Next: Combined PCT Symmetry Up: Dirac Equation Previous: Charge Conjugation

# Time Reversal

Consider time-reversal invariance, . This time we start with the Dirac equation in Hamiltonian form

 (5.253)

Define the transformation such that and , we have

 (5.254)

Since is generated by currents which reverse sign when the sense of time is reversed,

 (5.255) (5.256)

Also , since . The transformation must cause to get the correct form, therefore can be defined as:

1. take complex conjugate,
2. multiply by constant matrix .

 (5.257)

Therefore

 (5.258)

This implies must commute with and and anticommute with and . Therefore we can try

 (5.259)

The phase factor is arbitrary.

We apply to a plane-wave solution for a free particle of positive energy. Since

 (5.260)

and

 (5.261)

where ; Therefore projects out a free-particle solution with reversed direction of momentum and spin . This is known as Wigner time reversal''.

Next: Combined PCT Symmetry Up: Dirac Equation Previous: Charge Conjugation
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18