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InTech eBooks

Time as Quantum Observable, Canonical Conjugated to Energy

Vladislav Sergeevich Strongin, Erasmo Recami, P. Sergei

Published Feb 22, 2012• 3 citations
Publisher

Abstract

Time, as well as 3-position, sometimes is a parameter, but sometimes is an observable that in quantum theory would be expected to be associated with an operator. However, almost from the birth of quantum mechanics (cf., e.g., Ref.(Pauli, 1926; 1980)), it is known that time cannot be represented by a selfadjoint operator, except in the case of special systems (such as an electrically charged particle in an infinite uniform electric field)1. The list of papers devoted to the problem of time in quantum mechanics is extremely large (see, for instance, Refs. (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Goźdź A & Dȩbicki, 2007; Grot et al., 1996; Holevo, 1978; 1982; Kijowski, 1997; Kobe et al., 1994; Kocha’nski & Wo’dkievicz, 1999; Leo’n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky, 1998; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Olkhovsky, 2009; 2011; Recami, 1976; 1977; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007), and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (see, for instance, Refs.(Olkhovsky & Recami, 1968; 1969; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)). As to quantum mechanics, the very first relevant articles are probably Refs. (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), and refs. therein. A second set of papers on time in quantum physics (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Goźdź A & Dȩbicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al.,

DOI
10.5772/35264

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Contrary to claims contained in papers by Grot, Rovelli, and Tate [Phys. Rev. A 54, 4676 1996)] and Delgado and Muga [Phys. Rev. A 56, 3425 (1997)], the ``time operator,'' which I have constructed [Rep. Math. Phys. 6, 361 (1974)] in an axiomatic way, is a self-adjoint operator existing in a usual Hilbert space of (nonrelativistic or relativistic) quantum mechanics.

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Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, with the introduction of Time and Space operators; and with (iii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-selfadjoint (and even non-hermitian) operators are briefly discussed. We mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a "chronon". [This chapter is largely based on work developed, along the years, in collaboration with V.S.Olkhovsky, and, in smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk]. PACS numbers: 03.65.Ta; 03.65.-w; 03.65.Pm; 03.70.+k; 03.65.Xp; 03.65.Yz; 11.10.St; 11.10.-z; 11.90.+t; 02.00.00; 03.00.00; 24.10.Ht; 03.65.Yz; 21.60.-u; 11.10.Ef; 03.65.Fd; 02.40.Dr; 98.80.Jk. Keywords: time operator, space-time operator, non-selfadjoint operators, non-hermitian operators, bilinear operators, time operator for discrete energy spectra, time-energy uncertainty relations, Klein-Gordon equation, chronon, quantum dissipation, decoherence, nuclear optical model, cosmology, projective relativity.

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Time, as well as 3-position, sometimes is a parameter, but sometimes is an observable that in quantum theory would be expected to be associated with an operator. However, almost from the birth of quantum mechanics (cf., e.g., Ref.(Pauli, 1926; 1980)), it is known that time cannot be represented by a selfadjoint operator, except in the case of special systems (such as an electrically charged particle in an infinite uniform electric field)1. The list of papers devoted to the problem of time in quantum mechanics is extremely large (see, for instance, Refs. (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Goźdź A & Dȩbicki, 2007; Grot et al., 1996; Holevo, 1978; 1982; Kijowski, 1997; Kobe et al., 1994; Kocha’nski & Wo’dkievicz, 1999; Leo’n, 1997; Muga et al., 1999; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; Olkhovsky & Recami, 1992; Olkhovsky et al., 1995; Olkhovsky & Agresti, 1997; Olkhovsky, 1998; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007; Olkhovsky, 2009; 2011; Recami, 1976; 1977; Srinivas & Vijayalakshmi, 1981; Toller, 1999; Wang & Xiong, 2007), and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (see, for instance, Refs.(Olkhovsky & Recami, 1968; 1969; Olkhovsky et al., 2004; Olkhovsky & Recami, 2007)). As to quantum mechanics, the very first relevant articles are probably Refs. (Holevo, 1978; 1982; Olkhovsky & Recami, 1968; 1969; 1970; Olkhovsky, 1973; Olkhovsky et al., 1974; Olkhovsky, 1984; 1990; 1992; 1998; Recami, 1976; 1977), and refs. therein. A second set of papers on time in quantum physics (Aharonov et al., 1998; Atmanspacher & Amann, 1998; Blanchard P & Jadczyk, 1996; Busch et al., 1994; Delgado, 1999; Egusquiza & Muga, 1999; Giannitrapani, 1997; Goźdź A & Dȩbicki, 2007; Grot et al., 1996; Kijowski, 1997; Kobe et al.,
Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, with the introduction of Time and Space operators; and with (iii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-selfadjoint (and even non-hermitian) operators are briefly discussed. We mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a "chronon". [This chapter is largely based on work developed, along the years, in collaboration with V.S.Olkhovsky, and, in smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk]. PACS numbers: 03.65.Ta; 03.65.-w; 03.65.Pm; 03.70.+k; 03.65.Xp; 03.65.Yz; 11.10.St; 11.10.-z; 11.90.+t; 02.00.00; 03.00.00; 24.10.Ht; 03.65.Yz; 21.60.-u; 11.10.Ef; 03.65.Fd; 02.40.Dr; 98.80.Jk. Keywords: time operator, space-time operator, non-selfadjoint operators, non-hermitian operators, bilinear operators, time operator for discrete energy spectra, time-energy uncertainty relations, Klein-Gordon equation, chronon, quantum dissipation, decoherence, nuclear optical model, cosmology, projective relativity.