Feynman Formulas and Path Integrals for Some … · Feynman Formulas and Path Integrals for Some...

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ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 18, No. 4, 2011, pp. 387–399. c Pleiades Publishing, Ltd., 2011. Feynman Formulas and Path Integrals for Some Evolution Semigroups Related to τ -Quantization B. B¨ ottcher,* Ya. A. Butko,** R. L. Schilling,*** and O. G. Smolyanov**** *,*** Institut f¨ ur Mathematische Stochastik, Technische Universit¨at Dresden Zellescher Weg, 12-14, D-01069 Dresden, Germany, E-mail: * [email protected], *** [email protected] ** Department of Fundamental Sciences, Bauman Moscow State Technical University 105005, 2nd Baumanskaya str., 5, Moscow, Russia, E-mail:[email protected] **** Department of Mechanics and Mathematics, Lomonosov Moscow State University 119992, Vorob’evy gory, 1, Moscow, Russia, E-mail:[email protected] Received October 9, 2011 Abstract. This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n →∞) and their connections with phase space Feyn- man path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamilton- ian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of sto- chastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals. DOI 10.1134/S1061920811040017 1. INTRODUCTION, NOTATION, AND PRELIMINARIES The notion of path integral has been introduced by R. Feynman (at a heuristic level, [19]), and Feynman’s three main observations are the following. First, the solution of the Cauchy problem for the Schr¨odinger equation can be represented as a limit of finite-dimensional integrals over the nth Cartesian power of a configuration space as n tends to infinity. Second, this limit can be interpreted as an integral over a set of paths in the configuration space. Finally, it is noted that the integrand contains an exponential of the classical action. Feynman’s definition of path integral over trajectories in a phase space, introduced in [20], has the same structure; however, the classical action is expressed by a Hamiltonian function rather than by a Lagrangian. A formalization of the first observation is referred to as a Feynman formula, whereas the second observation leads to Feynman path integrals over trajectories in a configuration space or in a phase space. Representations of solutions of evolution equations by path integrals are also referred to as Feynman–Kac formulas, although Kac himself considered the heat equation only and used only an integral over paths in the configuration space in his formula. Currently, path integrals are one of the main tools in the mathematical apparatus of theoretical physics. They are important objects in quantum field theory, especially in the theory of gauge fields. On one hand, they enable one to represent any quantum quantity as a sum of inputs of virtual classical paths. A simple dependence on the Planck constant shows that, if 0, then the dominant input is given by a real classical trajectory, i.e., by the trajectory satisfying the principle of least action. On the other hand, path integrals are technically convenient to study semiclassical asymptotics, to construct series in perturbation theory, etc. In many problems, it is reasonable to consider the Hamiltonian formulation of quantum mechanics, and hence to work with the phase space (or Hamiltonian) Feynman path integrals. 387

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ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 18, No. 4, 2011, pp. 387–399. c© Pleiades Publishing, Ltd., 2011.

Feynman Formulas and Path Integrals for SomeEvolution Semigroups Related to τ -Quantization

B. Bottcher,* Ya. A. Butko,** R. L. Schilling,*** and O. G. Smolyanov****

*,*** Institut fur Mathematische Stochastik, Technische Universitat DresdenZellescher Weg, 12-14, D-01069 Dresden, Germany,

E-mail: * [email protected], *** [email protected]** Department of Fundamental Sciences, Bauman Moscow State Technical University

105005, 2nd Baumanskaya str., 5, Moscow, Russia,E-mail:[email protected]

**** Department of Mechanics and Mathematics, Lomonosov Moscow State University119992, Vorob’evy gory, 1, Moscow, Russia,

E-mail:[email protected]

Received October 9, 2011

Abstract. This note is devoted to Feynman formulas (i.e., representations of semigroups bylimits of n-fold iterated integrals as n → ∞) and their connections with phase space Feyn-man path integrals. Some pseudodifferential operators corresponding to different types ofquantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamilton-ian Feynman formulas for semigroups generated by these operators are obtained. Further, aconstruction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to thisconstruction, the Hamiltonian Feynman formulas obtained here and in our previous papersdo coincide with Hamiltonian Feynman path integrals. This connects phase space Feynmanpath integrals with some integrals with respect to probability measures. These connectionsenable us to make a contribution to the theory of phase space Feynman path integrals, toprove the existence of some of these integrals, and to study their properties by means of sto-chastic analysis. The Feynman path integrals thus obtained are different for different types ofquantization. This makes it possible to distinguish the process of quantization in the languageof Feynman path integrals.

DOI 10.1134/S1061920811040017

1. INTRODUCTION, NOTATION, AND PRELIMINARIES

The notion of path integral has been introduced by R. Feynman (at a heuristic level, [19]), andFeynman’s three main observations are the following. First, the solution of the Cauchy problemfor the Schrodinger equation can be represented as a limit of finite-dimensional integrals over thenth Cartesian power of a configuration space as n tends to infinity. Second, this limit can beinterpreted as an integral over a set of paths in the configuration space. Finally, it is noted that theintegrand contains an exponential of the classical action. Feynman’s definition of path integral overtrajectories in a phase space, introduced in [20], has the same structure; however, the classical actionis expressed by a Hamiltonian function rather than by a Lagrangian. A formalization of the firstobservation is referred to as a Feynman formula, whereas the second observation leads to Feynmanpath integrals over trajectories in a configuration space or in a phase space. Representations ofsolutions of evolution equations by path integrals are also referred to as Feynman–Kac formulas,although Kac himself considered the heat equation only and used only an integral over paths inthe configuration space in his formula.

Currently, path integrals are one of the main tools in the mathematical apparatus of theoreticalphysics. They are important objects in quantum field theory, especially in the theory of gauge fields.On one hand, they enable one to represent any quantum quantity as a sum of inputs of virtualclassical paths. A simple dependence on the Planck constant � shows that, if � → 0, then thedominant input is given by a real classical trajectory, i.e., by the trajectory satisfying the principleof least action. On the other hand, path integrals are technically convenient to study semiclassicalasymptotics, to construct series in perturbation theory, etc. In many problems, it is reasonable toconsider the Hamiltonian formulation of quantum mechanics, and hence to work with the phasespace (or Hamiltonian) Feynman path integrals.

387

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388 BOTTCHER et al.

