The existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form aA+bB (a,b∈R) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of f(aA+bB) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions f:R→C. In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.

An operational construction of the sum of two non-commuting observables in quantum theory and related constructions / Drago, Nicolo; Mazzucchi, Sonia; Moretti, Valter. - In: LETTERS IN MATHEMATICAL PHYSICS. - ISSN 0377-9017. - 2020:110(2020), pp. 3197-3242. [10.1007/s11005-020-01332-7]

An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

Drago, Nicolo;Mazzucchi, Sonia;Moretti, Valter
2020-01-01

Abstract

The existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form aA+bB (a,b∈R) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of f(aA+bB) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions f:R→C. In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.
2020
110
Drago, Nicolo; Mazzucchi, Sonia; Moretti, Valter
An operational construction of the sum of two non-commuting observables in quantum theory and related constructions / Drago, Nicolo; Mazzucchi, Sonia; Moretti, Valter. - In: LETTERS IN MATHEMATICAL PHYSICS. - ISSN 0377-9017. - 2020:110(2020), pp. 3197-3242. [10.1007/s11005-020-01332-7]
File in questo prodotto:
File Dimensione Formato  
Drago2020_Article_AnOperationalConstructionOfThe.pdf

accesso aperto

Tipologia: Versione editoriale (Publisher’s layout)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 647.77 kB
Formato Adobe PDF
647.77 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/273637
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact