In a minimalistic view, the use of noncommutative coordinates can be seen
just as a way to better express non-local interactions of a special kind:
1-particle solutions (wavefunctions) of the equation of motion in the presence
of an external field may look simpler as functions of noncommutative
coordinates. It turns out that also the wave-mechanical description of a system
of n such bosons/fermions and its second quantization is simplified if we
translate them in terms of their deformed counterparts. The latter are obtained
by a general twist-induced *-deformation procedure which deforms in a
coordinated way not just the spacetime algebra, but the larger algebra
generated by any number n of copies of the spacetime coordinates and by the
particle creation and annihilation operators. On the deformed algebra the
action of the original spacetime transformations looks twisted. In a
non-conservative view, we thus obtain a twisted covariant framework for QFT on
the corresponding noncommutative spacetime consistent with quantum mechanical
axioms and Bose-Fermi statistics. One distinguishing feature is that the field
commutation relations remain of the type "field (anti)commutator=a
distribution". We illustrate the results by choosing as examples interacting
non-relativistic and free relativistic QFT on Moyal space(time)s.Comment: Latex file 16 pages. Talk given at the conference "Noncommutative
Structures in Mathematics and Physics" (Satellite Conference to the 5th
European Congress of Mathematics), Brussels 22-26/7/2008. Appeared in the
Proceedings, Ed. S. Caenepeel, J. Fuchs, S. Gutt, C. Schweigert, A. Stolin,
F. Van Oystaeyen, Royal Flemish Academy of Belgium for Sciences and Arts,
brussels, 2010, pp. 163-17
