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Journal of Physics A Mathematical and General

New class of function-dependent deformations of creation and annihilation operators

Małgorzata Klimek

Published Jan 7, 1992• 2 citations
Publisher

Abstract

The author gives formulae for point spectra of Hermitian operators built from creation and annihilation type operators satisfying general algebraic relation which he calls (g, f, F) equivalence. By choosing the parameters in a specific way the description of the quantum q-oscillator is given in a model built from the q-derivative and dilatation operator. He shows the explicit form of eigenfunctions of the model from L 2 (R). Then he constructs the transformation between his model and the Q-oscillator with arbitrary Q real.

DOI
10.1088/0305-4470/25/1/002

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The author gives formulae for point spectra of Hermitian operators built from creation and annihilation type operators satisfying general algebraic relation which he calls (g, f, F) equivalence. By choosing the parameters in a specific way the description of the quantum q-oscillator is given in a model built from the q-derivative and dilatation operator. He shows the explicit form of eigenfunctions of the model from L 2 (R). Then he constructs the transformation between his model and the Q-oscillator with arbitrary Q real.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + -qa + a = 1) are studied when q > 1.These operators are symmetric but not self-adjoint.They have a one-parameter family of selfadjoint extensions.These extensions are derived explicitly.Their spectra and eigenfunctions are given.Spectra of different extensions do not intersect.The results show that the creation and annihilation operators a + and a of the q-oscillator for q > 1 cannot determine a physical system without further more precise definition.In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.
A displacement operator d_ζis introduced, verifying commutation relations [d_ζ, a_f^\dagger]=[d_ζ, a_f]=ζ(f)d_ζwith field creation and annihilation operators that verify [a_f,a_g]=0, [a_f,a_g^\dagger]=(g,f), as usual. f and g are test functions, ζis a Poincare invariant real-valued function on the test function space, and (g,f) is a Poincare invariant Hermitian inner product. The *-algebra generated by all these operators, and a state defined on it, nontrivially extends the *-algebra of creation and annihilation operators and its Fock space representation. If the usual requirement for linearity is weakened, as suggested in quant-ph/0512190, we obtain a deformation of the free quantum field.
The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.