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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Path integrals and conservation laws

Cécile DeWitt-Morette, Tian-Rong Zhang

Published Nov 15, 1983• 16 citations
Publisher

Abstract

If the initial and the final states of a system have classical limits, and if the classical limits cannot be chosen independently from each other without violating classical conservations laws, the JWKB approximation of the probability amplitude for the transition between such initial and final states requires special care. We compute it explicity for momentum to momentum transitions and for angular momentum to angular momentum transitions. It is shown how the classical conservation laws make their appearance in the limit $\ensuremath{\hbar}=0$. The statements are illustrated by applications to potential scattering theory.

DOI
10.1103/physrevd.28.2503

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If the initial and the final states of a system have classical limits, and if the classical limits cannot be chosen independently from each other without violating classical conservations laws, the JWKB approximation of the probability amplitude for the transition between such initial and final states requires special care. We compute it explicity for momentum to momentum transitions and for angular momentum to angular momentum transitions. It is shown how the classical conservation laws make their appearance in the limit \ensuremathℏ=0\ensuremath{\hbar}=0\ensuremathℏ=0. The statements are illustrated by applications to potential scattering theory.
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The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.
Quantum Theory's Uncertainty violates conservation of energy & momentum and invokes matter's unrealistic creation from, and dissolution into, nothing. In Unified Theory action is an evolute of sharmon with Planck constant h as its quantum. The inviolable conservations of energy and momentum ordain conservation of action, invalidation of the Uncertainty and introduction of the new of Null It has universal applications and is conceptually superior to the old Hamilton's of Least Action as the latter unrealistically associates turnover of action NOT with variations in but with constant energy & momentum. In Nature, the path chosen by an isolated closed system during a real physical change through a succession of intermediate states from the initial equilibrium state 'a' to the final equilibrium state 'b' is such that the summation or integration of action covering all variations of energy or momentum is zero or null. Hence the name: Principle of Null Action. The new principle has successfully derived the generally accepted equations for the following: Schwinger's Quantum Dynamical Principle, Klein-Gorden Field Equations, Schrodinger Wave Equation, Special and General Relativity, Euler's Equations, Lagrange's Equations, Maxwell Equations for Electromagnetic Radiation, Newton's laws of motion, Thermodynamic Equation of State, Discharge of Capacitance through Inductance & Resistance.