppose that the distribution µ E M (W) is normalized, moments of order 1, ..., p  1 are zero, and the moment of order p is equal to b, with b E Lp,j (L A). Then the limiting process W(w)(*µn1/v)n(dw)) f W(w)'Yb(dw) = nlim J
Chapter V. Probability Theory
234
is valid for any function cp E (D (W).
For Clinear locally convex spaces, the complete proof of this theorem can be found in [140, p. 123]. We can obtain a proof for a supercase by combining the proof of Theorem 1.1 with that given in [140]. If p = 2, then Theorems 1.1 and 1.3 coincide. Theorem 1.3 is a typical limit theorem for a fundamental solution of the Cauchy problem for a linear differential equation with constant coefficients. In particular, the heat conduction equation is associated with the ordinary CLT. If we consider the Cauchy problem for the equation 2q
then we obtain Krylov's theory [106]. Thus, Theorem 1.3 is an ordinary limit law which includes laws that are already known. This law was obtained on a superspace, the superspace being infinitedimensional.
It should be pointed out that in the limit theorem for Eq. (1.4) Krylov also considered a limiting process for integrals with respect to analytic functions of an infinitedimensional argument. Our class (D is considerably wider than that of Krylov. Here is an example of a countably additive measure on the aalgebra of Borel subsets of a Hilbert space satisfying the conditions of Theorem 1.3. 00
Example 1.3. Let the numbers )j E C, E )j = a# 0, the series j=1
converging absolutely. On a Hilbert space H we consider a plinear form 00 .1j (ej, v) P, v E H, i1 where {ej} is an orthonormal basis in H. Let the function f (t), t E R, be summable, and let
b(vP) _
+00
+00
ff(t) dt = a1, ftf(t) dt = 0
00
00
(n=1,...,p1),
1. Limit Theorems on a Superspace
235
+00
ftPf(t) dt = 1. 00
We denote by pj the measure concentrated on the onedimensional subspace Hj = {Aej} which is absolutely continuous relative to the Lebesgue measure corresponding to the scalar product on H and having a density f (xj), where xj = (ej, x), x E H. We set µ = E00Ajpj, and then this is a countably additive normalj=1
ized measure, and
f (v, w)mp(dw) = (E00 .1j (ej, v) m) H
E Aj(ej,
tm f(t) dt
00
7=1
1000
f
+00
M = 1, ..., p  1,
P, m = p.
j=1
In addition, +00
JH Ilxllmlµl(dx)
j=1
Aj)
f Itim If (t) I dt <
00
00
+00
if the integral f I t I m 1f (t) I dt is finite. The fulfilment of the last condition for all m is sufficient for the measure µ to be associated with the distribution from the space M(W). By analogy, we can introduce countably additive Avalued measures on the covering CSM Lye which satisfy the conditions of Theorem 1.3.
1.5. Superanalog of the Lyapunov theorem. Here, as before, the main difficulties arise already for signed measures. In particular, already for signed measures it is necessary to introduce an additional condition into the formulation of the Lyapunov theorem which was automatically fulfilled for probability measures.
Chapter V. Probability Theory
236
Definition 1.4. The distribution S E A'(V) is regular if there exists a compact subset B of the covering CSM Lv such that II(S, f) 11 < Cs maBx Ilf(v)II.
The space of regular distributions on the superspace V is denoted by AR(V); TR(W) = F'(AR(V)).
Proposition 1.1. If A = C, then A'(V)  AR(V). In order to prove this proposition, it suffices to use the Cauchy integral formula for analytic mappings of complex locally convex spaces.
Proposition 1.2. If A = C, then the space A'(V) consists of countably additive complexvalued measures with compact supports on the o algebra of Borel subsets of the locally convex space V.
Proof. The space A(V) is embedded into the space CC(V) of functions which are continuous on compact subsets of the locally convex space V. By the HahnBanach theorem, any distribution S E AI(V)
can be extended to the functional S E CC(V). Using proposition 5 from [56, p. 18] and [22], we find that S = p, where the measure p satisfies the conditions of the proposition. The uniqueness of the measure p follows from the triviality of the kernel of the operator of the Fourier transformation. It is not clear as yet what the situation is in a supercase. Furthermore, when proving the Lyapunov theorem, we shall use a logarithm on the CSA A. Suppose that in the CSA A there exists a body projector b consistent with the norm on A (see Chap. II). The logarithm of the element A of the CSA A is defined by the relation
lnA=InbA+ln(1+
c'O
ba
n1
(1) i1(c` Tt
This function is defined for all A satisfying the condition Ilcall < Ibal,
bA
0.
ba )
n
1. Limit Theorems on a Superspace
237
Note that if aj, j = 1, ..., n, are even elements, then n
n
exp{E In (1 + aj)} _ 11 (1 + aj), j=1
JJcaj JJ < 11 + bad l;
j=1
in this case In( f (1 + aj)) may be undetermined. 7=1
Theorem 1.4. Suppose that in the CSA A there exists a body projector consistent with the norm. Let µnk, k = 1, ..., n, n = 1, ..., be normalized evenvalued distributions belonging to M(W) and possessing covariance functionals ank and zero means. If there exists a form a E L2,k A) such that Orn
(1)
n = k ank 4 a; k=1
n
(2) E I1ank112 4 0; k=1
n (3)r
0 Sup I J pn = k=1 0
f O(W)'Ya(dw)
=1n
f W(W)µn1 * ... * Ann(dW)
is valid for any function cp E TR(W).
Proof. Using Taylor's formula, we obtain V, V)
unk (v) = .F(µ'nk) (v) = 1  ank (2
+ rnk (v),
where the estimate Ilrnk(v)II < Sup I f
(v)W)3eia(v,ur)Unk(d41)
O
holds for the remainder rnk. Here we used Lemma 1.2. Consequently, n the sum E IIrnkll tends to zero uniformly on the compact sets of the k=1
CSM Lv, and, in particular, max I l rnk l l 4 0. 1
Chapter V. Probability Theory
238
It follows from condition 2 that max I I ank I I + 0. 1
Furthermore, since Cunk = 21cank + Crnk, it follows that max I I conk l 10,
1
and since bunk = 1  2'bank + Crnk, it follows that I bunk l> 1  21 max I bank l max I brnk l 1
Thus, for any E > 0, there exists a number n such that 1min I bunk l> 1  E.
lmka n I l CUnk l l< E;
Therefore, for sufficiently large n we have an inequality (
Max IICUnkII)/( min Ibunkl) <
1
1
E
1E
< 1.
Consequently, the function n
n
gn(v) = 1: In unk = k=1 k=1
n
oo
+E
k=1 m=1
In bunk
(1)m1 Cunk) M ( m bunk J
is defined. The article [140, p. 128] contains a complete proof of the fact that 00
the sum of complexvalued functions k In bunk converges to (ba/2) k=1
uniformly on compact sets. Let us prove that Cgn + (ca/2) as n 4 oo uniformly on compact sets of the CSM L. Note that n
CU
n
1 m1
o0
nk + Cgn = k=1 bunk k=1 m=2
(
)
m
( CU nk) M = An + (nbunk
1. Limit Theorems on a Superspace
239
Next, we have n
n
1
Crnk
E bunk
`
k=1
E I I Crnk I I  0; 1 E k=1
n
Cank
 cank) <
1 1 E
(lmax
IlCankll
k=1(bunk n Ilbankll2)1/2)
XEIlbrnkll+ 2 k=1
1l<
4 0;
k=1 IlCankll' k=1 2
1 11 CUnk 11m2 I CUnk n k=1 I I bunk I I2 m=2 717 b7dnk I l
bn
<
I
? E) 2
(1
E(4'IlCankll'
00
X E 1( E m=2 7n
+
IICrnkll')
k=1
1E
m2
3. 0,
n 4 oo.
If µnk are countably additive Avalued measures, then Condition 3 in Theorem 1.4 can be replaced by a more common condition as compared with that from the ordinary probability theory, namely, n
Ef
(dw) + 0,
k=1
where Iµnkl are variations of vectorvalued measures. In the ordinary probability theory, there is no analog of condition 2. If condition 2 is not fulfilled, then the limiting process (1.5) is, in general, absent even for signed measures on the real line. If the measures µnk are not evenvalued, then the limiting process (1.5) is also, in general, absent even for Avalued measures on the real line.
Chapter V. Probability Theory
240
2.
Random Processes on a Superspace
In this section, we propose a formalism of the theory of random processes on a (finitedimensional) superspace in the framework of which a Wiener process was constructed and a representation of a solution of the heat conduction equation on a superspace was obtained as a probability mean (a probability version of the FeynmanKac formula).
2.1. Cylindrical distributions. Suppose that we are given a consisting of sequence of locally convex superspaces un,m = A. We assume that these CSM satisfy the followfunctions cp: ing condition of consistency: for any operator B E Gi,i(An+m, Ak+l) which maps the superspace C m into C and for any function W E l tk,1 the function 0 = cp o B belongs to the CSM Un,m 1
Let M=Mo®M1and R=Ra®R1,N=No®N1and S=So®S1 be two pairs of dual topological CSM, V = MO ® N1 and W = Ro ® S1 being dual superspaces over these CSM. We denote by II(W) a collection of finitedimensional Alinear projectors defined by the elements of the Aomodule MO ® No, i.e., maps
7r: W > Cn",," of the form 0 0 0 0 7rw = ((w, ml), ..., (w, mn,r); (w, nl), ..., (w, MOM,
where m?eMO) j=1,...,n,,;no ENo,i=1,...,m,,. We introduce a functional CSM consisting of cylindrical functions
U(W) = U { f (w) = cp o 7r(w): c' E aE11(W)
For any projector 7r: W > 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
in the superspace Cr'. This is a Banach superspace with a norm IlvlIv = Varvlo.
These are dual superspaces and the form of their A0duality is defined by the integral
(x, v) = E f x° (t) dv° (t) + > f x (t) dvi (t), 1
1
j=1
j=1
Chapter V. Probability Theory
242
where x(t) = x°(t) ® x'(t), v(t) = v°(t) ® v'(t).
The finitedimensional projectors 7r: Q * 7rx = (
have the form
f l x° (t) d ,k (t); j f x (t) da°t (t)), j
where k = 1 ,
...,n,
l = 1, ..., 7)Z,,,
vjOk, ail E V(l0, 1], A°), vjk(0) =
a°l (0) = 0.
Definition 2.3. A cylindrical random process x(t) = xµ(t) with with continuous trajectories is a normalvalues in the superspace ized cylindrical distribution p on a superspace Il (it is convenient for us to consider random processes emanating from zero).
2.3. QuasiGaussian cylindrical random processes. We set Un,,.n = E(C"m) which are spaces of Sentire functions of the first The cylindrical random process order of growth on superspaces x(t) all of whose finitedimensional distributions are quasiGaussian distributions on Cn"'"`x, is called a quasiGaussian cylindrical process.
Gaussian and Feynman random processes can be introduced by analogy.
Remark 2.2. The random processes introduced in this section are not random processes in the standard sense even if the algebra A = R. In this case, we deal with the generalization of the concept of random process under which generalized functions on C "` can be taken as finitedimensional distributions. Random processes of this kind are encountered, for instance, when we consider Schrodinger equations. The distribution on a space of trajectories that corresponds to these processes may not be a probability measure.
2.4. Random processes. Let W(1) be a topological CSM consisting of functions G: I  A and containing a space of cylindrical functions U(1). The continuous Alinear functional µ: 'I(S2) 4A is a IVcontinuation of the cylindrical distribution p if µ4u(n) = A. A random 'Yprocess x(t) = xµ(t) with continuous trajectories is a distribution µ. E W' (1) which is a 'Ycontinuation of the normalized cylindrical distribution p.
2. Random Processes on a Superspace
243
For any cylindrical quasiGaussian random process whose covariance functional is continuous on E x E there exists a continuation to
the 0process, 0 = 4)(0). We shall use the symbol of the mean value M to denote the integral (in the generalized sense) with respect to the distribution µ.
