Canonical, Squeezed and Fermionic Coherent States in a Right Quaternionic Hilbert Space with a Left Multiplication on It

In this paper, by using the nonlinear coherent states approach, we find a relation between the geometric structure of the physical space and the geometry of the corresponding projective Hilbert space. To illustrate the approach, we explore the quantum transition probability and the geometric phase in the curved space.

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MAGNETIC GROUP ON THE PSEUDOSPHERE

International Journal of Modern Physics B ◽

10.1142/s0217979292001559 ◽

1992 ◽

Vol 06 (21) ◽

pp. 3525-3537 ◽

Cited By ~ 3

Author(s):

V. BARONE ◽

V. PENNA ◽

P. SODANO

Keyword(s):

Magnetic Field ◽

Hilbert Space ◽

Quantum Mechanics ◽

Constant Magnetic Field ◽

Coherent States ◽

Compact Operators ◽

Point Of View ◽

Algebraic Point ◽

Planar Limit ◽

Discrete Part

The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.

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Groupoids and Coherent States

Open Systems & Information Dynamics ◽

10.1142/s1230161219500173 ◽

2019 ◽

Vol 26 (04) ◽

pp. 1950017 ◽

Cited By ~ 2

Author(s):

F. di Cosmo ◽

A. Ibort ◽

G. Marmo

Keyword(s):

Hilbert Space ◽

Harmonic Oscillator ◽

Coherent States ◽

Quantum Mechanical ◽

Natural Setting ◽

Invariant Subset ◽

Quantum Mechanical System ◽

Generalized Coherent States ◽

Invertible Elements ◽

Groupoid Algebra

Schwinger’s algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmonic oscillator as well as generalized coherent states for f-oscillators are exemplified in this picture.

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Examples of enhanced quantization: Bosons, fermions and anyons

International Journal of Modern Physics A ◽

10.1142/s0217751x1450119x ◽

2014 ◽

Vol 29 (22) ◽

pp. 1450119

Author(s):

T. C. Adorno ◽

J. R. Klauder

Keyword(s):

Hilbert Space ◽

Coherent States ◽

Canonical Quantization ◽

Action Functional ◽

Classical Action ◽

Unitary Transformations ◽

Classical Quantum ◽

Quantum Action ◽

Space Vectors ◽

Gaussian Vectors

Enhanced quantization offers a different classical/quantum connection than that of canonical quantization in which ℏ > 0 throughout. This result arises when the only allowed Hilbert space vectors allowed in the quantum action functional are coherent states, which leads to the classical action functional augmented by additional terms of order ℏ. Canonical coherent states are defined by unitary transformations of a fixed, fiducial vector. While Gaussian vectors are commonly used as fiducial vectors, they cannot be used for all systems. We focus on choosing fiducial vectors for several systems including bosons, fermions and anyons.

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