The Quaternionic Affine Group and Related Continuous Wavelet Transforms on Complex and Quaternionic Hilbert Spaces

By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its unitary irreducible representations. These representations are constructed both on a complex and a quaternionic Hilbert space. As in the real and complex cases, the representations for the quaternionic group also turn out to be square-integrable. Using these representations we constrct quaternionic wavelets and continuous wavelet transforms on both the complex and quaternionic Hilbert spaces.Comment: 15 page

A class of vector coherent states defined over matrix domains

A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable $z$. In the present scheme, the variable $z$ is replaced by a matrix valued function over appropriate domains. As particular examples, we analyze the quaternionic extensions of the the canonical coherent states and the Gilmore-Perelomov and Barut-Girardello coherent states arising from representations of SU(1,1).Comment: 14 page

Hermite Polynomials and Quasi-classical Asymptotics

We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated semi-classical asymptotics.Comment: 16 pages, no figure

SUSY Associated Vector Coherent States and Generalized Landau Levels Arising From 2-dimensional SUSY

We describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out.Comment: 28 page

Gabor-type Frames from Generalized Weyl-Heisenberg Groups

We present in this paper a construction for Gabor-type frames built out of generalized Weyl-Heisenberg groups. These latter are obtained via central extensions of groups which are direct products of locally compact abelian groups and their duals. Our results generalize many of the results, appearing in the literature, on frames built out of the Schr\"odinger representation of the standard Weyl-Heisenberg group. In particular, we obtain a generalization of the result in \cite{PO}, in which the product $ab$ determines whether it is possible for the Gabor system $\{E_{mb}T_{na}g \}_{m,n\in \mathbb Z}$ to be a frame for $L^2(\mathbb R)$. As a particular example of the theory, we study in some detail the case of the generalized Weyl-Heisenberg group built out of the $d$-dimensional torus. In the same spirit we also construct generalized shift-invariant systems.Comment: 22 page

Orthogonal polynomials, Laguerre Fock space and quasi-classical asymptotics

Continuing our earlier investigation of the Hermite case [J. Math. Phys. 55 (2014), 042102], we study an unorthodox variant of the Berezin-Toeplitz quantization scheme associated with Laguerre polynomials. In particular, we describe a "Laguerre analogue" of the classical Fock (Segal-Bargmann) space and the relevant semi-classical asymptotics of its Toeplitz operators; the former actually turns out to coincide with the Hilbert space appearing in the construction of the well-known Barut-Girardello coherent states. Further extension to the case of Legendre polynomials is likewise discussed.Comment: 25 pages, no figure

Berezin-Toeplitz quantization over matrix domains

We explore the possibility of extending the well-known Berezin-Toeplitz quantization to reproducing kernel spaces of vector-valued functions. In physical terms, this can be interpreted as accommodating the internal degrees of freedom of the quantized system. We analyze in particular the vector-valued analogues of the classical Segal-Bargmann space on the domain of all complex matrices and of all normal matrices, respectively, showing that for the former a semi-classical limit, in the traditional sense, does not exist, while for the latter only a certain subset of the quantized observables have a classical limit: in other words, in the semiclassical limit the internal degrees of freedom disappear, as they should. We expect that a similar situation prevails in much more general setups.Comment: 28 pages, no figure

A matrix-valued Berezin-Toeplitz quantization

We generalize some earlier results on a Berezin-Toeplitz type of quantization on Hilbert spaces built over certain matrix domains. In the present, wider setting, the theory could be applied to systems possessing several kinematic and internal degrees of freedom. Our analysis leads to an identification of those observables, in this general context, which admit a semi-classical limit and those for which no such limit exists. It turns out that the latter class of observables involve the internal degrees of freedom in an intrinsic way. Mathematically, the theory, being a generalization of the standard Berezin-Toeplitz quantization, points the way to applying such a quantization technique to possibly non-commutative spaces, to the extent that points in phase space are now replaced by $N\times N$ matrices.Comment: 15 pages, no picture

General construction of Reproducing Kernels on a quaternionic Hilbert space

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated

The Symmetry Groups of Noncommutative Quantum Mechanics and Coherent State Quantization

We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in $(2+1)$-space-time dimensions and the two-fold extension of the group of translations of $\mathbb R^4$. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.Comment: 27 page