art%3A10.1023%2FB%3AJOSS.0000019836.37663.d9.pdf

191

0022-4715/04/0400-0191/0 © 2004 Plenum Publishing Corporation

Journal of Statistical Physics, Vol. 115, Nos. 1/2, April 2004 (© 2004)

Some New Developments in the Theory of PathIntegrals, with Applications to Quantum Theory

S. Albeverio1–3 and S. Mazzucchi1 , 2

1 Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany.2 Dipartimento di Matematica, Università di Trento, 38050 Povo, Italy.3 BiBoS; IZKS; SFB611; CERFIM (Locarno); Acc. Arch. (Mendrisio).

Received April 18, 2003; accepted November 21, 2003

A survey of recent developments concerning rigorously defined infinite dimen-sional integrals, mainly of the type of ‘‘Feynman path integrals,’’ is given. Boththe theory and its applications, especially in quantum theory, are presented. Asfor the theory, general results are discussed including the case of polynomiallygrowing phase functions, which are handled by exploiting the connection withprobabilistic functional integrals. Also applications to continuous measurementtheory and the stochastic Schrödinger equation are given. Other applications ofprobabilistic methods in non relativistic quantum theory and in quantum fieldtheory, and their relations with statistical mechanics, are discussed.

KEY WORDS: Feynman path integrals; Schrödinger equation; euclideanquantum fields; convoluted Poisson noise; Poisson random fields.

4 These connections were discussed a bit more extensively in the lecture given by the firstnamed author.

It is a great pleasure to dedicate this paper to Gianni Jona-Lasinio on theoccasion of his 70th birthday—as a small sign of deep gratitude.The topic of the present work belongs to just one of several areas the first

named author had the opportunity and pleasure to discuss with Gianni over manyyears, always appreciating his genuine sense for what is essential in mathematicalphysics and receiving from him new motivations and ideas.The main body of this paper is Section 2, where we give a rather detailed dis-

cussion of some new developments concerning particular aspects of path integralmethods in quantum theory. In Sections 1 and 3 we take the opportunity tomention some other connections with work and interests of Gianni, even thoughtime and space do not permit to present them here in any details.4

1. PROBABILISTIC METHODS IN QUANTUM MECHANICS

Let us take as starting point the connection between quantum dynamics, asgiven by a family of unitary operators Ut=e−itH, t ¥ R being the time vari-able and H the (lower semibounded) Hamiltonian, and the correspondingheat equation evolution semigroup Uy=e−yH, y \ 0. Heuristically, but alsorigorously under some assumptions on H, Uy is the analytic continuation ofUt for ‘‘imaginary time’’ (i.e., for t replaced by −iy). The correspondingfunctional representation for H of the Schrödinger form − D2+V inL2(Rd, dx), with a suitable real-valued potential V, is, for the heat equa-tion, given by the Feynman–Kac formula

Uyf(x)=F e−>y

0 V(w(s)) dsf(w(y)) Px(dw), (1)

Px being Wiener measure giving the distribution of the Wiener (or Brownianmotion) process (w(s), 0 [ s [ t) started at x ¥ Rd at time 0, f the initialcondition. Let us stress that (1) is a well defined integral (e.g., for V, fbounded and continuous.

Heuristically we have

e−>y

0 V(w(s)) dsPx(dw)=‘‘Z−1e−SEy (w) Dw’’ — PxE(dw), (2)

with SEy the euclidean action functional,

SEy (w)=S0y(w)+F

y

0V(w(s)) ds, S0y(w) —

12 F

y

0|w(s)|2 ds (3)

Z is the normalization (‘‘heuristic free partition function’’)

Z=‘‘ F e−S0y (w) Dw’’ (4)

with Dw a ‘‘flat measure’’ (the E in PEx stands for ‘‘euclidean’’). One writes

Uyf(x)=EPxE f(w(y)) (5)

(with EQ — expectation with respect to the measure Q). We can write, in thesense of linear functionals

Uyf(x)=Of( · ), PxE( · )P (6)

(O , P being the dualization between a space of function f, say Cb(W), andthe space of finite measures on W — C([0, t]; R)).

192 Albeverio and Mazzucchi

Similarly, for the solutions of the corresponding Schrödinger evolutionone can expect, following Feynman (see Section 2), a formula of the type

Utf(x)=Of( · ), PxF( · )P, (7)

where the r.h.s. has to be understood as the evaluation at f of a (linearcontinuous) ‘‘Feynman path functional’’ (usually called Feynman pathintegral). PxF (where F stands for ‘‘Feynman’’) is heuristically obtained byreplacing the exponents e−S

Ey (w) in (2) and e−S

0y (w) in (4) with e

i(St(w) and

ei(S 0t (w) respectively, where St is the ‘‘classical action functional’’ defined asSEt in (3) but with V replaced by −V.

Whereas Feynman–Kac formula (1) holds essentially for any V havingnegative part with not too strong singularities and H lower bounded, theabove Feynman formula (7), holds essentially for any continuous V (notnecessarily such that H is lower bounded), see Section 2 for a further dis-cussion of Feynman path integrals. Let us remark that for historicalreasons (to conform to the original presentation of Feynman) (see, e.g.,ref. 44) (7) is usually rewritten, using the invariance of PxF(dw) under thetransformation induced in W by sW y−s, as Utf(x)=Of( · ), PxF( · )P, withPxF defined as PxF, but with the space of paths replaced by the one wherethe paths end at time t at x (and start anywhere at the initial time zero).In Section 2 we shall use this representation and denote the paths by c inorder not to confuse them with the previous paths.

One way to understand the principle of this, is to realize that for alarge class of V (see, e.g., ref. 69), H, as an operator on L2(Rd, dx), hasa strictly positive eigenfunction f in L2(Rd, dx), called ‘‘ground state,’’ suchthat H \ c, for some c >−., and Hf=cf. Then (H−c) is unitaryequivalent to a self-adjoint positive operator Hm in L2(Rd, m), withm(dx) — f2(x) dx, the unitary equivalence being given by f ¥ L2(Rd, dx)Y f

f ¥ L2(Rd, m), so that e−it(H−c)f=fe−itHmf−1f. A simple computation

shows that Hm is the self-adjoint operator uniquely associated with theclosed sesquilinear form Em(f, g) —

12 > Nf Ng dm, f, g ¥ D(N), N being the

natural gradient operator from L2(Rd, m) to L2(Rd, m) é Rd (see, e.g.,ref. 2). Em is the natural extension to the complex L2(Rd, m) of the classicalDirichlet form given by m. Under some weak regularity assumptions on f

(which, can be reinterpreted in terms of regularity assumptions on V), thereexists a measurable ‘‘drift vector,’’ bm (on Rd with values in Rd), such thatHm=−

12 D−bmN (on a dense domain) in L2(Rd, m). By the theory of

Dirichlet forms the corresponding semigroup (e−yHm)y \ 0 is the transitionsemigroup of a diffusion process X(y), y \ 0 on Rd, satisfying a stochasticdifferential equation of Langevin’s type: dX(y)=bm(X(y)) dy+dw(y), with