There are various approaches providing a mathematically rigorous meaning for phase spaceFeynman path integrals. Some phase space Feynman path integrals are defined in terms of theFourier transform and Parseval’s identity (see [41, 2, 39, 18, 14], and the references therein); someare defined by an analytic continuation of a Gaussian measure on the set of paths in a phasespace [41], some by regularization procedures, e.g., as limits of integrals with respect to Gaussianmeasures with diverging diffusion constant [17]; the integrands of some phase space Feynman pathintegrals are realized as Hida distributions in the setting of White Noise Analysis [4]. A varietyof approaches treats Feynman path integrals as limits of integrals over some finite-dimensionalsubspaces of paths as the dimension tends to infinity. Such path integrals are sometimes said to besequential. The general definition of a sequential Feynman pseudomeasure (Feynman path integral)on an abstract space (in particular, on a set of paths in a phase space) can be found in [41]. Someconcrete realizations are presented (e.g., in [43, 12, 1, 24, 30, 29, 28, 22]).

In this note, we extend the approach of [43] (introduced actually in [44, 45] to study surfacemeasures, see also [46, 48] on this subject). This method is based on the Chernoff theorem [15], whichis a generalization of the well-known Trotter formula (Trotter’s formula has been used to justifyFeynman’s heuristic result for the Schrodinger equation with a potential (e.g., in [32]) and to provethe original Feynman–Kac formula for the heat equation). This approach enables one to obtainFeynman formulas, i.e., representations of solutions of initial (and boundary) value problems forevolution equations (or, equivalently, representations of semigroups solving problems in question)in the form of limits of n-fold iterated integrals as n tends to infinity. Recently, this method hasbeen successfully used to obtain Feynman formulas for diverse classes of problems for evolutionequations on various geometric structures (see, e.g., [10, 35, 40, 39, 38, 7, 34]). In many cases, itis possible to obtain Feynman formulas with integrands that contain elementary functions only.Formulas of this kind can be used for direct computations and numerical modeling of an evolutionunder consideration.

In [3], Berezin posed the problem of distinguishing different procedures of quantization in thelanguage of Feynman path integrals. A partial answer to the problem is given in this note. Namely,some Hamiltonian Feynman formulas for semigroups generated by the τ -quantization of a quadraticHamiltonian function are obtained. These Feynman formulas give approximations to phase spaceFeynman path integrals with respect to the pseudomeasure Φ1

x defined in [12],1 and the integrandsare different for each τ . Therefore, the procedure of quantization is distinguishable.

In accordance with Berezin’s comment in [3], some (phase space) Feynman path integrals overdifferent sets of paths can be used to represent the same solution of a given evolution equation(cf. [42]). The quality (the degree of smoothness) of paths in the configuration space is inverselyproportional to the quality of paths in the momentum space. This reflects Heisenberg’s uncertaintyprinciple. Continuous paths in the configuration space and discontinuous paths in the momentumspace are treated in many works (cf. [4, 1, 29, 24]). In our approach, the picture is symmetric, thepaths have the same quality in both the spaces; they are piecewise constant, and their one-sidedcontinuity is different (and conjugate in a sense) for the configuration and momentum spaces.

1.1. Chernoff Theorem and Feynman Formulas

For a Banach space (X, ‖ · ‖X), the symbol L(X) always stands for the space of all continuouslinear operators on X equipped with the strong operator topology, ‖ · ‖ for the operator norm onL(X), and Id for the identity operator in X. If Dom(L) ⊂ X is a linear subspace and L : Dom(L) →X is a linear operator, then Dom(L) denotes the domain of L. A one-parameter family (Tt)t�0 ofbounded linear operators Tt : X → X is referred to as a strongly continuous semigroup if T0 = Id,Ts+t = Ts ◦ Tt for all s, t � 0, and limt→0 ‖Ttϕ − ϕ‖X = 0 for all ϕ ∈ X. If (Tt)t�0 is a stronglycontinuous semigroup on a Banach space (X, ‖ · ‖X), then the generator L of (Tt)t�0 is defined by

the rule Lϕ := limt→0(1/t)(Ttϕ− ϕ) with the domain Dom(L) :={ϕ ∈ X

∣∣∣∣ limt→0(1/t)Ttϕ− ϕ

exists as a strong limit}. Consider an evolution equation ∂f/∂t = Lf . If L is the generator of

1This pseudomeasure actually corresponds to the case of qp-quantization in [43], where a family of Feynman pseu-

domeasures Φτ , τ ∈ [0, 1], is defined and some Feynman formulas are obtained for Schrodinger groups generated by

the τ-quantization of some functions which are Fourier transforms of measures.

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FEYNMAN FORMULAS AND PATH INTEGRALS 389

a strongly continuous semigroup (Tt)t�0 on a Banach space (X, ‖ · ‖X), then the solution of theCauchy problem for this equation with initial value f(0) = f0 ∈ X is given by f(t) = Ttf0 for allf0 ∈ X. Therefore, to solve the evolution equation ∂f/∂t = Lf is to construct a semigroup (Tt)t�0

with the given generator L. If the desired semigroup is not known explicitly, it can be approximated.One of the tools to approximate a semigroup uses the Chernoff theorem [15] (here we present theversion of Chernoff’s theorem given in [43]).

Theorem 1.1. [Chernoff] Let X be a Banach space, let F : [0,∞) → L(X) be a strongly contin-uous mapping such that F (0) = Id, and let ‖F (t)‖ � eat for some a ∈ [0,∞) and all t � 0. Let Dbe a linear subspace of Dom(F ′(0)) such that the restriction of the operator F ′(0) to D is closable.Let (L,Dom(L)) be this closure. If (L,Dom(L)) is the generator of a strongly continuous semigroup(Tt)t�0, then, for any t0 > 0, the sequence (F (t/n))n)n∈N converges to (Tt)t�0 as n → ∞ in thestrong operator topology, uniformly with respect to t ∈ [0, t0]. Thus, Tt = limn→∞(F (t/n))n.

Here the derivative at the origin of a function F : [0, ε) → L(X), ε > 0, is a linear mappingF ′(0) : Dom(F ′(0)) → X such that F ′(0)g := limt→0(1/t)(F (t)g − F (0)g), where Dom(F ′(0)) isthe vector space of all elements g ∈ X for which the above limit exists.

