Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Free states of the gauge invariant canonical anticommutation relations. II


Author: B. M. Baker
Journal: Trans. Amer. Math. Soc. 254 (1979), 135-155
MSC: Primary 81D05; Secondary 46L10
DOI: https://doi.org/10.1090/S0002-9947-1979-0539911-9
MathSciNet review: 539911
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A class of representations of the gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is studied. These representations are induced by restricting the well-known pure, nongauge invariant generalized free states of the canonical anticommutation relations (henceforth CAR). Denoting a state of the CAR by $ \omega $, and the unique generalized free state of the CAR such that $ \omega \left( {a{{\left( f \right)}^{\ast}}a\left( g \right)} \right)\, = \,\left( {f,Tg} \right)$ and $ \omega \left( {a\left( f \right)a\left( g \right)} \right)\, = \,\left( {Sf,g} \right)$ by $ {\omega _{S,T}}$, it is shown that a pure, nongauge invariant state $ {\omega _{S,T}}$ induces a factor representation of the GICAR if and only if $ Tr\,T\left( {I - T} \right)\, = \,\infty $.


References [Enhancements On Off] (What's this?)

  • [1] H. Araki, On quasifree states of CAR and Bogoliubov automorphisms, Publ. Res. Inst. Math. Sci. Ser. A 6 (1970), 385-442. MR 0295702 (45:4768)
  • [2] B. M. Baker, Free states of the gauge invariant canonical anticommutation relations, Trans. Amer. Math. Soc. 237 (1978), 35-61. MR 479361 (80b:46081)
  • [3] E. Balslev, J. Manuceau and A. Verbeure, Representations of anticommutation relations and Bogoliubov transformations, Comm. Math. Phys. 8 (1968), 315-326. MR 0253646 (40:6860)
  • [4] O. Bratteli, Inductive limits of finite dimensional $ {C^{\ast}}$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195-234. MR 0312282 (47:844)
  • [5] J. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318-340. MR 0112057 (22:2915)
  • [6] R. Haag, The mathematical structure of the Bardeen-Cooper-Schrieffer model, Nuovo Cimento 25 (1962), 287-299. MR 0145921 (26:3449)
  • [7] R. T. Powers, Thesis, Princeton University, Princeton, N. J., 1967.
  • [8] -, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. 86 (1967), 138-171. MR 0218905 (36:1989)
  • [9] R. T. Powers and E. Stormer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 1-33. MR 0269230 (42:4126)
  • [10] G. Stamatopoulos, Thesis, University of Pennsylvania, Philadelphia, Pa., 1974.
  • [11] A. Van Daele and A. Verbeure, Unitary equivalence of Fock representations on the Weyl algebra, Comm. Math. Phys. 20 (1971), 268-278. MR 0286406 (44:3619)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 81D05, 46L10

Retrieve articles in all journals with MSC: 81D05, 46L10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0539911-9
Keywords: Anticommutation relations, gauge invariance, approximately finite $ {C^{\ast}}$-algebra, generalized free states factor representations
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society