On non-selfadjoint operators for observables in quantum mechanics and quantum field theory

TL;DR

This paper explores the significance and applications of non-selfadjoint operators in quantum mechanics and quantum field theory, including time operators, space-time operators, and quasi-hermitian Hamiltonians for unstable states.

Contribution

It provides a detailed analysis of non-selfadjoint operators' roles in quantum observables, extending their application to relativistic particles, unstable states, and quantum dissipation.

Findings

Non-selfadjoint time operators can be meaningful in quantum mechanics.

Quasi-hermitian Hamiltonians can describe decaying unstable states.

Applications include quantum dissipation and nuclear optical potentials.

Abstract

Aim of this paper is 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 maximal hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; and with: (ii) 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 physically important applications of non-selfadjoint (and even non-hermitian) operators are discussed: In particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential. [PACS numbers: 03.65.Ta; 03.65.-w; 03.65.Pm; 03.70.+k; 03.65.Xp; 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. Keywords: time operator in quantum mechanics; space-time operator; time-"Hamiltonian"; non-selfadjoint operators; non-hermitian operators; bilinear operators; time operator for discrete energy spectra; time-energy uncertainty relations; unstable states; quasi-hermitian Hamiltonians; Klein-Gordon equation; quantum dissipation; nuclear optical model].