Phase space path integral representation for the solution of a stochastic Schrödinger equation

2003 ◽  
pp. 237-252
Author(s):  
Luis A. Rincón
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Integral Representation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Path Integral Representation ◽  
Stochastic Schrödinger Equation

A new path integral representation for the solutions of the Schrödinger, heat and stochastic Schrödinger equations

2002 ◽  
Vol 132 (2) ◽  
pp. 353-375 ◽  
Author(s):  
VASSILI N. KOLOKOLTSOV
Keyword(s):  
Hilbert Space ◽  
Schrödinger Equation ◽  
Integral Representation ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Path Space ◽  
Schrödinger Equations ◽  
Path Integral Representation ◽  
Stochastic Schrödinger Equation ◽  
Finite Measures

Solutions to the Schrödinger, heat and stochastic Schrödinger equation with rather general potentials are represented, both in x- and p-representations, as integrals over the path space with respect to σ-finite measures. In the case of x-representation, the corresponding measure is concentrated on the Cameron–Martin Hilbert space of curves with L2-integrable derivatives. The case of the Schrödinger equation is treated by means of a regularization based on the introduction of either complex times or continuous non-demolition observations.

Continuous Quantum Measurement: Local and Global Approaches

1997 ◽  
Vol 09 (08) ◽  
pp. 907-920 ◽  
Author(s):  
S. Albeverio ◽  
V. N. Kolokol'tsov ◽  
O. G. Smolyanov
Keyword(s):  
Schrödinger Equation ◽  
Quantum System ◽  
Quantum Measurement ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Stochastic Equation ◽  
Point Of View ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Stochastic Schrödinger Equation

In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schrödinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menski's formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schrödinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator.

scholarly journals PHASE SPACE PATH INTEGRAL IN CURVED SPACE

2001 ◽  
Vol 16 (31) ◽  
pp. 5033-5042 ◽  
Author(s):  
RAFAEL FERRARO ◽  
MAURICIO LESTON
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Path Integral ◽  
Riemannian Geometry ◽  
Schrodinger Equation ◽  
Curved Space ◽  
Normal Coordinates ◽  
Equal Footing ◽  
Operator Ordering

Phase space path integral is worked out in a Riemannian geometry, by employing a prescription for the infinitesimal propagator that takes Riemann normal coordinates and momenta on an equal footing. The operator ordering induced by this prescription leads to the DeWitt curvature coupling in the Schrödinger equation.

scholarly journals THE NAMBU-GOTO STRING: ITS PHASE-SPACE PATH-INTEGRAL

1989 ◽  
Vol 04 (01) ◽  
pp. 173-216 ◽  
Author(s):  
JAN GOVAERTS
Keyword(s):  
Phase Space ◽  
Integral Representation ◽  
Path Integral ◽  
Scalar Particle ◽  
Constrained Systems ◽  
Modular Space ◽  
Path Integral Representation ◽  
Integration Measure ◽  
Phase Path ◽  
Space Phase

Using the Batalin-Fradkin-Vilkovisky formalism for constrained systems, the space-phase path-integral representation of the free bosonic string propagator is considered and compared with the Polyakov approach. A Gribov problem exists for this system with the consequence that the integration measure over modular space is left undetermined. We initially consider the case of the scalar particle with special emphasis on analogies with the string case. The question of Wick rotations in modular space is also discussed.

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