/ 94 11 06 6 v 2 1 8 M ay 1 99 5 UT - 688 , 1994 Phase Operator for the Photon Field and an Index Theorem Kazuo Fujikawa
نویسنده
چکیده
An index relation dim ker a†a− dim ker aa† = 1 is satisfied by the creation and annihilation operators a† and a of a harmonic oscillator. A hermitian phase operator, which inevitably leads to dim ker a†a − dim ker aa† = 0, cannot be consistently defined. If one considers an s + 1 dimensional truncated theory, a hermitian phase operator of Pegg and Barnett 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 a new class of quantum anomaly. From an anomaly view point ,it is not surprising that the phase operator of Susskind and Glogower, which carries a unit index, leads to an anomalous identity and an anomalous commutator.
منابع مشابه
ar X iv : h ep - t h / 95 06 00 3 v 1 1 J un 1 99 5 UT - 707 , 1995 Analytic Index and Chiral Fermions †
A recent application of an index relation of the form, dim ker M − dim ker M † = ν, to the generation of chiral fermions in a vector-like gauge theory is reviewed. In this scheme the chiral structure arises from a mass term with a non-trivial index.The essence of the generalized Pauli-Villars reg-ularization of chiral gauge theory,which is based on this mechanism,is also clarified.
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