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arXiv:1112.4198

"Time operator": the challenge persists

Bogdan Mielnik, Gabino Torres-Vega

Published Dec 18, 2011• 12 citations
arXivHTMLPDF

Abstract

Contrary to the conviction expressed by J. Kijowski [Phys. Rev. A 59, 897 (1999)] and shared in some other papers, the reasons to look for the 'time operator' in the context of the standard quantum doctrine of orthogonal projectors and self-adjoint observables are highly questionable. Some improved solutions in terms of POV measures invite critical discussions as well.

DOI
10.48550/arxiv.1112.4198

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Contrary to the conviction expressed by J. Kijowski [Phys. Rev. A 59, 897 (1999)] and shared in some other papers, the reasons to look for the 'time operator' in the context of the standard quantum doctrine of orthogonal projectors and self-adjoint observables are highly questionable. Some improved solutions in terms of POV measures invite critical discussions as well.
Approximate observables, their description in terms of POV measures and \nthe subsequent extension of the orthodox notion of observable are examined. \nThe sense in which the approximate observables are non-ideal observables \nis considered and a generalised concept of ideal observable is proposed. The \nconventional requirement of self-adjointness is relaxed to one of maximal \nsymmetry and the PV measures are replaced by the larger set of generalised \nspectral measures of maximal symmetric operators. The implications for \nthe quantisation problem are discussed and we give some new examples of \nobservables which have no orthodox representation, i.e. observables which \ncan be identified with non-self-adjoint maximal symmetric operators. Particular \nattention is paid to those aspects of the orthodox theory which rely \non specific properties of self-adjoint operators, such as their association with \nthe infinitesimal generators of groups of unitary operators, and we look at \nthe analogous properties in our generalised theory.
Abstract It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a D{\mathcal{D}}D D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of EEE E -connection for observables. Several examples are discussed.