Feynman Integral and Random Dynamics in Quantum Physics

1. Front Matter

   Pages i-xx

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2. Preliminaries

   • Zbigniew Haba

   Pages 1-11

3. Markov chains

   • Zbigniew Haba

   Pages 13-30

4. Stochastic differential equations

   • Zbigniew Haba

   Pages 31-40

5. Semi-groups and the Trotter product formula

   • Zbigniew Haba

   Pages 41-44

6. The Feynman integral

   • Zbigniew Haba

   Pages 45-78

7. Feynman integral and stochastic differential equations

   • Zbigniew Haba

   Pages 79-88

8. Random perturbations of the classical mechanics

   • Zbigniew Haba

   Pages 89-115

9. Complex dynamics and coherent states

   • Zbigniew Haba

   Pages 117-140

10. Quantum non-linear oscillations

    • Zbigniew Haba

    Pages 141-158

11. Feynman integral on analytic submanifolds

    • Zbigniew Haba

    Pages 159-167

12. Interaction with the environment

    • Zbigniew Haba

    Pages 169-184

13. Lindblad equation and stochastic Schrödinger equation

    • Zbigniew Haba

    Pages 185-249

14. Hamiltonian time evolution of the density matrix

    • Zbigniew Haba

    Pages 251-255

15. Stochastic representation of the Lindblad time evolution

    • Zbigniew Haba

    Pages 257-276

16. Decoherence and estimates on dissipative dynamics

    • Zbigniew Haba

    Pages 277-282

17. Diffusive behaviour of the Wigner function and decoherence

    • Zbigniew Haba

    Pages 283-290

18. Scattering and tunnelling in an environment

    • Zbigniew Haba

    Pages 291-301

19. The Feynman integral in quantum field theory

    • Zbigniew Haba

    Pages 303-311

20. The phase space methods in QFT

    • Zbigniew Haba

    Pages 313-329

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.

• Markov chain
• Rang
• Theoretical physics
• computational physics
• mathematical physics
• quantum field theory
• quantum mechanics
• quantum physics
• random dynamical system
• stochastic differential equation
• stochastic processes

• Institute of Theoretical Physics, University of Wrocław, Wrocław, Poland

  Zbigniew Haba

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