Phase Operator for the Photon, an Index Theorem and Quantum Deformation

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.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.

arXiv:hep-th/9603130To be published in the proceedings ofMar 19, 19960 citations

On the Weyl Correspondence of the Photon Phase Operator

展德会 ZHAN De-hui, 卢道明 Lu Daoming, 范洪义 FAN Hong-yi

Acta Sinica Quantum OpticaJan 1, 20180 citations

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

Kazuo Fujikawa

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.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.

arXiv:hep-th/9509128Sep 23, 19950 citations

Phase operator for the photon field and an index theorem

Kazuo Fujikawa

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 ${\mathit{u}}_{\mathit{n}}$ that satisfy ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ ${\mathit{au}}_{\mathit{n}}$ =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.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 ${\mathit{u}}_{\mathit{n}}$ that satisfy ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{au}}_{\mathit{n}}$=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.

arXiv:hep-th/9411066Physical Review AOct 1, 199533 citations

Phase operator in quantum optics

Masashi Ban

Physics Letters AMay 1, 199326 citations

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.

{\mathit{u}}_{\mathit{n}}

that satisfy ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$

{\mathit{au}}_{\mathit{n}}

=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.