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

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 $\mathbf{C}$, $\mathbf{P}$, $\mathbf{T}$ and $\mathbf{CPT}$ 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 $\mathbf{C}$ and $\mathbf{T}$ invariant while the $S$-matrix violates $\mathbf{P}$ and $\mathbf{CPT}$. In non-abelian gauge theories with matter fields such as the electro-weak and $QCD$ sectors of the standard model of particle physics, $\mathbf{C}$, $\mathbf{P}$, $\mathbf{T}$ and the product of any pair of them are broken while $\mathbf{CPT}$ 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 ${\theta }^{0i}\ne 0$, it contributes to breaking also $\mathbf{P}$ and $\mathbf{CPT}$. 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 ${\theta }^{0i}{P}_i^{\text{in}}=0$, but not otherwise. $\mathbf{P}$ and $\mathbf{CPT}$ are good symmetries of the theory in this special case.

18 pages, 1 figure, revised, 2 references added

References in corpus (12)

• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• A Gravity Theory on Noncommutative Spaces
• New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications
• General Properties of Noncommutative Field Theories
• Spin and Statistics on the Groenewold-Moyal Plane: Pauli-Forbidden Levels and Transitions
• Untwisting Noncommutative R^d and the Equivalence of Quantum Field Theories
• Discrete Symmetries (C,P,T) in Noncommutative Field Theories
• Statistics and UV-IR Mixing with Twisted Poincare Invariance
• Why is CPT fundamental?
• UV-IR Mixing in Non-Commutative Plane
• Unitary Quantum Physics with Time-Space Noncommutativity
• Deformed Kac-Moody and Virasoro Algebras

Cited by in corpus (14)

• Quantum Spacetime Phenomenology
• Wedge-Local Quantum Fields and Noncommutative Minkowski Space
• Twisted Gauge and Gravity Theories on the Groenewold-Moyal Plane
• Cosmic Rays and the Search for a Lorentz Invariance Violation
• Twisted Poincaré Invariant Quantum Field Theories
• Particle phenomenology on noncommutative spacetime
• S-Matrix on the Moyal Plane: Locality versus Lorentz Invariance
• Causality and statistics on the Groenewold-Moyal plane
• Quantum particles in non-commutative space-time: an identity crisis
• Wedge-Local Quantum Fields on a Nonconstant Noncommutative Spacetime
• Thermofield Dynamics for Twisted Poincare-Invariant Field Theories: Wick Theorem and S-matrix
• Lehmann-Symanzik-Zimmermann S-Matrix elements on the Moyal Plane
• Black Holes Thermodynamics in a new kind of Noncommutative Geometry
• Towards an Axiomatic Formulation of Noncommutative Quantum Field Theory. II