An abstract derivation of the inequality related to Heisenberg uncertainty principle
type: Item
instance of: scholarly article
title: An abstract derivation of the inequality related to Heisenberg uncertainty principle
zbMATH Open document ID: 0588.46045
DOI: 10.1016/0034-4877(85)90065-5
published in: Reports on Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: Heisenberg's uncertainty principle concerns a reciprocal relationship between the dispersions of certain incompatible magnitudes A and B as the statistical state varies between a dispersion-free state for A and a dispersion-free state for B. It is well-known that this principle can be derived in the Hilbert space formalism of quantum mechanics from some properties of inner products and operators. In the present paper it is shown that an abstract form of Heisenberg's principle can be derived in a more general mathematical framework without involving the Hilbert space. Namely, if m(A,\(\phi)\) denotes the statistical second moment of the observable A and we have \(m([A,B],\phi)=-m([A,-B],\phi)\) for all A, B and \(\phi\), then \(| m([A,B],\phi)|^ 2\leq m(A,\phi)m(B,\phi)\). If, moreover, there is a canonical pair of observables A, B such that \(m([A,B],[\phi,-\psi])=- m([A,B],[\phi,\psi])\) for all \(\phi\), \(\psi\), then the classical uncertainty principle follows. Here for a real function f on a vector space X f([x,y]) denotes \((f(x+y)-f(x)-f(y))\) for all x,y\(\in X\).
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 81S05
zbMATH DE Number: 3944569
zbMATH Keywords: Heisenberg's uncertainty principle
zbMATH Keywords: dispersions
zbMATH Keywords: statistical state
zbMATH Keywords: dispersion-free state
zbMATH Keywords: statistical second moment
author: Maciej Mączyński
MaRDI profile type: Publication
full work available at URL: 0034-4877(85)90065-5
OpenAlex ID: W2053001360
cites work: Q4739908
cites work: Algebraic conditions for a function on an Abelian group
cites work: On inner products in linear, metric spaces
cites work: Q5723020
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description: scientific article; zbMATH DE number 3944569