Ebook: Feynman Integral and Random Dynamics in Quantum Physics: A Probabilistic Approach to Quantum Dynamics

The Feynman integral is considered as an intuitive representation of quantum mechanics showing the complex quantum phenomena in a language comprehensible at a classical level. It suggests that the quantum transition amplitude arises from classical mechanics by an average over various interfering paths. The classical picture suggested by the Feynman integral may be illusory. By most physicists the path integral is usually treated as a convenient formal mathematical tool for a quick derivation of useful approximations in quantum mechanics. Results obtained in the formalism of Feynman integrals receive a mathematical justification by means of other (usually much harder) methods. In such a case the rigour is achieved at the cost of losing the intuitive classical insight. The aim of this book is to formulate a mathematical theory of the Feynman integral literally in the way it was expressed by Feynman, at the cost of complexifying the configuration space. In such a case the Feynman integral can be expressed by a probability measure. The equations of quantum mechanics can be formulated as equations of random classical mechanics on a complex configuration space. The opportunity of computer simulations shows an immediate advantage of such a formulation. A mathematical formulation of the Feynman integral should not be considered solely as an academic question of mathematical rigour in theoretical physics.

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Content:
Front Matter....Pages i-xx
Preliminaries....Pages 1-11
Markov chains....Pages 13-30
Stochastic differential equations....Pages 31-40
Semi-groups and the Trotter product formula....Pages 41-44
The Feynman integral....Pages 45-78
Feynman integral and stochastic differential equations....Pages 79-88
Random perturbations of the classical mechanics....Pages 89-115
Complex dynamics and coherent states....Pages 117-140
Quantum non-linear oscillations....Pages 141-158
Feynman integral on analytic submanifolds....Pages 159-167
Interaction with the environment....Pages 169-184
Lindblad equation and stochastic Schr?dinger equation....Pages 185-249
Hamiltonian time evolution of the density matrix....Pages 251-255
Stochastic representation of the Lindblad time evolution....Pages 257-276
Decoherence and estimates on dissipative dynamics....Pages 277-282
Diffusive behaviour of the Wigner function and decoherence....Pages 283-290
Scattering and tunnelling in an environment....Pages 291-301
The Feynman integral in quantum field theory....Pages 303-311
The phase space methods in QFT....Pages 313-329
Computer simulations of quantum random dynamics....Pages 331-352
Back Matter....Pages 353-367

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Content:
Front Matter....Pages i-xx
Preliminaries....Pages 1-11
Markov chains....Pages 13-30
Stochastic differential equations....Pages 31-40
Semi-groups and the Trotter product formula....Pages 41-44
The Feynman integral....Pages 45-78
Feynman integral and stochastic differential equations....Pages 79-88
Random perturbations of the classical mechanics....Pages 89-115
Complex dynamics and coherent states....Pages 117-140
Quantum non-linear oscillations....Pages 141-158
Feynman integral on analytic submanifolds....Pages 159-167
Interaction with the environment....Pages 169-184
Lindblad equation and stochastic Schr?dinger equation....Pages 185-249
Hamiltonian time evolution of the density matrix....Pages 251-255
Stochastic representation of the Lindblad time evolution....Pages 257-276
Decoherence and estimates on dissipative dynamics....Pages 277-282
Diffusive behaviour of the Wigner function and decoherence....Pages 283-290
Scattering and tunnelling in an environment....Pages 291-301
The Feynman integral in quantum field theory....Pages 303-311
The phase space methods in QFT....Pages 313-329
Computer simulations of quantum random dynamics....Pages 331-352
Back Matter....Pages 353-367
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