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arXiv:hep-th/9208017 · Journal of Physics A Mathematical and General

Coherent states of SU(N) groups

Dmitri M Gitman, A. L. Shelepin

Published Jan 21, 1993• 39 citations
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Abstract

An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.

DOI
10.1088/0305-4470/26/2/018

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An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.
This work can be considered as a continuation of our previous study, in which an explicit form of coherent states (cs) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l, 1) group and construct explicitly corresponding cs. The cs are parametrized by dots of a coset space, which is, in that particular case, the open complex ball CD l . This space together with the projective space CP l , which parametrizes the cs of the SU(l + 1) group, exhaust all complex spaces of constant curvature. Thus, both sets of cs provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature. This is why the cs of the SU(N) and SU(l, 1) groups are of importance in connection with quantization theory. The constructed cs form an overcompleted system in the representation space and, as quantum states possessing a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is investigated in terms of symbols of operators; the limit of the so called star commutator of the symbols generates the Poisson bracket in CD l , the latter plays the role of the phase space for the corresponding classical mechanics.
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.
Getting coherent state superpositions to stay put in phase space: QQQ functions and one dimensional integral representations of generator eigenstates
We study quantum mechanics in the phase space associated with the coherent state (CS) manifold of Lie groups. Eigenstates of generators of the group are constructed as one dimensional integral superpositions of CS along their orbits. We distinguish certain privileged orbits where the superposition is in phase. Interestingly, for closed in phase orbits, the geometric phase must be quantized to 2πZ2π\mathbb{Z}2πZ, else the superposition vanishes. This corresponds to exact Bohr-Sommerfeld quantization. The maximum of the Husimi-Kano QQQ quasiprobability distribution is used to diagnose where in phase space the eigenstates of the generators lie. The QQQ function of a generator eigenstate is constant along each orbit. We conjecture that the maximum of the QQQ function corresponds to these privileged in phase orbits. We provide some intuition for this proposition using interference in phase space, and then demonstrate it for canonical CS (H4H_4H4​ oscillator group), spin CS (SU(2)SU(2)SU(2)) and SU(1,1)SU(1,1)SU(1,1) CS, relevant to squeezing.
In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.
We construct the system of generalized coherent states for the quantum Kepler problem corresponds to the homogeneous domain SU(2,2)/S(U(2)×U(2))SU(2,2)/S(U(2)\times U(2))SU(2,2)/S(U(2)×U(2)). We show that the SU(2,2)-equivariant momentum map for this domain yields the momentum map for the classical Kepler problem via appropriate limiting passage. We also show that under this passage the average values of quantum observables in this system of coherent states pass into the functions on classical phase space and −i/h-i/h−i/h times commutator pass into the Poisson bracket.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.