Journal of Physics A Mathematical and General

Extended coherent states and path integrals with auxiliary variables

John R. Klauder, BERNARD F. WHITING

Published Apr 7, 1993• 6 citations

Abstract

The usual construction of coherent states allows a wider interpretation in which the number of distinguishing state labels is no longer minimal; the label measure determining the required resolution of unity is then no longer unique and may even be concentrated on manifolds with positive co-dimension. Paying particular attention to the residual restrictions on the measure, the authors choose to capitalize on this inherent freedom and in formally distinct ways, systematically construct suitable sets of extended coherent states which, in a minimal sense, are characterized by auxiliary labels. Interestingly, they find these states lead to path integral constructions containing auxiliary (essentially unconstrained) path-space variables. The impact of both standard and extended coherent state formulations on the content of classical theories is briefly examined, the latter showing the existence of new, and generally constrained, classical variables. Some implications for the handling of constrained classical systems are given, with a complete analysis awaiting further study.

DOI: 10.1088/0305-4470/26/7/025

Related papers

6. Coherent quantum physics

Arnold Neumaier

Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.

De Gruyter eBooksOct 24, 20190 citations

GENERAL PROBLEMS AND DIRECTIONS On the Role of Coherent States in Quantum Foundations 1

John R. Klauder

Coherent states, and the Hilbert space representations they generate, provide ideal tools to discuss classical/quantum relationships. In this paper we analyze three separate classical/quantum problems using coherent states, and show that useful connections arise among them. The topics discussed are: (1) a truly nat� ural formulation of phase space path integrals; (2) how this analysis implies that the usual classical formalism is simply a subset of the quantum formalism, and thus demonstrates a universal coexistence of both the classical and quantum formalisms; and (3) how these two insights lead to a complete analytic solution of a formerly insoluble family of nonlinear quantum field theory models.

Jan 1, 20110 citations

Selecta from a Life-Long Obsession with Path Integrals

John R. Klauder, Christopher Casenas Bernido, M. Victoria Carpio-Bernido

The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.

AIP conference proceedingsJan 1, 20080 citations

On the role of coherent states in quantum foundations

J. Klauder

Coherent states, and the Hilbert space representations they generate, provide ideal tools to discuss classical/quantum relationships. In this paper we analyze three separate classical/quantum problems using coherent states, and show that useful connections arise among them. The topics discussed are: (1) a truly natural formulation of phase space path integrals; (2) how this analysis implies that the usual classical formalism is “simply a subset” of the quantum formalism, and thus demonstrates a universal coexistence of both the classical and quantum formalisms; and (3) how these two insights lead to a complete analytic solution of a formerly insoluble family of nonlinear quantum field theory models.

arXiv:1008.4307Optics and SpectroscopyOct 1, 20114 citations

Note on the Real Coherent-State for Many-Fermion Systems

Hiroshi Kuratsuji

Recently, the coherent-state labeled by the real parameters (RCS) has been introduced in connec tion with the numerical path integration for many-fermion systems.!) This is a complement of the coherent-state labeled by complex vari ables (CCS) which gives a basis of the path inte gral in the generalized phase space leading to the quantum version of the time-dependent mean field. 2 ),3) The central formula of both types of coherent states is the relation of of unity. For the case of RCS, Dobacewski and Koonin 4 ) have very recently proved this relation based on the commutability of the projection operator with the algebra generating the coher ent-state. The proof is rather general, since one uses only the Lie algebraic concept of the highest weight vectors. However it presupposes a specific form of the integration measure from the outset. In this note, apart from such a Lie-alge braic approach, we construct the RCS for many fermion systems based on the geometric device of the Lie group and associated label space (the so called quotient space),S) thereby the partition of unity is proved through a natural construction of the invariant measure. We also point out a specific role of the RCS for path integration in comparison with the path integral in the represen tation of CCS. In the following we are concerned with the RCS of determinantal states.. We firstly show that the underlying group becomes an orthogonal group instead of the unitary group for the CCS. 2 ) Let (aI', ai) be annihilation operators of the ~'particle and hole states with It = 1 ~ m, i = 1 ~ n with respect to the initial determinantal state 1 ([}o> = IL~lai +10>. Consider the linear canonical transformation where T(g) is an irreducible unitary representa tion of a Lie group G and A, B, C and Dare m x m, m x n, n x m and n x n matrices. Now consider the real restriction of (1). Hence the canonical nature leads to g'gT=J<m+n) (2) or

Progress of Theoretical PhysicsJan 1, 19840 citations

Coherent State Path Integrals at (Nearly) 40

John R. Klauder

Coherent states can be used for diverse applications in quantum physics including the construction of coherent state path integrals. Most definitions make use of a lattice regularization; however, recent definitions employ a continuous-time regularization that may involve a Wiener measure concentrated on continuous phase space paths. The introduction of constraints is both natural and economical in coherent state path integrals involving only the dynamical and Lagrange multiplier variables. A preliminary indication of how these procedures may possibly be applied to quantum gravity is briefly discussed.

arXiv:quant-ph/9810043Oct 14, 19984 citations