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.

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].

STATIONARY PHASE, QUANTUM MECHANICS AND SEMI-CLASSICAL LIMIT

1996 ◽  
Vol 08 (08) ◽  
pp. 1161-1185 ◽  
Author(s):  
JORGE REZENDE
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Stationary Phase ◽  
Finite Number ◽  
Path Integral ◽  
Critical Points ◽  
Classical Limit ◽  
Oscillatory Integral ◽  
Feynman Path Integral ◽  
Feynman Path

A method of stationary phase for the normalized-oscillatory integral on Hilbert space is developed in the case where the phase function has a finite number of critical points which are non-degenerate. Applications to the Feynman path integral and the semi-classical limit of quantum mechanics are given.

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