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arXiv:quant-ph/0009022 · Journal of Physics A Mathematical and General

Decomposition of Hilbert space in sets of coherent states

Nuno Barros e Sá

Published May 31, 2001• 3 citations
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Abstract

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.

DOI
10.1088/0305-4470/34/23/304

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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.
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.
Generalized coherent states are developed for SU ( n ) systems for arbitrary n . This is done by first iteratively determining explicit representations for the SU ( n ) coherent states and then determining parametric representations useful for applications. For SU ( n ), the set of coherent states is isomorphic to a coset space SU ( n )/ SU ( n -1), and thus shows the geometrical structure of the coset space. These results provide a convenient (2 n -1)-dimensional space for the description of arbitrary SU ( n ) systems. We further obtain the metric and measure on the coset space and show some properties of the SU ( n ) coherent states.
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.
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.
The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).