Citation:
Reviews in Mathematical Physics, 2000, vol. 12, n. 9, p. 1159-1218

ISSN:
0129-055X (Print) 1793-6659 (Online)

DOI:
10.1142/S0129055X00000459

Sponsor:
The authors thank the sfb 288 for support during the visit to University of Potsdam. We also benefitted from an ARC grant which funded a
visit of F.Ll. to the University of New South Wales. Finally, F.Ll. would like to thank Sergio Doplicher for his kind hospitality at the ‘Dipartamento di Matematica dell’Università di Roma "La Sapienza" in March 2000, when the final version of this paper was prepared. The visit was supported by a EU TMR network "Implementation of concept and methods from Non–Commutative Geometry to Operator Algebras and its applications", contract no. ERB FMRX-CT 96-0073.

Keywords:
Quantum system with constraints
,
Algebraic Quantum Field Theory (AQFT)
,
Dirac state
,
Gupta-Bleuler electromagnetism
,
Field C*-algebra
,
Constraint set
,
Maximal C*-algebra of physical observables
,
Weak Haag-Kastler axioms
,
Local quantum constraints

We analyze the situation of a local quantum field theory with constraints, both indexed by the same set of space-time regions. In particular we find "weak" Haag–Kastler axioms which will ensure that the final constrained theory satisfies the usual Haag–KastlerWe analyze the situation of a local quantum field theory with constraints, both indexed by the same set of space-time regions. In particular we find "weak" Haag–Kastler axioms which will ensure that the final constrained theory satisfies the usual Haag–Kastler axioms. Gupta–Bleuler electromagnetism is developed in detail as an example of a theory which satisfies the "weak" Haag–Kastler axioms but not the usual ones. This analysis is done by pure C*-algebraic means without employing any indefinite metric representations, and we obtain the same physical algebra and positive energy representation for it than by the usual means. The price for avoiding the indefinite metric, is the use of nonregular representations and complex valued test functions. We also exhibit the precise connection with the usual indefinite metric representation.[+][-]

We conclude the analysis by comparing the final physical algebra produced by a system of local constrainings with the one obtained from a single global constraining and also consider the issue of reduction by stages. For the usual spectral condition on the genWe conclude the analysis by comparing the final physical algebra produced by a system of local constrainings with the one obtained from a single global constraining and also consider the issue of reduction by stages. For the usual spectral condition on the generators of the translation group, we also find a "weak" version, and show that the Gupta–Bleuler example satisfies it.[+][-]

Description:

60 pages, no figures.-- MSC2000 codes: 46L60, 81T05, 81R15.