EAN: 9783662140789

TRANSLATION GROUP AND PARTICLE REPRESENTATIONS ...

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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