Can a wavefunction be written as an active transformation of the metric tensor?

Question

Let's examine the Dirac Lagrangian of quantum mechanics:

$$\bar{\psi}\left(i\gamma^{\mu}\overrightarrow{\partial}_{\mu}-mc^{2}\right)\psi=0$$

Where the arrow implies direction of action for the derivative. The barred wavefunction can be written as:

$$\bar{\psi}=\psi^{\dagger}\gamma^{0}$$

Where the dagger denotes the conjugate transpose. Then we have:

$$\psi^{\dagger}\gamma^{0}\left(i\gamma^{\mu}\overrightarrow{\partial}_{\mu}-mc^{2}\right)\psi=0$$

It is very common in discussions of conservecd current to also examine the Adjoint Dirac equation:

$$\psi^{\dagger}\gamma^{0}\left(i\gamma^{\mu}\overleftarrow{\partial}_{\mu}+mc^{2}\right)\psi=0$$

Where the derivative is now acting to the left. Adding both of these together we obtain a conservation of current equation:

$$\partial_{\mu}\left(\psi^{\dagger}\gamma^{0}\gamma^{\mu}\psi\right)=0$$

This is all standard, my question here is can we define another, equivalent Dirac Lagrangian of the form:

$$\psi^{\dagger}\left(i\gamma^{\mu}\overrightarrow{\partial}_{\mu}-mc^{2}\right)\psi^{,}=0$$

Where:

$$\psi^{,}=\gamma^{0}\psi$$

Which together with it's adjoint equation:

$$\psi^{\dagger}\left(i\gamma^{\mu}\overleftarrow{\partial}_{\mu}+mc^{2}\right)\gamma^{0}\psi=0$$

$$\psi^{\dagger}\left(i\gamma^{\mu}\overrightarrow{\partial}_{\mu}-mc^{2}\right)\gamma^{0}\psi=0$$ also defines a conserved current

$$\partial_{\mu}\left(\psi^{\dagger}\gamma^{\mu}\gamma^{0}\psi\right)=0$$ Now if we combine all four of our expressions we obtain:

$$\partial_{\mu}\left(\psi^{\dagger}\gamma^{0}\gamma^{\mu}\psi\right)+\partial_{\mu}\left(\psi^{\dagger}\gamma^{\mu}\gamma^{0}\psi\right)=0$$ or, perhaps more familiar:

$$\partial_{\mu}\left(\psi^{\dagger}g^{0\mu}\psi\right)=0$$

Which is just the condition that:

$$\partial_{\mu}\left(\psi^{\dagger}Ig^{\nu\mu}\psi\right)e_{0}=0$$

Where I is the identity $4x4$ matrix. From this standpoint, $\psi$ seems to simply be playing the role of a coordinate transformation, or more properly an active transformation, changing the physical properties of the metric.

Note that the current is an integrand, and one can apply Gauss's law here to obtain a second order equation. So here's my question again:

Are we changing anything physically by making the switch:

$$\psi\Longrightarrow\gamma^{0}\psi$$

instead of:

$$\psi^{\dagger}\Longrightarrow\psi^{\dagger}\gamma^{0}$$

In the Dirac Lagrangian?

Answer 1

First of all, as Triatticus commented, a Lagrangian is not equal to zero in general. Dirac Lagrangian is zero only conditional on the equation of motion is satisfied. It happens for A Lagrangian that only involves bilinears.

Secondly, your equation $$ \psi^{\dagger}\left(i\gamma^{\mu}\overrightarrow{\partial}_{\mu}-mc^{2}\right)\psi^{,}=0 $$ is wrong, since $\gamma^0$ and $\gamma^\mu$ do NOT commute.

Thirdly, the conservation rule as you derived $$ \partial_{\mu}\left(\psi^{\dagger}\gamma^{0}\gamma^{\mu}\psi\right)=0 $$ is based purely on classical equation of motion (Dirac equation). Such conservation laws may NOT survive in a full quantum theory. The axial current in ABJ anomaly is a perfect example.