## Free states of the gauge invariant canonical anticommutation relations

HTML articles powered by AMS MathViewer

- by B. M. Baker PDF
- Trans. Amer. Math. Soc.
**237**(1978), 35-61 Request permission

## Abstract:

The gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is viewed as an inductive limit of finitedimensional ${C^\ast }$-algebras, and a study is made of a simple class of its representations. In particular, representations induced by restricting the wellknown gauge invariant generalized free states from the entire canonical anticommutation relations (henceforth CAR) are considered. Denoting (a) a state of the CAR by $\omega$ and its restriction to the GICAR by ${\omega ^ \circ }$, (b) the unique gauge invariant generalized free state of the CAR such that $\omega (a{(f)^\ast }a(g)) = (f,Ag)$ by ${\omega _A}$, it is shown that $(1)\;\omega _A^ \circ$ induces (an impure) factor representation of the GICAR if and only if ${\text {Tr}}\;A(I - A) = \infty$, (2) two (impure) GICAR factor representations $\omega _A^ \circ$ and $\omega _B^\circ$ are quasi-equivalent if and only if ${A^{1/2}} - {B^{1/2}}$ and ${(I - A)^{1/2}} - {(I - B)^{1/2}}$ are Hilbert-Schmidt class operators.## References

- Huzihiro Araki,
*On quasifree states of $\textrm {CAR}$ and Bogoliubov automorphisms*, Publ. Res. Inst. Math. Sci.**6**(1970/71), 385โ442. MR**0295702**, DOI 10.2977/prims/1195193913 - Huzihiro Araki and Walter Wyss,
*Representations of canonical anticommutation relations*, Helv. Phys. Acta**37**(1964), 136โ159. MR**171521** - B. M. Baker,
*Free states of the gauge invariant canonical anticommutation relations. II*, Trans. Amer. Math. Soc.**254**(1979), 135โ155. MR**539911**, DOI 10.1090/S0002-9947-1979-0539911-9 - Ola Bratteli,
*Inductive limits of finite dimensional $C^{\ast }$-algebras*, Trans. Amer. Math. Soc.**171**(1972), 195โ234. MR**312282**, DOI 10.1090/S0002-9947-1972-0312282-2 - E. Balslev, J. Manuceau, and A. Verbeure,
*Representations of anticommutation relations and Bogolioubov transformations*, Comm. Math. Phys.**8**(1968), 315โ326. MR**253646** - James G. Glimm,
*On a certain class of operator algebras*, Trans. Amer. Math. Soc.**95**(1960), 318โ340. MR**112057**, DOI 10.1090/S0002-9947-1960-0112057-5 - Sergio Doplicher, Rudolf Haag, and John E. Roberts,
*Fields, observables and gauge transformations. I*, Comm. Math. Phys.**13**(1969), 1โ23. MR**258394** - Sergio Doplicher, Rudolf Haag, and John E. Roberts,
*Fields, observables and gauge transformations. II*, Comm. Math. Phys.**15**(1969), 173โ200. MR**260294** - Rudolf Haag and Daniel Kastler,
*An algebraic approach to quantum field theory*, J. Mathematical Phys.**5**(1964), 848โ861. MR**165864**, DOI 10.1063/1.1704187
R. T. Powers, Thesis, Princeton Univ., 1967.
- Robert T. Powers,
*Representations of uniformly hyperfinite algebras and their associated von Neumann rings*, Ann. of Math. (2)**86**(1967), 138โ171. MR**218905**, DOI 10.2307/1970364 - Robert T. Powers and Erling Stรธrmer,
*Free states of the canonical anticommutation relations*, Comm. Math. Phys.**16**(1970), 1โ33. MR**269230** - David Shale and W. Forrest Stinespring,
*States of the Clifford algebra*, Ann. of Math. (2)**80**(1964), 365โ381. MR**165880**, DOI 10.2307/1970397
G. Stamatopoulos, Thesis, Univ. of Pennsylvania, 1974.

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**237**(1978), 35-61 - MSC: Primary 46L60; Secondary 81E05
- DOI: https://doi.org/10.1090/S0002-9947-1978-0479361-6
- MathSciNet review: 479361