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The Double Cover of the Real Symplectic Group and a Theme from Feynman’s Quantum Mechanics

Michael C. Berg

Published Jan 1, 2012• 1 citation
OpenAlex

Abstract

Abstract. We present a direct connection between the 2-cocycle defining the double cover of the real symplectic group and a Feynman path integral describing the time evolution of a quantum mechanical system.

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It was an interested and important problem to give the description of quantization, i.e. of passing from classical physical systems to the corresponding quantum ones, from the moment that quantum mechanics came into existence. In the end we succeeded in giving it as follows: Let L be a Lagrangian function. Then the Hamiltonian function H is defined through the Legendre transformation of L. The Hamiltonian operator ( ) at time in quantum mechanics is defined from H. It should be noted that ( ) has ordering ambiguities (cf. [14]). Let be a probability amplitude at time . Then its temporal evolution can be given by the solution of the Schrodinger equation
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable τττ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 1-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.