Dirac Spinor Transformation (Ryder)

[physstudent.4]

Homework Statement

This complies when I type it in my Latex editor, but not on here. If you could either let me know how to fix that or copy and paste what I have into your own editor to help, that'd be great. Thanks!

While Ryder is setting up to derive a transformation rule for Dirac spinors, I have failed to follow one step that is crucial for a subsequent derivation for an infinitesimal rotation. He has (with t representing conjugate transpose):

[ tex ]\bar{\psi} \gamma \psi=( \phi_R^t \phi_L^t ) \left( \begin{array}{ccc}
0 & -\sigma \\
\sigma & 0 \end{array} \right) \left( \begin{array}{ccc}
\phi_R \\
\phi_L
\end{array} \right) [ /tex ]

But I was under the impression the adjoint spinor was represented as
[ tex ]
\bar{\psi}=\psi^t \gamma^0=( \phi_L^t \phi_R^t )
[ /tex ]
since
[ tex ]
\gamma^0=\left( \begin{array}{ccc}
0 & 1 \\
1 & 0 \end{array} \right)
[ /tex ]
If this is the case, the proceeding derivation does not follow.

Homework Equations

Above

The Attempt at a Solution

Above the above

 

[TSny]

Homework Statement

This complies when I type it in my Latex editor, but not on here. If you could either let me know how to fix that or copy and paste what I have into your own editor to help, that'd be great. Thanks!

While Ryder is setting up to derive a transformation rule for Dirac spinors, I have failed to follow one step that is crucial for a subsequent derivation for an infinitesimal rotation. He has (with t representing conjugate transpose):

$$\bar{\psi} \gamma \psi=( \phi_R^t \phi_L^t ) \left( \begin{array}{ccc} 0 & -\sigma \\ \sigma & 0 \end{array} \right) \left( \begin{array}{ccc} \phi_R \\ \phi_L \end{array} \right)$$

But I was under the impression the adjoint spinor was represented as
$$\bar{\psi}=\psi^t \gamma^0=( \phi_L^t \phi_R^t )$$
since
$$\gamma^0=\left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right)$$
If this is the case, the proceeding derivation does not follow.

Homework Equations

Above

The Attempt at a Solution

Above the above

Just leave out the spaces in [ tex ] and [ /tex ]

I think you are right that Ryder has not written $\bar{\psi} \vec{\gamma} \psi$ correctly. However, if you continue his derivation of how the expression transforms under an infinitesimal spatial rotation using your correct expression, you will still be led to the same final result.