Feynman path integrals as infinite-dimensional oscillatory integrals: Some new developments

1994 ◽  
Vol 35 (1-2) ◽  
pp. 5-26 ◽  
Author(s):  
Sergio Albeverio ◽  
Zdzisław Brzeźniak
Keyword(s):  
Path Integrals ◽  
Oscillatory Integrals ◽  
Feynman Path ◽  
New Developments ◽  
Infinite Dimensional ◽  
Feynman Path Integrals

scholarly journals A unified approach to infinite-dimensional integration

2016 ◽  
Vol 28 (02) ◽  
pp. 1650005 ◽  
Author(s):  
S. Albeverio ◽  
S. Mazzucchi
Keyword(s):  
Parabolic Equations ◽  
Path Integrals ◽  
Linear Functionals ◽  
Unified Approach ◽  
Feynman Path ◽  
Infinite Dimensional ◽  
Feynman Path Integrals ◽  
Theoretical Structure ◽  
Probabilistic Type ◽  
And Diffusion

An approach to infinite-dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite-dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Feynman path integrals, Schrödinger and diffusion equations, as well as higher order hyperbolic and parabolic equations.

A polynomial invariant of integral homology 3-spheres

1995 ◽  
Vol 117 (1) ◽  
pp. 83-112 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Asymptotic Formula ◽  
Quotient Space ◽  
Path Integrals ◽  
Feynman Path Integral ◽  
Polynomial Invariant ◽  
Feynman Path ◽  
Flat Connections ◽  
Gauge Transformations ◽  
Infinite Dimensional ◽  
Feynman Path Integrals

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].

Generalized measure in Feynman path integrals

1976 ◽  
Vol 28 (3) ◽  
pp. 793-805 ◽  
Author(s):  
V. P. Maslov ◽  
A. M. Chebotarev
Keyword(s):  
Path Integrals ◽  
Feynman Path ◽  
Feynman Path Integrals

A REPRESENTATION OF THE BELAVKIN EQUATION VIA PHASE SPACE FEYNMAN PATH INTEGRALS

2004 ◽  
Vol 07 (04) ◽  
pp. 507-526 ◽  
Author(s):  
S. ALBEVERIO ◽  
G. GUATTERI ◽  
S. MAZZUCCHI
Keyword(s):  
Phase Space ◽  
Continuous Measurement ◽  
Quantum Particle ◽  
Oscillatory Integral ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Infinite Dimensional ◽  
Feynman Path Integrals ◽  
Characteristics Method ◽  
Stochastic Characteristics

The Belavkin equation, describing the continuous measurement of the momentum of a quantum particle, is studied. The existence and uniqueness of its solution is proved via analytic tools. A stochastic characteristics method is applied. A rigorous representation of the solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the phase space is also given.

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