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Journal of Mathematics

COMMUTATOR IN ABSTRACT HEISENBERG'S INEQUALITY

Tian Gu-ji

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

In abstract Heisenberg's inequality (see~[2]),the commutator [A,B] may have too small adomain and be not closed. By changing the usual definition of commutator into that in a kind of weak sense,this paper gives a sufficient condition,which is more convenient than and completely different from that in [4],for some class of self-adjoint operators to avoid these shortcomings.

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•Nov 27, 2012•0 citations
In abstract Heisenberg's inequality (see~[2]),the commutator [A,B] may have too small adomain and be not closed. By changing the usual definition of commutator into that in a kind of weak sense,this paper gives a sufficient condition,which is more convenient than and completely different from that in [4],for some class of self-adjoint operators to avoid these shortcomings.
【The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space (H, B) with an abstract Wiener measure p 1_11​ . For ϕ\phiϕ ∈\in∈ L 2^22 (p 1_11​ ) and T ∈\in∈ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) ϕ\phiϕ is the Fourier-Wiener transform of ϕ\phiϕ . The conditions when the equality holds also discussed.】
Abstract In this paper we discuss some aspects of the Heisenberg uncertainty relation, mostly from the point of view of non self-adjoint operators. Some equivalence results, and some refinements of the inequality, are deduced, and some relevant examples are discussed. We also begin a sort of dynamical analysis of the relation, in connection with what has been recently called γ-dynamics and γ-symmetries, and we discuss in some details the role of different scalar products in our analysis. The case of self-adjoint operators is recovered as a special case of our general settings.
Abstract Real Hilbert space is a Euclidean space of infinite dimensionality. Complex Hilbert space is less amenable to elementary geometrical intuitions, but these do serve to give structure to the present argument. The familiar process of finding, relative to a vector x, the expectation Ā of a self-adjoint linear operator A, and then using the deviation A — Ā to obtain a dispersion ΔA, may be interpreted as a projection leading to a unit vector e orthogonal to x. If we have n self-adjoint operators Ar, we thus form n unit vectors er orthogonal to x. From a basic property of Hilbert space, the vector c1e1 + c2e2 + ... + cnen has a non-negative norm for arbitrary complex coefficients. This demands the positive-indefinite character of a real quadratic form in 2n variables; equivalently, all its eigenvalues must be non-negative. This leads to inequalities on the dispersions ΔAr, the expectations of the commutators i[Ar, As] and other expectations. For n = 2, this process leads to Schrödinger’s improvement of the Heisenberg inequality. The case n = 3 is here worked out explicitly.
Abstract. In [1] we defined symmetry classes, commutation symmetries and symmetry operators in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigated them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In the present paper we consider symmetry operators a ∈ C[SN] on H which fulfil a ∗ = a, where a denotes the complex conjugate of a and the element a ∗ ∈ C[SN] of a = P p app is defined by a ∗ = P p app−1. Such symmetry operators are self-adjoint because of the relation 〈au|v 〉 = 〈u|a ∗ v 〉 which holds true for all a ∈ C[SN] and u, v ∈ H. They yield consequently observables of the quantum mechanical Heisenberg model. We prove the following results: