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arXiv:hep-th/9603130 · To be published in the proceedings of

Phase operator problem and an index theorem for Q-deformed oscillator

Kazuo Fujikawa

Published Mar 19, 1996• 0 citations
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Abstract

The notion of index is applied to analyze the phase operator problem associated with the photon. We clarify the absence of the hermitian phase operator on the basis of an index consideration. We point out an interesting analogy between the phase operator problem and the chiral anomaly in gauge theory, and an appearance of a new class of quantum anomaly is noted. The notion of index, which is invariant under a wide class of continuous deformation, is also shown to be useful to characterize the representations of Q-deformed oscillator algebra.

DOI
10.48550/arxiv.hep-th/9603130

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•Jan 1, 1996•22 citations
The notion of index is applied to analyze the phase operator problem associated with the photon. We clarify the absence of the hermitian phase operator on the basis of an index consideration. We point out an interesting analogy between the phase operator problem and the chiral anomaly in gauge theory, and an appearance of a new class of quantum anomaly is noted. The notion of index, which is invariant under a wide class of continuous deformation, is also shown to be useful to characterize the representations of Q-deformed oscillator algebra.
An index relation dim ker a−dim ker a†=1dim\ ker\ a - dim\ ker\ a^{\dagger} = 1dim ker a−dim ker a†=1 is satisfied by the creation and annihilation operators a†a^{\dagger}a† and aaa of a harmonic oscillator. Implications of this analytic index on the possible form of the phase operator are discussed. A close analogy between the present phase operator problem and chiral anomaly in gauge theory, which is associated with Atiyah-Singer index theorem, is emphasized.
An index relation dim ker ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a-dim ker ${\mathrm{aa}}^{\ifmmode^\circ\else\textdegree\fi{}}$=1 is satisfied by the creation and annihilation operators ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ and a of a harmonic oscillator, where dim ker ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a, for example, stands for the number of normalizable basis vectors un{\mathit{u}}_{\mathit{n}}un​ that satisfy ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$aun{\mathit{au}}_{\mathit{n}}aun​=0. A Hermitian phase operator, which inevitably leads to dim ker ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a-dim ker ${\mathrm{aa}}^{\ifmmode^\circ\else\textdegree\fi{}}$=0, cannot be consistently defined. If one considers an (s+1)-dimensional truncated theory, a Hermitian phase operator of Pegg and Barnett [Europhys. Lett. 6, 483 (1988)] which carries a vanishing index can be defined. However, for arbitrarily large s, we show that the vanishing index of the Hermitian phase operator of Pegg and Barnett causes a substantial deviation from minimum uncertainty in a characteristically quantum domain with small average photon numbers. We also mention an interesting analogy between the present problem and the chiral anomaly in gauge theory which is related to the Atiyah-Singer index theorem. It is suggested that the phase operator problem related to the above analytic index may be regarded as an alternative class of quantum anomaly. From an anomaly viewpoint, it is not surprising that the phase operator of Susskind and Glogower [Physics 1, 49 (1964)], which carries a unit index, leads to an anomalous identity and an anomalous commutator.
The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.
The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite-dimensional cyclic representations. As a realistic physical system of a GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the Hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg - Barnett-type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit .
This paper addresses a construction of new q-Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q-deformed boson system is studied.
The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q → 1.