Some New Developments in the Theory of Path Integrals 193

(w(s), s ¥ [0, y]) a standard Brownian motion on Rd. The initial distribu-tion can be taken to be any point in Rd, m is an invariant measure for X(y).One has in particular for any g ¥ Cb(Rd):

(e−yHmg)(x)=Ex|X(g(X(y)))=Ex[e−12>y0 b(w(s))

2 dse−>y

0 b(w(s)) dw(s)g(w(y))]

(with Ex|X the expectation with respect to the distribution of X started atx and Ex the one with respect to the Wiener measure Px). The densitye−>

y0 b(w(s))

2 dse−>y0 b(w(s)) dw(s) with respect to the Wiener measure Px(dw) is called

Girsanov functional. Combining this formula with

e−yHm=f−1e−y(H−c)f

we also obtain

(e−yHmg)(x)=f−1(x) Ex[e−>y

0 V(w(s)) dsf(w(y)) g(w(y))].

These are just a few examples of transformation formulae which constitutesthe basis of stochastic analysis. In particular the representations relatingthe Dirichlet operator Hm and X extend to more general situations, wherethe state space and the coefficients can be ‘‘singular’’ and the state space Rd

is replaced by some possibly infinite-dimensional state space. See, e.g.,refs. 1, 2, 54, and references therein for these connections. The work ofJona-Lasinio and his coworkers has played an essential role in pointing outthe relations between processes like the above X and quantum theory pro-viding new insights in vast areas of stochastic analysis and quantum theory(in the non relativistic theory as well as the relativistic one, in particular inconnection with Nelson’s quantum mechanics and the Euclidean approach,see also Section 3). The latter has also permitted to exploit probabilistic aswell as classical statistical mechanical methods in quantum (field) theory).Let us point out that stochastic processes have been used in quantumtheory both in an instrumental way (e.g., for deriving spectral esti-mates (54, 55, 71)) and in a conceptual way (e.g., in stochastic mechanics or inthe theory of Schrödinger processes, see, e.g., refs. 1–49, 52, 69, 70, 77, andreferences therein).

2. SOME BASIC ELEMENTS AND NEW DEVELOPMENTS IN THE

THEORY OF FEYNMAN PATH INTEGRALS

The representation (7) in Section 1 of the solution of Schrödingerequation was already presented heuristically by Feynman in his thesis inthe forties, following a suggestion by Dirac. Let us rewrite it with the


(reduced) Planck’s constant ( inserted and with k0 instead of f for initialcondition. Schrödinger equation is then

˛ i(“

“t k=−(2

2m Dk+Vk

k(0, x)=k0(x)(8)

(7) is understood by Feynman as a way to compute the wave function k attime t evaluated at the point x ¥ Rd as an ‘‘integral over histories,’’ that isas an integral over all possible paths with finite energy arriving at time t atthe point x:

k(t, x)=‘‘ F{c | c(t)=x}

ei(St(c)k0(c(0)) Dc ’’ (9)

St(c) being the classical action (introduced in Section 1) evaluated alongthe path c, and Dc an heuristic ‘‘flat’’ measure on the space of paths. Weremark that (7) resp. (9) are entirely similar to (6) resp. (5) (taking intoaccount (2)). In (9) we write c instead of w as used in (5), just to stress thatwe are in the framework of Feynman rather than in the one of probabilisticintegrals.

Even if more than 50 years have passed, Feynman’s formula is stillfascinating, as it shows the link between the classical description of thephysical world and the quantum one. In fact it provides a quantizationmethod, allowing to associate, at least heuristically, a quantum evolutionto each classical Lagrangian. Moreover it allows the study of the ‘‘semi-classical limit,’’ that is the study of the behavior of the solution of theSchrödinger equation taking into account that ( is small. Indeed since ( issmall the integrand is strongly oscillating and the main contribution to theintegral should come from those paths c which make stationary the phasefunction S. These, by Hamilton’s least action principle, are the classicalorbits of the system.

Formula (9), as it stands, has not of course a well defined mathemati-cal meaning. Indeed neither the normalization constant, nor the ‘‘flatmeasure’’ Dc on the space of paths are well defined. Formula (1) in Sec-tion 1 was derived by Kac in 1949 precisely as a rigorous replacement,valid for the corresponding heat equation, of the heuristic expression (9).As we already mentioned in Section 1, (1) is a well defined integral on thespace of continuous paths with respect to a s-additive measure Px (forpaths starting at x). Such an interpretation is not possible for the heuristic‘‘Feynman measure’’ e

i(St(c) Dc. Indeed Cameron (33) proved that the latter

cannot be realized as a complex s-additive measure, even on very nicesubsets.


Nevertheless, under suitable hypothesis on the potential V and on theinitial datum k0, one can indeed give to the integral (9) a rigorous mathe-matical meaning as a linear continuous functional on a suitable class offunctions, as is expressed by (7) (whose probabilistic counterpart is (6)). Inthe literature several realizations of the ‘‘Feynman functional’’ can be found,for instance by means of analytic continuation, (33, 35, 41, 52, 60, 64, 66, 67, 73, 74) bynon standard analysis (7) or as an infinite dimensional distribution in theframework of Hida calculus, (40, 49) or via ‘‘complex Poisson measures,’’ (1, 65)

or as a infinite dimensional oscillatory integral. (4, 18, 42, 50, 51) The lattermethod is particularly interesting as it is the only one by which a develop-ment of an infinite dimensional version of the stationary phase method andthe corresponding study of the semiclassical limit (( a 0) of the solution hasbeen performed.

In the following we shall denote by H a (finite or infinite dimen-sional) real separable Hilbert space, whose elements, resp. scalar product,will be denoted by x, y ¥H, resp by Ox, yP. f:HQ C will be a functionon H and Q: D(Q) ıHQH an invertible, densely defined and self-adjoint operator.

We shall denote by M(H) the Banach space of the complex boundedvariation measures on H, endowed with the total variation norm, that is:

m ¥M(H), ||m||=sup Ci|m(Ei)|,

where the supremum is taken over all sequences {Ei} of pairwise disjointBorel subsets of H, such that 1i Ei=H. M(H) is a Banach algebra,where the product of two measures m f n is by definition their convolution:

m f n(E)=FH

m(E−x) n(dx), m, n ¥M(H)

and the unit element is the vector d0.We will denote by F(H) the space of complex functions on H which

are Fourier transforms of measures belonging to M(H), that is:

f:HQ C f(x)=FHe iOx, bPmf(db) — mf(x).

F(H) is a Banach algebra of functions, where the product is the pointwiseone, the unit element is the function 1, i.e., 1(x)=1 -x ¥H and the normis given by ||f||=||mf ||.


