On Non-Self-Adjoint Operators for Observables in Quantum Mechanics and Quantum Field Theory

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.

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

“…[10][11][12][13] and [24][25][26][27][28]38] the operatort (in the t-representation) had the property that any averages over time, in the one-dimensional (1D) scalar case, were to be obtained by use of the following measure (or weight):…”

“…In Refs. [24,25,26,27,38], there were defined the mean time durations for the particle 1D transmission from x i to x f > x i , and reflection from the region (x i , +∞) back to the interval x f ≤ x i . Namely…”

“…. ., and of f (t) , quantity f (t) being any analytical function of time, can be found in [27,41,38], where single-valued expressions have been explicitly written down.…”

“…denotes the average over t with the measures W (x, t) dt or W ± (x, t) dt in the t-representation) can be derived also for operators which are simply hermitian, by a straightforward generalization of the procedures which are common in the case of selfadjoint (canonically conjugate) quantities, like coordinatex and momentum p x . Moreover, relation (8) satisfies [27,41,38] the Dirac "correspondence" principle, since the classical Poisson brackets {q 0 , p 0 }, with q 0 = t and p 0 = −E, are equal to 1. In Refs.…”

Time operator in QFT with Virasoro constraints

“…For example, the non-self-adjoint (but hermitian, or better maximal hermitian) operator proposed in Ref. [1,2,[4][5][6][7][8][9], as an operator for the observable time in quantum mechanics, is fully acceptable, mathematically and physically, without 3 having to modify or split the ordinary Hilbert space.…”