Editing CPT symmetry (section)

==Derivation of the CPT theorem==

Consider a [[Lorentz boost]] in a fixed direction ''z''. This can be interpreted as a rotation of the time axis into the ''z'' axis, with an [[imaginary number|imaginary]] rotation parameter. If this rotation parameter were [[real number|real]], it would be possible for a 180° rotation to reverse the direction of time and of ''z''. Reversing the direction of one axis is a reflection of space in any number of dimensions. If space has 3 dimensions, it is equivalent to reflecting all the coordinates, because an additional rotation of 180° in the ''x-y'' plane could be included.

This defines a CPT transformation if we adopt the [[Antiparticle#Feynman–Stueckelberg interpretation|Feynman–Stueckelberg interpretation]] of antiparticles as the corresponding particles traveling backwards in time. This interpretation requires a slight [[analytic continuation]], which is well-defined only under the following assumptions:
#The theory is [[Lorentz invariant]];
#The vacuum is Lorentz invariant;
#The energy is bounded below.

When the above hold, [[quantum field theory|quantum theory]] can be extended to a Euclidean theory, defined by translating all the operators to imaginary time using the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. The [[commutation relation]]s of the Hamiltonian, and the [[Poincaré group|Lorentz generator]]s, guarantee that [[Lorentz invariance]] implies [[rotational invariance]], so that any state can be rotated by 180 degrees.

Since a sequence of two CPT reflections is equivalent to a 360-degree rotation, [[fermion]]s change by a sign under two CPT reflections, while [[boson]]s do not. This fact can be used to prove the [[spin-statistics theorem]].