Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory |
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Authors: | Hendrik Grundling |
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Institution: | (1) Department of Mathematics, Research School of Physical Sciences, Australian National University, Canberra, Australia |
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Abstract: | Since there are some important systems which have constraints not contained in their field algebras, we develop here in aC*-context the algebraic structures of these. The constraints are defined as a groupG acting as outer automorphisms on the field algebra , :G Aut ,
G
Inn , and we find that the selection ofG-invariant states on is the same as the selection of states onM(G
) by (U
g)=1 g G, whereU
g M (G
)/ are the canonical elements implementing
g
. These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics inM(G
), and in particular the maximal constraint free physical algebra . A nontriviality condition is given for to exist, and we extend the notion of a crossed product to deal with a situation whereG is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next theC*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory. |
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Keywords: | |
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