Let us assume first of all that H is finite dimensional, i.e., H=Rn,and define the ‘‘Fresnel integral’’

F2 ei2(

Ox, QxPf(x) dx

in the following way

Definition 1. A function f: RnQ C is Fresnel integrable withrespect to Q if and only if for each f ¥S(Rn) such that f(0)=1 the limit

limEQ 0(2pi()−n/2 F e

i2(

Ox, QxPf(x) f(Ex) dx (10)

exists and is independent of f. In this case the limit is called the Fresnelintegral of f with respect to Q and denoted by

F2 ei2(

Ox, QxPf(x) dx. (11)

The description of the full class of Fresnel integrable functions is noteasy, but one can find some interesting subsets of it. Indeed the followingresult holds: (18, 42)

Theorem 1. Let f ¥F(Rn). Then f is Fresnel integrable and itsFresnel integral with respect to Q is given by:

F2 ei2( Ox, QxPf(x) dx=(det Q)−1/2 F e

−i(2 Oa, Q−1aP

mf(da). (12)

The definition of oscillatory integral can be extended to the case whereH is infinite dimensional. (4, 42)

Definition 2. A function f:HQ C is Fresnel integrable withrespect to Q if and only if for each sequence Pn of projectors onton-dimensional subspaces of H, such that Pn [ Pn+1 and Pn Q 1 stronglyas nQ., (1 being the identity operator in H), the finite dimensionalapproximations of the Fresnel integral of f with respect to Q

(2pi()−n/2 FPnHei2(

OPnx, QPnxPf(Pnx) d(Pnx),


are well defined and the limit

limnQ.(2pi()−n/2 F

PnHei2(

OPnx, QPnxPf(Pnx) d(Pnx) (13)

exists and is independent of the sequence {Pn}.In this case the limit is called the Fresnel integral of f with respect to

Q and is denoted by

F2 ei2(

Ox, QxPf(x) dx.

One can prove (4, 42) that if f ¥F(H) then f p Pn ¥F(Pn(H)) and f isFresnel integrable. Moreover, if Q−I is trace class, the following Cameron-Martin–Parseval type formula holds:

F2 ei2(

Ox, QxPf(x) dx=(det Q)−1/2 FH

e−i(2Oa, Q−1aP

mf(da) (14)

where det Q=|det Q| e−pi Ind Q is the Fredholm determinant of the operatorQ,|det Q| is its absolute value and Ind(Q) is the number of negative eigen-values of the operator Q, counted with their multiplicity.

In this setting one can give a rigorous mathematical interpretationof formula (9) in terms of an infinite dimensional oscillatory integral ona suitable Hilbert space of paths. Let us consider the Sobolev spaceH=H1, 2(t) ([0, t], R

d), that is the space of absolutely continuous functionsc: [0, t]Q Rd, c(t)=0, such that > t0 |c(s)|2 ds <., endowed with thefollowing scalar product

Oc1, c2P=Ft

0c1(s) · c2(s) ds.

H is essentially the ‘‘Cameron-Martin space.’’Let us consider, on the other hand, the Schrödinger equation in

L2(Rd)

i(“

“tk=Hk (15)

with initial datum k|t=0=k0, where H=− (2

2 D+12 xW2x+VŒ(x), where

x ¥ Rd, W2 \ 0 is a d×d matrix, VŒ ¥F(Rd) and k0 ¥F(Rd) 5 L2(Rd) (weset m=1 for simplicity of notation).


By considering the operator L on H1, 2(t) ([0, t], Rd) given by

Oc, LcP — Ft

0c(s) W2c(s) ds,

and the functionW: Ht Q C

W(c) — Ft

0VŒ(c(s)+x) ds+2xW2 F

t

0c(s) ds, c ¥H1, 2(t) ,

formula (9)

‘‘const F{c | c(t)=x}

ei(>t0 (

12c(s)2− 1

2c(s) W2c(s)−VŒ(c(s))) ds

k0(c(0)) Dc’’

can be interpreted as the rigorously defined infinite dimensional oscillatoryintegral on H1, 2(t) ([0, t], R

d)

F2 ei2(

Oc, (I+L) cPe−i(W(c)

k0(c(0)+x) dc. (16)

Moreover one can prove (4, 42) that (16) is a representation of the solution of(15) evaluated in x ¥ Rd at time t.

2.1. Application to the Quantum Theory of Measurement

It is possible to extend the definition of infinite dimensional oscilla-tory integral in order to consider complex-valued phase functions. Thisallows to give Feynman path integrals representations to a larger class ofSchrödinger equations, with applications to the quantum theory of mea-surement (see refs. 13 and 14).

Theorem 2. Let H be a real separable Hilbert space, let l ¥H be avector in H and let L1 and L2 be two self-adjoint, trace class commutingoperators on H such that I+L1 is invertible and L2 is non negative. Letmoreover f:HQ C be the Fourier transform of a complex boundedvariation measure mf on H:

f(c)=mf(c), f(c)=FH

e iOc, aPmf(da).


Then the infinite dimensional oscillatory integral (with complex phase)

F2H

ei2(

Oc, (I+L) cPeOl, cPf(c) dc

is well defined and it is given by

F2H

ei2(

Oc, (I+L) cPeOl, cPf(c) dc=det(I+L)−1/2 FH

e−i(2

Oa−il, (I+L)−1 (a−il)Pmf(da)

(17)

(L being the operator on the complexification HC of the real Hilbert spaceH given by L=L1+iL2 ).

Equation (17) can be recognized as an analytic continuation of theCameron-Martin–Parseval formula (14).

The latter theorem can be applied to the solution of a particularstochastic Schrödinger equation describing the continuous non-demolitionmeasurement of the position of a quantum particle: the Belavkin equation.5

5 Related equations have been proposed also by other physicists, for instance in work byDiosi, Ghirardi, Rimini and Weber, Gisin, Mensky (see the references in refs. 13 and 14).

It is well known that the continuous time evolution described by the ordi-nary Schrödinger equation is valid if the quantum system is ‘‘undisturbed,’’but if it is submitted to the measurement of one of its observables andinteracts with the measuring apparatus this is no longer true. Indeed thestate of the system after the measurement is the result of a random anddiscontinuous change: the so-called ‘‘collapse of the wave function.’’ Bymodeling the measuring apparatus with a one-dimensional bosonic field, byassuming the interaction Hamiltonian has a particular form and by meansof the quantum stochastic calculus of Hudson and Parthasarathy, in 1987V. P. Belavkin (30) proposed a stochastic partial differential equationdescribing the selective dynamics of a d-dimensional particle submitted tothe measurement of one of its (possibleM-dimensional vector) observables,described by the self-adjoint operator R on L2(Rd)

˛dk(t, x)=−i(Hk(t, x) dt− l2 R

2k(t, x) dt+`l Rk(t, x) dW(t)

k(0, x)=k0(x) (t, x) ¥ [0, T]×Rd(18)

where H is the quantum mechanical Hamiltonian, W is an M-dimensionalBrownian motion on a probability space (W,F, P), dW(t) is the Ito dif-ferential and l > 0 is a coupling constant, which is proportional to theaccuracy of the measurement. In the particular case of the description of


the continuous measurement of position one has that R is the operatormultiplication by x, so that Eq. (18) assumes the following form:

˛dk(t, x)=−i(Hk(t, x) dt− l2 x

2k(t, x) dt+`l xk(t, x) dW(t)

k(0, x)=k0(x) (t, x) ¥ [0, T]×Rd,(19)

while in the case of momentum measurement, (where R=−i(N) one has:

˛dk(t, x)=−i(Hk(t, x) dt+l(

2

2 Dk(t, x) dt− i`l ( Nk(t, x) dW(t)

k(0, x)=k0(x) (t, x) ¥ [0, T]×Rd.(20)

Even before Belavkin, M. B. Mensky had proposed a heuristic formulafor the selective dynamics of a particle whose position is continuouslyobserved. According to Mensky the state of the particle at time t if theobserved trajectory is the path [a] is given by the ‘‘restricted pathintegrals’’

k(t, x, [a])=‘‘ F{c(t)=x}

ei(St(c)e−l >

t0 (c(s)−a(s))

2 dsf(c(0)) Dc’’. (21)

One can see that, as an effect of the correction term e−l >t0 (c(s)−a(s))

2 ds due tothe measurement, the paths c giving the main contribution to the integral(21) are those closer to the observed trajectory [a]. By means of Theorem 2one can prove a Feynman path integral representation to the solution ofBelavkin equation and give a rigorous mathematical meaning to Mensky’sheuristic formula. Indeed in the case of position measurement, the follow-ing holds: (13)

Theorem 3. Let V and f be Fourier transform of finite complexmeasures on Rd. Then there exists a solution of the stochastic Schrödingerequation (19) and it can be represented by the following infinite dimen-sional oscillatory integral with complex phase on the Hilbert space H=H1, 2(t) ([0, t], R

d)

k(t, x)=F2 ei(St(c)−l >

t0 (c(s)+x)

2 ds

· e >t0 `l(c(s)+x) dW(s) dsf(c(0)+x) dc

=e−l |x|2 t+`l x ·w(t) F2

H

ei2(

Oc, (I+L) cPeOl, cPe−2l( >t0 x · c(s) ds

· e−i >t0 V(x+c(s)) dsf(c(0)+x) dc


where l ¥H, l(s)=`l > ts w(y) dy and

L:HC QHC, Oc1, Lc2P=−2il( Ft

0c1(s) c2(s) ds.

In the case of the Belavkin equation describing momentum measure-ment the stochastic term plays the role of a complex random potentialdepending on the momentum of the particle. In this case one has to use a‘‘phase space Feynman path integral,’’ that is an infinite dimensional oscil-latory integral on the Hilbert space H1, 2(t) ([0, t], R

d)×L2([0, t]) —H×Lt(see refs. 12 and 14).

Theorem 4. Let V and f be Fourier transform of finite complexmeasures on Rd. Then there exist a solution of the Cauchy problem (20),which can be represented by the following ‘‘phase space Feynman pathintegral:’’

k(t, 0, w)=F2H×Lt

ei(>t0 (p(s) q(s)−

p2(s)2) dse−l >

t0 p(s)

2 ds

· e−i(>t0 V(q(s)) dse `l >

t0 p(s) dW(s)f(q(0)) dq dp

=F2Ht ×Lt

ei(O(q, p), A(q, p)PeO(q, p), lPe−

i(>t0 V(q(s)) dsf(q(0)) dq dp (22)

where by (q, p) we denote a generic element of H×Lt, q ¥H, p ¥ Lt, A isthe operator on the complexification of H×Lt given by

O(q, p), A(q, p)P=2 Ft

0

1 p(s) q(s) ds−(1−2il() F t0p2(s)2 ds

and l ¥Ht×Lt.

2.2. Extension of the Theory of Feynman Path Integrals to the Case

of Polynomially Growing Phase Functions

The examples discussed in 3.1 show that the infinite dimensional oscil-latory integrals are a powerful tool, with application to a larger class of(deterministic resp. stochastic) Schrödinger equations. The main restrictionof such integrals has been so far the fact that the potentials V for which aFeynman path integral representation of the solution of the correspondingSchrödinger equation can be defined are of the type ‘‘quadratic plusbounded perturbation.’’ (4, 18, 42) Indeed, in order to define the infinite


dimensional oscillatory integral, the correction VŒ to the harmonic oscilla-tor potential 12 xW

2x has to belong to F(Rd), so that it is bounded. Anextension to unbounded potentials which are Laplace transforms ofbounded measures has been developed in refs. 6 and 61. It includes someexponentially growing potentials but does not cover the case of potentialswhich are polynomials of degree larger than 2. In fact the problem is notsimple, as it has been proved (75) that in one dimension, if the potential istime independent and super-quadratic in the sense that V(x) \ C(1+|x|)2+E

at infinity, C > 0 and E > 0, then the fundamental solution of time depen-dent Schrödinger equation is nowhere C1. In refs. 20 and 21 a solution tothe problem of providing a direct rigorous Feynman path integral defini-tion for such potentials (without going to the tool of analytic continuationfrom a representation of the heat equation as for instance in refs. 31, 41,and 67) is given and a Feynman path integral representation for the solu-tion of the Schrödinger equation for an anharmonic oscillator potentialV(x)=1

2 xW2x+lx4, l > 0, is developed. The first step is the definition and

the computation of the oscillatory integral >H ei(F(x)f(x) dx, when H is

finite dimensional and the phase function F(x)=P(x) is an arbitrary evenpolynomial with positive leading coefficient. In this case a generalization ofTheorem 1 can be proved. The main tool is the following lemma, which canbe proved by using the analyticity of ekz+

i(P(z), z ¥ C, and a change of inte-

gration contour (see ref. 20).

Lemma 1. Let P: RNQ R be an even polynomial with positiveleading coefficient. Then the Fourier transform of the distribution e

i(P(x):

F(k)=FRNe ik · xe

i(P(x) dx, ( ¥ R0{0} (23)

is an entire bounded function and admits the following representation:

F(k)=e iNp/4M FRNe ie

ip/4Mk ·xei(P(eip/4Mx) dx, ( > 0 (24)

or

F(k)=e−iNp/4M FRNe ie

−ip/4Mk ·xei(P(e−ip/4Mx) dx, ( < 0. (25)

Theorem 5. Let f ¥F(RN), f=mf. Then the generalized Fresnelintegral

I(f) — F ei(P(x)f(x) dx, ( ¥ D0{0}


is well defined and it is given by the formula of Parseval’s type:

F ei(P(x)f(x) dx=F F(k) mf(dk), (26)

where F(k) is given by (24) or (25)

F(k)=F e ikxei(P(x) dx.

The integral on the r.h.s. of (26) is absolutely convergent (hence it can beunderstood in Lebesgue sense).

It is particularly interesting to examine the case in which P(x)=12 x(I−B) x−lA(x, x, x, x), where ( > 0, l [ 0, I, B are N×N matrices,I being the identity, (I−B) is symmetric and strictly positive andA: RN×RN×RN×RNQ R is completely symmetric and positive fourthorder covariant tensor on RN.