Remark 2.3. If the algebra A = R, then the standard theory of random processes results if we take a space of bounded Borel functions on the space of trajectories SZ as the space W(Q).
2.5. Wiener process. We consider a finitedimensional superspace C,,2k. On the corresponding superspace E we consider a quadratic form
c(v, v) = f f
min(t, s)(Mdv(t), dv3)),
where the covariation matrix M satisfies the constraints indicated in Sec. 3.5 of Chap. I. A random quasiGaussian process with a covariance functional c(v, v), v E E, is called a quasiWiener random process. Suppose that there exists a body projector in the CSA A. A quasiWiener random process for which bA > 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 <_ 1 for all
elements n from the ring generated in K by its unit element. If JXJK =
1 for all x E K*, then the absolute value is said to be trivial. The absolute value of the field K is a homomorphism of the multiplicative group K* of the field K into a multiplicative group R' of the field R. We denote the image of this homomorphism by F. Everywhere below we denote by K a complete nonArchimedean field (Char K) = 0 with a nontrivial absolute value.
Example 1.1 (fields of padic numbers). Any rational number x 0 can be uniquely represented as x = p" m, where (p, n) = (p, m) = 1. Here p is a fixed prime number, p = 2,3,5... Each p is associated with its own field of padic numbers. The absolute value for the rational
number x is defined by the relations jx 1P = p", x # 0, 101p = 0. This absolute value is nonArchimedean. The completion of the field of rational numbers relative to the metric p(x, y) = Ix  ylP is called a field of padic numbers and is denoted by Q,,. This field is locally compact. It was pointed out that InIK < 1 in a nonArchimedeand field, and therefore 1 / ln! I K does not decrease but increases with the growth of n. For what follows, we must know the order of growth of 1 / I n! I K. It is known (see, e.g., Borevich and Shafarevich [11]) that the growth is exponential for K = QP 1/In!IP < pn/(P1).
(1.1)
It should also be pointed out that there are not very many possibilities for the construction of nonArchimedean absolute values on the field of rational numbers Q. By Ostrovskii's theorem (see, e.g.,
1. Differentiable and Analytic Functions
259
[11, 49, 72]), any absolute value of the field of rational numbers Q is equivalent either to one of the padic absolute values or to a real absolute value. Consequently, there are no other completions of the field of rational numbers except for the field of real numbers and fields
of padic numbers (if we consider only completions with respect to metrics defined by absolute values). It stands to reason that there exist nonArchimedean fields which are finite and infinite extensions of the fields of padic numbers. Using inequality (1.1) and Ostrovskii's theorem, we find that 1 / n! I K grows exponentially in any K, 1/In!IK = 1/In!IP < pInl(p1)
Suppose that the quadratic equation x2  T = 0, T E K, has no solutions in the field K. We denote the quadratic extension K(f) by Z. The elements of Z can be represented as z = x + fry, x, y E K; the conjugation operation in Z is defined by the relation z = x  Vf'ry; the absolute value on Z will also be denoted by I IK, it is defined by the relation JzI K = Iz21K = 11x2 7y2I K. We shall use the symbol z 12 to denote the square of length of the element z E Z: I z 12 = zz (it belongs to the field K).

Example 1.2. Let K = Qp. Then there exist seven different quadratic extensions of the field Qp if p = 2 and three if p # 2. As distinct from an Archimedean case where the quadratic extension R(i), which is a field of complex numbers, is algebraically closed, the quadratic extensions of field of padic numbers are not algebraically closed. Extensions of any finite order are not algebraically closed either. A nonArchimedean norm on a Klinear space E is a map II . E + R+ satisfying the following conditions: (1) IIPxII = IAIKIIxII, A E K, x E E, (2) Ilx + yll < max(Ilxll, Ilyli), II
(3) IIxiI
=0s.x=0.
A complete linear normed space whose norm is a nonArchimedean is known as a nonArchimedean Banach space. A nonArchimedean
Chapter VI. NonArchimedean Superanalysis
260
Banach algebra and a nonArchimedean Banach commutative superalgebra (CSA) are defined in the same way, cf. Chap. I. A nonArchimedean Banach CSA is denoted by A = AO ® A1. In order to construct a rich nonArchimedean superanalysis on a CSA A, we must impose the following natural constraints 1. IIx®Oil =max(IIxII,II0II),xEAo,0EA1. 2. Either the algebra A is commutative (i.e., A = Ao and Al = 0) or the A1annihilator in the algebra A is trivial. 3. The set rno = { I I A I I: A E Ao } coincides with r.
The removal of these constraints complicates considerably the development of nonArchimedean superanalysis.
Example 1.3. Let B be a commutative nonArchimedean Banach algebra with a unit element and a norm 11 The CSA G,,.(B) = A°°B can be introduced by analogy with an Archimedean case (see Chap. I). Different subalgebras of the CSA GA(B) are the main model examples of a CSA which are used in nonArchimedean superanalysis. In Archimedean superanalysis, a significant role was played by the algebras G00' (B). However, the 11norms do not satisfy the strong triangle inequality and are not Archimedean. We must consider oo
I
I
.
norms of the type of a supremum or maximum. We denote by GOO (B) the subalgebra of the CSA Goo(B): GOO 00
(B) = if c Go.(B): If Il,,. = Sup If Ili < oo}.
The CSA GOO (B) is a nonArchimedean Banach CSA satisfying conditions 1 and 200for any algebra B. Condition 3 for the CSA G00" (B) is satisfied if rB = r and the algebra B possesses the following property: for any bounded sequence {b,,} of elements of this algebra sup IlbnIl E FB n
Let, for instance, B = K = Qp. Then sup,, IIbnIIP = supnp "n E r = {p": v = 0, ±1, ±2,...}, i.e., the CSA G'(Qp) satisfies condition 00 3. Now if B = K = Qp is an algebraic closure of the field Qp, then r = {pr: r E Q} and, consequently, supra IIbnIIQ; = Supra pr^ does not, in general, belong to r, i.e., the CSA GOO 00 (QP) does not satisfy condition 3. Similar arguments hold for the field K = CP = Qp which is a completion of the algebraic closure.
1. Differentiable and Analytic Functions
261
We denote by G° (B) the subalgebra of the algebra G'00 (B): G°00 (B) = {f E Gm(B): lim Ilfill = 0, IiI = i1 + ... + in}. Let us verify, for instance, that Go00 (B) is an algebra. Let f, g E Go 00 (B) and cp = f g, and then
f
21 < ... <
Zn, 2s
= I'm U at;
where a7Q is a signum function. Using the fact that B is a nonArchimedean Banach algebra, we obtain Il
i1...in ll < Max 11f71...7k II yua=i
ll'
Furthermore, since f , g E Go (B), there exists N = NE : I I f7 , I I g,, I I < El max(llf II., ll MIA) if I'yl > 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
Chapter VI. NonArchimedean Superanalysis
262
We denote by Ap = A(UP x Am, A) the space of functions which are Sanalytic at zero and for which Taylor's series
f (x, 0) = E
1
ack+vf
a! axaaOA (O)x'O
converges on Up x Am, i.e., lim
1a1+00
f
PI«1
axa800 (0)
la!lx
=
0.
In the space Ap we introduce a norm ac,+'O f
llfllp = max axctaeo (0) This is a nonArchimedean Banach space.
In the same way, we introduce spaces of Sanalytic functions A(Up', x ... x Upn x Am, A') with norms aa1+...+ak+fl f
llf Ilpi...pk = ma x
P1
...Pk°kf
axl1...8xkkao (0) lal!...ak!lK
We use the symbol Ao  AO(KA'm, A) to denote the space of all functions which are Sanalytic at zero, the space being endowed with an inductive topology Ao (KA'm, A) = lim ind A(U x Am, A). p,0
By the symbol A  A(KA'm, A) we denote the space of Sentire functions f : KA'm  A which is endowed with a projective topology: A(KA'm, A) = lim proj A(UP x A', A). A nonArchimedean prenorm is a function ll ll satisfying conditions 1 and 2 from the definition of a nonArchimedean norm. A topological
1. Differentiable and Analytic Functions
263
Klinear space in which the topology is defined by a system of nonArchimedean prenorms is called a nonArchimedean locally convex space. A metrizable locally convex space is known as a Frechet space (the topology on a Frechet space is defined by a countable system of prenorms).
NonArchimedean Banach, topological and locally convex nonArchimedean CSM are defined as in an Archimedean case.
Proposition 1.1. The functional spaces A0 and A are complete nonArchimedean locally convex CSM; A is a Frechet CSM.
Proposition 1.2. The differentiation operators az , Ap are continuous, and the inequalities
akf axe
< I!IKIIfIIP, p
of
aoj
a:
AP +
< Ilfllp p
hold true.
Proposition 1.3. The space Ap is a nonArchimedean Banach CSA, and the inequality IIf911P <_ llf llPll9llp
(1.4)
holds true.
The proof of these propositions is based on the strong triangle inequality for a nonArchimedean norm and on conditions 13 imposed on the norms on the nonArchimedean CSA A (when considering higherorder derivatives, we use the fact that l Cn l K <_ 1). The reader can find complete proofs in [160].
Corollary 1.1. Differentiation operators are continuous in the functional spaces AO and A.
Corollary 1.2. Functional spaces AO and A are topological algebras.
Example 1.4. (an infinitely Sdifferentiable function of anticommuting variables which is not a polynomial). Let K = Qp and let f (t) be a Dieudonne function. It is only important for us that this function
Chapter VI. NonArchimedean Superanalysis
264
is differentiable on it E Qp: ItIp < 1} with values in Qp and its deriva
tive is zero (and the function is not a constant). Let A = Go (Qp). Then 0 = 01Q1 + ... + 0123g1g2q3 + ..., Oil... ,I E Qp. We set W(0) = f (01),
and then this function is infinitely Sdifferentiable (it is Frechet differentiable, and the derivative is determined by the multiplication of the CSA A by zero), but this function is not a polynomial.
2.
Generalized Functions
We choose functional spaces AP, A0, and A as spaces of test functions on a nonArchimedean superspace. The conjugate spaces A'P, Ao, and A' obtained in the same way as in an Archimedean case by means of the identification of the spaces of Alinear functionals which are right and left continuous are spaces of generalized functions on a superspace KA''" For an infiniteorder differential operator
P
_

P°x,
p
0,3
9
axaoa
P,,E A '
Q
where 6(x, 0) is a Dirac bfunction on KA'm, we set IIPIIP = sup IIPaA11 Ia!JxpH0I, a$
P E F.
We introduce a space of infiniteorder differential operators DP = {P: IIPIIP < 00}Proposition 2.1. The equalities A' = lim ind DP, Ao = lim proj DP hold true.
Proposition 2.2. The equalities A" = A, Ao = Ao hold true. The proof of these propositions is based on the fact that Taylor's series for Sanalytic functions converge in the corresponding spaces of test functions. The reader can find the complete proof in [160, 163]. As in an Archimedean case, the differentiation operators in the space of generalized functions are defined by the relations ag aLSP aR9 W)
(g,
W)
= (g, a0j
2. Generalized Functions
265
Corollary 1.1 implies
Proposition 2.3. Differentiation operators are continuous in spaces of generalized functions.
Corollary 1.2 implies
Proposition 2.4. The operation of multiplication by a test function is continuous in spaces of generalized functions.
In what follows, we shall use an integral notation for the action of a generalized function on a test function. As usual (see Chap. II), we introduce the convolution operation and the operation of direct multiplication of generalized functions as well as the operation of convolution of a generalized and a test function.
Proposition 2.5. The operations of direct multiplication and convolution of generalized functions are continuous and supercommutative.
Proof. Let us verify the continuity of a direct product. Let gg E A(UP' x Am', A), j = 1, 2, f E A(U,,1+n2 x Am' +M2 A), and then f (xi, x2) 0l, 02) aa1+a2+Q1+Q2 f
1
a10201,62
Ofl!GY2! axllax22ae1
al a2 Vl qq
22
qq
(0)x1 x2 Bl e2
,
with 0')a1+a2+p1+p2 f
PI a11+1a21
llm 1a1!IKIa2!IK
axl 1 ax22 ae21 ae22
(0)
= 0.