A family of operators (F (t))t�0 is said to be Chernoff equivalent to a semigroup (Tt)t�0 if thisfamily satisfies the assertions of the Chernoff theorem; i.e., if the limit

Tt = limn→∞

(F (t/n))n (1.1)

is achieved locally uniformly with respect to t in L(X). In many cases, the operators F (t) areintegral operators, and hence the limit on the right-hand side of (1.1) is a limit of iterated integrals.In this setting, we obtain the so-called Feynman formula.

Definition 1.2. A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semi-group solving the problem) by the limit of n-fold iterated integrals as n → ∞.

We use this term, because it was Feynman ([19, 20]) who introduced a functional (path) integralas a limit of iterated finite-dimensional integrals. The limits in Feynman formulas coincide with (orin some cases define) some functional integrals with respect to probability measures or Feynman-type pseudomeasures on a set of paths of a physical system. A representation of a solution of aninitial (or initial-boundary) value problem for an evolution equation (or, equivalently, a represen-tation of the semigroup resolving the problem) by a functional integral is usually referred to as aFeynman–Kac formula. Hence, the iterated integrals in a Feynman formula for some problem giveapproximations to a functional integral in the Feynman–Kac formula representing the solution ofthe same problem. The integrands of these approximations contain only elementary functions inmany cases, and therefore, these approximations can be used for direct calculations and simulations.

The notion of Feynman formula was introduced in [43], and the way to obtain Feynman formulaswith the help of the Chernoff theorem has been developed in a series of papers [43–48]. Recently,this approach has been successfully applied to obtain Feynman formulas for different classes ofproblems for evolution equations on different geometric structures (see, e.g., [11, 13, 7–9, 33, 34,39]), and also to construct some surface measures on infinite-dimensional manifolds (see [44–49]).

An identity of the form Tt = limn→∞(F (t/n))n is referred to as a Lagrangian Feynman formulaif all F (t), t > 0, are integral operators with elementary kernels (see, e.g., formula (2.6) below);if all F (t) are pseudodifferential operators (for the definition, see Section 1.2), we speak of Hamil-tonian Feynman formulas (see, e.g., formula (2.4)). This terminology is inspired by the fact thata Lagrangian Feynman formula gives approximations to a functional integral over a set of pathsin the configuration space of a system (whose evolution is described by the semigroup (Tt)t�0),whereas a Hamiltonian Feynman formula corresponds to a functional integral over a set of pathsin the phase space of some system.

1.2. Pseudodifferential Operators, Their Symbols, and τ -Quantization

Let H : Rd × Rd → C be a measurable function, and let τ ∈ [0, 1]. Define a pseudodifferential

operator (ΨDO) Hτ (·,D) with τ -symbol H(q, p) on a Banach space (X, ‖ · ‖X) of some functions

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on Rd by the rule

Hτ (q,D)ϕ(q) = (2π)−d

Rd

Rd

eip·(q−q1)H(τq + (1− τ)q1, p)ϕ(q1) dq1 dp, (1.2)

where the domain Dom(Hτ (·,D)) is the set of all ϕ ∈ X for which the right-hand side of (1.2) iswell defined as an element of (X, ‖ · ‖X). Below, we consider pseudodifferential operators on thespace C∞(Rd) := {ϕ ∈ C(Rd) : lim|x|→∞ ϕ(x) = 0} with the sup-norm ‖ϕ‖∞ := supx∈Rd |ϕ(x)|and on the space L2(R

d). We always assume that the set of test functions C∞c (Rd) belongs to the

domain of the operator Hτ (·,D).

The mapping H �→ Hτ (·,D) from a space of functions on Rd×R

d to the space of linear operators

on (X, ‖ · ‖X) is referred to as τ -quantization, and the very operator Hτ (·,D) is referred to be theτ -quantization of the function H. Note that, if the symbol H is a sum of functions depending

only on one of the variables q or p, then the ΨDOs Hτ (·,D) coincide for all τ ∈ [0, 1]. If we have

H(q, p) = qp = pq for q, p ∈ R1, then Hτ (q,D)ϕ(q) = −iτq ∂

∂qϕ(q)− i(1 − τ) ∂∂q (qϕ(q)). Therefore,

different numbers τ correspond to different orderings of noncommuting operators such that we havethe “qp”-quantization for τ = 1, the “pq”-quantization for τ = 0, and the Weyl quantization forτ = 1/2. A function H(q, p) is usually regarded as a Hamiltonian function of a classical system. In

this case, the operator Hτ (·,D) is referred to as the Hamiltonian of the quantum system obtainedby τ -quantization of the classical system with the Hamiltonian function H.

1.3. Hamiltonian Feynman Formula for Feller Semigroups

Let (Tt)t�0 be a strongly continuous contraction semigroup (i.e., ‖Tt‖ � 1 for all t � 0) on(C∞(Rd), ‖ · ‖∞) which is positivity preserving (i.e., ϕ � 0 yields Ttϕ � 0 for all t � 0). Then(Tt)t�0 is called a Feller semigroup. The generator of a Feller semigroup is called a Feller generator.A strong Markov process (Xt)t�0 on R

d is called a Feller process if the family of operators (Tt)t�0

defined on (C∞(Rd), ‖ · ‖∞) by Ttϕ(q) = Eq[ϕ(Xt)] ≡

∫Rd ϕ(y)P (0, q, t, dy) is a Feller semigroup.

All Levy processes and many diffusions on Rd are Feller processes.

Let (Xt)t�0 be a Feller process with the semigroup (Tt)t�0. Let us consider a function λt(q, p) =

Eq[eip(Xt−q)

], i.e., the characteristic function of the random variable Xt − q under the assumption

thatX0 = q almost surely. In this case, for all t � 0, the restriction of Tt to the set C∞c (Rd) is a ΨDO

with the 1-symbol λt(q, p), and hence the (Feller) generator L of the semigroup (Tt)t�0 (restrictedto a proper subspace2) is also a ΨDO with the 1-symbol −H(q, p) := limt→0(1/t)(λt(q, p)− 1).Note that λt(q, p) = e−tH(q,p) in general.3 Some possibilities of constructing and approximatingthe symbol of the semigroup starting from the symbol of the generator can be found in [6, 13, 37].Many sufficient conditions for a function −H(q, p) to be the 1-symbol of a Feller generator areknown. However, these conditions are not unified, and it is an open question what are the mostgeneral ones. Some necessary conditions can be summarized in the following version of Courrege’sresult [16].