Lemma 2. Under the assumptions above the Fourier transform of

the distribution ei2(x · (I−B) x

(2pi()N/2e−il(A(x, x, x, x):

F(k)=FRNe ik ·xei2(x · (I−B) x

(2pi()N/2e−il(A(x, x, x, x) dNx (27)

is a bounded complex-valued entire function on RN admitting the followingrepresentation

F(k)=FRNe ie

ip/4k ·x e− 12(x · (I−B) x

(2p()N/2eil(A(x, x, x, x) dNx

=E[e ieip/4k ·xe

il(A(x, x, x, x)e

12(x ·Bx] (28)

where E denotes the expectation value with respect to the centered Gaussianmeasure on RN with covariance operator (I.

Theorem 6 (Parseval equality). Let f ¥F(RN), f=mf. Then,under the assumptions above, the generalized Fresnel integral

I(f) — F2 ei2(x · (I−B) xe

−il(A(x, x, x, x)f(x) dx


is well defined and it is given by:

F2 ei2(x · (I−B) xe

−il(A(x, x, x, x)f(x) dx=F F(k) mf(dk), (29)

where F(k) is given by Eq. (28). Moreover if mf is such that -x ¥ RN the

integral > e−`22kx |mf | (dk) is convergent and the positive function g: RnQ R,

defined by g(x)=e12(x ·Bx > e−

`22kx |mf | (dk) is summable with respect to the

centered Gaussian measure on RN with covariance (I, then f extends to ananalytic function on CN and the corresponding generalized Fresnel integralis given by:

F2RN

ei2(x · (I−B) x

(2pi()N/2e−il(P(x)f(x) dx=E[e

il(P(x)e

12(x ·Bxf(e ip/4x)]. (30)

Remark 1. In ref. 20 general polynomial phase functions are alsodiscussed and analytic resp. Borel summable expansions of (24) in powersof ( are proved

The result in Theorem 6 has been generalized to the infinite dimen-sional case.

Let H be a real separable infinite dimensional Hilbert space, withinner product O , P and norm | |. Let n be the finitely additive cylindermeasure on H, defined by its characteristic functional n(x)=e−

(

2|x|2. Let || ||

be a ‘‘measurable’’ norm on H, that is || || is such that for every E > 0 thereexist a finite-dimensional projection PE:HQH, such that for all P + PEone has n({x ¥H | ||P(x)|| > E}) < E, where P and PE are called orthogonal(P + PE ) if their ranges are orthogonal in (H, O , P). One can easily verifythat || || is weaker than | |. Denote by B the completion of H in the|| ||-norm and by i the continuous inclusion of H in B, one can prove thatm — n p i−1 is a countably additive Gaussian measure on the Borel subsetsof B. The triple (i,H, B) is called an abstract Wiener space. (48, 62) Giveny ¥Bg one can easily verify that the restriction of y to H is continuouson H, so that one can identify Bg as a subset of H and each elementy ¥Bg can be regarded as a random variable n(y) on (B, m). Given anorthogonal projection P in H, with

P(x)=Cn

i=1Oei, xP ei


for some orthonormal e1,..., en ¥H, the stochastic extension P of P on B iswell defined by

P( · )=Cn

i=1n(ei)( · ) ei.

Given a function f:HQB1, where (B1, || ||B1 ) is another real separableBanach space, the stochastic extension f of f to B exists if the functionsf p P: BQB1 converge to f in probability with respect to m as P con-verges strongly to the identity in H. Let A:H×H×H×HQ R be acompletely symmetric positive covariant tensor operator on H such thatthe map V:HQ R+, xW V(x) — A(x, x, x, x) is continuous in the || ||norm. As a consequence V is continuous in the | |-norm, moreover it can beextended by continuity to a random variable V on B, with V|H=V.Moreover given a self-adjoint trace class operator B:HQH, the quadra-tic form on H×H:

x ¥HW Ox, BxP

can be extended to a random variable on B, denoted again by O · , B · P. Inthis setting one can prove the following generalization of Theorem 6. (21)

Theorem 7. Let B be self-adjoint trace class, (I−B) strictly posi-tive, l [ 0 and f ¥F(H), f — mf, and let us suppose that the boundedvariation measure mf satisfies the following assumption

FH

e(

4Ok, (I−B)−1 kP |mf | (dk) <+.. (31)

Then the infinite dimensional oscillatory integral

F2H

ei2(

Ox, (I−B) xPe−il

(A(x, x, x, x)f(x) dx (32)

exists and is given by:

FH

E[e in(k)(w) eip/4e12(

Ow, BwPe il

(V(w)] mf(dk).


It is also equal to:

E[e il

(V(w)e

12(

Ow, BwPf(e ip/4w)] (33)

E denotes the expectation value with respect to the Gaussian measure m

on B.

Such a theory allows an extension of the class of potentials for whichan infinite dimensional oscillatory integral representation of the solution ofthe corresponding Schrödinger equation can be defined. Let us consider theSchrödinger equation

i(ddt

k=Hk (34)

on L2(Rd) for an anharmonic oscillator Hamiltonian H of the followingform:

H=−(2

2D+12xW2x+lC(x, x, x, x), (35)

where C is a completely symmetric positive fourth order covariant tensoron Rd, W is a positive symmetric d×d matrix, l \ 0 a positive constant. Itis well known, see ref. 69, that H is essentially self-adjoint on C.0 (R

d). Weare going to show the way to give mathematical meaning to the ‘‘Feynmanpath integral’’ representation of the solution of Eq. (34):

k(t, x)=‘‘ Fc(0)=x

ei(>t0c(s)2

2ds− i

(>t0 [

12c(s) W2c(s)+lC(c(s), c(s), c(s), c(s))] ds

k0(c(t)) Dc’’,

as the analytic continuation (in the parameter l) of an infinite dimensionalgeneralized oscillatory integral on a suitable Hilbert space.

Let us consider the Cameron-Martin space Ht, that is the Hilbertspace of absolutely continuous paths c: [0, t]Q Rd, with c(0)=0 and innerproduct Oc1, c2P=> t0 c1(s) c2(s) ds. The cylindrical Gaussian measure on Htwith covariance operator the identity extends to a s-additive measure onthe Wiener space Ct={w ¥ C([0, t]; Rd) | c(0)=0}: the Wiener measureW.(i, Ht, Ct) is an abstract Wiener space.

Let us consider moreover the Hilbert space H=Rd×Ht, and theBanach space B=Rd×Ct endowed with the product measureN(dx)×W(dw),N being the Gaussian measure on Rd with covariance equal to the d×didentity matrix. (i,H, B) is an abstract Wiener space.


