The quantum theory of second class constraints: Kinematics (Q582543)

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.