Algebraic quantum mechanics

Algebraic quantum mechanics is an abstraction and generalization of the Hilbert space formulation of quantum mechanics due to von Neumann [5]. In fact, von Neumann himself played a major role in developing the algebraic approach. Firstly, his joint paper [3] with Jordan and Wigner was one of the first attempts to go beyond Hilbert space (though it is now mainly of historical value). Secondly, he founded the mathematical theory of operator algebras in a magnificent series of papers [4, 6]. Although his own attempts to apply this theory to quantum mechanics were unsuccessful [18], the operator algebras that he introduced (which are now aptly called von Neumann algebras) still play a central role in the algebraic approach to quantum theory. Another class of operator algebras, now called C∗-algebras, introduced by Gelfand and Naimark [1], is of similar importance in algebraic quantum mechanics and quantum field theory. Authoritative references for the theory of C∗-algebras and von Neumann algebras are [14] and [21]. Major contributions to algebraic quantum theory were also made by Segal [7, 8] and Haag and his collaborators [2, 13].