# Symmetries II: Discrete spacetime symmetries

This chapter shows that the combination of these three discrete operations — parity, time reversal, and charge conjugation — is an exact symmetry of any local relativistic quantum field theory, even when none of the individual operations represents an exact symmetry of the theory. This result — commonly referred to as the TCP theorem — is first obtained for the case of theories with a dynamics specified by a Lagrangian (or Hamiltonian) density; and secondly, in a more general framework, by appealing to the irreducible fundamental properties of local field theory as incorporated in the Wightman axioms of Section 9.2. As a natural concomitant of the axiomatic proof of the TCP theorem, the chapter also indicates how the Spin-Statistics connection arises from the spectral and locality axioms of the Wightman formulation of field theory, with no commitment needed to a specific Lagrangian/Hamiltonian dynamics.

*Keywords:*
parity, time reversal, charge conjugation, symmetry, relativistic quantum field theory, TCP theorem, Spin-Statistics theorem

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