Quantum Fields on the Groenewold-Moyal Plane: C, P, T and CPT

TL;DR

This paper investigates the symmetry properties of quantum field theories on the Groenewold-Moyal plane, revealing conditions under which ${f C}$, ${f P}$, ${f T}$, and ${f CPT}$ are preserved or violated, especially in gauge theories and the standard model.

Contribution

It identifies classes of noncommutative quantum field theories that maintain certain discrete symmetries and analyzes how these symmetries are broken in non-abelian gauge theories with non-zero $ heta^{0i}$.

Findings

${f C}$, ${f P}$, ${f T}$, and ${f CPT}$ invariance depends on the type of fields and noncommutativity parameters.

In non-abelian gauge theories, ${f C}$, ${f P}$, ${f T}$, and their products are broken when $ heta^{0i} eq 0$.

The $S$-matrix depends on $ heta^{ ext{0i}}$ and is frame-dependent, with effects vanishing in specific frames.

Abstract

We show that despite the inherent non-locality of quantum field theories on the Groenewold-Moyal (GM) plane, one can find a class of ${\bf C}$, ${\bf P}$, ${\bf T}$ and ${\bf CPT}$ 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 ${\bf C}$ and ${\bf T}$ invariant while the $S$-matrix violates ${\bf P}$ and ${\bf CPT}$. In non-abelian gauge theories with matter fields such as the electro-weak and $QCD$ sectors of the standard model of particle physics, ${\bf C}$, ${\bf P}$, ${\bf T}$ and the product of any pair of them are broken while ${\bf CPT}$ remains intact for the case $\theta^{0i} =0$. (Here $x^{\mu} \star x^{\nu} - x^{\nu} \star x^{\mu} = i \theta^{\mu \nu}$, $x^{\mu}$: coordinate functions, $\theta^{\mu \nu} = -\theta^{\nu \mu}=$ constant.) When $\theta^{0i} \neq 0$, it contributes to breaking also ${\bf P}$ and ${\bf CPT}$. 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 $\theta^{0i}P^{\textrm{in}}_{i} = 0$, but not otherwise. ${\bf P}$ and ${\bf CPT}$ are good symmetries of the theory in this special case.