- #1
facenian
- 436
- 25
On page 25 of his book "Electrodynamics and classical theory of fields and particles" he presents this identity
[tex]\sigma_\mu\sigma_\nu-\frac{i}{2}\epsilon_{\mu\nu\beta\alpha}\sigma^\beta\sigma^\alpha=\delta_{\mu\nu}[/tex]
where [itex]\sigma^\mu:(\mathbf{I},-\mathbf{\sigma})[/itex] and [itex]\sigma_\mu:(\mathbf{I},\mathbf{\sigma})[/itex] , [itex]\sigma=(\sigma_1,\sigma_2,\sigma_3)\,and\,\sigma_i[/itex] are the Pauli matrices, [itex]\sigma_0=\mathbf{I}[/itex]
It seems that this can't be right because if we put [itex]\mu=1\,\sigma=2[/itex] then we have [itex]\sigma_1\sigma_2=0[/itex] while the correct result is [itex]\sigma_1\sigma_2=i\sigma_3[/itex]
[tex]\sigma_\mu\sigma_\nu-\frac{i}{2}\epsilon_{\mu\nu\beta\alpha}\sigma^\beta\sigma^\alpha=\delta_{\mu\nu}[/tex]
where [itex]\sigma^\mu:(\mathbf{I},-\mathbf{\sigma})[/itex] and [itex]\sigma_\mu:(\mathbf{I},\mathbf{\sigma})[/itex] , [itex]\sigma=(\sigma_1,\sigma_2,\sigma_3)\,and\,\sigma_i[/itex] are the Pauli matrices, [itex]\sigma_0=\mathbf{I}[/itex]
It seems that this can't be right because if we put [itex]\mu=1\,\sigma=2[/itex] then we have [itex]\sigma_1\sigma_2=0[/itex] while the correct result is [itex]\sigma_1\sigma_2=i\sigma_3[/itex]