Unbounded Local Observables in Quantum Statistical Mechanics
- 1 June 1970
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 11 (6), 1868-1884
- https://doi.org/10.1063/1.1665338
Abstract
The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*‐algebra of quasilocal bounded observables in Fock space , an arbitrary, locally normal state φ on , and a corresponding Gel'fand‐Naimark‐Segal (GNS) representation Rφ of in a Hilbert space . We then construct a set of local, closed operators whose domains are dense in , such that includes all the local observables of the system. The representation Rφ is then extended so as to provide a *‐homomorphism of into the closed, densely defined operators in . Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo‐Martin‐Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space‐correlation functions, and the temporally ergodic properties of time‐correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of are thermodynamical limits of the corresponding correlations for finite systems.
Keywords
This publication has 11 references indexed in Scilit:
- Asymptotically abelian systemsCommunications in Mathematical Physics, 1967
- On the equilibrium states in quantum statistical mechanicsCommunications in Mathematical Physics, 1967
- Range of forces and broken symmetries in many-body systemsCommunications in Mathematical Physics, 1967
- A theorem on canonical commutation and anticommutation relationsCommunications in Mathematical Physics, 1966
- Conserved currents and associated symmetries; Goldstone's theoremCommunications in Mathematical Physics, 1966
- Invariant states in statistical mechanicsCommunications in Mathematical Physics, 1966
- States of physical systemsCommunications in Mathematical Physics, 1966
- Covariance algebras in field theory and statistical mechanicsCommunications in Mathematical Physics, 1966
- Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of SuperconductorsReviews of Modern Physics, 1962
- Bose-Einstein Condensation and Liquid HeliumPhysical Review B, 1956