Generalized Bargmann Space Research Articles

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

Loop gravity in terms of spinors

We show that loop gravity can equally well be formulated in in terms of spinorial variables (instead of the group variables which are commonly used), which have recently been shown to provide a direct link between spin network states and discrete geometries. This results in a new, unitarily equivalent formulation of the theory on a generalized Bargmann space. Since integrals over the group are exchanged for straightforward integrals over the complex plane we expect this formalism to be useful to efficiently organize practical calculations.

Spinor representation for loop quantum gravity

We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.

Coherent States Quantization for Generalized Bargmann Spaces with Formulae for Their Attached Berezin Transforms in Terms of the Laplacian on ℂ N

While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of such formulae is discussed.

Coherent state transforms attached to generalized Bargmann spaces on the complex plane

AbstractWe construct a family of coherent states transforms attached to generalized Bargmann spaces [C. R. Acad. Sci. Paris, t. 325, 1997] in the complex plane. This constitutes another way of obtaining the kernel of an isometric operator linking the space of square integrable functions on the real line with the true‐poly‐Fock spaces [Oper. Theory, Adv. Appl. v. 117, 2000]. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

An Elementary Construction on Nonlinear Coherent States Associatedto Generalized Bargmann Spaces

Consider the space L2(ℂ, dμ(z)), where is the Gaussian measure, and its generalized Bargmann subspaces Em which are the null kernels of the operator ; m = 0,1, …. In this work, we present an other construction of Em following the Hermite functions which allows us to define a family of generalized Bargmann transform Bm which maps isometrically Em into L2(ℝ). The generalized coherent states ∣z〉 m associated to Em are constructed and some properties of them are given.

Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics

We employ the recently established basis (the two-variable Hermite-Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A303, 311 (2002)] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.

Resolution of Unity of Generalized Bargmann Spaces by means ofDisplaced Fock States

We resolve the unity of generalized Bargmann spaces in Cn [J.Math. Phys. 41 (2000) 3507] by means of displaced Fock states.

Explicit formulas for reproducing kernels of generalized Bargmann spaces on Cn

We introduce a class of generalized Bargmann spaces on Cn for which we establish explicit formulas of their reproducing kernels. Some applications of the obtained formulas are given.

Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants

We establish explicit formulas for the reproducing kernels of a class of generalized Bargmann spaces on C n.

A Possible Generalization of the Bargmann Space11The project supported in part by National Natural Science Foundation of China

By virtue of the technique of integration within an ordered product of operators we show that the ordinary Bargmann space basis function can be extended to two-variable Hermite polynomials, the latter can span a generalized Bargmann space. The corresponding two-mode Hermite polynomial states are thus introduced and their properties are investigated.

SU(3) Clebsch-Gordan coefficients with definite permutation symmetry

Multiquark wave functions representing physical particles must not only have the correct transformation properties under the Poincaré, SU(3) flavor and SU(3) color groups, but must also be antisymmetric under the interchange of quark labels induced by these groups. Under the interchange of just space-time, flavor, or color labels, the multiquark wave functions may be of mixed permutation symmetry, as long as the effect of interchanging all types of labels is antisymmetric. It is shown how to construct multiquark wave functions as irreducible representations of SU(3) (or more generally any compact group) that have a definite permutation symmetry. The multiquark wave functions are realized as polynomials in a generalized Bargmann space, and it is shown that the Clebsch-Gordan coefficients arising in the n-fold tensor product decomposition can be obtained by differentiating these polynomials in a prescribed way. Tables are given for two to six multiquark states with definite SU(3) and permutation symmetry. The question of decoupling these multiquark states into quark-antiquark or three-quark states times ocean states is also discussed.