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Georgian Mathematical Journal

Roots of the Phase Operators

G. Jorjadze, I. Sarishvili

Published Jan 1, 1995• 0 citations
Publisher

Abstract

The k th 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 .
DOI
10.1515/gmj.1995.37

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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
To find an operator representation of the phase variable of a single-mode electromagnetic field, the Schrodinger representation is extended to a larger Hilbert space including states with infinite excitation by nonstandard analy- sis, so that the self-adjoint phase operator is acting on the extended Hilbert space.It is shown that this phase operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter already known in the quantum estimation theory, and is naturally considered as the controversial limit of the phase operators on finite dimensional spaces recently proposed by Pegg and Barnett.Eventually, two of recent promising attempts to obtain the correct statistics of the phase variable in quantum mechanics is synthesized in the present framework based on new mathematical ideas concerning the number system including infinite and infinitesimal numbers.
Identical quantum particles exhibit only two types of statistics: bosonic and fermionic. Theoretically, this restriction is commonly established through the symmetrization postulate or (anti)commutation constraints imposed on the algebra of creation and annihilation operators. The physical motivation for these axioms remains poorly understood, leading to various generalizations by modifying the mathematical formalism in somewhat arbitrary ways. In this work, we take an opposing route and classify quantum particle statistics based on operationally well-motivated assumptions. Specifically, we consider that a) the standard (complex) unitary dynamics defines the set of single-particle transformations, and b) phase transformations act locally in the space of multi-particle systems. We develop a complete characterization, which includes bosons and fermions as basic statistics with minimal symmetry. Interestingly, we have discovered whole families of novel statistics (dubbed transtatistics) accompanied by hidden symmetries, generic degeneracy of ground states, and spontaneous symmetry breaking – effects that are (typically) absent in ordinary statistics.