3.3 Discrete symmetries: Parity, Time reversal, and Charge conjugation

Discrete symmetries like parity, time reversal, and charge conjugation are crucial in quantum field theory. They shape how particles and fields behave, leading to conservation laws and constraints on interactions. Understanding these symmetries is key to building consistent theories.

Violations of these symmetries, especially CP violation, have big implications. They help explain the matter-antimatter imbalance in the universe and guide research into physics beyond the Standard Model. These symmetries connect the microscopic world to cosmic mysteries.

Discrete Symmetries of Parity, Time Reversal, and Charge Conjugation

Definitions and Properties

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• Parity (P) is a spatial reflection that inverts the sign of spatial coordinates, effectively creating a mirror image of the system
• Time reversal (T) reverses the direction of time, causing a system to evolve backwards in time
• Charge conjugation (C) transforms a particle into its antiparticle, reversing the sign of all internal quantum numbers (electric charge, baryon number, lepton number) while leaving mass, momentum, and spin unchanged
• Discrete symmetries provide insight into the fundamental properties of particles and their interactions in quantum field theory

Importance in Quantum Field Theory

• Discrete symmetries constrain the possible forms of Lagrangians and dictate the allowed interactions between particles and fields
• Invariance under discrete symmetries leads to conservation laws (parity conservation, time-reversal invariance, charge conjugation symmetry)
• Violations of discrete symmetries (P, CP) have been observed experimentally and provide crucial tests of the Standard Model and theories beyond it
• Understanding discrete symmetries is essential for constructing consistent and physically meaningful quantum field theories

Transformation Properties of Fields

Scalar Fields

• Scalar fields are invariant under parity and time reversal transformations
• Scalar fields may transform under charge conjugation depending on their charge (neutral scalar fields are invariant, while charged scalar fields acquire a phase factor)
• Examples of scalar fields include the Higgs field and the pion field

Vector Fields

• Vector fields (electromagnetic field) change sign under parity and time reversal but are invariant under charge conjugation
• The electric field ($\vec{E}$) is a polar vector, while the magnetic field ($\vec{B}$) is an axial vector, leading to different transformation properties
• The photon, the quantum of the electromagnetic field, is its own antiparticle and is invariant under charge conjugation

Spinor Fields

• Spinor fields (fermions) transform non-trivially under parity, time reversal, and charge conjugation, with their components mixing and acquiring phase factors
• Left-handed and right-handed spinor components transform differently under parity, leading to parity violation in weak interactions
• Dirac spinors (electrons, quarks) transform into their charge-conjugated counterparts under charge conjugation, while Majorana spinors are invariant
• The transformation properties of spinor fields are crucial for understanding the chiral nature of weak interactions and the existence of neutrino masses

CPT Theorem and its Construction

Statement and Consequences

• The CPT theorem states that any Lorentz-invariant, local quantum field theory must be invariant under the combined action of charge conjugation (C), parity (P), and time reversal (T) transformations
• CPT invariance implies that particles and their corresponding antiparticles have equal masses and lifetimes but opposite charges and magnetic moments
• The theorem is a consequence of the fundamental assumptions of quantum field theory (Lorentz invariance, locality, unitarity)
• CPT invariance holds regardless of the order in which C, P, and T transformations are applied, as they commute with each other

Proof and Mathematical Formulation

• The proof of the CPT theorem relies on the properties of the Lorentz group and the structure of quantum field theory
• The theorem can be derived using the Wightman axioms, which formalize the assumptions of quantum field theory in terms of vacuum expectation values of field operators
• The CPT transformation is represented by an anti-unitary operator $\Theta$ that acts on the Hilbert space of states and satisfies $\Theta^2 = 1$
• The action of $\Theta$ on field operators is determined by their transformation properties under C, P, and T, ensuring the invariance of the theory

Implications of CPT Invariance in Quantum Field Theory

Constraints on Particle Properties and Interactions

• CPT invariance requires that particles and antiparticles have identical masses, lifetimes, and decay widths, as observed experimentally with high precision
• Any violation of CPT invariance would imply a breakdown of one or more fundamental assumptions of quantum field theory (Lorentz invariance, locality)
• CPT invariance restricts the possible forms of Lagrangians and determines the allowed interactions between particles and fields, ensuring the consistency of the theory
• Tests of CPT invariance (comparing particle and antiparticle properties, searching for CPT-violating processes) provide stringent constraints on extensions of the Standard Model and theories beyond quantum field theory

Relation to Matter-Antimatter Asymmetry

• The observed matter-antimatter asymmetry in the universe cannot be explained by CPT invariance alone, as it predicts equal amounts of matter and antimatter
• Additional mechanisms, such as CP violation or baryon number violation, are required to account for the dominance of matter over antimatter
• CP violation has been observed in the weak interactions of quarks (kaon and B meson systems) and is a necessary condition for baryogenesis
• The search for CP violation in the lepton sector (neutrino oscillations) and the study of baryon number violating processes (proton decay) are active areas of research in particle physics