 #1
 353
 167
Homework Statement:
 I want to check the parity invariance of QED
Relevant Equations:

$$\mathscr{L}=\bar{\psi}\left(i\!\!\not{\!\partial}m\right)\psi  \frac{1}{4}F^{\mu\nu}F_{\mu\nu}  J^\mu A_\mu$$
$$\psi(x) \rightarrow \eta^*\gamma^0\psi(\mathscr{P}x)$$
$$A^\mu(x) \rightarrow \eta^* \mathscr{P}^{\mu}{}_{\nu}A^\nu(\mathscr{P}x)$$
Hi, I'm trying to check that the QED Lagrangian
$$\mathscr{L}=\bar{\psi}\left(i\!\!\not{\!\partial}m\right)\psi  \frac{1}{4}F^{\mu\nu}F_{\mu\nu}  J^\mu A_\mu$$
is parity invariant, I'm using the general transformations under parity given by
$$\psi(x) \rightarrow \eta^*\gamma^0\psi(\mathscr{P}x), \qquad A^\mu(x) \rightarrow \eta^* \mathscr{P}^{\mu}{}_{\nu}A^\nu(\mathscr{P}x)$$
where ##\mathscr{P}## is the Parity Lorentz transformation ##\mathscr{P}=\text{diag}(1,1,1,1)##.
I have proved that
$$\bar{\psi}(x)\psi(x)\rightarrow \bar{\psi}(\mathscr{P}x)\psi(\mathscr{P}x), \qquad \bar{\psi}(x)\gamma^\mu\psi(x)\rightarrow \bar{\psi}(\mathscr{P}x)\mathscr{P}^{\mu}{}_{\nu}\gamma^\nu\psi(\mathscr{P}x)$$
independently of ##\eta_\psi##. But when I try to use that to check the invariance of ##\bar{\psi}\!\!\not{\!\partial}\psi## I find with a problem, I find
$$\bar{\psi}(x)\!\!\not{\!\partial}\psi(x) \rightarrow \bar{\psi}(\mathscr{P}x)\gamma^\mu \frac{\partial}{\partial x^\mu}\psi(\mathscr{P}x)$$
but, shouldn't be ##\bar{\psi}(\mathscr{P}x)\gamma^\mu \frac{\partial}{\partial (\mathscr{P}x)^\mu}\psi(\mathscr{P}x)##?
On the other hand, the term ##J^\mu A_\mu## seems to be invariant only if ##\eta_\gamma = 1## (which I know it's the case), but how can I prove that this is true?
$$\mathscr{L}=\bar{\psi}\left(i\!\!\not{\!\partial}m\right)\psi  \frac{1}{4}F^{\mu\nu}F_{\mu\nu}  J^\mu A_\mu$$
is parity invariant, I'm using the general transformations under parity given by
$$\psi(x) \rightarrow \eta^*\gamma^0\psi(\mathscr{P}x), \qquad A^\mu(x) \rightarrow \eta^* \mathscr{P}^{\mu}{}_{\nu}A^\nu(\mathscr{P}x)$$
where ##\mathscr{P}## is the Parity Lorentz transformation ##\mathscr{P}=\text{diag}(1,1,1,1)##.
I have proved that
$$\bar{\psi}(x)\psi(x)\rightarrow \bar{\psi}(\mathscr{P}x)\psi(\mathscr{P}x), \qquad \bar{\psi}(x)\gamma^\mu\psi(x)\rightarrow \bar{\psi}(\mathscr{P}x)\mathscr{P}^{\mu}{}_{\nu}\gamma^\nu\psi(\mathscr{P}x)$$
independently of ##\eta_\psi##. But when I try to use that to check the invariance of ##\bar{\psi}\!\!\not{\!\partial}\psi## I find with a problem, I find
$$\bar{\psi}(x)\!\!\not{\!\partial}\psi(x) \rightarrow \bar{\psi}(\mathscr{P}x)\gamma^\mu \frac{\partial}{\partial x^\mu}\psi(\mathscr{P}x)$$
but, shouldn't be ##\bar{\psi}(\mathscr{P}x)\gamma^\mu \frac{\partial}{\partial (\mathscr{P}x)^\mu}\psi(\mathscr{P}x)##?
On the other hand, the term ##J^\mu A_\mu## seems to be invariant only if ##\eta_\gamma = 1## (which I know it's the case), but how can I prove that this is true?