Hi,
I have a 4x4 Hamiltonian describing a part of my system. To get a holistic view of the situation I need to do a charge inversion on the matrix. What is the 4x4 charge inversion operator? And what is the logic behind it (e.g. how is it built up, for example -i (\tau_0 direct product \sigma_y))?
Biswajoy Brahmachari
For Dirac spinors, charge conjugation matrix is defined by,
C-1 γμ C = - γμ T
Now in Dirac-Pauli representation
γ0= ((1,0),(0,-1)) and γi=((0,σi),(-σi,0))
So we get
C= γ2γ0
ψc=ηψ C ψ̅ T
|ηψ|=1
This is the 4x4 matrix form of the charge conjugation matrix, which acts upon 4 component spinors. In the Hamiltonian, however, spinorial indices are all contracted. After all Hamiltonian should be a Lorentz scalar.
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