Let us consider two vectors f, k0 ¥ L2(Rd) 5F(Rd).We are going todefine the following infinite dimensional oscillatory integral on H:

‘‘ FRd×Ht

f(x) ei2(

>t0 c(s)2 dse−

i2(

>t0 [(c(s)+x) W2(c(s)+x) ds

· eil(C(c(s)+x, c(s)+x, c(s)+x, c(s)+x) ds

k0(c(t)+x) dx Dc’’. (36)

Let us consider the operator B:HQH given by:

(x, c)0 (y, g)=B(x, c),

y=tW2x+W2 Ft

0c(s) ds, g(s)=W2x 1 ts−s

2

22−F s

0Fu

tW2c(r) dr du

(37)

and the fourth order tensor operator A given by:

A((x1, c1), (x2, c2), (x3, c3), (x4, c4))

=Ft

0C(c1(s)+x1, c2(s)+x2, c3(s)+x3, c4(s)+x4) ds. (38)

Let us consider moreover the function f:HQ C given by

f(x, c)=(2pi()d/2 e−i2(|x|2

f(x) k0(c(t)+x) (39)

with this notation expression (36) can be written in the following form:

F2H

ei2((|x|2+|c|2)e−

i2(

O(x, c), B(x, c)Pe−il(A((x, c), (x, c), (x, c), (x, c))f(x, c) dx dc. (40)

In the following we will denote by Wi, i=1,..., d, the eigenvalues ofthe matrix W.

Theorem 8. Let us assume that l [ 0, and that for each i=1,..., dthe following inequalities are satisfied

Wit <p

2, 1−Wi tan(Wit) > 0. (41)

Let f, k0 ¥ L2(Rd) 5F(Rd). Let m0 be the complex bounded variationmeasure on Rd such that m0=k0. Let mf be the complex bounded variation


measure on Rd such that mf(x)=(2pi()d/2 e− i2(|x|2f(x). Assume in addition

that the measures m0, mf satisfy the following assumption:

FRdFRde(

4xW−1 tan(Wt) x

e (y+cos(Wt)−1 x)(1−W tan(Wt))−1 (y+cos(Wt)−1 x) |m0 | (dx) |mf | (dy) <..

(42)

Then the function f:HQ C, given by (39) is the Fourier transform of abounded variation measure mf on H satisfying

FH

e(

4O(y, g), (I−B)−1 (y, g)P |mf | (dy dg) <. (43)

(B being given by (37)) and the infinite dimensional oscillatory integral (40)is well defined and is given by:

FRd×Ht

1FRd×Cte ie

ip/4(x ·y+`( n(c)(w))e12(

>t0 (`( w(s)+x) W2(`( w(s)+x) ds

· e il

(>t0 C(`( w(s)+x,`( w(s)+x,`( w(s)+x,`( w(s)+x) dsW(dw)

e−|x|2

2(

(2p()d/2dx2 mf(dy dc).

(44)

This is also equal to

(i)d/2 FRd×Cte il

(>t0 C(`( w(s)+x,`( w(s)+x,`( w(s)+x,`( w(s)+x) ds

· e12(

>t0 (`( w(s)+x) W2(`( w(s)+x) ds

f(e ip/4x) k0(e ip/4`( w(t)+e ip/4x) W(dw) dx.(45)

The oscillatory integral (40) can heuristically be written in the follow-ing form:

(f, k(t))=‘‘ FRd

f(x) F{c | c(t)=x}

ei(St(c)k0(c(0)) Dc dx’’

and interpreted as a rigorous realization of the Feynman path integralrepresenting the inner product between the vector f ¥ L2(Rd) and the solu-tion of the Schrödinger equation (34) with initial datum k0. We remarkthat the infinite dimensional oscillatory integral (40) is well defined only ifl [ 0, but its expressions in terms of the absolutely convergent integrals(44) and (45) are well defined also for each l ¥ R. In fact one can verify


directly that the absolutely convergent integrals (44) and (45) are analyticfunctions of the complex variable l if Im(l) > 0, continuous in Im(l)=0and coinciding with (40) if l [ 0. Moreover one can prove that when l \ 0the Gaussian integrals (44) and (45) represent the inner product Of, k(t)P,where k(t) is the solution of the Schrödinger equation.

Theorem 9. Let f, k0 ¥S(Rd) satisfy assumption (42). Then (44)and (45) represent, for l \ 0, the scalar product between f and the solutionof the Schrödinger equation (34).

3. A NOTE ON PROBABILISTIC METHODS IN QUANTUM FIELD

THEORY

Extensions of the methods discussed in Sections 1 and 2 to infinitedimensional state spaces have also been studied, particularly in relations toapplications to quantum field theory.

As for the Feynman path integrals we simply mention the rigorousconstruction of the Chern–Simons model of gauge fields in 3 space-timedimensions and the corresponding rigorous computation of topologicalinvariants using the theory of infinite dimensional oscillatory integrals(Fresnel integrals), see refs. 15, 27, 28, 63, and references therein.

As for the heat semigroup evolutions, associated infinite dimensionalprocesses and their connections to Euclidean quantum fields let us justmention that the measure which plays a corresponding role to the onementioned in Section 1, e−>

y

0 V(w(s)) dsPx(dw), is (for Euclidean scalarquantum fields over the Euclidean space-time Rd)

mv(dw)=‘‘Z−1e−l >Rd v(w(y)) dym(dw)’’ (46)

where Z — ‘‘> e−l >Rd v(w(y)) dym(dw)’’, l > 0, v: RQ R being an ‘‘interactiondensity,’’ m being Nelson’s free field measure (describing free Euclideanquantum fields over Rd of mass m > 0). w is the Euclidean coordinateprocess, representing the Euclidean quantum field (over Rd).

For d=2, mv can be given a rigorous meaning (for suitable v), see,e.g., refs. 19, 45, 49, and 71.

mv was given a further meaning as ‘‘equilibrium measure’’ for astochastic evolution equation, the ‘‘stochastic quantization equation,’’ inwork by Jona-Lasinio and coworkers (56) (see also refs. 22–25). This hasgiven also a considerable momentum to the theory of stochastic partialdifferential equations, see also, for instance, refs. 58 and 59. m itself can be


looked upon as ‘‘convolution transform’’ of the Gaussian white noisemeasure mGWN, (49) i.e.,

m=mGWN((`−D+m2)−1.), (47)

D being the Laplacian in L2(Rd) (see refs. 10 and 49).Models constructed by replacing in (46), (47) mGWN by mPWN, with

mPWN a Poisson-type white noise measure have been constructed in recentyears and shown to satisfy, together with their vector-valued analogues, allMorchio-Strocchi axioms for indefinite metric quantum fields, see refs. 9and 11, and references therein.

Relativistic models with local interaction and non-trivial scattering(even in 4-space-time dimensions!) have been constructed. (8) For recentwork on extending these models to curved space-times see ref. 47.

New connections with models of (classical) statistical mechanics (ofparticles) have emerged from this work, see refs. 9 and 46.

Previous work on the use of Poisson measures in the study of quantummodels is in ref. 39.

ACKNOWLEDGMENTS

It is a great pleasure to thank the organizers for the kind invitation togive a lecture at the Symposium in Honor of G. Jona-Lasinio and contri-bute to the Proceedings. We acknowledge the warm hospitality of theMathematics Institutes in Bonn and Trento, where this work was written.The second named author gratefully acknowledges financial support by theEU-project ‘‘Quantum Probability with Applications to Physics, Informa-tion Theory and Biology.’’

REFERENCES

1. S. Albeverio, Wiener and Feynman path integrals and their applications, in Proceedings ofSymposia in Applied Mathematics, Vol. 52 (1997), pp. 163–194.

2. S. Albeverio, M. Schachermayer, and M. Talagand, Lectures on Probability Theory andStatistics, Proceedings of Ecole d’Eté de Probabilités de Saint-Flour 2000, P. Bernard, ed.,Lecture Notes in Mathematics 1816 (Springer, Berlin, 2003).

3. S. Albeverio, A. M. Boutet de Monvel-Berthier, and Z. Brzezniak, The trace formula forSchrödinger operators from infinite dimensional oscillatory integrals, Math. Nachr. 182:21–65 (1996).

4. S. Albeverio and Z. Brzezniak, Finite-dimensional approximation approach to oscillatoryintegrals and stationary phase in infinite dimensions, J. Funct. Anal. 113:177–244 (1993).

5. S. Albeverio and Z. Brzezniak, Oscillatory integrals on Hilbert spaces and Schrödingerequation with magnetic fields, J. Math. Phys. 36:2135–2156 (1995).


6. S. Albeverio, Z. Brzezniak, and Z. Haba, On the Schrödinger equation with potentialswhich are Laplace transforms of measures, Potential Anal. 9:65–82 (1998).

7. S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, and T. Linstrøm, Pure and Applied Math-ematics, Vol. 122 (Academic Press, Orlando, FL, 1986).

8. S. Albeverio and H. Gottschalk, Scattering theory for quantum fields with indefinitemetric, Comm. Math. Phys. 216:491–513 (2001).

9. S. Albeverio, H. Gottschalk, and M. Yoshida, Representing Euclidean quantum fields asscaling limit of particle systems, J. Stat. Phys. 108:361–369 (2002).

10. S. Albeverio, H. Gottschalk, and J. L. Wu, Convoluted generalized white noise, Schwingerfunctions, and their analytic continuation, Rev. Math. Phys. 8:763–817 (1996).

11. S. Albeverio, H. Gottschalk, and J. L. Wu, Models of local relativistic quantum fieldswith indefinite metric (in all dimensions), Comm. Math. Phys. 184:509–531 (1997).

12. S. Albeverio, G. Guatteri, and S. Mazzucchi, Phase space Feynman path integrals,J. Math. Phys. 43:2847–2857 (2002).

13. S. Albeverio, G. Guatteri, and S. Mazzucchi, Representation of the Belavkin equation viaFeynman path integrals, Probab. Theory Related Fields 125:365–380 (2003).

14. S. Albeverio, G. Guatteri, and S. Mazzucchi, Representation of the Belavkin equation viaphase space Feynman path integrals, to appear in IDAQP.

15. S. Albeverio, A. Hahn, and A. Sengupta, Rigorous Feynman path integrals, with applica-tions to quantum theory, gauge fields, and topological invariants, SFB 611, Preprint,Bonn, No. 58, 2003, to appear in Proc. Conf. Mathematical Legacy of Feynman’s PathIntegral, Lisbone 2003.

16. S. Albeverio, V. N. Kolokol’tsov, and O. G. Smolyanov, Représentation des solutions del’équation de Belavkin pour la mesure quantique par une version rigoureuse de la formuled’intégration fonctionnelle de Menski, C. R. Acad. Sci. Paris Sér. I Math. 323:661–664(1996).

17. S. Albeverio, V. N. Kolokol’tsov, and O. G. Smolyanov, Continuous quantum measure-ment: Local and global approaches, Rev. Math. Phys. 9:907–920 (1997).

18. S. Albeverio and R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals,Lecture Notes in Mathematics, Vol. 523 (Springer-Verlag, Berlin, 1976).

19. S. Albeverio, R. Høegh-Krohn, J. E. Fenstad, and T. Lindstrøm, Nonstandard Methods inStochastic Analysis and Mathematical Physics (Academic Press, New York, 1986), [alsotranslated into Russian by A. K. Svonskin and M. A. Shubin (Mir, Moscow, 1990)].

20. S. Albeverio and S. Mazzucchi, Generalized Fresnel Integrals, Preprint of the University ofBonn, No. 59 (2003).

21. S. Albeverio and S. Mazzucchi, Feynman Path Integrals for Polynomially Growing Poten-tials, Preprint of the University of Trento, UTM 638 (2003).

22. S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions:Solutions via Dirichlet forms, Probab. Theory Related Fields 89:347–386 (1991).

23. S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces—The construction of the associated diffusion process, Probab. Theory Related Fields83:405–434 (1989).

24. S. Albeverio and M. Röckner, Dirichlet forms, quantum fields, and stochastic quantiza-tion, in Stochastic Analysis, Path Integration, and Dynamics, Pitman Res. Notes Math.Ser., Vol. 200 (Longman Sci. Tech., Harlow, 1989), pp. 1–21.

25. S. Albeverio and B. Rüdiger, Infinite dimensional Stochastic differential equations obtainedby subordination and related Dirichlet forms, J. Funct. Anal. 204:122–156 (2003).

26. S. Albeverio, B. Rüdiger, and J. L. Wu, Analytic and Probabilistic Aspects of Lévy Pro-cesses and Fields in Quantum Theory, O. E. Bandorff-Nielsen, T. Mikosch, and S. L.Resnick, eds. (Birkhäuser Boston, Boston, MA, 2001), pp. 187–224.


27. S. Albeverio and J. Schäfer, Abelian Chern–Simons theory and linking numbers via oscil-latory integrals, J. Math. Phys. 36:2157–2169 (1995).

28. S. Albeverio and A. Sengupta, A mathematical construction of the non-Abelian Chern–Simons functional integral, Comm. Math. Phys. 186:563–579 (1997).

29. R. Azencott and H. Doss, L’équation de Schrödinger quand h tend vers zéro: Uneapproche probabiliste (French) [The Schrödinger equation as h tends to zero: A proba-bilistic approach], in Stochastic Aspects of Classical and Quantum Systems (Marseille,1983), Lecture Notes in Math., Vol. 1109 (Springer, Berlin, 1985), pp. 1–17.

30. V. P. Belavkin, A new wave equation for a continuous nondemolition measurement, Phys.Lett. A 140:355–358 (1989).

31. G. Ben Arous and F. Castell, A probabilistic approach to semi-classical approximations,J. Funct. Anal. 137:243–280 (1996).

32. L. Bertini, G. Jona-Lasinio, and C. Parrinello, Stochastic quantization, stochastic cal-culus, and path integrals: Selected topics, Progr. Theoret. Phys. Suppl. 111:83–113 (1993).

33. R. H. Cameron, A family of integrals serving to connect the Wiener and Feynmanintegrals, J. Math. Phys. 39:126–140 (1960).

