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arXiv:hep-th/0006150

Path Integral Method with Symmetry

Shogo Tanimura

Published Jun 20, 2000• 0 citations
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Abstract

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.

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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.
Applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered. Analysis of the symmetry of the Lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group. Identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold. The integration is performed using the orthogonality of the representations. Compact and noncompact rotation groups are considered, where the corresponding path integral is embedded in Euclidean and pseudo-Euclidean spaces, respectively. The unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient. In the first case Fourier analysis leads to an expansion in group characters. In the second case an expansion in zonal spherical functions is obtained. As examples the groups SO(n), SU(2), SO(n−1,1), and SU(1,1) are explicitly discussed. The path integral on SO(n+m) and SO(n,m) in bispherical coordinates is also treated.
Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold MMM is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of MMM with the quotient space G/HG/HG/H, where HHH is the isotropy group of an arbitrary fixed point of MMM, we show that quantum mechanics on G/HG/HG/H possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle G(G/H,H)G(G/H, H)G(G/H,H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.
The quantization of physical systems moving on group and symmetric spaces has been an area of active research over the past three decades. This book shows that it is possible to introduce a representation-independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables.Furthermore, the book shows that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations, and although the configuration space is in general a multidimensional curved manifold, it is shown that the resulting lattice phase space path integral has the form of a lattice phase space path integral on a multidimensional flat manifold. Hence, a novel and extremely natural phase space path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phase space path integral quantization is (a) exact, (b) more general than, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds.To illustrate the general theory, a representation-independent propagator is explicitly constructed for SU(2) and the affine group
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite number of the usual exponentials. This procedure leads to a set of stochastic differential equations on the group manifold, which can be further formulated in terms of the supersymmetric effective action. This action has the form of the Witten topological field theory in the continuum limit. As a consequence, we show how it can be used to obtain the exact results for a specific quantum many-body system which can be otherwise solved only by the Bethe ansatz. This represents an example of a many-body system treated exactly using the path-integral formulation. Moreover, our method can deal with time-dependent parameters, which we demonstrate explicitly.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.