Theorem 1.3. [Courrege] Let (L,Dom(L)) be a Feller generator such that C∞c (Rd) ⊂ Dom(L).

Then L∣∣C∞

c (Rd)is a pseudodifferential operator,

Lϕ(q) = −H1(q,D)ϕ(q) = −(2π)−d

Rd

Rd

eip·(q−q1)H(q, p)ϕ(q1) dq1 dp, ϕ ∈ C∞c (Rd), (1.3)

with the 1-symbol −H : Rd × Rd → C which is measurable, locally bounded in both variables (q, p),

and satisfies the following Levy–Khintchine representation for any fixed q:

2The generator can admit no restriction to the test functions, since they need not belong to the domain in general.

Usually, one has to assume that the test functions belong to the domain.3We have H(q, p) = H(p) and λt(q, p) ≡ λt(p) = e−tH(p) only in the case of Levy processes.

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FEYNMAN FORMULAS AND PATH INTEGRALS 391

H(q, p) = c(q) + ib(q) · p+ p ·A(q)p +∫

y �=0

(1− eiy·p +

iy · p1 + |y|2

)N(q, dy), (1.4)

where (c(q), b(q), A(q), N(q, ·)) is the Levy characteristics of H(q, ·) for each q ∈ Rd, i.e., for every

q ∈ Rd, A(q) is a symmetric positive semi-definite d× d matrix, c(q) � 0, b(q) ∈ R

d, and N(q, ·) isa Radon measure on R

d \ {0} with

Rd

|y|21 + |y|2N(q, dy) < ∞.

Even if a Feller semigroup with the given generator is not known explicitly, it sometimes can beapproximated by means of the Chernoff formula (cf. [13, 6, 11]).

Theorem 1.4. [13] Let a function H : Rd×Rd → C be measurable and locally bounded with re-

spect to the pair of variables (q, p). Assume that H(q, ·) satisfies the Levy–Khintchine representation(1.4) for every chosen q ∈ R

d. Assume that

supq∈Rd

|H(q, p)| � κ(1 + |p|2) for all p ∈ Rd and some κ > 0, (1.5)

p �→ H(q, p) is uniformly continuous (with respect to q ∈ Rd) at p = 0, (1.6)

q �→ H(q, p) is continuous for all p ∈ Rd. (1.7)

Assume also that the function H(q, p) is such that −H1(·,D) is closable, the closure is the generatorof a strongly continuous semigroup on C∞(Rd), and the set C∞

c (Rd) of test functions is an operatorcore for this generator. Let F (t) be a ΨDO with the 1-symbol e−tH(q,p), i.e., if ϕ ∈ C∞

c (Rd), then

F (t)ϕ(q) = (2π)−d

Rd

Rd

eip·(q−q1)e−tH(q,p)ϕ(q1) dq1 dp. (1.8)

In this case, the family (F (t))t�0 is Chernoff equivalent to a strongly continuous semigroup (Tt)t�0

generated by the closure of the ΨDO −H1(·,D) with the 1-symbol −H(q, p), and the Chernoff

formula Tt = limn→∞[F ( t

n )]n

is valid in L(C∞(Rd)) locally uniformly with respect to t � 0.

Remark 1.5. (i) Note that condition (1.5) actually means that the ΨDO −H1(·,D) is anoperator with bounded “coefficients” (c(q), b(q), A(q), N(q, ·)). The Levy–Khintchine representation(1.4) also implies that the function H(q, p) is continuous with respect to the variable p.

(ii) Some conditions on the function H(q, p) under which −H1(·,D) is closable and the closureis the generator of a strongly continuous semigroup on C∞(Rd) can be found, for example, in [26,Vol. 2, Chaps. 2.6–2.8] and in [27]. For all these constructions, C∞

c (Rd) is always an operator core.Note that C∞

c (Rd) is also an operator core for generators of Levy processes (see [36, Th. 31.5]).

(iii) If the function H(q, p) satisfies the assumptions of Theorem 1.4, then the semigroup (Tt)t�0

is in fact a positivity preserving contraction semigroup, i.e., a Feller semigroup.

Remark 1.6. Let us assume in addition that H : Rd × Rd → C satisfies the condition

∃C > 0 such that∥∥∂α

q ∂βp e

−tH∥∥L∞(Rd×Rd)

� C, (1.9)

where α, β ∈ Nd0, α = 0 or 1, β = 0 or 1, and ∂α

q ∂βp are derivatives in the distributional sense. Note

that this condition is satisfied, e.g., if H satisfies the inequality |H(q, p)| � c|p|r for |p| � 1, forsome c > 0, and for some r ∈ (0, 2). Then F (t) : L2(R

d) → L2(Rd) by [23, Th. 2]. In this case, the

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formula obtained in Theorem 1.4 for each ϕ ∈ C∞(Rd)∩L2(Rd) becomes the following Hamiltonian

Feynman formula:

(Ttϕ)(q0) = limn→∞

(2π)−dn

(Rd)2nexp

(

i

n∑

k=1

pk · (qk−1 − qk)

)

× exp

(

− t

n

n∑

k=1

H(qk−1, pk)

)

ϕ(qn) dq1 dp1 · · · dqn dpn.(1.10)

We refer to [23] for further conditions on H ensuring that F (t) takes L2(Rd) to L2(R

d). If thefunction H satisfies sufficient conditions for F (t)ϕ to be in S(Rd) for each ϕ ∈ S(Rd), then, for anyϕ ∈ S(Rd), the Hamiltonian Feynman formula (1.10) becomes valid at every point q0 ∈ R

d (see[13] for further discussion).

Remark 1.7. The same Chernoff formula can be proved for symbols with unbounded coefficientssuch that lim|q|→∞ sup|p|� 1

|q|

∣∣H(q, p)∣∣ = 0, due to [5]. Using the perturbation results of [13, Th. 5.1],

one can show that the Hamiltonian Feynman formula (1.10) is also valid, say, for the quantizationof a harmonic oscillator whose symbol is H(q, p) = p2 + q2.

