On a Time–Space Operator (and other Non-Self-Adjoint Operators) for Observables in QM and QFT

Abstract

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, the introduction of Time and Space operators; and with (iii) the problem of the four-position and fourmomentum 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 briefly 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 paper 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.

Conjugate variables in quantum field theory and a refinement of Pauli’s theorem

“…(cp. (10). The definition of Q nc and the commutation relation (159) hold on the function space described before for Q and P and likewise they have the same domain of Hermiticity.…”

“…It turns out that due to the presence of the polarization vectors this commutator does not vanish. When searching for non-commutative coordinates one may thus rely on preconjugate pairs [28], one may introduce Θ's like in (10) or one can employ the operators Q (eff ) , (45). We hope to come back to this question in the near future.…”

“…It has been admitted as a Hermitian but not self-adjoint operator. A wealth of further literature has been provided in [10]. On the more abstract level time operators are understood as positiveoperator-valued measures [11][12][13][14], or affiliated to C * -algebras [15].…”

Time of arrival operator in the momentum space

“…Despite all these difficulties, several authors [22,23,24,25,26,27] have attempted to construct the time operator within non-relativistic and relativistic quantum mechanics. However, none of these papers listed provides a construction of the time operator that satisfies the conditions required from this object in this work.…”