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arXiv:hep-th/9509128

Phase operator for the photon, an index theorem, and quantum anomaly

Kazuo Fujikawa

Published Sep 23, 1995• 0 citations
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Abstract

An index relation $dim\ ker\ a - dim\ ker\ a^{\dagger} = 1$ is satisfied by the creation and annihilation operators $a^{\dagger}$ and $a$ 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.

DOI
10.48550/arxiv.hep-th/9509128

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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 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.
Abstract. The kth root taken of the bosonic phase operator is considered. This leads to the extension of the Hilbert space. In the case k = 2 an ordinary fermionic extension arises. Particles whose statistics depends on k are introduced for other values of k. 1. Phase operators The problem of polar decomposition of the creation and annihilation operators (a ∗ , a) of a harmonic oscillator goes back to Dirac [1]. In this section some aspects of this problem will be reviewed [2]. Let h be the oscillator hamiltonian h = (p 2 + q 2)/2 (1.1) on the phase plane Γ with coordinates p, q and Poisson brackets (PB) If the polar angle ϕ is introduced by then for the complex variables a and a ∗ {q, p} = 1. (1.2) p = √ 2h sin ϕ, q = √ 2h cos ϕ, (1.3) a = q + ip
We introduce the inverse of the harmonic-oscillator creation and annihilation operators by their actions on the number states. We then show that three of the two-photon annihilation operators, a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^, a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$, and a^ 2^{2}2, possess eigenstates that are analogous to the often-used coherent and squeezed states. A family of the eigenstates of the operator a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^ is the customary squeezed vacuum S(\ensuremath{\sigma})\ensuremath{\Vert}0〉 while another family of the eigenstates of a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$ is the state S(\ensuremath{\sigma})\ensuremath{\Vert}n=1〉. We find some interesting properties related to squeezing, bunching, and antibunching exhibited by these states. An alternative method of summing some series by using these states is also considered. It is hoped that these states will find applications in quantum optics and quantum mechanics in general.
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A general definition of ``cosine'' and ``sine'' operators, C and S, for harmonic oscillator phase is proposed and its consequences examined. An important feature of the spectral analysis is the ``chain sequence'' condition which ensures that C and S have unit norm. The (nonunitary) operator U = C + iS is shown to be an annihilation-type operator whose spectral properties bear a remarkable analogy to those of the standard annihilation operator, although its spectrum fills the unit circle rather than the entire complex plane. Statistical properties of the eigenstates of U are discussed briefly.