© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Observables, maximal symmetric operators, POV measures and their applications in quantum mechanics

Robert H. Fountain

Published Jan 1, 1996• 0 citations

Abstract

Approximate observables, their description in terms of POV measures and \nthe subsequent extension of the orthodox notion of observable are examined. \nThe sense in which the approximate observables are non-ideal observables \nis considered and a generalised concept of ideal observable is proposed. The \nconventional requirement of self-adjointness is relaxed to one of maximal \nsymmetry and the PV measures are replaced by the larger set of generalised \nspectral measures of maximal symmetric operators. The implications for \nthe quantisation problem are discussed and we give some new examples of \nobservables which have no orthodox representation, i.e. observables which \ncan be identified with non-self-adjoint maximal symmetric operators. Particular \nattention is paid to those aspects of the orthodox theory which rely \non specific properties of self-adjoint operators, such as their association with \nthe infinitesimal generators of groups of unitary operators, and we look at \nthe analogous properties in our generalised theory.

Related papers

Observables, maximal symmetric operators and POV measures in quantum mechanics

K. Kong Wan, Robert H. Fountain, Z Y Tao

There has been a lot of interest in generalizing orthodox quantum mechanics to include POV measures as observables, namely as unsharp observables. Such POV measures are related to symmetric operators. The present paper examines how symmetric operators arise and how they can represent observables.

Journal of Physics A Mathematical and General•Apr 21, 1995•9 citations

Observables in Quantum Mechanics and the Importance of Self-Adjointness

Tajron Jurić

We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.

arXiv:2103.01080•Universe•Feb 17, 2022•15 citations

ON NON-SELF-ADJOINT OPERATORS FOR OBSERVABLES IN QUANTUM MECHANICS AND QUANTUM FIELD THEORY

Erasmo Recami, V. S. Olkhovsky, Sergei P. Maydanyuk

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.

arXiv:0903.3187•International Journal of Modern Physics A•Apr 10, 2010•14 citations

Non-self-adjoint operators as observables in quantum theory and nuclear physics

V. S. Olkhovsky, Sergei P. Maydanyuk, Erasmo Recami

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.

Physics of Particles and Nuclei•Jul 1, 2010•6 citations

Complete sets of non-self-adjoint observables: An unbounded approach

Giuseppina Epifanio, Todor S. Todorov, Camillo Trapanı

The notion of completeness of a set 𝒮 of compatible observables represented by maximal symmetric operators is discussed directly in terms of unbounded operators. In contrast with what happens for self-adjoint observables, the present framework forces us to involve some partial algebraic structures such as the partial GW*-algebra generated by 𝒮. In this way the previous approaches based on von Neumann algebras and on O*-algebras are generalized.

Journal of Mathematical Physics•Mar 1, 1996•2 citations

On non-self-adjoint operators for observables in Quantum Mechanics and Quantum Electrodynamics

Erasmo Recami, V. S. Olkhovsky, Sergei P. Maydanyuk

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.

International Journal of Modern Physics A•Jan 1, 2010•0 citations

Approximate observables, their description in terms of POV measures and \nthe subsequent extension of the orthodox notion of observable are examined. \nThe sense in which the approximate observables are non-ideal observables \nis considered and a generalised concept of ideal observable is proposed. The \nconventional requirement of self-adjointness is relaxed to one of maximal \nsymmetry and the PV measures are replaced by the larger set of generalised \nspectral measures of maximal symmetric operators. The implications for \nthe quantisation problem are discussed and we give some new examples of \nobservables which have no orthodox representation, i.e. observables which \ncan be identified with non-self-adjoint maximal symmetric operators. Particular \nattention is paid to those aspects of the orthodox theory which rely \non specific properties of self-adjoint operators, such as their association with \nthe infinitesimal generators of groups of unitary operators, and we look at \nthe analogous properties in our generalised theory.

We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.

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.

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.

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.