Asymmetry in momentum space: Restoring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="script">C</mml:mi> <mml:mi mathvariant="script">P</mml:mi> <mml:mi mathvariant="script">T</mml:mi> </mml:math> inva

The positive and negative energy modes of a field theory in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>κ</a:mi> </a:math> -Minkowski/ <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>κ</c:mi> </c:math> -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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi mathvariant="script">C</e:mi> <e:mi mathvariant="script">P</e:mi> <e:mi mathvariant="script">T</e:mi> </e:math> -invariance is restored. We prove that this new proposal works by studying the transformation properties under deformed discrete symmetries of the new relativistic charges.