Representations of coherent states in non-orthogonal bases

We perform the momentum-space quantization of a spin-less particle moving on the SU (2) group manifold, that is, the three-dimensional sphere S 3 , by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations.

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“…Therefore the Hilbert spaces L 2K α (R d ), P W and L 2K d−α (R d ) define a Gelfand triple of (rigged) Hilbert spaces [48].…”

Section: Digression On Hilbert Spaces Dirac's Bra-ket Formulation mentioning

confidence: 99%

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Guerrero

López-Ruiz

Aldaya

2020

J. Phys. A: Math. Theor.

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“…Non-hermitian Hamiltonians have been used for a long time as effective Hamiltonians (think, for instance, of the optical potential in nuclear physics [1]). With the introduction of P T -symmetric Hamiltonians [2] in Quantum Mechanics they became very popular, however only a few papers have been devoted to coherent states (CS) for non-Hermitian systems (see [3], where Gazeau-Klauder CS are constructed using the definition of scalar product in terms of the CPT norm, [4,5,6,7,8,9] where the notion of pseudo-bosons and bi-coherent states are introduced, or [10,11]) as compared with the huge amount of papers devoted to usual coherent states.…”

Section: Introduction and Physical Motivationmentioning

confidence: 99%

Non-Hermitian Coherent States for Finite-Dimensional Systems

Guerrero

2018

Springer Proceedings in Physics

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We introduce Gilmore-Perelomov coherent states for non-unitary representations of non-compact groups, and discuss the main similarities and differences with respect to ordinary unitary Gilmore-Perelomov coherent states. The example of coherent states for the non-unitary finite dimensional representations of SU (1, 1) is considered and they are used to describe the propagation of light in coupled PT-symmetric optical devices.1 The first theoretical proposal of P T -symmetry in optics was given in [12].

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“…It is worthwhile to mention that, there exist many generalized coherent states categorizing in this special class of quantum states, which exhibit the nonclassicality features of light, i.e., 'nonclassical' light [29,30,31]. Based on the above explanations, regarding the DNSs as well as the nonlinear coherent states, one may motivate to establish a direct connection between DNSs and nonlinear coherent states, which is led to the concept of 'nonlinear displaced number states' (NDNSs).…”

Section: Introductionmentioning

confidence: 99%

Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features: A new approach

Tavassoly

Faghihi

2015

Chinese Phys. B

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Recently, nonlinear displaced number states (NDNSs) have been manually introduced, in which the deformation function f (n) has been artificially added to the well-known displaced number states (DNSs). In this paper, after expressing enough physical motivation of our procedure, four distinct classes of NDNSs are presented by applying algebraic and group treatments. To achieve this purpose, by considering the DNSs and recalling the nonlinear coherent states formalism, the NDNSs are logically defined through an algebraic consideration. In addition, by using a particular class of Gilmore-Perelomov-type of SU (1, 1) and a class of SU (2) coherent states, the NDNSs are introduced via group theoretical approach. Then, in order to examine the nonclassical behaviour of these states, sub-Poissonian statistics by evaluating Mandel parameter and Wigner quasi-probability distribution function associated with the obtained NDNSs are discussed, in detail.

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Representations of coherent states in non-orthogonal bases

Cited by 46 publications

References 18 publications

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Non-Hermitian Coherent States for Finite-Dimensional Systems

Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features: A new approach

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Representations of coherent states in non-orthogonal bases

Cited by 46 publications

References 18 publications

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S 3

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S 3

Non-Hermitian Coherent States for Finite-Dimensional Systems

Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features: A new approach

Contact Info

Product

Resources

About

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³