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Chinese Physics C

PATH INTEGRAL OF A SYMMETRY SYSTEM WITH CONSTRAINTS

Qiang Ma

Published Jan 1, 1985• 0 citations
OpenAlex

Abstract

When constraints of a system form a function group, the path integral quantum amplitude can exactly be deduced without adding the restrictions {χ_i,χ_j}=0 which are imposed on the gauge conditions or the weak concept. It can be shown that the contraints produced by symmetry just form a function group. Most of the interesting systems in physics are such symmetrical systems.

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When constraints of a system form a function group, the path integral quantum amplitude can exactly be deduced without adding the restrictions {χ_i,χ_j}=0 which are imposed on the gauge conditions or the weak concept. It can be shown that the contraints produced by symmetry just form a function group. Most of the interesting systems in physics are such symmetrical systems.
In both classical and quantum mechanics, a dynamical system with symmetry is reduced to a system which has a smaller number of degrees of freedom. In this paper we formulate path integrals for the quantum system reduced by symmetry. The original system is defined in terms of a Riemannian manifold M, on which a Lie group G acts by isometric transformations. Then we show that the path integral on M is reduced to a family of path integrals on a quotient space Q = M/G and that the reduced path integrals are classified by irreducible unitary representations of G. Stratification geometry of the set of spaces (M, Q), which is a generalization of the concept of principal fiber bundle, is used to describe the reduced path integral and consequently the path integral is expressed as a product of three factors; the rotational energy amplitude, which represents integration over the fiber directions, the vibrational energy amplitude, which represents integration over the directions perpendicular to the fiber, and the holonomy factor, which is caused by non-integrability of the vibrational directions. An attention is also paid for a singular point in Q, which arises when the group action of G on M is not free. We determine the boundary condition of the path integral kernel for a path which runs through the singularity.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.