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Journal of Mathematical Physics

Definition of Commutators via the Uncertainty Principle

Bryce Seligman DeWitt

Published Jul 1, 1962• 32 citations
Publisher

Abstract

A general demonstration is given of the fact that the quantization of any system, as expressed by the uncertainty principle, implies the quantization of all other systems to which it can be coupled. It is also shown that the precise form which the uncertainty principle takes is uniquely specified. That is to say, the uncertainty principle effectively defines the commutator of any pair of observables. The argument serves to detach completely from its canonical origins a previously given definition of the commutator (which is a generalization of one given by Peierls) applicable to systems with infinite-dimensional invariance groups, for which the identification of canonical variables is an inconvenience. The measurements of a single observable and of a pair of observables are analyzed according to the Bohr-Rosenfeld scheme.

DOI
10.1063/1.1724265

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A general demonstration is given of the fact that the quantization of any system, as expressed by the uncertainty principle, implies the quantization of all other systems to which it can be coupled. It is also shown that the precise form which the uncertainty principle takes is uniquely specified. That is to say, the uncertainty principle effectively defines the commutator of any pair of observables. The argument serves to detach completely from its canonical origins a previously given definition of the commutator (which is a generalization of one given by Peierls) applicable to systems with infinite-dimensional invariance groups, for which the identification of canonical variables is an inconvenience. The measurements of a single observable and of a pair of observables are analyzed according to the Bohr-Rosenfeld scheme.
Given two or more non-commuting observables, it is generally not possible to simultaneously assign precise values to each. This quantum mechanical uncertainty principle is widely understood to be encapsulated by some form of uncertainty relation, expressing a trade-off between the standard deviations or other measures of uncertainty of two (or more) observables, resulting from their non-commutativity. Typically, such relations are coarse, and miss important features. It was not until very recently that a broader perspective on quantum uncertainty was envisaged and explored, one that utilises the notion of an uncertainty region. Here we review this new approach, illustrating it with pairs or triples of observables in the case of qubit and qutrit systems. We recall some of the shortcomings of traditional uncertainty relations, and highlight their inability to identify the full uncertainty region. These shortcomings suggest a precautionary note that, we surmise, ought to accompany the presentation of the uncertainty principle in introductory quantum mechanics courses.
We demonstrate two points: 1) the formalism of quantum mechanics can be understood simply as a structure for the expression of the physical notion that not all observables can have values simultaneously; 2) the specific uncertainty relations can be derived (rigorously) by combination of the invariance principle with a general uncertainty relation based only on the existence of unspecified pairs of conjugate observables. For this purpose, we present a formulation of quantum mechanics based strictly on the invariance principle and a “weak” statement of the uncertainty principle that asserts only the existence of incompatible (conjugate) observables without specifying which observables are incompatible. We go on to show that the invariance principle can be used to develop the equations of motion of the theory, including the Klein-Gordon and Schrodinger equations.
Quantum uncertainty lies at the heart of quantum mechanics, fundamentally distinguishing it from classical physics. Traditionally quantified through variance-based measures, these approaches often rely on the expectation values of observables, providing a macroscopic statistical view rather than an intrinsic characterization of the wavefunction's inherent indeterminacy. This paper introduces a novel commutator-induced metric designed to directly quantify quantum uncertainty within the wavefunction itself, independent of specific measurement outcomes or a pair of conjugate variables. By leveraging the fundamental non-commutativity of quantum operators, this metric provides a more fundamental and intrinsic measure of the delocalization and inherent indeterminacy encoded in the quantum state. We define this metric based on the expectation value of an operator derived from the self-commutator of the wavefunction's defining operators, mapping the non-commutativity inherent in the Hilbert space onto a scalar value that reflects the degree of quantum uncertainty. This approach offers a new perspective on interpreting the uncertainty principle, providing a generalized framework that extends beyond canonical conjugate pairs and potentially offering insights into the structure of quantum information. The paper details the theoretical construction of this metric, explores its mathematical properties, and discusses its potential applications in quantum information theory, quantum measurement, and the foundational understanding of quantum mechanics.
Abstract In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables A^\hat{A}A^ and B^\hat{B}B^, with eigenvalues ana_nan​ and bmb_mbm​, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable Q2Q_2Q2​ conditioned on the first {\em probe} measurement yielding Q1Q_1Q1​ we obtain information on the statistical distribution of the {\em system} variable bmb_mbm​ conditioned on having found the system variable ana_nan​ in the interval δaδaδa around a(n)a^{(n)}a(n). The width of this statistical distribution as function of δaδaδa constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.