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arXiv:1403.3033 · Journal of Mathematical Physics

Coherent-state overcompleteness, path integrals, and weak values

Fernando Parisio

Published Mar 1, 2016• 1 citation
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Abstract

In the Hilbert space of a quantum particle the standard coherent-state resolution of unity is written in terms of a phase-space integration of the outer product zz. Because no pair of coherent states is orthogonal, one can represent the closure relation in non-standard ways, in terms of a single phase-space integration of the “unlike” outer product z′z, z′≠z. We show that all known representations of this kind have a common ground and that our reasoning extends to spin coherent states. These unlike identities make it possible to write formal expressions for a phase-space path integral, where the role of the Hamiltonian H is played by a weak energy value Hweak. Therefore, in this context, we can speak of weak values without any mention to measurements. The quantity Hweak appears as the ruler of the phase-space dynamics in the semiclassical limit.

DOI
10.1063/1.4943014

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In the Hilbert space of a quantum particle the standard coherent-state resolution of unity is written in terms of a phase-space integration of the outer product zz. Because no pair of coherent states is orthogonal, one can represent the closure relation in non-standard ways, in terms of a single phase-space integration of the “unlike” outer product z′z, z′≠z. We show that all known representations of this kind have a common ground and that our reasoning extends to spin coherent states. These unlike identities make it possible to write formal expressions for a phase-space path integral, where the role of the Hamiltonian H is played by a weak energy value Hweak. Therefore, in this context, we can speak of weak values without any mention to measurements. The quantity Hweak appears as the ruler of the phase-space dynamics in the semiclassical limit.
Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present work studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead, a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.
Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise, e.g., in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.
Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed. I.
We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.
By using the overcompleteness of coherent states we find an alternative form of the unit operator for which the ket and the bra appearing under the integration sign do not refer to the same phase-space point. This defines a new quantum representation in terms of Bargmann functions, whose basic features are presented. A continuous family of secondary reproducing kernels for the Bargmann functions is obtained, showing that this quantity is not unique for representations based on overcomplete sets. We illustrate the applicability of the presented results by deriving a semiclassical expression for the Feynman propagator that generalizes the well-known van Vleck formula and seems to point a way to cope with long-standing problems in semiclassical propagation of localized states.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in \cite{Bar01} for the two corresponding path integral forms suggested by Klauder and Skagerstan in \cite{Klau85}. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.