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arXiv:hep-th/9308157 · Journal of Physics A Mathematical and General

Coherent states of SU(l, 1) groups

D M Gitman, A L Shelepin

Published Dec 7, 1993• 18 citations
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Abstract

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.

DOI
10.1088/0305-4470/26/23/039

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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.
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.
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.
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.
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.
Abstract The problem of quantizing properly the canonical pair “angle and action variables”, φ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1,2): The crucial point is that the phase space has the global structure (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1,2) acts transitively, effectively and Hamilton‐like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space has the conic structure of an orbifold . That structure is closely related to a Z2 gauge symmetry which corresponds to the center of a 2‐fold covering of SO(1,2), the symplectic group . The basic variables on the phase space are the functions h0 = I, h1 = I cos φ and h2 = ‐I sin φ the Poisson brackets of which obey the Lie algebra . In the quantum theory they are represented by the self‐adjoint Lie algebra generators K0, K1, and K2 of a unitary representation, where K0 has the spectrum {k + n, n = 0, 1, …; k > 0}. A crucial prediction is that the classical Pythagorean relation h12 + h22 = h02 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be “measured” by means of the operators K1 and K2 alone without introducing any new phase operators! The SO(1,2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The additional coherent states can be used for the introduction of (Husimi type) “Q” distributions and (Sudarshan‐Glauber type) “P” representations of the density operator. The three operators K0, K1, and K2 are fundamental in the sense that one can construct composite position and momentum operators out of them! The new framework poses quite a number of fascinating experimental and theoretical challenges.