Formulas for commutators of functions of non-commuting operators

Jean-Marc Lévy-Leblond

Alexei Daletskii, Alexander Kalyuzhny, Eugene Lytvynov +1 more

Let [Formula: see text] be a separable Hilbert space and [Formula: see text] be a self-adjoint bounded linear operator on [Formula: see text] with norm [Formula: see text], satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator [Formula: see text] determines a [Formula: see text]-deformed Fock space [Formula: see text]. We start with reviewing and extending the known results about the structure of the [Formula: see text]-particle spaces [Formula: see text] and the commutation relations satisfied by the corresponding creation and annihilation operators acting on [Formula: see text]. We then choose [Formula: see text], the [Formula: see text]-space of [Formula: see text]-valued functions on [Formula: see text]. Here [Formula: see text] and [Formula: see text] with [Formula: see text]. Furthermore, we assume that the operator [Formula: see text] acting on [Formula: see text] is given by [Formula: see text]. Here, for a.a. [Formula: see text], [Formula: see text] is a linear operator on [Formula: see text] with norm [Formula: see text] that satisfies [Formula: see text] and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function [Formula: see text] in the case [Formula: see text] determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its [Formula: see text]-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

Nicolò Drago, Sonia Mazzucchi, Valter Moretti

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 A + b B ( $$a,b\in {{\mathbb {R}}}$$ 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 $$f(aA+bB)$$ f ( a A + b B ) 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 $$f: {{\mathbb {R}}}\rightarrow {{\mathbb {C}}}$$ 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.

Mark K. Transtrum, Jean-François S. Van Huele

We derive an expression for the commutator of functions of operators with constant commutations relations in terms of the partial derivatives of these functions. This result extends the well-known commutation relation between one operator and a function of another operator. We discuss the range of applicability of the formula with examples in quantum mechanics.