2. FEYNMAN FORMULAS FOR THE τ -QUANTIZATIONOF A QUADRATIC HAMILTON FUNCTION

Throughout this section, we use the following notation: Cm(Rd) stands for the space of m-timescontinuously differentiable functions, C0,λ(Rd) for the space of Holder continuous functions withparameter λ ∈ (0, 1], Cb(R

d) for the space of bounded continuous functions, Cmb (Rd) for the space

{f ∈ C(Rd); ∂αf ∈ Cb(Rd), |α| � m}, and Cm,λ

b (Rd) for the space Cmb (Rd) ∩ {f ∈ C(Rd); ∂αf ∈

C0,λ(Rd), |α| = m }.

2.1. Relationships between Different Quantizations of the Same Hamiltonian Function

Let us consider a Hamiltonian function H(·, ·) : Rd × Rd → C such that

H(q, p) = p · A(q)p+ ib(q) · p+ c(q), (2.1)

where A(q) ≡ 0 is a positive semidefinite symmetric matrix, b(q) ∈ Rd, c(q) � 0 for all q ∈ R

d, and

A(·), b(·), and c(·) are continuous and bounded. Consider an operator Hτ (·,D) with the τ -symbolH(q, p), τ ∈ [0, 1], as in (1.2). In the case of τ = 1, the Hamiltonian function H(q, p) is the 1-symbol

(or “qp”-symbol) of an operator H1(q,D), which can be extended to the set C2(Rd) by the formula

H1(q,D)ϕ(q) = − tr(A(q)Hessϕ(q)) + b(q) · ∇ϕ(q) + c(q)ϕ(q). (2.2)

Lemma 2.1. Let A(·) ∈ C2(Rd), b(·) ∈ C1(Rd), and c(·) ∈ C(Rd). Then, for every point τ ∈[0, 1], the operator Hτ (q,D) can be extended to C2(Rd) by Hτ (q,D)ϕ(q) = − tr(A(q)Hessϕ(q)) +[b(q)− 2(1− τ) divA(q)] · ∇ϕ(q) + [c(q) + (1− τ) div b(q)− (1− τ)2 tr(HessA(q))]ϕ(q). Therefore,

Hτ (q,D)ϕ(q) = Hτ1 (q,D)ϕ(q), where Hτ

1 (q,D)ϕ(q) is a pseudodifferential operator with 1-symbolHτ (q, p) = p · A(q)p + ibτ (q) · p+ cτ (q), and we have bτ (q) = b(q)− 2(1 − τ) divA(q) and cτ (q) =c(q) + (1− τ) div b(q)− (1− τ)2 tr(HessA(q)).

Proof. Let us consider the case d = 1 andH(q, p) = A(q)p2 to simplify the calculations. Assumefor a moment that A(·) ∈ C∞(R). Let BR = [−R,R], let χBR

be the indicator of BR, let δ be theDirac delta function, and let

⟨f, g

⟩mean the action of a generalized function distribution f ∈ S′(R)

on the test function g ∈ S(R). For any ϕ ∈ C∞c (R), we have

Hτ (q,D)ϕ(q) =1

R

R

eip(q−y)H(τq + (1 − τ)y, p)ϕ(y) dy dp

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FEYNMAN FORMULAS AND PATH INTEGRALS 393

= limR→+∞

1

BR

suppϕ

eip(q−y)H(τq + (1− τ)y, p)ϕ(y) dy dp

= limR→+∞

1

suppϕ

A(τq + (1− τ)y)ϕ(y)

BR

eip(q−y)p2 dp dy

= limR→+∞

suppϕ

A(τq + (1− τ)y)ϕ(y)F−1[p2χBR(p)](q − y) dy

= limR→+∞

⟨F−1[p2χBR

(p)](q − ·), A(τq + (1− τ)·)ϕ⟩

= limR→+∞

⟨−∇2F−1[χBR

(p)](q − ·), A(τq + (1− τ)·)ϕ⟩

= limR→+∞

⟨F−1[χBR

(p)](q − ·),−∇2A(τq + (1− τ)·)ϕ⟩=⟨δ(q − ·),−∇2A(τq + (1− τ)·)ϕ

=−[A(τq+(1−τ)y)ϕ′′(y)−2(1−τ)A′(τq+(1−τ)y)ϕ′(y)−(1−τ)2A′′(τq+(1−τ)y)ϕ(y)

]

y=q

= −A(q)ϕ′′(q)− 2(1 − τ)A′(q)ϕ′(q)− (1− τ)2A′′(q)ϕ(q).

For A(·) ∈ C∞(R), we have −∇2A(τq + (1− τ)·)ϕ ∈ S(R) and F−1[χBR(p)] ∈ C∞(R) ⊂ S′(R).

If A(·) ∈ C2(R), then there is a sequence of infinitely differentiable functions {An(·)}∞n=1 thatconverges to A locally uniformly4 on R together with their derivatives up to the second order. Inthis case, the distribution (the generalized function) A(τq + (1 − τ)·)F−1[p2χBR

(p)](q − ·) can beapproximated by distributions An(τq + (1− τ)·)F−1[p2χBR

(p)](q − ·) for which the above manip-ulations remain valid.

Using similar calculations, we can see for H(q, p) = ib(q)p+c(q) that Hτ (q,D)ϕ(q) = b(q)ϕ′(q)+(1− τ)b′(q)ϕ(q) + c(q)ϕ(q). The case of higher dimensions, d > 1, can be treated similarly.

2.2. Hamiltonian Feynman Formula for the τ -Quantization of Quadratic Hamilton Functions

Using the relationship between different quantizations of the same Hamiltonian function and theHamiltonian Feynman formula (1.10) for τ = 1, we obtain the following assertion.

Theorem 2.2. Let A(q) be a positive semidefinite symmetric matrix, b(q) ∈ Rd, c(q) � 0 for

all q ∈ Rd and let A(·) ∈ C2

b (Rd), b(·) ∈ C1

b (Rd), and c(·) ∈ Cb(R

d). Consider a Hamiltonian

function H(·, ·) : Rd × Rd → C such that H(q, p) = p·A(q)p + ib(q) · p + c(q). Let Hτ (·,D) be a

pseudodifferential operator with τ -symbol H(q, p) for τ ∈ [0, 1]. Assume that −Hτ(·,D) generates astrongly continuous semigroup (T τ

t )t�0 on the space C∞(Rd) and the set C∞c (Rd) is a core for the

generator. For each τ ∈ [0, 1], consider a family of pseudodifferential operators (F τ (t))t�0 defined

for all ϕ ∈ S(Rd) by the formulas F τ (t)ϕ(q) = (2π)−d∫Rd

∫Rd e

ip·(q−y)e−tHτ (q,p)ϕ(y) dy dp, where

Hτ (q, p) = p · A(q)p+ ibτ (q)· p + cτ (q),

bτ (q) = b(q)− 2(1− τ) divA(q),

cτ (q) = c(q) + (1− τ) div b(q)− (1− τ)2 tr(HessA(q)).

