Eur. Phys. J. C 20, 767-772 (2001)
DOI: 10.1007/s100520100664
Space-time noncommutativity, discreteness of time and unitarity
M. Chaichian1, 2, A. Demichev1, 2, P. Presnajder1, 2 and A. Tureanu1, 21 High Energy Physics Division, Department of Physics, University of Helsinki, Helsinki, Finland
2 Helsinki Institute of Physics, P.O. Box 64, 00014 Helsinki, Finland
(Received: 19 March 2001 / Published online: 29 June 2001 -© Springer-Verlag / Società Italiana di Fisica 2001 )
Abstract
Violation of unitarity for noncommutative field theory on compact
space-times
is considered. Although such theories are
free of ultraviolet divergences they
still violate unitarity, while in a usual field theory such a
violation occurs
when the theory is nonrenormalizable. The
compactness of space-like coordinates
implies discreteness of the time variable which leads to
the appearance of
unphysical modes and violation of unitarity even in the absence
of a
star-product in the interaction terms. Thus, this conclusion
holds also for
other quantum field theories with discrete time. Violation of
causality, among
others, occurs also in the case of
the nonvanishing of the commutation relations between
observables at space-like distances with a typical scale of
noncommutativity.
While this feature allows for a possible violation of the
spin-statistics
theorem, such a violation does not rescue the situation but makes
causality violation scale as the inverse of the mass appearing
in the considered
model, i.e., it becomes even more severe. We also stress
the role of smearing over the
noncommutative coordinates entering the field operator symbols.
© Società Italiana di Fisica, Springer-Verlag 2001
