At the time I learned quantum field theory it was considered a folk
theo rem that it is easy to construct field theories fulfilling
either the locality or the spectrum condition. The construction of
an example for the latter case is particularly easy. Take for
instance an irreducible representation of the Poincare group with
positive energy, and as an algebra of observables all compact
operators in that representation space. This algebra of observables
is even an asymptotically Abelian algebra. Since it has only a
single repre sentation - except for multiples of this one - it is
hardly possible to replace locality in order to obtain a theory
with a reasonable physical structure. This example shows that it is
not sufficient to replace locality by asymptotic Abelian-ness. The
construction of a theory fulfilling locality without a pos itive
energy representation was first done by Doplicher, Regge, and
Singer [DRS]. However, modern investigations on the locality ideal
in the algebra oftest functions, started by Alcantara and Yngvason
[AY], seem to indicate that this is a general feature; this means
that most of the algebras of ob servables fulfilling the locality
condition will not have representations that also fulfil the
spectrum condition. This discussion shows that quantum field theory
becomes a subject of interest only if both conditions are satisfied
at the same time.
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