(2.3)

Then the family (F τ (t))t�0 is Chernoff equivalent to the semigroup (T τt )t�0, and hence the Chernoff

formula T τt = limn→∞

[F τ ( t

n)]n

is valid in L(C∞(Rd)) locally uniformly with respect to t � 0.

Theorem 2.2 is a straightforward consequence of Lemma 2.1 and Theorem 1.4, since the functionHτ (q, p) satisfies all assumptions of Theorem 1.4.

Remark 2.3. (i) Note that Hτ : Rd × Rd → C satisfies condition (1.9) of Remark 1.6. Hence,

F τ (t) : L2(Rd) → L2(R

d), and the Chernoff formula obtained in Theorem 2.2 for each function

4The convergence is uniform on compacta only; however, this condition is sufficient, since the test functions are

either compactly supported or rapidly decreasing.

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394 BOTTCHER et al.

ϕ ∈ C∞(Rd) ∩ L2(Rd) becomes the Hamiltonian Feynman formula,

(T τt ϕ)(q0) = lim

n→∞(2π)−dn

(Rd)2nexp

(

in∑

k=1

pk · (qk−1 − qk)

)

× exp

(

− t

n

n∑

k=1

Hτ (qk−1, pk)

)

ϕ(qn) dq1 dp1 · · · dqn dpn

= limn→∞

(2π)−dn

(Rd)2nexp

(

in∑

k=1

pk · (qk−1 − qk)

)

exp

(

− t

n

n∑

k=1

pk ·A(qk−1)pk

)

× exp

(

− t

n

n∑

k=1

[c(qk−1) + (1− τ) div b(qk−1)− (1− τ)2 tr(HessA(qk−1))

])

× exp

(

−it

n

n∑

k=1

[b(qk−1)− 2(1 − τ) divA(qk−1)

]· pk

)

ϕ(qn) dq1 dp1 · · · dqn dpn.

(2.4)

(ii) If, moreover, the coefficients A(·), b(·), c(·) are infinitely differentiable, then F (t)ϕ ∈ S(Rd)for each ϕ ∈ S(Rd). Hence, the pointwise equation in the Hamiltonian Feynman formula (1.10)holds for any ϕ ∈ S(Rd) and at every point q0 ∈ R

d.

2.3. Lagrangian Feynman Formula for the τ -Quantization of a Quadratic Hamiltonian Function

Consider now an operator L given by formula (2.2). This operator corresponds to the 1-quanti-zation of the Hamiltonian function (2.1). Assume that A(q) is a symmetric positive-definite matrixfor every q ∈ R

d, A−1(·), b(·) ∈ C2,β(Rd) for some 0 < β � 1, and c(·) ∈ Cb(Rd). Assume also that

there is an α, 0 < α � 1, such that C2,αc (Rd) is a core for the closed extension (L,Dom(L)) of

(L,C2,αc (Rd)) which generates a strongly continuous semigroup (T 1

t )t�0 on C∞(Rd). In this case,due to results of [10], we have the following Lagrangian Feynman formula for the semigroup (T 1

t )t�0.

Theorem 2.4. (cf. [10]) Under the above assumptions, the following Lagrangian Feynman for-mula holds for all ϕ ∈ C∞(Rd), q0 ∈ R

d, and t > 0:

T 1t ϕ(q0) = lim

n→∞

Rd

· · ·∫

Rd

exp

⎝ t

n

n∑

j=1

V (qj−1)

⎠ exp

⎝−1

2

n∑

j=1

A−1(qj−1)b(qj−1) · (qj − qj−1)

× pA(t/n, q0, q1) · · · pA(t/n, qn−1, qn)ϕ(qn) dq1 . . . dqn, (2.5)

where V (q) = −c(q)− (1/4)b(q) ·A−1(q)b(q), a(q) = detA(q), and pA(t, q, y) = (1/√

a(q)(4πt)d)×exp

(−A−1(q)(q − y) · (q − y)/(4t)

), and the convergence is locally uniform in R

d × [0,∞).

Combining Lemma 2.1 and Theorem 2.4, we obtain a Lagrangian Feynman formula for thesemigroup generated by the τ -quantization of the Hamiltonian function (2.1).

Theorem 2.5. Let A(q) be a positive definite symmetric matrix, b(q) ∈ Rd, c(q) � 0 for all

q ∈ Rd, let A(·), A−1(·), b(·) ∈ C2,β

b (Rd) for some 0 < β � 1, and let c(·) ∈ Cb(Rd). Consider a

Hamiltonian function H(·, ·) : Rd × Rd → C such that H(q, p) = p · A(q)p + ib(q) · p + c(q). Let

Hτ (q,D) be a pseudodifferential operator with τ -symbol H(q, p) for τ ∈ [0, 1]. Assume that the

closure Lτ of −Hτ (q,D) generates a strongly continuous semigroup (T τt )t�0 on the space C∞(Rd)

and there exists an ατ , 0 < ατ � 1, such that C2,ατc (Rd) is a core for the closure (Lτ ,Dom(Lτ ))

of (Lτ , C2,ατc (Rd)). Then the following Lagrangian Feynman formula holds for all ϕ ∈ C∞(Rd),

q0 ∈ Rd, t > 0:

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FEYNMAN FORMULAS AND PATH INTEGRALS 395

T τt ϕ(q0) = lim

n→∞

Rd

· · ·∫

Rd

exp( t

n

n∑

j=1

Vτ (qj−1))exp

(− 1

2

n∑

j=1

A−1(qj−1)bτ (qj−1) · (qj − qj−1))

× pA(t/n, q0, q1) · · · pA(t/n, qn−1, qn)ϕ(qn) dq1 . . . dqn, (2.6)

where bτ (q) = b(q) − 2(1 − τ) divA(q), Vτ (q) = −c(q) − (1 − τ) div b(q) + (1 − τ)2 tr(HessA(q)) −14A

−1(q)bτ (q) · bτ (q), a(q) = detA(q), pA(t, q, y) = 1√a(q)(4πt)d

exp(−A−1(q)(q−y)·(q−y)

4t

), and the

convergence is locally uniform in Rd × [0,∞).

