Recently, the coherent-state labeled by the real parameters (RCS) has been introduced in connec tion with the numerical path integration for many-fermion systems.!) This is a complement of the coherent-state labeled by complex vari ables (CCS) which gives a basis of the path inte gral in the generalized phase space leading to the quantum version of the time-dependent mean field. 2 ),3) The central formula of both types of coherent states is the relation of of unity. For the case of RCS, Dobacewski and Koonin 4 ) have very recently proved this relation based on the commutability of the projection operator with the algebra generating the coher ent-state. The proof is rather general, since one uses only the Lie algebraic concept of the highest weight vectors. However it presupposes a specific form of the integration measure from the outset. In this note, apart from such a Lie-alge braic approach, we construct the RCS for many fermion systems based on the geometric device of the Lie group and associated label space (the so called quotient space),S) thereby the partition of unity is proved through a natural construction of the invariant measure. We also point out a specific role of the RCS for path integration in comparison with the path integral in the represen tation of CCS. In the following we are concerned with the RCS of determinantal states.. We firstly show that the underlying group becomes an orthogonal group instead of the unitary group for the CCS. 2 ) Let (aI', ai) be annihilation operators of the ~'particle and hole states with It = 1 ~ m, i = 1 ~ n with respect to the initial determinantal state 1 ([}o> = IL~lai +10>. Consider the linear canonical transformation where T(g) is an irreducible unitary representa tion of a Lie group G and A, B, C and Dare m x m, m x n, n x m and n x n matrices. Now consider the real restriction of (1). Hence the canonical nature leads to g'gT=J<m+n) (2) or
Recently, the coherent-state labeled by the real parameters (RCS) has been introduced in connec tion with the numerical path integration for many-fermion systems.!) This is a complement of the coherent-state labeled by complex vari ables (CCS) which gives a basis of the path inte gral in the generalized phase space leading to the quantum version of the time-dependent mean field. 2 ),3) The central formula of both types of coherent states is the relation of of unity. For the case of RCS, Dobacewski and Koonin 4 ) have very recently proved this relation based on the commutability of the projection operator with the algebra generating the coher ent-state. The proof is rather general, since one uses only the Lie algebraic concept of the highest weight vectors. However it presupposes a specific form of the integration measure from the outset. In this note, apart from such a Lie-alge braic approach, we construct the RCS for many fermion systems based on the geometric device of the Lie group and associated label space (the so called quotient space),S) thereby the partition of unity is proved through a natural construction of the invariant measure. We also point out a specific role of the RCS for path integration in comparison with the path integral in the represen tation of CCS. In the following we are concerned with the RCS of determinantal states.. We firstly show that the underlying group becomes an orthogonal group instead of the unitary group for the CCS. 2 ) Let (aI', ai) be annihilation operators of the 'particle and hole states with It = 1 ~ m, i = 1 ~ n with respect to the initial determinantal state 1 ([}o> = ILlai +10>. Consider the linear canonical transformation where T(g) is an irreducible unitary representa tion of a Lie group G and A, B, C and Dare m x m, m x n, n x m and n x n matrices. Now consider the real restriction of (1). Hence the canonical nature leads to g'gT=J<m+n) (2) or
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).
In this paper we study the generalized coherent states (g.c.s.) in n -dimensional space. Their annihilation operators consist of linear combinations of the annihilation operators a l , and their coherent states are the product of the simple coherent states. All the known annihilation operators which have been studied as yet, and also the magnetic ones, are special cases of these operators. The new operators A k + A m form the Lie algebra U ( n ) of the Lie group U ( n ). Finally, instead of the operators a l and a l + we can use the new operators, which depend on a complex parameter.
The n-dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path-integral representation involving novel first-order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.
In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,..,λ1+n/2〉, encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the annihilation-operator coherent states, i.e., the eigenstates of the noncompact lowering generators corresponding to complex eigenvalues. These states generalize the coherent states introduced by Barut and Girardello for Sp(2,R), and later on extended by Deenen and Quesne to the Sp(2d,R) irreducible representations of the type 〈(λ+n/2)d〉. When λ1,...,λd are not all equal, it was shown by Deenen and Quesne that the eigenvalues do not completely specify the eigenstates of the noncompact lowering generators. In the present work, their characterization is completed by a set of continuous labels parametrizing the (unitary-operator) coherent states of the maximal compact subgroup U(d). The resulting coherent states are therefore of mixed type, being annihilation-operator coherent states only as regards the noncompact generators. A realization in a subspace of a Bargmann space of analytic functions shows that such coherent states satisfy a unity resolution relation in the representation space of 〈λd +n/2,...,λ1+n/2〉, and therefore may be used as a continuous basis in such space. The analytic functions and the differential operators representing the representation space discrete bases and the Sp(2d,R) generators, respectively, are found in explicit form. It is concluded that the annihilation-operator coherent state representation provides the mathematical foundation for the use of differentiation operators with respect to the noncompact raising generators in symbolic expressions of the Sp(2d,R) generators. This is to be compared with the habit of replacing a boson annihilation operator by a symbolic differentiation with respect to the corresponding creation operator, accounted for by the Bargmann representation of such operators.
We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.
Abstract. We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere S d. The coherent states are labeled by points in the associated phase space T ∗ (S d). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of B. Hall and M. Stenzel, we give here a substantially different description based on ideas of T. Thiemann and of K. Kowalski and J. Rembieliński. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.