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Physical Review A

Covariance and uniqueness of the canonical fermion coherent state

Chang Jae Lee

Published Nov 1, 1992• 2 citations
Publisher

Abstract

A unitary Fock-space operator that leaves the canonical quantization rule for fermions covariant is introduced. The defining equation of a canonical fermion coherent state retains the symmetry under this unitary transformation. This fact mandates that the coherent state must accompany a partner state, and that the form of a state in the pair is related to the partner state by the symmetry. A careful definition of the phase in the extended fermion Fock space is crucial for the uniqueness of the normalized fermion coherent state. Also the proper choice of the phase renders the canonical fermion coherent state equal to a displaced-vacuum state.

DOI
10.1103/physreva.46.6049

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A unitary Fock-space operator that leaves the canonical quantization rule for fermions covariant is introduced. The defining equation of a canonical fermion coherent state retains the symmetry under this unitary transformation. This fact mandates that the coherent state must accompany a partner state, and that the form of a state in the pair is related to the partner state by the symmetry. A careful definition of the phase in the extended fermion Fock space is crucial for the uniqueness of the normalized fermion coherent state. Also the proper choice of the phase renders the canonical fermion coherent state equal to a displaced-vacuum state.
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Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.
This chapter presents an analogous formalism which is developed for systems composed of many identical particles. Another extremely useful basis of the Fock space is the basis of coherent states which is analogous to the basis of position eigenstates in quantum mechanics. A crucial property of the coherent states is their overcompleteness in the Fock space, that is, the fact that any vector of the Fock space can be expanded in terms of coherent states. Boson coherent states are the physical states which emerge naturally when taking the classical limit of quantum mechanics or of a quantum field theory. In contrast, Fermion coherent states are not contained in the Fermion Fock space, they are not physically observable, and there are no classical fields of Fermions. In the subsequent development, this physical difference will give rise to some significant differences in the treatment of Bosons and Fermions.
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.
The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.