4 Mathematical Description of Infinitely Extended Systems

Franco Strocchi

Franco Strocchi

From the discussion of the previous Chapter, it appears that the description of infinite systems looks much more difficult than in the finite-dimensional case, above all because of the existence of (too) many possible representations of the algebra of canonical variables.

Federico Zalamea

Alok Sinha

Jean-Pierre Antoine

Forty years ago, Haag and Kastler [27] introduced the algebraic approach to Quantum Field Theory, and soon their method was extended to statistical mechanics. The rationale was to free the theory from particular realizations, linked to specific models, and to focus on the basic structure, namely, the algebra of observables and states on it. This more abstract language, exploiting the deep mathematical theories of C*-algebras and von Neumann algebras, soon became the standard approach to the mathematically rigorous description of physical systems with infinitely many degrees of freedom. Textbooks flourished and, among many authors, Gérard Emch argued forcefully for the new approach [6].