Mathematical Theory of Feynman Path Integrals

Albeverio, Sergio; Høegh-Krohn, Raphaël; Mazzucchi, Sonia

doi:10.1007/978-3-540-76956-9

Cited by 119 publications

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“…2), 4) by the invariance of the infinite dimensional oscillatory integrals under unitary transformation on paths space [AHK76], and by the infinite dimensional oscillatory integral representation for the solution of the Schrödinger equation with a potential of the type " harmonic oscillator plus Fourier transform of measure" (see [ABHK82, ET84, AB93] for more details).…”

Section: The Feynman-vernon Influence Functionalmentioning

confidence: 99%

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A rigorous approach to the Feynman-Vernon influence functional and its applications. I

Albeverio

Cattaneo

Mazzucchi

et al. 2007

Journal of Mathematical Physics

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Abstract. A rigorous representation of the Feynman-Vernon influence functional used to describe open quantum systems is given, based on the theory of infinite dimensional oscillatory integrals. An application to the case of the density matrices describing the Caldeira-Leggett model of two quantum systems with a quadratic interaction is treated. IntroductionOne of the crucial problems of modern physics consists in understanding the behaviour of an open quantum system, i.e. of a quantum system coupled with a second system often called reservoir or enviroment. One is interested in the dynamics of the first system, taking into account the influence of the enviroment on it. A typical example is the study of a quantum particle submitted to the measurement of an observable. In fact, from a quantum mechanical point of view, the interaction with the measuring apparatus cannot be neglected and modifies the dynamics of the particle. On the other hand the evolution of the measuring instrument is not of primary interest. A particularly intriguing approach to this problem was proposed in 1963 by Feynman and Vernon ( see [FH65,FV63]) within the path integral formulation of quantum mechanics. In 1942 R.P. Feynman [Fey42], following a suggestion by Dirac (see [Dir33,Dir47], proposed an alternative (Lagrangian) formulation of quantum mechanics (published in [Fey48]), that is an heuristic, but very suggestive representation for thesolution of the Schrödinger equationdescribing the time evolution of the state ψ of a d−dimensional quantum particle. The parameter is the reduced Planck constant, m > 0 is the mass of the particle and F = −∇V is an external force. According to Feynman's proposal the wave function of the system at time t evaluated at the point x ∈ R d is heuristically given as an "integral over histories", or as an integral over all possible paths γ in the configuration space of the system with finite energy passing in the point x at time t:(2) where S t (γ) is the classical action of the system evaluated along the path γ, i.e. :Dγ is an heuristic Lebesgue "flat" measure on the space of paths and ( {γ|γ(t)=x} e i S • t (γ) Dγ) −1 is a normalization constant. Feynman and Vernon (see [FH65,FV63]) generalized this idea to the study of the time evolution of the reduced density operator of a system in interaction with an enviroment. Let denote ρ A , ρ B , respectively, the initial density matrices of the system and of the enviroment, S A , S B , respectively, the action functionals of the system and of the enviroment and S I the contribution to the total action due to the interaction. Then the kernel of the reduced density operator of the system ρ R (obtained by tracing over the environmental coordinates) is heuristically given by:(5) where F is the formal influence functional (IF): FEYNMAN-VERNON INFLUENCE FUNCTIONAL 3The number of spin-offs originated by the seminal work [FV63] is so large that it is nearly impossible to give here a complete list, and we limit ourselves to shortly mention some of them. Probably the most inf...

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Section: The Feynman-vernon Influence Functionalmentioning

confidence: 99%

“…Our aim is to fill this gap following the ideas introduced in [AHK76,AHK77] in connection with the rigorous mathematical definition of Feynman path integrals (2) and in order to realize formulae (5) and (6) as well defined infinite dimensional oscillatory integrals on a suitable Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

A rigorous approach to the Feynman-Vernon influence functional and its applications. I

Albeverio

Cattaneo

Mazzucchi

et al. 2007

Journal of Mathematical Physics

Self Cite

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“…First, the semi-classical approximation can be validated easier in a phase space formulation and second that quantum mechanics are founded on the phase space, i.e., every quantum mechanical observable can be expressed as a function of the space and momentum. A discussion about phase space path integrals can be found in the monograph [2] and in the references therein. In the last fifty years there have been many approaches for giving a mathematically rigorous meaning to the phase space path integral by using, e.g., analytic continuation; see [13,14] or Fresnel integrals [2,1].…”

Section: Introductionmentioning

confidence: 99%

“…A discussion about phase space path integrals can be found in the monograph [2] and in the references therein. In the last fifty years there have been many approaches for giving a mathematically rigorous meaning to the phase space path integral by using, e.g., analytic continuation; see [13,14] or Fresnel integrals [2,1]. Here we choose a white noise approach.…”

Section: Introductionmentioning

confidence: 99%

A white noise approach to phase space Feynman path integrals

Böck

Grothaus

2013

Theor. Probability and Math. Statist.

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Abstract. The concepts of phase space Feynman integrals in White Noise Analysis are established. As an example the harmonic oscillator is treated. The approach perfectly reproduces the right physics. I.e., solutions to the Schrödinger equation are obtained and the canonical commutation relations are satisfied. The latter can be shown, since we not only construct the integral but rather the Feynman integrand and the corresponding generating functional. IntroductionAs an alternative approach to quantum mechanics Feynman introduced the concept of path integrals ([4, 5, 6]), which was developed into an extremely useful tool in many branches of theoretical physics. In this article we develop the concepts for realizing Feynman integrals in phase space in the framework of White Noise Analysis. The phase space Feynman integral for a particle moving from y 0 at time 0 to y at time t under the potential V is given byHere is Planck's constant, and the integral is thought of as being over all position paths with x(0) = y 0 and x(t) = y and all momentum paths. The missing restriction on the momentum variable at time 0 and time t is an immediate consequence of the Heisenberg uncertainty relation, i.e., the fact that one cannot measure momentum and space variable at the same time. The path integral to the phase space has several advantages. First, the semi-classical approximation can be validated easier in a phase space formulation 2010 Mathematics Subject Classification. Primary 60H40, 46F25.

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“…Notable among these would be the prodistributions of DeWitt-Morette [10], the oscillatory integrals of Albeverio and Høegh-Krohn [1] which explores the Fresnel integrals, and the white noise analysis approach of Hida and Streit [27]. In this paper, we shall consider the HidaStreit approach which, through the years, has solved various classes of potentials [2,3,5,6,7,8,13,18,19,21,27] in nonrelativistic quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%