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

Abstract

The positive and negative energy modes of a field theory in $κ$-Minkowski/$κ$-Poincaré 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 Poincaré 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.

Figures (2)

• Figure 1: (1+1)D depiction of $\kappa$-momentum space. The portion of the hyperboloid covered by the momentum space coordinates is the (yellow) one over the $p_+=p_0+p_4$ (orange) plane and for positive $p_4$ (l.h.s.). The on-shell orbits are the one sectioned by the plane $p_4=\sqrt{m^{2}+\kappa^{2}}$, respectively the blue (positive frequency) and red (negative frequency) curves on the r.h.s.. The blue and red dots depict two particular on-shell modes connected by the antipode map.
• Figure 2: A pictorial description of the action of ${\cal P}$ and ${\cal T}$ in the momentum space of a field.