We shall show that the function (92, f)(xl, 01) belongs to the space A(U,, l x Amt, A). Note that 92 = g2 E) 921, where Ig2I = 0, Ig2I = 1. Furthermore, al+a2+r1+Q2 f
PIa11
Ial!IK 1a 1: a2! aa1+a2+/31+r2
+aI
axi1ax22a013qq1ae22
f
axl 1 ax22 ae101 ae22
(0)
0 xa2 eAz )
(g2,
2
2
(0))(gl,x22e22)](1)101111621
Chapter VI. NonArchimedean Superanalysis
266
'U1+a2+Q1+#2f
Phil+Ik2I
!5 sup A la1! Kla2! K axi1ax22aepla0 2 11
x sup 11(92, x22e2
(O)M
IIPI012
°2132
(we have used the fact that the parity operator a is an isometry of the nonArchimedean Banach CSA A, see property 1 for the norm on A). Note that 119211, = sup 1192, x'01) IIPI'l Next, we have V E > 0 3 N aQ
IalI+Ia21>N a01+02+fl1+A2 f
PIQ11+1021
<E
lal! Kla2!IK axi1ax22aep1ae22 (0) II
Let 1 a1 I > N. Then I a1 I + 1 a2 I > N for all a2 and, consequently, aOl+02+01+132 f
P101I+I02I
sup
A la1!IKI&2!IK axilax22agA1ae22 11
(0) 11
< E.
Thus we have lim
PIQ
a0l+rl (92, f)
ll
Iall+00 Ia1!IK
11
axilaoAl
(0)
0,
with II (91®92,f)11
sup
a°1+Q2+p1+02 f
1
(0) < 01a A A2 Ia1!IKIa2! K axilax2zae21ae02 2 I
x11(91,x°10")II II(92,x22e22)II < II91IIPII92IIPIIfIIP
A similar proposition is valid for the convolution of a test and a generalized function.
3. Laplace Transformation
267
Laplace Transformation
3.
In a nonArchimedean case, it is more convenient to use a Laplace transformation rather than a Fourier transformation. Here, in contrast to an Archimedean case, the quadratic extension is not unique and it is more convenient not to employ quadratic extensions at all.
Definition 3.1. The Laplace transform (twosided) of the generalized function g E AD(KK'm, A) is defined by the relation
L(g)(y, ) = J
g(dxd9)e(x'y)+(B,E)
Theorem 3.1. The Laplace transformation L: .A' (Kn'm, A) + A(Kn'm, A) is an isomorphism of a Frechet CSM. Proof. In the proof we shall use estimate (1.2). Let g c A' (Kn'm). Then L(g)(y,0 = E
1
C,p
Y.
and
IIL(g)M1 = scup Ia
11(g,XQ9 )IIP1'1 <
Let f c A(Kn'm). We set (g, x`Bp) = 8
,90
(0). Then
a°+p f
f
1
1P'a, = Ilf HP p Ia1!IK ax0aep (°) The proof of the fact that Ker L = {0} repeats the proof for an 11
1
Archimedean case. By virtue of this theorem, we have a harmonic analysis
AD(Kn'm, A)  A(Kn'm, A), A0(KK'm) A) 4 A' (Kn'm, A).
(3.1)
Chapter VI. NonArchimedean Superanalysis
268
By definition, the Parseval equality holds (see Chap. II). The Laplace transformation possesses the usual properties, namely, it transforms a convolution and a direct product into a product and derivatives into a multiplication by variables. belongs to the space Note that the function fx®B(y, l;) = Ap and A' = U A'',. Ap if 1I'M < (ppl/(p1))1. Furthermore, A = e(x,y)+(e,0
pE
Let M E A' and let p E r be such that p E A',. By virtue of the Parseval equality, we have L'(p)(x, 0) = (bxeo, L'(p)) = (L(5x®B), p) = (fx®e, p)
=
r e(X'Y)+(e,0p(dydC)
(3.2)
for any point x ® 0 E KA'm, IIxII < (pp'/(p We introduce one more functional space aa+Of
G6 = {f E Ao: Illflll6 = su Note that G6 C
n
ax aae
Ibici (O)
< °O}.
AP and, in particular, G6 C Ao. (p<6p'/(1P))
Theorem 3.2. The adjoint Laplace transformation L': A'' 4 G6i b = p' is an isomorphism of Banach CSM. Note that the space Ap is not reflexive if the absolute value I IK is discrete (for instance, for padic numbers). Therefore the operator L" transforms the space G'P_1 not into Ap but into a wider space A'P. Then we have a Laplace calculus
A'AA'' GP1 CAa, APCA'P G''1 Ao.
(3.1a)
By virtue of the standard properties of adjoint operators, we have L"(g) = L(g) for the element g E Ao fl Gp_1. The Parseval equality is
f g(dxd9)L'(p)(x, B) = f L"(g)(x, 0)p(dxd6),
4. Gaussian Distributions
269
g c G'1,
p c A'*
(3.3)
Note that L" (g) (x, 0) is not simply a symbolical notation. The elements of the space A'P can be realized as functions on an openandp, and then 6.,®o E A'P, closed subset of a superspace. Indeed, let x and, by relation (3.3), L"(g)(x, 9) = (L" (g), Sx®e) = (g, L'(8x®e)) =
f
g(dy<)e(y,x)+(E,B)
Thus, the function L"(g) is defined on the set UP x Am. It is easy to verify that this function os Sanalytic on the set (Up )" x Am, where Up = {x c Ao; JIxjj < p}. Remark 3.1. Concerning the reflexivity theory for nonArchimedean locally convex spaces see, for instance, [72].
4.
Gaussian Distributions Suppose that the matrix B =
Boo
Blo
B°
,
E A, satisfies
the same constraints as those used in the definition of a Gaussian distribution on an Archimedean superspace, the vector a E A. The Gaussian distribution on K""' is defined as a generalized function from the space A'(KK'm, A) for which the Laplace transform L'('ya,B)(w) _ exp{ (Bw, w) + (a, w) }. 2 Under certain constraints on the covariation matrix B and the
mean a, the Gaussian distribution 'ya,B can be extended from the space of Sentire functions A(K;,'m, A) to wider spaces A(U' x Am, A). Let us use the Laplace calculus (3.1a). If the Laplace transform L'(ya,B) of the Gaussian distribution ya,B belongs to the class GP1 C A0i then ya,B can be extended to the space AP. A detailed description of these
extensions and the discussion of the problem of normalization of a Gaussian integral can be found in [163] (see also [160]).
Chapter VI. NonArchimedean Superanalysis
270
Duhamel nonArchimedean Integral. Chronological Exponent
5.
Everywhere in this section we denote by E a Banach module over a (commutative) Banach algebra A0; the symbol LA, ,(E) is used for the space of A0linear continuous operators in E. As usual, we define Sdifferentiable and Sanalytic mappings from Ao into E or LAD(E). The spaces of Sanalytic functions on a ball Up, p E r, are denoted by A(UP,E) and A(UP,Lno(E)). The integral fa f (t) dt for the analytic function f : U,,  E is defined by the relation 16 a
tndt = (bn+1 
an+1)/(n
+ 1),
a = aP,
b = bp.
Lemma 5.1. Let the function cp(t, s) belong to the class A(U,,, E). Then the function u(t) = f0 W(t, s)ds belongs to the class A(U6, E),
b < p, and IIUIIo < C(p, b, K)IIkIIP.
Proof. We shall prove that in the space A(U6, E) the series u(t) n,m=0
a
(0)/n!(m + 1)! _ E Unm(t) n,m=0
converges, for which purpose we shall estimate the general term of the series an+m W11 Ilunmllb = 6n+m+l &tnasm
(0)II/In!(rn + 1)!IK
I)n+m/Im+1IK.
It remains to note that 1/InIK < ne, a = a(K). Recall that an evolution operator of a linear differential equation in the Banach A0module E (with evolution parameter t E A0) Wi(t) = a(t)x(t),
a(t) E LAD(E),
(5.2)
5. Chronological Exponent
271
is a family of operators V (t, T) E LA, ,(E) which is a solution of the Cauchy problem
V(t,T) = a(t)V(t,T),
V(T,T) = I.
Theorem 5.1 (Duhamel integral). Let the functions f (t) E,A(U",E), a(t) E A(UO, Lno (E)); V (t, T) constitutes an evolution family of the class A(UP, LA,,(E)) of Eq. (5.2). Then there exists a unique solution of the class A(U6i E), b < p, of the Cauchy problem for the nonhomogeneous equation
x(t) = a(t)x(t) + f (t),
x(O) = xo,
(5.3)
x(t) = V (t' 0)xo + f V (t, r) f (T)dT.
(5.4)
defined by the Duhamel integral t
Theorem 5.1 is a direct corollary of Lemma 5.1.
Corollary 5.1 (a linear homogeneous equation with constant coefficients). Let the operator b E Lno(E); the function f (t) E A(Up, E), where p < (IIbIIp1/(p1))'. Then there exists a unique solution of the class A(U6i E), 5 < p, of the Cauchy problem 1(t) = bx(t) + f (t),
1(0) = xo,
(5.5)
defined by the Duhamel integral x(t) = ebtxo + f t e(ts)b f(s)ds.
Theorem 5.2 (chronological exponent). Let the operatorvalued function a(t) belong to the class A(Up, Lno(E)). Then there exists a unique evolution family V (t, T) of the class A(u6 , Lna (E)), where
Chapter VI. NonArchimedean Superanalysis
272
b<
pi/(P1)
min(p, IIaIIp 1), of Eq. (5.1) defined by the chronological
exponent V (t, T) = T exp (
f
00
_ Ef
t
a(s)ds}
'n1 a(s1)ds1...
n=0 Tit
(sn)dsn.
JrT
(5.7)
Proof. We shall show that the function V (t, T) defined by the relation (5.7) belongs to the class A(UU, Gno (E)). Note that V (t, T) 00
E Vn (t, T) n=0 00
00
J1=0
in=0
Vn(t,T) = E ... E ra(il)(T)...a(in)(T)/jl!...in!] X( t  T) )1+...+in+n /(in + 1) (in +jn1 + 2)...(jn + ... +
1*1 + n)
Vnj (t,
i Using inequalities (1.3) and (1.4), we obtain I I VnjI I <
IIa(j1)
I
I 6 . . . II
a(in)
II6II (t
7)j1+...+in+n
II61
Ij1!...jn!IK I (in + 1)...(jn + ... + jl + n)I K < IlaIIPPP
1+...+7n+np(31+...+in)/I(jn + 1)...(j1 + ... + in + n)I K
<
(U \ )1+...+Jn J1 (IIaIIPb)nII(j1+...+jn+n)!IK
Using estimate (1.2), we get (ap`/(P1)
)il+...+7n(IIaIIP5p`/(P1))
II Vnj 116 <
1
n.
P
Thus, lim I I Vn; I I6 = 0. Consequently, the series > Vni converges in Ii1+00 i the space .A(U2, LAO(E)), and we have the estimate for the sum of the series IIVnII6 <
(6p'1(P1)IIaIIP)n.
6. Cauchy Problem
273
This estimate implies the convergence of the series E Vn in the space n
A(1
,
LA. (E) )
Theorem 5.3. Suppose that the operatorvalued function a(t) belongs to the class .A(U'1, ICA,(E)), and the vectorfunction f (t) belongs to the class A(UPZ, E). Then there exists a unique solution of the Cauchy problem (5.3) of the class A(U , E), where
6 < min(p2, defined by the Duhamel integral
x(t) = T exp{ fo t a(s)ds}xo +
p1pLl(1P), IIaIIp1p!1(1P)),
f T exp{ f t a(s)ds t
} f (7) d7.
