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arXiv:1708.03578 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Coordinate representation for non-Hermitian position and momentum operators

F. Bagarello, F. Gargano, S. Spagnolo, S. Triolo

Published Sep 1, 2017• 14 citations
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Abstract

In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q ^ and p ^ similar, in a suitable sense, to the self-adjoint position and momentum operators q ^ 0 and p ^ 0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions , and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q ^ and p ^ , based on the so-called quasi *-algebras .

DOI
10.1098/rspa.2017.0434

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In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q ^ and p ^ similar, in a suitable sense, to the self-adjoint position and momentum operators q ^ 0 and p ^ 0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions , and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q ^ and p ^ , based on the so-called quasi *-algebras .
The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (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-self-adjoint (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.
Aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint 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.
A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.
The aim of this paper is to show the importance and applicability of non-self-adjoint operators for description of observables both in relativistic and nonrelativistic quantum mechanics, as well as in quantum electrodynamics. The paper considers (i) a maximally Hermitian (but non-self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics and (ii) the problem of defining the four-dimensional coordinate and momentum operators describing relativistic particles with zero spin (each of these operators contains both Hermitian and anti-Hermitian parts). Some other applications of non-selfadjoint and non-Hermitian operators in physics are analyzed as well. The paper ends with an analysis of a few cases of T -non-invariant interactions, including quantum dissipation and nuclear optical potentials.
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.