Remark 2.6. Under the assumptions of Theorem 2.5 and some growth conditions on A and bτ ,see [31, Th. 5.7.6], the action of the semigroup (T τ

t )t�0 can also be represented by the Feynman–Kacformula

T τt ϕ(q) = E

q

[exp

(−∫ t

0

cτ (Xτs ) ds

)ϕ(Xτ

t )

], q ∈ R

d, t ∈ (0, T ), (2.7)

where Eq is the law of the diffusion process (Xτt )0<t<T in R

d corresponding to the variable diffusion

coefficient√

A(·) and the drift −bτ (·), starting at the point q ∈ Rd; here bτ and cτ are as in Theo-rem 2.2. Hence, the Lagrangian Feynman formula (2.6) gives an approximation for the functionalintegral in (2.7).

3. HAMILTONIAN FEYNMAN PATH INTEGRALS FOR FELLER SEMIGROUPS

A Hamiltonian Feynman mapping (a pseudomeasure) is a linear functional Φ on a vector spaceof functionals whose common domain is a vector space of functions defined on a segment [0, t],t > 0, and taking values in the phase space E = Q × P of a classical Hamiltonian system. Thevalue Φ(G) of the Feynman mapping Φ at a functional G is called the Hamiltonian Feynman pathintegral of G or the Feynman integral of G over the set of paths in the phase space, and it is oftendenoted by

Φ(G) =

∫G(q(·), p(·))ei

∫t

0p(τ)q′(τ)dτ

t∏

τ=0

dq(τ) dp(τ) or

∫G(q(·), p(·))Φ(dq, dp).

Under some additional assumptions (see [41] and the references therein), the function

(t, x) �→∫

e−i∫

t

0H(q(τ)+x,p(τ))dτ

f0(q(t) + x)ei∫

t

0p(τ)q′(τ)dτ

t∏

τ=0

dq(τ) dp(τ)

gives a solution for the Schrodinger equation i∂f∂t

= Hf , where H is a pseudodifferential operatorwith the symbol H.

There are different ways to define a Hamiltonian Feynman mapping (pseudomeasure). One ofthem (originally going back to Feynman) is to define Φ(G) as a limit of some finite-dimensionalintegrals. Let E = Q × P , where Q and P are locally convex spaces, Q = P ∗, P = Q∗ (as vectorspaces), and let the space P×Q be identified with a space of linear functionals on E in the followingway: for any l = (pl, ql) ∈ P ×Q and any z = (q, p) ∈ E, we have l(z) = pl(q) + p(ql).

Definition 3.1. Let {En = Qn×Pn}n∈N be an ascending sequence of finite-dimensional vectorsubspaces of E = Q × P , where Qn and Pn are vector subspaces of Q and P , respectively. Thevalue Φ{En}(G) of the Feynman pseudomeasure Φ{En} associated with the sequence {En}n∈N at afunction G : E → C, i.e., the Feynman path integral of G, is defined by the formula

Φ{En}(G) = limn→∞

(∫

En

ei〈p,q〉 dq dp)−1

En

G(q, p)ei〈p,q〉 dq dp, (3.1)

if this limit exists. In this formula, we use the notation 〈p, q〉 = p(q), and all integrals must be

regarded in a suitably regularized sense; for example,∫En

G(z)dz = limε→0

∫En

G(z)e−ε|z|2dz.

Below we use the following construction of Hamiltonian Feynman pseudomeasure (cf. [43, 12]). Forany t > 0, let PC([0, t],Rd) be the vector space of all functions on [0, t] taking values in R

d all

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396 BOTTCHER et al.

of whose first distributional derivatives are finitely supported measures. Let PC l([0, t],Rd) be thespace of all left continuous functions in PC([0, t],Rd), and let PCr([0, t],Rd) be the space of all rightcontinuous functions in PC([0, t],Rd). For any x ∈ R

d, let Qxt = {f ∈ PCr([0, t],Rd) : f(0) = x},

Pt = PC l([0, t],Rd), and Ext = Qx

t × Pt. The spaces Qxt and Pt are taken in the duality with

respect to the form 〈q(·), p(·)〉 �→∫ t

0p(s)q′(s) ds, where q′(s) ds stands for the measure which is

the distributional derivative of q(·). We treat the elements of Ext as functions taking values in

E = Q×P = Rd × R

d.

Let t0 = 0, and let tk = kn t for any n ∈ N and any k ∈ N, k � n. Let Ln ⊂ PC l([0, t],Rd) be

the space of functions whose restrictions to any interval ( k−1n

t, knt] are constant functions, and let

Rn ⊂ PCr([0, t],Rd) be the space of functions that are constant on any interval [k−1n t, k

n t). LetQn = Rn ∩Qx

t , Pn = Ln ∩ {f : f(0) = lim0<t→0 f(t)}. Let Jn be the mapping of En = Qn × Pn to(Rd ×Rd)n defined by

J(q, p) =(q( t

n ), p(tn ), q(

2tn ), p(2tn ), . . . , q(ntn ), p(ntn )

)≡(q1, p1, . . . , qn, pn

).

The mapping Jn is a one-to-one correspondence between En and (Rd × Rd)n. Therefore, in this

particular case, Definition 3.1 can be represented as follows.

Definition 3.2. The Hamiltonian Feynman path integral

Φ1x(G) ≡

Ext

G(q, p)Φ1x(dq, dp) ≡

Ext

G(q(·), p(·))ei∫ t

0p(τ)q′(τ)dτ

t∏

τ=0

dq(τ) dp(τ)

of a function G : Qxt × Pt → R is defined as the limit

Φ1x(G) = lim

n→∞

1

(2π)mn

(Rd×Rd)nG(J−1

n (q1, p1, . . . , qn, pn))

× exp

[

i

n∑

k=1

pk · (qk − qk−1)

]

dq1 dp1 . . . dqn dpn,

(3.2)

where q0 = x.

Remark 3.3. For the pseudomeasure Φ1x, the corresponding generalized density can be defined:

Ext

G(q(τ), p(τ))Φ1x(dq dp) = lim

n→∞Cn

Qn×Pn

G(q(τ), p(τ)) exp

[i

∫ t

0

p(τ)q′(τ) dτ

]νn(dq) νn(dp),

where (Cn)−1 =

∫Qn×Pn

exp[i∫ t

0p(τ)q′(τ) dτ

]νn(dq) νn(dp) and νn stands for the Lebesgue mea-

sure.

Remark 3.4. The construction described above was used in [43] and [34] to represent solutionsof some Schrodinger-type equations by taking this very kind of Hamiltonian Feynman path integrals;the same has been done for the heat equation in [12]. The proofs use Chernoff’s theorem.

Due to Definition 3.2, the Hamiltonian Feynman formula (1.10) for a Feller semigroup (Tt)t�0 canbe interpreted as a Hamiltonian Feynman path integral with respect to the Feynman pseudomeasureΦ1

x. Therefore, the following theorem holds.

Theorem 3.5. Let a function H : Rd ×Rd → C be measurable and locally bounded with respect

to the pair of variables (q, p). Assume that H(q, ·) satisfies the Levy–Khintchine representation (1.4)for any chosen q ∈ R

d and that the conditions (1.5), (1.6), (1.7), and (1.9) hold. Assume also that

the function H(q, p) is such that −H1(·,D) is closable, the closure is the generator of a stronglycontinuous semigroup on C∞(Rd), and the set C∞

c (Rd) of test functions is an operator core for thisgenerator. Then the strongly continuous semigroup (Tt)t�0 generated by the closure of the ΨDO

−H1(·,D) with the 1-symbol −H(q, p) can be represented by the Hamiltonian Feynman path integralwith respect to the Feynman pseudomeasure Φ1

x,

Ttϕ(x) =

Ext

e−∫

t

0H(q(s),p(s))ds

ϕ(q(t))Φ1x(dq dp). (3.3)

Similarly, the Hamiltonian Feynman formula (2.4) can be interpreted as a Hamiltonian Feynmanpath integral with respect to the Feynman pseudomeasure Φ1

x as follows.

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FEYNMAN FORMULAS AND PATH INTEGRALS 397

Corollary 3.6. [Hamiltonian Feynman path integral for the τ -quantization of a quadraticHamiltonian function] Let A(q) be a positive-semidefinite symmetric matrix, b(q) ∈ R

d, c(q) � 0for all q ∈ R

d, and let A(·) ∈ C2b (R

d), b(·) ∈ C1b (R

d), and c(·) ∈ Cb(Rd). Consider a Hamiltonian

function H(·, ·) : Rd × Rd → C given by H(q, p) = p · A(q)p + ib(q) · p + c(q). Let Hτ (·,D) be

a pseudodifferential operator with τ -symbol H(q, p), τ ∈ [0, 1]. Assume that −Hτ(·,D) generatesa strongly continuous semigroup (T τ

t )t�0 on the space C∞(Rd), and the set C∞c (Rd) is a core for

the generator. Then the semigroup (T τt )t�0 can be represented by the Hamiltoni an Feynman path

integral with respect to the Feynman pseudomeasure Φ1x,

T τt ϕ(x) =

Ext

exp

[−∫ t

0

Hτ (q(s), p(s))ds

]ϕ(q(t))Φ1

x(dq dp)

=

Ext

exp

[−∫ t

0

p(s) ·A(q(s))p(s) ds]exp

[−i

∫ t

0

[b(q(s))− 2(1− τ) divA(q(s)) · p(s)

]ds

]

× exp

[−∫ t

0

[c(q(s)) + (1 − τ) div b(q(s))− (1− τ)2 tr(HessA(q(s)))

]ds

]ϕ(q(t))Φ1

x(dq dp).

Remark 3.7. (i) Note that the Feynman path integrals in Corollary 3.6 are different for differ-ent τ . Thus, it is possible to distinguish the process of quantization in the language of the Feynmanpath integrals under consideration.

(ii) The results of Theorem 3.5 and Corollary 3.6 can also be understood in the following way.A Hamiltonian Feynman path integral with respect to the Feynman pseudomeasure Φ1

x exists andcoincides with the action of some strongly continuous semigroup on C∞(Rd) for a certain classof integrands. Moreover, if this semigroup is a Feller semigroup, then the Hamiltonian Feynmanpath integral with respect to the Feynman pseudomeasure Φ1

x coincides with a functional integral(expectation) with respect to a probability measure generated by the underlying Feller process(Xt)t�0. Therefore, we have a kind of change-of-variable formula

Ext

exp

(−∫ t

0

H(q(s), p(s))ds

)ϕ(q(t))Φ1

x(dq dp) = Ex[ϕ(Xt)

]≡ Ttϕ(x).

Example 3.8. Due to the perturbation results in [13, Th. 5.1], the Hamiltonian Feynmanformula (1.10) and the Hamiltonian Feynman path integral (3.3) are also valid for the semigroup

(Tt)t�0 generated by the 1-quantization of a Hamiltonian function H(q, p) =√

p2 +m2−m+V (q)corresponding to a relativistic spinless particle in a potential field V : Rd → [0,∞), V ∈ C(Rd).Therefore, we have the following representations for the semigroup (Tt)t�0:

Ttϕ(x) = limn→∞

(2π)−dn

(Rd)2nexp

(

i

n∑

k=1

pk · (qk−1 − qk)

)

× exp

(

− t

n

n∑

k=1

√p2k +m2 −m+ V (qk−1)

)

ϕ(qn) dq1 dp1 · · · dqn dpn

=

Ext =Qx

t ×Pt

exp

(−∫ t

0

(√

p(s)2 +m2 −m+ V (q(s)))ds

)ϕ(q(t))Φ1

x(dq dp)

= Ex

[exp

(−∫ t

0

V (Xs)ds

)ϕ(Xt)

],

where (Xt)t�0 is a Levy process with 1-symbol H(p) =√

p2 +m2 −m. The last equation is dueto Ichinose and Tamura [25].

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ACKNOWLEDGMENTS

Financial support by the Russian Foundation for Basic Research via the project 10-01-00724-a,by the grant of the President of Russian Federation via the project MK-943.2010.1, and by theErasmus Mundus Action 2 Programme of the European Union via the project EMA 2 MULTIC10-865 is gratefully acknowledged.

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