In order to prove this theorem, it suffices to note that the function f (t) and the evolution family V (t, T) = T exp{ f1t a(s)ds} are Sanalytic on a ball of radius J. We have thus constructed an analog of the chronological exponent in the additive representation (5.7) of the Peano chronological calculus [117]. There is, evidently, no nonArchimedean analog of a multiplicative chronological integral (Volterra chronological calculus [94]) in a nonArchimedean case.
6.
Cauchy Problem for Partial Differential Equations with Variable Coefficients It should be pointed out, in the first place, that we can regard
the functional spaces A(UU x Am, A) as modules over a commutative Banach algebra A0. Thus, the results of the preceding section are applicable to spaces of operators LAo (A(UP x A', A)).
Lemma 6.1. Suppose that the function a(x, t, 0) belongs to the class A(U61 x U x Am, A). Then the operatorvalued function t p a(t) E £A0(A(U x Am, A)), a(t)(co)(x, B) = a(t, x, 9)lp(x, 8), belongs to the class A(u6 , LAo (A(U x A', A)) )
Chapter VI. NonArchimedean Superanalysis
274
Proof. Note that 00 to
0)
an+a+Aa
 E n! L ff
a(t' x,
00 to
00
(0) a!
J=
 un(x' e).
By virtue of estimate (1.4), the operators of multiplication by the functions un (x, B) are continuous, and we have an estimate for the norm of the multiplication operator IlunllGAO(A(uP xA,A)) < IlUnllp
It remains to note that 6nllunllp/In!IK
li
< lim
(0)11
bn
In!IK
max
la!IK
QQ
=
a1
II acnazaae
0,
PI
and we have an inequality Ilall A(u6,rAO(A(uP (Ar,A))) <_ Ilallip
It should next be pointed out that by virtue of estimate (1.3) the differentiation operator as aA
axe a o
: A(U x Am, A)  A(L[P x Am, A)
is continuous, and as ap 11
ax, ao8
I.cAO(A(UP xAr,A))
< PI I l a!l K.
(6.2)
Estimates (6.1) and (6.2) imply Proposition 6.1. Suppose that the coefficients acO (t, x, 0) of the infiniteorder differential operator P (t, x, 0,
a
a
ax ae =
00
E a,,p (t, x, e) iai=o,A
as ao ax° aBp
6. Cauchy Problem
275
belong to the class A(U X LAP x Am, A) and satisfy the condition 0.
(6.3)
Then the operatorvalued function t + P(t, x, 8, az, B) belongs to the class A(U6, LA,(A(UP x Am, A))), and 8
IP(t,x,0' a'
8x aB )I A(u6GAO(A(uv XAm,A)))
< up(P) 
jp.
Theorem 6.1. Suppose that the elements of the matrix differential operator P = (P3(t, x, 0, ax, ae)) satisfy condition (6.3) and the function cp(x, 8) belongs to the class A(U x ZAP x Am, Ak). Then there exists a unique solution of the Cauchy problem
zl(t, x, 0) = P(t, x, 8,
8x'
a8)u(t, x, 8) + f (t, x, 0),
(6.5) (6.6)
u(0, x, 0) = (P (X, 0),
of the class A(U; X Up x Am, A), where s; < min(a, Sph/(1n), o.(P)lph/(1p)), a(P) = maxcr(P1j) =,i
In order to prove this theorem, it suffices to use Proposition 6.1 (estimate (6.4)) and Theorem 5.3. The solution of the Cauchy problem (6.5), (6.6) is defined by the
relation
r
t
u(t, x, 0) = T exp{ f P(s, x) 0,
+f
c
T exp l f
c
8x'
l P (s, x, 0,
aa ,
a8)ds}co(x, 0)
l ) ds } f (T, x, 8) dT.
In particular, condition (6.3) is satisfied by all finiteorder differential operators.
Chapter VI. NonArchimedean Superanalysis
276
In particular, Theorem 6.1 implies the solvability of the Cauchy problem for the diffusion equation on a nonArchimedean superspace
au at
B) _ E
(t
i,j=1
82z aij (t, x, 9) axiaxj (t, x, 9)
+ E bij (t, x, 9) 1
a2 U
a9iaoj
(t, x, 9).
Theorem 6.1 also ensures the existence of a solution for the equation of a nonArchimedean superdifferentiation with a locally Sanalytic potential V(t, x, 9). Suppose that T is a finite extension of the field K and E is a nonArchimedean Banach CSA over the field T constructed on the basis of the CSA A. The theory developed in this section can be immediately generalized to the functions f : U x Am + E.
In particular, let T = K(J) be a quadratic extension of the field K. The analog of Theorem 6.1 for maps with values in E implies a local solvability of the Cauchy problem for the Schrodinger equation on a nonArchimedean superspace for locally Sanalytic coefficients and potentials
at
(t, x, 9)
h2
 2 [i
32
n
l aij (t, x, B) axiaxj
2
+
bij (t, x, 9) 1
a9i a97
+ V (t' x, 0)1,0(t, x, 9). J
NonArchimedean Supersymmetrical Quantum Mechanics
7.
We can apply the results of the preceding section to the Schrodinger
equation for the nonArchimedean supersymmetrical quantum mechanics on a Riemannian surface. The formalism of this quantum mechanics develops according to the same scheme as in an Archimedean case.
7. Quantum Mechanics
277
On a flat space in Cartesian coordinates a supersymmetric Hamiltonian is defined by means of conjugate supercharges Q and Q': H = {Q', Q}/h, where
f VG
[Pa
+
1
Q=
T
VG
1dPd [Pd + v
adVJ
the operators W'° and Wa are, respectively, fermion operators of creation and annihilation which satisfy the commutation relations {XY`°, fib} = hbab,
Pa is the operator of the momentum !a,, and 171d is a plane metric. For a Riemannian surface with metric gµ"(x), x E Ao we have
Q' _
[ica
+
1T
Q=V '
DaV
[co
1T + oaV
where r,µ = Pµ  fF 'I' pxF,, are covariant derivatives. We have commutation relations Jai,
[P., x°] _ X01

'o] _
[P., 4``Q] = 0,
h
*3
rJA0qfµ,
_
[P., 41,6] = 0,
h
A
*k&
[tea, #0] = hR6«o'p'o`yµ
(for detailed computations see [163]). It should be emphasized that every quadratic extension Z = K(V 9r is associated with its own supersymmetrical quantum mechanics. In addition, we can consider higherorder extensions. Of the prime interest are Galois extensions. Every element of a Galois group is associated with its own superchange. A supersymmetrical quantum mechanics with superchanges arises where n may be odd. For infinite Galois groups we obtain a nonArchimedean supersymmetrical quantum mechanics with an infinite number of superchanges.
Chapter VI. NonArchimedean Superanalysis
278
The formalism described above can be generalized to number fields without an absolute value. The corresponding operators are realized in the space of formal power series over the CSA over this field. We can choose the nonArchimedean analog of the De Witt CSA A as the CSA G,,.K.
The Hamiltonian H = {Q*, Q}/h of the supersymmetrical quantum mechanics (h E K) is an ordinary differential operator on a nonArchimedean superspace. To this operator we can apply Theorem 6.1 which implies a solvability (local) of the Cauchy problem for the corresponding Schrodinger equation.
8.
Trotter Formula for nonArchimedean Banach Algebras
Different versions of Trotter's formula play an important part in the investigations and construction of evolution families of operators. Here we propose a nonArchimedean version of the Trotter formula. Everywhere in this section we use the symbol B to denote a commutative nonArchimedean Banach algebra over a field K. Since B is nonArchimedean, the exponent eA for the arbitrary element A E B does not exist (eA exists if IIAII < pl/(1P)). Therefore the ordinary notation of the Trotter formula eA+C
= llm (eA/neC/n)n n+oo
is meaningless (even if IIAII, IIAII < p1/(1P))
Theorem 8.1 (Trotter formula). Suppose that A, C E B, IIAII, IIAII < Pl/(1P), and the sequence cn E K, cn j4 0, is such that liM I cnI K = n+oo oo. Then we have a nonArchim.edean version of Trotter's formula eA+C
= lim (eA/C"eC/`^)`°.
(8.1)
A complete (rather cumbersome) proof of relation (8.1) can be found in [160].
9. Volkenborn Distribution
279
As a consequence of relation (8.1) for a wide class of differential operators (see Sec. 6) we obtain
exp{t(P(x,0,
= limmo( p{ tP(x,
,
B) +V(x,9))}
w0'ax, A 1 8B)
}exp{tVM}).
Consider the Schrodinger operator in a scalar case, A = K = QP. We set cn = pn, and then we have
exp{t(2 =L + V) MX) h
=lim n oo
pn
n
exp{t pn } exp{t
hn
V }co(x)
= n4oo lira o (x)
The function cpn (x) gives an approximate solution of the Schrodinger
equation with an accuracy to within e = nn (see the estimate of the rate of convergence for (8.1) in [160]). It is not difficult to calculate the operators a ,pn6 and eOp°v. Thus, everything reduces to the extraction of a root of degree pn. This algorithm can be used for solving Schrodinger equation over a field of padic numbers by numerical methods.
9.
Volkenborn Distribution on a nonArchimedean Superspace
Let K = Qp be a field of padic numbers. A unit ball U1 = ZP = < 11 of this field is known as a set of integer padic numbers (it is a ring). Any integer padic number can be uniquely represented as { x lp
a series convergent in Qp: X = xo +xlp+x272 +... +xnpn +...,
xj=0,1,...,p1.
Chapter VI. NonArchimedean Superanalysis
280
The Volkenborn integral over a ring of integer padic numbers is defined by the relation p^1
f
f (s)v(ds)
nlim
pn V' f (1)
Generally speaking, this limit does not exist for continuous functions. For this limit to exist, the function f (s) must be sufficiently smooth (see Schikhov [72, p. 167]). For our consideration, it is sufficient that this limit exists for entire analytic function on Qp and that the linear functional defined by this integral is continuous. Thus, the Volkenborn integral is associated with a generalized function v E A'(Qp, Qp) In order to define the superanalog of the Volkenborn integral, we shall do as we did in the case of a Gaussian integral. A padic Gaussian distribution was introduced as a generalized function whose Fourier transform is a quadratic exponent. We shall introduce a Volkenborn distribution on a superspace as a generalized function from the space A'(KK''n, A) whose Laplace transform can be obtained by means of a superextension of the Laplace transform for the Volkenborn distribution on Qp. This Laplace transform is known (see [72, p. 172]), it is
V (t) = fzPet'v(ds) =
to
°O
t
t _ = n=o E Bn n e 1
where Bn are Bernoulli numbers. Note that IBnlp < p and, consequently, I Bn I K < pt. By this estimate, the function V (t) can be con
tinued to the Sanalytic function on the ball LIP, where p < p and, in particular, V (t) E Ao (Ao, A). Furthermore, the function n
V
(t1i ..,.tn)
= V (t)...V (tn) _ 11
et' t'
is an Sanalytic function on the ball U n' p < pt/(1p), and, in particular, V(t1i ..., tn) E Ao(Ao, A).
9. Volkenborn Distribution
281
Let us consider the continuation of the function V (t) to the anticommuting variables
f e°8(9+2)dB. Definition 9.1. A Volkenborn distribution on a nonArchimedean superspace KA'm is a generalized function v c A'(KK''n, A) whose Laplace transform is defined by the relation ((t, E KK''n): V(t,
[n
et'tj
exp{2(e1+...+lm}.
11
Proposition 9.1. The Volkenborn distribution can be extended to the space A(U x Am, A). In order to prove this statement, it suffices to note that sup IBnIK < 00n
It follows immediately from the definition that f f (x, O)v(dxdO) =
f (ff(x,o)v(do))v(dx) = ff(x,_)v(dx).
Everywhere in what follows we shall use the notation u for the generalized function L1(u), where u E A. The properties of the Volkenborn integral over a superspace (cf. Schikhov [72]) :
Proposition 9.2. Let the function f E A(KA 1, A). Then
f f (x, O)v(dxdO) = f f (x + 1, 0 + 1)v(dxd9).
