ABSTRACT
We have discussed rotations and Lorentz transformations in an earlier chapter. These are continuous spacetime symmetries shown by classical and quantum field theories. They give rise to conserved charges with a continuous spectrum of values. But there are also a few discrete symmetries, some associated with spacetime, which are seen in many systems: parity or space inversion and time reversal, as also charge conjugation, which has nothing to do with spacetime. Parity corresponds to ~x→ −~x, while time reversal corresponds to x0 → −x0. One considers the behavior of the action under field transformations
φ(x0, ~x)→ φ(x0,−~x) (7.1) and
φ(x0, ~x)→ φ(−x0, ~x). (7.2) It is clear that the free scalar field theory action is invariant under these transformations. One can also insert ±1 in the right sides of these equations without spoiling the invariance. The sign becomes important in interacting theories. In the case of parity, the factor ±1 is called the intrinsic parity of the field. The +1 factor corresponds to a scalar and the −1 factor to a pseudoscalar, which changes sign under a reflection.
