The quantum theory of second class constraints: Kinematics
type: Item
instance of: scholarly article
title: The quantum theory of second class constraints: Kinematics
zbMATH Open document ID: 0691.46044
DOI: 10.1007/BF01218261
author: C. A. Hurst
author: H. Grundling
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: In the \(C^*\)-algebra formulation of quantum mechanics, one open problem has been the rigorous definition of the imposition of constraints, analogous to the treatment of degenerate systems by the Dirac procedure [see \textit{M. J. Gotay}, \textit{J. M. Nester} and \textit{G. Hinds}, J. Math. Phys. 19, 2388-2399 (1978; Zbl 0418.58010)]. This paper describes a method for doing so, and gives some examples. The data of the problem are a unital C*-algebra \({\mathcal F}\), its set of states \({\mathcal S}\), and a set \({\mathcal U}\) of unitary elements of \({\mathcal F}\). The physical structures should be such that \({\mathcal U}\) acts trivially upon them. We say that \({\mathcal U}\) is first class if the C* algebra generated by \(\{\) (U-I)\(| U\in {\mathcal U}\}\) does not contain I. In this case, we select from \({\mathcal S}\) the states which are 0 on that C* algebra, and we factor out of \({\mathcal F}\) the elements which are annihilated by those states. If \({\mathcal U}\) is not first class, we say that it contains second class constraints. The paper defines a canonical subset of \({\mathcal U}\), denoted \({\mathcal U}_{I_ 0}\), which is first class. The ``Dirac states'' are those selected by \({\mathcal U}_{I_ 0}\). Then, let \({\mathcal D}_ 0\) be the algebra of elements which are ``weakly zero'', that is, annihilated by Dirac states; formally, \[ {\mathcal D}_ 0:=[{\mathcal F}C*({\mathcal U}_{I_ 0}-I)]\cap [C*({\mathcal U}_{I_ 0}-I){\mathcal F}]. \] The ``quantum observable algebra'' consists of the elements which are ``weakly gauge invariant''; formally, it is \[ {\mathcal O}:=\{A\in {\mathcal F}| UAU^{- 1}-A\in {\mathcal D}_ 0\quad for\quad all\quad U\in {\mathcal U}\}. \] Finally, the maximal physical algebra is \[ {\mathcal R}:={\mathcal O}/({\mathcal O}\cap {\mathcal D}_ 0). \] Examples include a linear boson field with linear Hermitian constraints and a linear Boson field with quadratic constraints.
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 81Q99
zbMATH DE Number: 4131048
zbMATH Keywords: \(C^*\)-algebra formulation of quantum mechanics
zbMATH Keywords: imposition of constraints
zbMATH Keywords: first class
zbMATH Keywords: second class constraints
zbMATH Keywords: Dirac states
zbMATH Keywords: quantum observable algebra
zbMATH Keywords: weakly gauge invariant
zbMATH Keywords: linear boson field with linear Hermitian constraints
zbMATH Keywords: linear Boson field with quadratic constraints
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4131048