Proof. Note that
f (x, 0) =
ff
(dyde)ey"a
=
f
A'(f)(dydC)eyx+fe,
Chapter VI. NonArchimedean Superanalysis
282
where a: Ao(KA"', A) + Ao(KA"', A), 9(y, e) ' g(y, e), is a Alinear continuous operator. Using the Parseval equality, we obtain
f f (x, 9)v(dxd9) = fA'(f)(dyde)L'(v)(y,e) =
f f (dyde) [e_y y 11 el{ = fic'(f)(dYde)L'(v)(y,e),
where rc: Ao(KA"', A) + Ao(KA"', A), 9(y, C) H ey+Eg(y, C), is a A
linear continuous operator. It remains to note that
L(rc'(f))(x,0) = f(x+1,9+1). Similarly, for the function f E A(K;,'m, A) we have
f f (xl) ..., xn, 01,..., 9m)v(dxd9) = ff(xi + 1,...,x+ 1,0k + 1,...,Om + 1)v(dxd0). This relation is also valid for the function f from the class A(ul x Am, A).
Proposition 9.3. The relation
f f (X,,..., xj + S + 1, ..., xn, 9) f (xl, ,
af
fa
xj + S, ..., xn, 0))v(dxd9)
(xl, ..., Si ...)xn, 9)v(dxl...dxjldxj+l...dxnd9).
holds for any function f E A(KAn'm, A) and s E Ao.
Proof. We shall restrict the consideration to a onedimensional case (n = 1, m = 0), namely f (f(x + s + 1)  f(x + s))v(dx)
10. InfiniteDimensional Superanalysis
=
ff
(dy)(eyls+1)
 ey')
y ey
1=
283
f f (dy)yey' = of (s).
The relation that we have proved is also valid for the function f c A(U x Am, A) for shifts by vectors s from a unit ball. Proposition 9.4. For any function f E A(KA'm, A) and for 77 E Al we have a relation
f(f(X,9i,...,Oj+ii+1,...,0m) f (x, 01, ..., Oj + 77, ...,
fR a0j
(x, 01 i ..., ii, ..., Om)v(dxdOi ...dOj1d0j+1...dO,,,).
The proof is similar to the proof of the preceding proposition.
Remark 9.1. We have determined the Volkenborn integral only for Sanalytic functions. This integral can apparently be extended to some classes of Ssmooth functions.
10.
InfiniteDimensional nonArchimedean Superanalysis
The analysis on an infinitedimensional nonArchimedean superspace is developed along with an analysis on an infinitedimensional Archimedean superspace. The main idea is the same, namely, we consider superspaces over a pair of topological CSM and develop the theory of Sdifferentiable and Sanalytic functions on a pair (a superspace and a covering supermodule). Here we expose an infinitedimensional nonArchimedean superanalysis over specific CSM, namely, spaces A(KA'm, A) or A'(KAn'm, A).
We choose these spaces because of their good topological properties. It stands to reason that we can develop an infinitedimensional superanalysis over arbitrary locally convex nonArchimedean CSM. All general constructions can be generalized to this case without changes. However, it is necessary to use here the fine results from the theory of
Chapter VI. NonArchimedean Superanalysis
284
nonArchimedean locally convex spaces, but in the framework of this book we cannot do this. We set ecp = x°OA = 41 ...Xnn 0"1...00 . Note that these monomials form a topological basis in the CSM A(KA'm, A), with the parity These Ie,vl = I/0I (mod 2). Furthermore, we set e' O = monomials form a topological basis in the CSM A'(KA'm, A), with the parity Ieapl = IQI Proposition 10.1. 1. Let b E Gn,r (An, A). Then there exists azaae,0
REF, IIbIIR =ai,.. sup ,an
IIba1A1...".OnIIRII
< oo,
(10.1)
$11 ,...9n R q where ba1Q1...anpn
b(ea1#1I
eanOn)
2. For any sequence b = (ba1#1...'InQn) of elements of the CSA A that satisfies condition (10.1) for a certain R E IF, the form b=Eba1Q1...QnOnel
®...®e'
(10.2)
belongs to the CSM Gn,r(An, A).
In the CSM Gn,r (An, A) we introduce an inductive topology Gn,r (An A) = l irn ind Cn,r (AP, A).
This is a strict inductive limit of nonArchimedean Banach CSM. We set IbIR= sup IIb(f,...,f)II, IIfIIR<1
and then c = 2p'AP1). Consequently, the topologies defined by the norms I IR and II  IIR are IbIR < IIbIIR < cnIbIR,
equivalent.
Proposition 10.2. 1. Let b c Gn,r((A')n, A). Then, for all R E F, we have
IIbIIR =al,.. sup ,an 01 ... ,Sn
RI'I/Ia!IK < 00,
(10.3)
10. InfiniteDimensional Superanalysis where
qq b,,1131...cfnpp
_b
285
eon Qn
e'IP1)
2. For any sequence b = {b,,,1#1...'In On} of elements of the CSA A that satisfies condition (10.3) for all R E F, the form b
®... ® e
b01A1...OnAn
(10.4)
nQn
a.
belongs to the CSM £,((A')", A). In the CSM Cn,r((A')", A) we introduce a projective topology Ln,r((A,)", A) = lim proj Ln,,, ((AR)' , A).
This is a nonArchimedean Frechet CSM. The explicit description of spaces of continuous polylinear map
pings on the CSM A and A' (relations (10.2) and (10.3)) makes it possible to obtain an explicit description of spaces of Sanalytic functions on A and A'. Let us now construct an infinitedimensional nonArchimedean superanalysis. We set M = A(KK q, A), N = A(K;;S, A) and introduce an infinitedimensional superspace over a pair of nonArchimedean Frechet CSM
(M, N): X = Mo ® Ni =
A(KAP,q, Ao)
® A(Kns) Ai)
The covering CSM Lx = M ® N = A(KAq, A) ® A(K;;S, A). We shall use the symbol UR,P to denote a ball of radius p E F with respect to the norm II IIR, R E F in the superspace X:
UR,P={f EX: f0 IIf 11R = maX[sup 11
axi1...ax°Ta0Q1...aeg9 (O) I ai!...a,,!IK
fl
ap ax11...ax°*ae1 ...aep 11
O 0
11
ial!...ar!I K]

By UR,P we denote the corresponding ball in the covering CSM Lx: UR,P = {f E Lx: If IIR < P} (with UR,P = UR,P n X).
Chapter VI. NonArchimedean Superanalysis
286
Just as in an Archimedean case (see Chap. III) we define the Sdifferentiability and Sanalyticity on an infinitedimensional superspace.
We denote by A(UR,P, A) a space of maps Sanalytic on the ball UR,P. Every map of this kind can be expanded in a power series 00
(10.5)
F(f) = > 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
Chapter VII. Noncommutative Analysis
302
_
{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
2. Differential Calculus on Algebras
303
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;
2. Differential Calculus on Algebras
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.
Chapter VII. Noncommutative Analysis
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 <
in
oo}
Then, as before, we introduce a direct sum of ordered tensor products (n+1)
E(m)
(A) = ®(®A)
.
or, r
Let us consider a canonical map
j: ,( )1)(A) 31
32
7n
ii
t2
in
,Cn((Am)n, A),
j (a ® b ® ... 0 c) (hi, ..., hn) = ah7til bh72 ;2 ...h7inn c, 1
h; E Am
.
We set Gn,A((Am)n, A) = Im j ' S2( )1) (A) = E( )1) (A)/Ker j.
The map f : G  A, G c Am is said to be n times NCdifferentiable if it is n times Frechet differentiable as a map of the Banach space Am into A and the derivatives f (k) (x) belong to the classes SZ( )1 (A).
Example 2.10. Considre a function of two noncommuting variables
f (x, y) = axbxcyd,
a, b, c, d E A.
Then we have 1
1
1
1
Oxf = a®bxcyd+axb0cyd,
Dixf = a®b®cyd+a®b®cyd, 1
2
2
1
2. Differential Calculus on Algebras
f=
V
307
b®c®d.
a®b®c®d+a®1
1
2
2
1
1
1
By analogy, we can consider a more general case of functions of several noncommuting variables with values in the Banach Amodule. We shall begin with introducing an ordered tensor product of n copies of the algebra A and one copy of the module M. As before, we assign two indices, an upper index and a lower one, to every tensor product and, in addition, assign an upper index to the module M. Thus we have k+1
®(A, M) O,K
71
72
7k
2k+1
jn
A®A®...®M ®...®A. it ik in ik+1
i2
Next, we again introduce a direct sum of ordered tensor products with respect to or, rc, and with respect to the number k of the position of the module M: n+1
k
®(A, M). E 1) (A, M) = k=1 ® o,w o,K
Then we consider a canonical map j: E( 1)(A, M) 4 Ln ((Am) n' M). We set
Gn A((Am)", M) = IM j 1(n+1) E(m)1) (A, M)/Ker j. The map f : G p M, where G is an open subset of Am, is said to be n times NCdifferentiable if it is n times Frechet differentiable as a map of the Banach spaces Am and M into M and its derivatives belong to the classes SZ( 1) (A, M). By analogy, we can define NCanalytic functions of several noncommuting variables with values in the module M.
Suppose that there exists a topological basis {En} in the module M (as in a Banach space); r m, and rnm, are the right and the left structural constants:
en Em = > rnRnsES, 9
Enem =
rnmsE.'.
9
The function f (x) with values in M can be expressed in terms of the basis and its NCderivative Vf(x) = E(VfnmL(X)En 0 em + VfnmR(x)en ® Em) n,m
Chapter VII. Noncommutative Analysis
308
Theorem 2.4. Let the functions g, f : G + M be NCdifferentiable. Then the Alinear combination cp = a f Q + Agp, a, 0, A, p E A is also NCdifferentiable, with Vco = aV f /3 + \Vgµ and R
V cslL = > 1'mrlrkns(akV fnmLNr + \kV gnmLpr)) VcSIR =
L
/3r + )kVgnmRµr) E7knsrmrl(akVfnmR/
Theorem 2.5. Let the functions f : G > M and g: G + A be NCdifferentiable. Then the product co(x) = f (x)g(x) is also NCdifferentiable, and the Leibniz formula V W = V f g + f Vg is valid and we have R
VcOnpL = >(1'mkpVfnmLgk + rkmn {kVgmpL),
VcnpR = >('ykmnfkVgmpR + rmkpVfnmRgk)
Theorem 2.6. Suppose that f : G + Am, where G is an open subset of An, and g: W > Ak, where W is an open subset of Am, are NCdifferentiable at points x E G and y = f (x) E W respecAk is also tively. Then the composite function cp = g o f : G NCdifferentiable at a point x and Vcp = VgV f . For the module M = Am we can identify M ®A ®A ®M and (A ® A)m and realize the NCderivatives as matrices with elements from
1Z(2)(A). The functions f (x) and g(y) are vectorfunctions: f (x) = (f 1(x), ..., fm(x)) and g(y) = (gl(y),..., gk(y)). As in the ordinary analysis, we obtain matrices of the derivatives V f (x) = (V., f' (x) ), Vg(y) = (V g2(x)), V ;fi(x), V :g'(y) E Q(2) (A). The matrix U = (U;3 = a13 ®b;3) acts on the vector h = (hl, ..., hn) according to the law Uh = (n a;jhjb;j). The matrices U = (U, = a13 0 b13), V = (Vj = 1
czj (9 d13) are multiplied according to the law UV = ( a;,cjk 0 d3kb;3).
3. Generalized Functions
309
Example 2.11. Consider the Clifford algebra A2 with two generators Q1 and Q2: o1a2 + o2u1 = 0, tr = o2 = 1. This is a fourdimensional Klinear space with basis e1 = 1, e2 = Q1, e3 = a2, e4 = 5102 Using (2.1), we find that the kernel of the canonical map j: A2 ®A2 + C(A2i A2) is zero and the algebras A2 ®A2 and 1(2) (A2) coincide. Let f (x) = a1xo1 and g(y) = or,ya1. Then co(x) = g(f (x)) = x,
VW = 10 1. Using the chain rule, we get the same answer, namely, Ocp = VgVf = (Q1 0 a,) x (Q1 ® a1) = of ® o
3.
.
Generalized Functions of Noncommuting Variables The function f : G + M, where G is an open subset of A'", is
said to be p times continuously NCdifferentiable if f is n times NC(A, M) is continudifferentiable and the derivative V" f : G ous. The space of p times continuously NCdifferentiable functions f : G p M will be denoted by NP(G, M). We use the symbol N"(G, M) to denote the space of NCanalytic functions f : G + M. In the space NP(G, M) we introduce a topology of uniform convergence on compact subsets D C G together with all NCderivatives. This topology is defined by a system of prenorms 11f II Dj =SUP J V3f (x) Ijn(i+1)(A M), xED
= 0, 1'...) p.
(m)
The topology in the space N°° (G, M) of infinitely NCdifferentiable functions is defined by a system of prenorms {11 I lD,j };°_o. The space NW(G, M) of NCanalytic functions can be topologized by means of a system of prenorms (2.4). The spaces N°° (G, M) and NW (G, M) are taken as spaces of test functions of noncommuting variables. It is natural to introduce generalized functions of noncommuting variables as linear continuous functionals of these spaces. Here we have a rather difficult problem of defining the concept of linearity which would correspond to noncommutative analysis.
Chapter VII. Noncommutative Analysis
310
In the projective tensor product A ® M we introduce a structure
of the right A ® Amodule by setting (a ® m) (b 0 c) = ab 0 cm, a, b, c E A, m E M. Next, we consider the space Gr (A 0 M, A ® A) of
right A 0 Alinear continuous functionals TR: A 0 M + A 0 A. The map SR from 4(A 0 M, A 0 A) into the space £(M, A) of Klinear continuous functionals is defined by the relation SR(TR) (m) = trA TR(e ®m),
where trA, which is a trace on the projective tensor product A 0 A, is defined by the relation
trA X unmen ® em) = > unmenem n,m
n,m
We denote the image of this map by GA(M, A). This space is precisely the space of functionals which are linear in the noncommutative sense, i.e., NClinear functionals. We have obtained a right realization of these functionals. By analogy, we can obtain their left realization proceeding from the A ® Amodule M 0 A and the space Gi (M 0 A, A 0 A) of left A 0 Alinear continuous functionals. Both the right and the left construction lead to the same space GA(M, A) of NClinear functionals. We shall call the space GA(M, A) a topological conjugate of the module M and denote it by M'. The spaces (N°°(G, M))' and (N" (G, M))' are spaces of generalized functions of noncommuting variables. In conclusion, we shall outline a scheme of constructing a theory of noncommuting manifolds. It is natural to regard as a noncommuting manifold a Banach manifold with a model Banach space A'n in which the functions of transitions from a chart to a chart are NCdifferentiable (a finite or infinite number of times) or are NCanalytic. It is possible to consider noncommutative manifolds over Clifford algebras, algebras of matrices and operators, algebras of pseudodifferential operators.
It is obvious that the exposed formalism can be generalized to locally convex algebras and to other types of tensor products (for instance, to inductive products). When the concept of NClinearity is
3. Generalized Functions
311
defined, it is not difficult to construct an infinitedimensional noncommutative analysis, i.e., the theory of mapping Amodules (the theory of generalized functions of an infinite number of noncommuting variables and, in particular, Feynman and Gauss continual integrals over noncommuting spaces).
Remarks The results of Sec. 1 were published in [65, 68]. The results of Secs. 2 and 3 are announced in [165].
Apparently, the theory of analytic functions of several matrices constructed by LappoDanilevskii [41] was the first version of noncommutative
functional analysis. It should be pointed out that LappoDanilevskii also considered analytic functions of a countable number of matrices, i.e., constructed a version of an infinitedimensional noncommutative functional analysis.
Chapter VIII
Applications in Physics
In this book we tried to explain a new approach to superanalysis. We hope that this approach will be widely used in applications to physics (especially in quantum field theory, quantum string theory, theory of gravitation), the more so as the majority of physicists have intuitively used functional rather than algebraic approach to superanalysis. Speaking about a superspace, physicists usually mean a set of points endowed with a superstructure and not a ringed space. The language of structural bundles is marvellous, but it is too powerful a tool for studying such a simple structure as superanalysis. In any event, studying the works by Salam, Strathdee, Wess, Zumino, Schwinger's pioneer work, I have realized that in these works the functional approach to superanalysis was used at the physical level of strictness. It was not my intent to study serious physical supermodels in the framework of functional superanalysis. This book is a monograph in mathematics, and the main goal was to expose the mathematical apparatus. Any physicist who reads this monograph will be able to use the apparatus of functional superanalysis in the investigations in which there arise fermion degrees of freedom. In this chapter we propose two new physical formalisms. In Sec. 1 we consider quantization in Hilbert supermodules. The main difference of this quantization from the standard quantization in a Hilbert space consists in the application of the theory of Avalued probabilities (Chap. V). However, we can restrict the consideration to the
314
Chapter VIII. Applications in Physics
De Witt formalism and regard as physical only those states which are associated with probabilities belonging to the interval [0, 1] C A. In Sec. 2 we try to give a correct mathematical definition of the amplitudes of the quantum field theory with real interactions of the type of (w4)4 with the aid of the distribution theory on infinitedimensional spaces. When we consider transition amplitudes for quantum fermion fields, superfields, and gauge fields with the FaddeevPopov ghosts, we have an infinitedimensional superspace over a pair of CSM. We prove the convergence of a series from the perturbation theory in a space of distributions on an infinitedimensional superspace.
1.
Quantization in Hilbert Supermodules
We propose to carry out quantization of systems containing boson and fermion degrees of freedom in Hilbert CSM (or in Hilbert superspaces). This approach to quantization differs from the quantization in Hilbert spaces that was used before (or, in particular, from the quantization of bosonfermion systems that was considered by Berezin in the framework of an algebraic approach to superanalysis [3]). When quantization is carried out in a Hilbert space, the appearance of fermion degrees of freedom does not lead to a serious change in the procedure of quantization. Just as in a pure boson case, we consider a complex Hilbert space H and selfadjoint operators in H. The quantum states have the form
f = E .fneni
(1.1)
n
where fn E C, {en} is an orthonormal basis in H. These states admit an ordinary probabilistic interpretation, namely, { Ifn l2 } are frequencies of realization of pure states {en}. Everything is much more complicated and interesting in the case of quantization in a Hilbert supermodule M. De Witt was the first to consider this kind of quantization at a physical level of strictness [27]. The quantum states have the form (1.1), but the coefficients fn belong to the CSA A rather than to the number field C. However, De Witt regards as physical states only states with numerical coefficients.
2. Transition Amplitudes
315
A more general approach is possible under which all vectors of the Hilbert CSM M (defined at the mathematical level in Chap. III) are regarded as physical states. For the probabilistic interpretation of the
state f = (1.1), f,, E A, use is made of the spectrally probabilistic formalism exposed in Chap. V. The main advantage of the quantization in the Hilbert CSM M is the availability of a structure of a module over A in M. This structure makes it possible to consider the transformations of the state space with parameters from A and, in particular, with anticommuting parameters (infinitedimensional analogs of SUSY transformations). The main drawback is the absence of a spectral theory of selfadjoint operators in Hilbert CSM. However, this is a mathematical rather than physical problem.
2.
Transition Amplitudes and Distributions on the Space of Schwinger Sources
One of the main problems of mathematical physics is a strict mathematical definition of a continual integral for the amplitude of transition from vacuum to vacuum in the presence of a source fi(x): Z(S)
f exp{ 2 J (aµ(o(x)aµW(x)  m2cp2(x)) dx
i f V (cp(x)) dx + i f W(x)e(x) dx} 11 dcp(x).
(2.1)
X
By now, the only correct definition of symbol (2.1) is Slavnov's definition in the framework of the perturbation theory (see [126]). Apparently, no other definitions of the Feynman path integrals can be applied in the quantum field theory for real interactions. The class of functionals c(W) constructed in Chap. III (as well as all classes of Feynman integrable functionals known to me, see [3, 5, 24, 26, 45, 50, 53, 55, 59, 6568, 133]) cover only model interactions of the quantum field theory. The main difficulties in the definition of symbol (2.1) are, evidently, of a computational rather than of ideological nature.
Chapter VIII. Applications in Physics
316
The definition in the framework of perturbation theory gives an answer in the form of a formal series about whose convergence nothing
is known. This cannot be considered to be a satisfactory solution of the problem either. I suggest the definition of integral (2.1) based on the theory of infinitedimensional distributions. Let us first consider a onedimensional example +00
ZW =
J
2
exp{i (2  V (cp) +
}dV.
00
The integrand function g(W) = f (co)e" is not summable, and the integral is understood as a Fourier transform in the sense of generalized functions of the function f (c,). Recall (see, e.g., [15]) that a generalized function Z(C) is defined by the Parseval equality (Z, u) = (Z, u),
Z(AP) = f M.
(2.2)
The situation is the same in an infinitedimensional case. We shall make meaningful not the values of Z at fixed points but the distribution Z(dC) on an infinitedimensional space. Thus, we suggest that the continual integral (2.1) be realized not as a function on an infinitedimensional space of sources but as an infinitedimensional distribution.
2.1. Boson fields. Let V and W be infinitedimensional dual modules over a commutative Banach algebra AO (in particular, linear spaces). For any Sentire function f (cp) on the space V the symbol f f (cp)e'('P,{>dcp is correctly defined as the distribution
Z(0de on the dual space W. By virtue of the Parseval equality (4.2), Chap. III, the expression
(v, Z) = f u(e)Z(de) is defined as
r
(u, Z) = f
u(di7)2(?7),
2. Transition Amplitudes
317
where 2(77) = P(Z)(r7).
In order to make this notation consistent with the notation from physics, it is convenient to change the sign in the exponent when defining the Fourier transform, i.e., f µ(d77) exp{i(r7,
and then we have Z(77)
=
f ei(1'0zw)
f ei('' (f f
(cP)e'(w,t)dw)
de) dco = f f ((p)S((p
= f f (cP) (f
d1

17)dW
= f (rl).
Thus, the distribution Z(<) E M (W) is correctly defined by the relation
fu(e)1) = Ju(dii)f(ri),
uE
(W)
For a scalar boson field (see (2.1)) we set V =!9(R'), W = G'(R4). The function f (co) has the form fB(co) = exp{ 2 f (aµcP(x)8µcp(x)
 m2cp2(x))dx  i f V (w(x))dx}.
The function f B (cp) is Sentire on the infinitedimensional space V for any polynomial potential V(x). Consequently, the distribution for a scalar boson field Z(dC) = Z(e)de, Z(C) = (2.1) belonging to the space M(C'(R4)) is correctly defined. The variational derivatives of the distribution Z(dC) (derivatives in the sense of the distribution theory)
U(xl, ..., xn) (<)
 i 6C(xl) ... i 6C(xn) Z('K)
are also correctly defined. This distribution belongs to the space M(C'(R4)) and acts on the test function according to the law
f u(e)U(xl, ..., X.)(de) 4'(R4)
Chapter VIII. Applications in Physics
318
f
u(dW)co(xl)...(p(xn) fB((p).
g(R4))
Note that the function co H cp(xl)...cp(xn) fB(cp) belongs to the space .A(9(R4)) 2.2. Fermion fields. We set V = G(R4, A8), W = 9'(R4, A8) and
write the elements of the superspace V as a(x) (x), Vij (x))'=o; the elements of the superspace W are Q(x) = (e3 (x), tj The function
fF(i. V)) = exp{z f (Y'(x)i7µ
,
(x)
M (x)0(x)) + V (fi(x), V) (x))dx},
where V is a polynomial, is Sentire on the superspace V. As in a boson case, the distribution
Z(<<) = Cf fF(
dz)(x)dib(x))
)e'(E,+G)+i(E,+V)
x
x [J de(x)de(x)
is correctly defined on the superspace of sources W. This distribution belongs to the space M(G'(R4, A8))
Ju(e,)Z(ded) = fu(ddib)fF(b). By analogy, we can consider the case of interaction of a fermion and a boson field. Here the distribution Z(dcpdede) is defined on the superspace of sources G'(R4, RA8).
In the theory of gauge fields, a generating functional can be realized as a distribution on a superspace of sources Ja(x), ea(x), ea(x), where Ja are commuting sources of the gauge fields Aa; a and Ea are anticommuting sources of the FaddeevPopov ghosts ca and c (see [53]).
2.3. Superfields. We set V
g+(RU4,4,
Ao) x g(Ru4, Ao),
2. Transition Amplitudes
319
W = 9+(Ru4, Ao x G' (Ru4, Ao)
(see Example 5.2 in Chap. III). The distribution Z(dJ+dJ_)
(r fs(w+,
w_)e'(j+,w+)+i(J,w)
\J
11
dw+(x, O)dw_(x, 9))
(x,e)ER. 4
x
II
dJ_(x, 9)dJ+(x, B),
(x,B)ER44
where
fs(w+, w) = exp{ i f (g (DD)2(w+w_)
1M DD (2 (W+ +W2))  V (w+, w_)) dx } ,
2
V is a polynomial, is correctly defined on the superspace of sources W.
2.4. Some physical consequences. The formalism exposed above makes it possible to assign a mathematical meaning to source functionals that appear in the quantum field theory. The part played by the theory of distributions on a space of sources in mathematical physics is similar, in many respects, to the part played by the theory of generalized functions on R". A strict mathematical meaning is assigned to expressions whose definition as functions of a point and an (infinitedimensional) space of sources did not met with success (compare with the definition of the Dirac 6function). In the framework of the distribution theory on a space of sources, the quantum field theory is not a "pathological" theory from a point of view of mathematics. It is an ordinary theory of generalized functions, but only in an infinitedimensional space. For fermion fields, superfields, and for boson fields with anticommuting FaddeevPopov ghosts, an infinitedimensional space is endowed with a superstructure. As an unexpected physical consequence we find that the quantum field theory is a statistical theory with respect to source functions.
Chapter VIII. Applications in Physics
320
For instance, for an electromagnetic field with a polynomial selfaction the amplitude of transition from vacuum to vacuum Z(J) is not defined when the external electromagnetic current J' (x) is fixed (and, in particular, when there is no current at all). However, the averagings of the amplitude over the space of external currents are correctly defined. Thus, under the proposed approach to the quantization of an electromagnetic field, we have nothing to say about the amplitudes for a fixed external current, only the mean amplitudes with respect to the fluctuations of external currents are correctly defined. In particular, there is no state without external currents here. In any event, we cannot get any physical quantities for such a state. We can only compute the means with respect to the fluctuations about the state without external currents.
2.5. The relationship with the Schwinger theory of sources. The proposed formalism is a further development of the Schwinger theory of sources [70]. In the Schwinger formalism, a particle is described by means of a switching in vacuum of a source K2(x) which occupies a finite space
time domain followed by switching of a stoke Kl (x) which registers the effect of the action of K2 (x) on vacuum (Kl (x) and K2 (x) are combined to form a single source K(x) = Ki(x) + K2(x)) The action of the source K(x) on vacuum is of a probabilistic nature, and therefore we must perform a large number of switchings of the source K(x) in order to obtain the probability characteristics of a particle. Schwinger supposes that in the process of a large number of experiments it is possible to obtain completely similar sources K(x). However, since the number of experiments is very large, the sources fluctuate, and these fluctuations have not been taken into account in Schwinger's formalism.
We propose to complement Schwinger's formalism with assumptions concerning the probabilistic nature of the source. The physical answers in this formalism are means with respect to the fluctuations of the sources.
2.6. Gaussian packets of sources. We restrict our considera
2. Transition Amplitudes
321
tion to a boson field: V = G(R4) and W = g'(R4). Let us consider the scheme, given in Chap. III, for constructing a space of functions of an infinitedimensional argument which are integrable with respect to the Feynman distribution. Everywhere in this book, we took the space A(V) as the space 1(V). Restricting the space 'P(V) and providing it with a stronger topology, we can achieve a situation where the space oP (W) = .F(xF'(V)) contains the Fourier transform of all countably additive measures on V [140] and, in particular, the Fourier transforms of Gaussian measures, i.e., all Gaussian packets of the form p(C) = exp{2B(e, )},
where B is a continuous quadratic form on the space W (the space W is nuclear, and therefore the continuity is sufficient for the countable additivity, see, e.g., [26]). For the Gaussian mean of the amplitude of transition from vacuum to vacuum we have (Z)B = f Z(e)p(e) dC
= f exp{2B(C,e)}Z(de) = ffB(co)'yB(d), where 'yB is a Gaussian measure with a covariance functional B. Substituting the expression for the function fB(cp), we obtain
(Z)B = f exp
l2
f(aw(x)a(x)  m2co2(x))dx
i f V(W(x)) dx}7B(dc2)
2.7. Continual integral in the framework of the perturbation theory. In his article [126] (see also [53]) Slavnov proposed a definition of a continual integral in the quantum field theory and in the
framework of the perturbation theory. With the aid of the Parseval equality Slavnov defined the quasiGaussian integrals
f
exp{i f w(x)e(x) dx}
Chapter VIII. Applications in Physics
322
x expi i f cp(x)K(x  y)cp(y) dxdy} 11 dcp(x) l
y
n
1
i 4(x1)...4(xn)
exp{
f
2
e(x)K1(x
 y)e(y) dxdyJ
(the function f (cp) = cp(xl)...cp(xn) exp{i f w(x)e(x) dx} is a Fourier transform of the distribution n
1
IL (d77) = in
where b{ is a Dirac measure concentrated at the point , the function g(x) =
exp{2
f
e(x)K1(x

dxdyJ
is, by definition, a Fourier transform of the Feynman distribution 7(dco)
= exp{ f cp(x)K(x  y)co(y) dxdy} II dcp(x), 2 x
and relation (2.2) is rewritten as (f, ry) = (µ, g)). The expression for the generating functional (2.1) is defined in the framework of the perturbation theory. A connection constant is introduced before the potential in (2.1) and exp{ig f V(cp(x)) dx is expanded in a series in the powers of g which is integration termwise with the use of the relation (2.2). As a result, we obtain
ZW _
E(
n=0
n9 )n
(f V(
'
n
)dx
/
x exp{ 2 f e(x)D'(x  y)e(y) dxdy where Dc(x) is a Feynman propagator (Green's causal function)
1
/
Dc(p) =
p2
 7n2 + io.
(2.3)
2. Transition Amplitudes
323
Note that even in a finitedimensional case series (2.3) does not converge. Let us consider, in the framework of the perturbation theory, the integral +00
Z( ) if=o = 0000
k
1 era
f exp{ 2 02  29cp2n + icpe}dcp to

00
1 (i9)k a2nk +00 cP2 k! \ i2nk a2nk f e"°{ eXp{ 22 }
dco
00
_
0
1 [in+19] (2kn)!
00

2ir2
k=O
k!
2kn
(2.4)
(kn)!
This series diverges for n > 2. How can we interpret the formalism of distributions on a space of sources in the framework of the perturbation theory? We multiply series (2.3) by the formal expression rj de(x): X
Z(C) ll de (x) =
E 00
(
n!
)n
(Jv(_8))dx)
n
x exp{  f e(x)D`(x  y) e(y) dxdy } 11 de (x) 2
l
(2.5)
J x
and consider
7(dk) = expS  2 f e(x)D`(x 
dxdy } 11 de (x)
l
J
x
as a Feynman distribution with a covariance operator B = Then we can write series (2.5) in the form Z(dC)

'
n=0
( n9)n
(f V ( (x) )dx)
+ m2.
(2.6)
where the variational derivatives of the Feynman distribution are understood as generalized derivatives of a distribution on an infinitedimensional space.
Chapter VIII. Applications in Physics
324
Theorem 2.1 (convergence of a series from the perturbation theory of the quantum field theory in a space of distributions on an infinitedimensional space). For any polynomial V (x) series (2.6) converges in the space of distributions M(9'(R4)).
Similar theorems hold for fermion fields, superfields, gauge fields. The corresponding series of the perturbation theory consist of generalized derivatives of Feynman distributions on an infinitedimensional superspace.
Remark 2.1. The following example illustrates the situation with a series of the perturbation theory. Consider the series Z(l;)
_
e27rin 00
n=0
This series (consisting of "good" ordinary functions) does not converge pointwise but only converges in the framework of the theory of J (l;) being a periodic sfunction. Actugeneralized functions, converges. Note that we must ally, the series Z(<) = E n=0
distinguish between two problems, namely, the problem of assigning a mathematical meaning to the sums of series of the perturbation theory in the quantum field theory and the problem of calculating the sums of series of the perturbation theory at specific points. We have solved only the first problem, i.e., we have found that the sums of series of the perturbation theory are distributions on an infinitedimensional superspace. The question concerning the assigning of meaning to the values of generalized functions at specific points has not yet been answered even in a finitedimensional case. The difficulties encountered in this direction are not connected with infinite dimensionality of quantum models. Indeed, let us try to calculate the value of a periodic bfunction at
a point e = 0. A formal substitution of
= 0 into a trigonometric series gives a series > 1. Is it possible to obtain a finite number from this infinity? Formulas that are of the same mystical nature as those from the field theory do exist. Let us sum up a trigonometric series
2. Transition Amplitudes
325
as a geometric progression: 1/(1  e2ai{). The function Z(C) can be continued up to a meromorphic function in a complex plane. We expand this function in a Laurent series at the point C = 0,
Z() _  27
1
+
1
2
+ajC+a2e2+...
and regard the value of the regular part of the Laurent series at the point C = 0 as a regularization of the sum of the series00E 1. Thus we n=0
have Zreg(0) = 2
Using the same rule to calculate the value at the point C = 0 for w the series > (which converges only in the space of generalized n=0 functions), we obtain
0
(E n=0
ne2nintll
_ =0
1
12'
and this is consistent with the answer n =  which we obtain n=0 for the main state of a boson string with the aid of the Riemann (function. By the way, even the negativity of the sum of the series consisting of positive numbers is a result that admits a physical interpretation. The main state of a boson string is a tachyon (see [25]). One of the possible points of view concerning the nature of divergences that appear in the process of the calculation of values of generalized functions at specific points (and, possibly, concerning the nature of generalized functions itself) is the following. It is possible that divergences appear because when calculating physical quantities we use only the field of real numbers (or its quadratic extension, i.e., the field of complex numbers).
2.8. NonArchimedean hopes. The field of real numbers is used in physics for such a long time (from the time of Newton) that many researchers regard it as something given by God or something inherent in the nature of the world around us. However, real numbers are only the creation of our mind (for instance, Poincare said the
Chapter VIII. Applications in Physics
326
following: "As a result, we can say that our mind is capable of creating
symbols; due to this ability, it constructed mathematical continuity (i.e., a field of real numbers) which is only a particular system of symbols," see [51].
It should be pointed out that far from always the symbols called real numbers were regarded as something real.
