Asymmetry in momentum space: restoring ${\cal C}{\cal P}{\cal T}$ invariance of $\kappa$-field theory

Abstract

The positive and negative energy modes of a field theory in $\kappa$-Minkowski/$\kappa$-Poincar\'e noncommutative spacetime have very different symmetry properties. This can be understood geometrically by considering that they span two distinct sectors of a curved momentum space. By performing an explicit direct computation of the relativistic Noether charges and their algebra within the canonical formalism, we identify a striking consequence of this asymmetry in momentum space: charge conjugation and Poincar\'e invariance are incompatible. We then notice how the structure of momentum space suggests that time reversal could be deformed so that the overall ${\cal C}{\cal P}{\cal T}$-invariance is restored. We prove that this new proposal works by studying the transformation properties under deformed discrete symmetries of the new relativistic charges.