Quantum fields on the Groenewold-Moyal plane: C, P, T and CPT

We show that despite the inherent non-locality of quantum field theories on the Groenewold-Moyal (GM) plane, one can find a class of C, P, T and CPT invariant theories. In particular, these are theories without gauge fields or with just gauge fields and no matter fields. We also show that in the presence of gauge fields, one can have a field theory where the Hamiltonian is C and T invariant while the S-matrix violates P and CPT. In non-abelian gauge theories with matter fields such as the electro-weak and QCD sectors of the standard model of particle physics, C, P, T and the product of any pair of them are broken while CPT remains intact for the case theta(0i) = 0. (Here x(mu) * x(nu) - x(nu) * x(mu) = i theta(mu nu) ,x(mu): coordinate functions, theta(mu nu) = -theta(nu mu) =constant.) When theta(0i)not equal 0, it contributes to breaking also P and CPT. It is known that the S-matrix in a non-abelian theory depends on theta(mu nu) only through theta(0i). The S-matrix is frame dependent. It breaks (the identity component of the) Lorentz group. All the noncommutative effects vanish if the scattering takes place in the center-of-mass frame, or any frame where theta(0i) P-i (in) = 0, but not otherwise. P and CPT are good symmetries of the theory in this special case.