"As early as Middle Ages, such combinations of symbols as f were called numeri ficti, "madeup numbers," or, in Liber abaci by Leonardo of Pisa written in 1202, they were called numeri surdi, "blanc numbers," and were not considered to be numbers at all." For the first time, in Arithmetica integra by Michel Stiefel published in Nuremberg
in 1544 they were given a conditional meaning of numbers and the corresponding name numeri irrationales. Stiefel stated: "irrationalis numerus non est verus numerus," i.e., that "irrational number is not a true number," see Florenskii [61, p. 507]. Possibly, in 400700 years many number fields which are now regarded only as abstract mathematical constructions will be regarded as real physical objects. Just as now we use the symbols of real numbers to denote a point in spacetime (actually, identifying them), in future, we shall, possibly, use the symbols of numbers from other fields to denote some physical quantities (actually, identifying them). Since Archimedean number fields (complete, normed) are exhausted by the field of real numbers (and its quadratic extension, i.e., the field of complex numbers), only nonArchimedean number fields deserve particular attention (see [21, 66]). Every process of measuring a physical quantity begins from a choice
of a unit of measurement 1 and a coefficient of increase of the measurement unit K = m (and, respectively, the coefficient of decrease k = 1/m). Actually, we can only measure numbers of the form
x=
an ,Mn
+...+ a1 +ao+alm+...ia,m'; in
aj = 0,..., m  1. And then our mind has two equally natural possibilities, namely, to admit that the process of decreasing the unit of measurement m times can be continued indefinitely and to admit that
2. Transition Amplitudes
327
the process of increasing the unit of measurement m times
can be
continued indefinitely. In the first case, we obtain the symbols of the form
X=...+an +ao+alm+...+a,.mr. Mn +...+a1 m These are real numbers in the madic representation. These numbers are well known in physics (although, it should be recalled that 700 years ago they were not regarded as numbers). In the second case, we obtain symbols of the form
x= an +ao+alm+...+armr{... Mn +...+a1 m These are madic numbers (see, e.g., [72]). In particular, if m = p is a prime number, then these numbers form a field, a field of padic numbers. Since the field of padic numbers appeared as a result of an infinite
increase of the unit of measuremens 1 p times, we shall try to use panic numbers for describing quantities that are understood as infinities in the field of real numbers. Let us consider, for instance, a trigonometric series 00
C
ZP(S)
_
E pne21rinf
n=0
which converges in a space of generalized functions. The value of this 00 generalized function at the point = 0 is a series E pn, p = 2, 3, ... n=0
diverging in the field of real numbers. However, this series converges in the field of padic numbers, and the sum of the series is a rational number ZP(0) = 1/(1  p). We can show that a onedimensional model of the perturbation theory, series (2.4), converges in the quadratic extension of the field of padic numbers for any p (if p  1 (mod 4), then the series converges even in a field of padic numbers).
We hope that the series from the perturbation theory for the padic quantum field theory converges not only in the space of infinite
328
Chapter VIII. Applications in Physics
padicvalued distributions, but also at every point of the padic space of sources.
See [66] for the quantum mechanics and the quantum field theory with padicvalued functions.
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Index Algebra ArefyevaVolovich 294 associative 7 Banach 9 Grassmann 8 KobayashiNagamashi 53 noncommutative 294 Rogers 10 De Witt 51 Amplitude of transition 315 Body of probability 248 of superspace 92
Cauchy problem 75 CauchyRiemann conditions 21 Chronological exponent nonArchimedean 270 Correspondence principle 197 Covariance 167 functional 228
Distribution cylindrical 240 exponential 233 Feynman 167 Gaussian 167, 269 quasiGaussian 167 Volkenborn 281
Duality of supermodules 128, 141 of superspaces 158 FaddeevPopov ghosts 145 Feynman integral of boson field 170 of fermion field 318 in perturbation theory 321 of spinor field 170 of superfield 169, 319 Field boson 170, 316 fermion 170 neutral chiral 170 nonArchimedean 258 Formula FeynmanKac 205 for integration by parts 172 NewtonLeibniz 36 Trotter 278 nonArchimedean 278 Frequency interpretation in Banach algebra 251 in quantum supertheories 315 Generalized functions analytic 63 nonArchimedean 264 noncommutative 309
346
Index Involution 136
Kolmogorov's axiomatics 245 in Banach algebras 245 Law of large numbers 232 Limit theorems 227, 254 Mean value 228 Mises theory 251 Nilpotent soul 99 subalgebra 99 Operator 86 adjoint 126 d'Alembert 86 evolution 205 heat conduction 79 Laplace 79 orthogonal 132 pseudodifferential 183 Schrodinger 86 selfadjoint 138 unitary 138 Pseudotopological superalgebra 94 superspace 96 Pseudotopology 94 Probability conditional 249 multivalued 247 Projective tensor product 301 Quantization 196 Random process 242 cylindrical 241 quasiGaussian 242 Wiener 243 Soul of probability 248
of superalgebra 92 of superspace 92 Spectrum of an event 247 Super algebra 7
with involution 136 Lie 199 locally convex 60
analyticity 18 conformality 20 differentiability 10 form 43 group 204 manifold 19 module 60 Banach 116 conjugate 61 covering 144 Hilbert 130 locally convex 60 topological 60 space 9 Banach 19 Hilbert 145 infinitedimensional 222 nonArchimedean 258 noncommutative 294 pseudotopological 92 symmetry 19 Theorem central limit 228, 229 of Lyapunov 235 of Riesz 131 Topological basis 142 Transformation Fourier 74 Laplace 267
Index
Transition amplitude 315 Wiener process 242
347
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NJa. Vilenkin and A.U. Klimyk: Representation of Lie Groups and Special Functions. Recent Advances. 1995, 497 pp. ISBN 0792332105 Yu. A. Mitropolsky and A.K. Lopatin: Nonlinear Mechanics, Groups and Symmetry. 1995, 388 pp. ISBN 079233339X
R.P. Agarwal and P.Y.H. Pang: Opial Inequalities with Applications in Differential and Difference Equations. 1995, 393 pp. ISBN 0792333659
A.G. Kusraev and S.S. Kutateladze: Subdifferentials: Theory and Applications. 1995, 408 pp. ISBN 0792333896 M. Cheng, D.G. Deng, S. Gong and C.C. Yang (eds.): Harmonic Analysis in China. 1995, 318 pp. ISBN 079233566X M.S. Livgic, N. Kravitsky, A.S. Markus and V. Vinnikov: Theory of Commuting Nonselfadjoint Operators. 1995, 314 pp. ISBN 0792335880
A.I. Stepanets: Classification and Approximation of Periodic Functions. 1995, 360 pp. ISBN 0792336038 C.G. Arnbrozie and FH. Vasilescu: Banach Space Complexes. 1995, 205 pp. ISBN 0792336305 E. Pap: NullAdditive Set Functions. 1995, 312 pp.
ISBN 0792336585
C.J. Colbourn and E.S. Mahmoodian (eds.): Combinatorics Advances. 1995, 338 pp. ISBN 0792335740 V.G. Danilov, V.P. Maslov and K.A. Volosov: Mathematical Modelling of Heat and Mass Transfer Processes. 1995, 330 pp. ISBN 0792337891 A. Laurin6ikas: Limit Theorems for the Riemann ZetaFunction. 1996, 312 pp. ISBN 0792338243
A. Kuzhel: Characteristic Functions and Models of NonselfAdjoint Operators. 1996, 283 pp. ISBN 0792338790 G.A. Leonov, I.M. Burkin and A.I. Shepeljavyi: Frequency Methods in Oscillation Theory. 1996, 415 pp. ISBN 0792338960
B. Li, S. Wang, S. Yan and C.C. Yang (eds.): Functional Analysis in China. 1996, 390 pp. ISBN 0792338804 P.S. Landa: Nonlinear Oscillations and Waves in Dynamical Systems. 1996, 554 pp. ISBN 0792339312 A.J. Jerri: Linear Difference Equations with Discrete Transform Methods. 1996, 462 PP. ISBN 0792339401
Other Mathematics and Its Applications titles of interest
I. Novikov and E. Semenov: Haar Series and Linear Operators. 1997, 234 pp. ISBN 079234006X L. Zhizhiashvili: Trigonometric Fourier Series and Their Conjugates. 1996, 312 pp. ISBN 0792340884
R.G. Buschman: Integral Transformation, Operational Calculus, and Generalized FuncISBN 079234183X tions. 1996, 246 pp. V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan: Dynamic Systems on Measure ISBN 0792341163 Chains. 1996, 296 pp.
D. Guo, V. Lakshmikantham and X. Liu: Nonlinear Integral Equations in Abstract Spaces. ISBN 0792341449 1996, 350 pp. Y. Roitberg: Elliptic Boundary Value Problems in the Spaces ofDistributions. 1996, 427 pp. ISBN 0792343034
Y. Komatu: Distortion Theorems in Relation to Linear Integral Operators. 1996, 313 pp. ISBN 0792343042 A.G. Chentsov: Asymptotic Attainability. 1997, 336 pp.
ISBN 0792343026
S.T. Zavalishchin and A.N. Sesekin: Dynamic Impulse Systems. Theory and Applications. 1997, 268 pp. ISBN 0792343948 U. Elias: Oscillation Theory of 7woTerm Differential Equations. 1997, 226 pp. ISBN 0792344472 D. O'Regan: Existence TheoryforNonlinear Ordinary Differential Equations. 1997, 204 pp. ISBN 0792345118 Yu. Mitropolskii, G. Khoma and M. Gromyak: Asymptotic Methods for Investigating QuaISBN 0792345290 siwave Equations of Hyperbolic 7jpe. 1997, 418 pp. R.P. Agarwal and P.J.Y. Wong: Advanced Topics in Difference Equations. 1997, 518 pp. ISBN 0792345215
N.N. Tarkhanov: The Analysis of Solutions of Elliptic Equations. 1997, 406 pp. ISBN 0792345312
B. RieLan and T. Neubrunn: Integral, Measure, and Ordering. 1997, 376 pp. ISBN 0792345665 N.L. Gol'dman: Inverse Stefan Problems. 1997, 258 pp.
ISBN 0792345886
S. Singh, B. Watson and P. Srivastava: Fixed Point Theory and Best Approximation: The KKMmap Principle. 1997, 230 pp. ISBN 0792347587 A. Pankov: GConvergence and Homogenization of Nonlinear Partial Differential Operators. 1997, 263 pp. ISBN 079234720X
Other Mathematics and Its Applications titles of interest
S. Hu and N.S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. 1997, 980 pp. ISBN 0792346823 (Set of 2 volumes: 0792346831)
L.A. Sakhnovich: Interpolation Theory and Its Applications. 1997, 216 pp. ISBN 0792348300 G.V. Milovanovid: Recent Progress in Inequalities. 1998, 531 pp.
ISBN 0792348451
V.V. Filippov: Basic Topological Structures of Ordinary Differential Equations. 1998, 530 pp. ISBN 0729349512 S. Gong: Convex and Starlike Mappings in Several Complex Variables. 1998, 208 pp. ISBN 0792349644 A.B. Kharazishvili: Applications of Point Set Theory in Real Analysis. 1998, 244 pp. ISBN 0792349792
R.P. Agarwal: Focal Boundary Value Problems for Differential and Difference Equations. 1998, 300 pp. ISBN 0792349784 D. PrzeworskaRolewicz: Logarithms and Antilogarithms. An Algebraic Analysis Approach. 1998, 358 pp. ISBN 0792349741
Yu. M. Berezansky and A.A. Kalyuzhnyi: Harmonic Analysis in Hypercompkx Systems. ISBN 0792350294 1998, 493 pp. V. Lakshmikantham and A.S. Vatsala: Generalized Quasilinearization for Nonlinear ProbISBN 0792350383 lems. 1998, 286 pp. V. Barbu: Partial Differential Equations and Boundary Value Problems. 1998, 292 pp. ISBN 0792350561 J. P. Boyd: Weakly Nonlocal Solitary Waves and BeyondAllOrders Asymptotics. Generalized Solitons and Hyperasymptotic Perturbation Theory. 1998, 610 pp. ISBN 0792350723
D. O'Regan and M. Meehan: Existence Theory for Nonlinear Integral and IntegrodifferenISBN 0792350898 tial Equations. 1998, 228 pp. A.J. Jerri: The Gibbs Phenomenon inFourierAnalysis, Splines and WaveletApproximations. ISBN 0792351096 1998, 364 pp.
C. Constantinescu, W. Filter and K. Weber, in collaboration with A. Sontag: Advanced ISBN 0792352343 Integration Theory. 1998, 872 pp. V. Bykov, A. Kytmanov and M. Lazman, with M. Passare (ed.): Elimination Methods in ISBN 0792352408 Polynomial Computer Algebra. 1998, 252 pp.
W.H. Steeb: Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum ISBN 0792352319
Mechanics. 1998, 234 pp.