For systems with constraints the issue of non-commutability of quantization and elimination of non-physical variables is studied in the framework of path integrals (PIs). It is shown that one should take into consideration the curvilinearity of physical variables and their phase space reduction in order to provide a one-to-one correspondence between the Dirac scheme and PI description. The latter leads to a modification of the standard PI (PI with a gauge condition). A general PI derivation is suggested for any method of picking out physical variables which corresponds to the Dirac scheme.
For systems with constraints the issue of non-commutability of quantization and elimination of non-physical variables is studied in the framework of path integrals (PIs). It is shown that one should take into consideration the curvilinearity of physical variables and their phase space reduction in order to provide a one-to-one correspondence between the Dirac scheme and PI description. The latter leads to a modification of the standard PI (PI with a gauge condition). A general PI derivation is suggested for any method of picking out physical variables which corresponds to the Dirac scheme.
The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, it is shown that there is a class of constrained systems, which can be quantized only by the Dirac method. A possible interpretation of the gauge degrees of freedom is given. Introduction There are two main methods to quantize the Hamiltonian systems with first class constraints: the Dirac quantization [1] and the reduced phase space quantization [2], whereas two other methods, the path integral method [3, 2] and the BRST quantization [4] being the most popular method for the covariant quantization of gauge-invariant systems, are based on and proceed from them [2, 5]. The basic idea of the Dirac method consists in imposing quantum mechanically the first class constraints as operator conditions on the states for singling out the physical ones [1]. The reduced phase space quantization first identifies the physical degrees of freedom at the classical level by the factorization of the constraint surface with respect to the action of the gauge group, generated by the constraints. Then the resulting Hamiltonian system is quantized as a usual unconstrained system [2]. Naturally, the problem of the relationship of these two methods arises. It was discussed in different contexts in literature [6], and there is an opinion that the differences between the two quantization methods can be traced out to a choice of factor ordering in the construction of various physical operators.
The constrained Hamiltonian systems admitting no gauge conditions are considered. The methods to deal with such systems are discussed and developed. As a concrete application, the relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of systems under consideration. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, a class of constrained systems, which can be quantized only by the Dirac method, is discussed. A possible interpretation of the gauge degrees of freedom is given.
The constrained Hamiltonian systems admitting no gauge conditions are considered. The methods for dealing with such systems are discussed and developed. As a concrete application, the relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of systems under consideration. It is found that the two quantization methods may give similar, or essentially different, physical results, and moreover a class of constrained systems, which can be quantized only by the Dirac method, is discussed. A possible interpretation of the gauge degrees of freedom is given.
The Hamiltonian treatment of constrained systems in Gu¨ler′s formalism leads us to the total differential equations in many variables. These equations are integrable if the corresponding system of partial differential equations is a Jacobi system. The main aim of this paper is to investigate the quantization of the finite dimensional systems with constraints using the canonical formalism introduced by Gu¨ler. This approach is applied for two systems with constraints and the results are in agreement with those obtained by Dirac's canonical quatization method and path integral quantization method.
In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R . The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action. We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ , rather than of order ħ 2 . Most importantly it is not unique , leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O( ħ ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.
The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.