34. P. Cartier and C. DeWitt-Morette, Functional integration, J.Math. Phys. 41:4154–4187 (2000).35. D. M. Chung, Conditional analytic Feynman integrals on Wiener spaces, Proc. AMS

112:479–488 (1991).36. K. L. Chung and J. C. Zambrini, Introduction to Random Time and Quantum Randomness,

Monographs of the Portuguese Mathematical Society (McGraw–Hill, Lisbon, 2001).37. A. B. Cruzeiro, L. Wu, and J. C. Zambrini, Bernstein processes associated with a Markov

process, in Stochastic Analysis and Mathematical Physics (Santiago, 1988), Trends. Math.(Birkhäuser Boston, Boston, MA, 2000), pp. 41–72.

38. I. Daubechies and J. R. Klauder, Quantum-mechanical path integrals with Wienermeasure for all polynomial Hamiltonians II, J. Math. Phys. 26:2239–2256 (1985).

39. G. F. De Angelis, G. Jona-Lasinio, and V. Sidoravicius, Berezin integrals and Poissonprocesses, J. Phys. A 31:289–308 (1998).

40. M. De Faria, H. H. Kuo, and L. Streit, The Feynman integrand as a Hida distribution,J. Math. Phys. 32:2123–2127 (1991).

41. H. Doss, Sur une résolution stochastique de l’équation de Schrödinger à coefficientsanalytiques, Comm. Math. Phys. 73:247–264 (1980).

42. D. Elworthy and A. Truman, Feynman maps, Cameron-Martin formulae, and anharmo-nic oscillators, Ann. Inst. H. Poincaré Phys. Théor. 41:115–142 (1984).

43. M. V. Fedoriuk and V. P. Maslov, Semi-Classical Approximation in Quantum Mechanics(D. Reidel, Dordrecht, 1981).

44. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill,New York, 1965).

45. J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View (Springer-Verlag, New York, 1987).

46. H. Gottschalk, Particle Systems with Weakly Attractive Interactions, Preprint of the Uni-versity of Bonn (Sept. 2002), (www.lqp.uni-goettingen.de/papers/02/08/).

47. H. Gottschalk and H. Thaler, Interacting Quantum Fields with Indefinite Metric on Glo-bally Hyperbolic Space-Times, Preprint of the University of Bonn, SFB 611 n.10.

48. L. Gross, Abstract Wiener spaces, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Vol. 2(1965), pp. 31–42.

49. T. Hida, H. H. Kuo, J. Potthoff, and L. Streit,White Noise (Kluwer, Dordrecht, 1995).50. K. Ito, Wiener integral and Feynman integral, in Proc. Fourth Berkeley Symposium onMathematical Statistics and Probability, Vol. 2 (California University Press, Berkeley,1961), pp. 227–238.


51. K. Ito, Generalized uniform complex measures in the Hilbertian metric space with theirapplications to the Feynman path integral, Proc. Fifth Berkeley Symposium on Mathemat-ical Statistics and Probability, Vol. 2, Part 1 (California University Press, Berkeley, 1967),pp. 145–161.

52. G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman’s OperationalCalculus (Oxford University Press, New York, 2000).

53. G. Jona-Lasinio, Invariant measures under Schrödinger evolution and quantum statisticalmechanics, in Stochastic Processes, Physics, and Geometry: New Interplays, I (Leipzig,1999), CMS Conf. Proc., Vol. 28 (Amer. Math. Soc., Providence, RI, 2000), pp. 239–242.

54. G. Jona-Lasinio, Stochastic processes and quantum mechanics, Colloquium in Honor ofLaurent Schwartz, Vol. 2 (Palaiseau, 1983), Astérisques, No. 132 (1985), pp. 203–216 .

55. G. Jona-Lasinio, G. Martinelli, and E. Scoppola, Tunneling in One Dimension: GeneralTheory, Instabilities, Rules of Calculation, Applications, Mathematics+physics, Vol. 2(World Scientific, Singapore, 1986), pp. 227–260.

56. G. Jona-Lasinio and P. K. Mitter, On the stochastic quantization of field theory, Comm.Math. Phys. 101:409–436 (1985).

57. G. Jona-Lasinio and P. K. Mitter, Large deviation estimates in the stochastic quantizationof f42, Comm. Math. Phys. 130:111–121 (1990).

58. G. Jona-Lasinio and R. Sénéor, On a class of stochastic reaction-diffusion equations intwo space dimensions, J. Phys. A 24:4123–4128 (1991).

59. G. Jona-Lasinio and R. Sénéor, Study of stochastic differential equations by constructivemethods I, J. Stat. Phys. 83:1109–1148 (1996).

60. G. Kallianpur, D. Kannan, and R. L. Karandikar, Analytic and sequential Feynmanintegrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin Formula, Ann.Inst. H. Poincaré, Prob. Th. 21:323–361 (1985).

61. T. Kuna, L. Streit, and W. Westerkamp, Feynman integrals for a class of exponentiallygrowing potentials, J. Math. Phys. 39:4476–4491 (1998).

62. H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. (Springer-Verlag, Berlin/Heidelberg/New York, 1975).

63. S. Leukert and J. Schäfer, A rigorous construction of Abelian Chern–Simons pathintegrals using white noise analysis, Rev. Math. Phys. 8:445–456 (1996).

64. V. Mandrekar, Some remarks on various definitions of Feynman integrals, in LecturesNotes in Math., K. Jacob Beck, ed. (1983), pp. 170–177.

65. V. P. Maslov,Méthodes opérationelles (Mir, Moscou, 1987).66. P. Mullowney, A General Theory of Integration in Function Spaces, Pitman Research

Notes in Mathematics, Vol. 153 (1987).67. E. Nelson, Feynman integrals and the Schrödinger equation, J. Math. Phys. 5:332–343

(1964).68. J. Rezende, The method of stationary phase for oscillatory integrals on Hilbert spaces,Comm. Math. Phys. 101:187–206 (1985).

69. M. Reed and B. Simon, Methods of Modern Mathematical Physics. Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975).

70. B. Simon, Functional Integration and Quantum Physics (Academic Press, New York/London, 1979).

71. B. Simon, The P(f)2 Euclidean (Quantum) Field Theory, Princeton Series in Physics(Princeton University Press, Princeton, NJ).

72. O. G. Smolyanov and E. T. Shavgulidze, Path Integrals (Moskov. Gos. Univ., Moscow,1990).

73. H. Thaler, Solution of Schrödinger equations on compact Lie groups via probabilisticmethods, to appear in Potential Analysis.


74. A. Truman, The Feynman maps and the Wiener integral, J. Math. Phys. 19:1742–1750(1978).

75. K. Yajima, Smoothness and non-smoothness of the fundamental solution of time depen-dent Schrödiger equations, Comm. Math. Phys. 181:605–629 (1996).

76. J. C. Zambrini, Feynman integrals, diffusion processes and quantum symplectic two-forms, J. Korean Math. Soc. 38:385–408 (2001).

77. T. J. Zastawniak, Equivalence of Albeverio–Høegh–Krohn–Feynman Integral for Anhar-monic Oscillators and the Analytic Feynman Integral, Univ. Iagel. Acta Math., Vol. 28(1991), pp. 187–199.


Documents

art%3A10.1023%2FB%3AJOSS.0000019836.37663.d9.pdf