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In playing with gamma matrices of the $\mathcal{\mathscr{C}}l_{1,3}(R)$ variety, it's not uncommon to hear allusions to $\gamma^5$ being related to the volume 4-form. To illustrate the similarities:

$$\gamma^{5}=\frac{i}{4!}\sqrt{-\eta}\epsilon_{0123}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$

$$dV=\frac{1}{4!}\sqrt{-\eta}\epsilon_{0123}dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}$$

Where $dV$ is the Minkowskian four-volume form. How exactly can we relate these to one another? I know we can represent an object transforming like a vector as:

$$V^{\mu}=\bar{\psi}\gamma^{\mu}\psi$$ Where \psi has the form of a Dirac spinor. It seems reasonable to suppose we can define some spinor then that acts as a map from the gamma matrix to it's equivalent vector basis:

$$e^{\mu}=\bar{\psi}\gamma^{\mu}\psi$$ considering how gamma matrices transform under Lorentz transformations, namely:

$$\gamma^{\mu}=S^{-1}\gamma^{\mu}S=\Lambda_{\bar{\mu}}^{\mu}\gamma^{\bar{\mu}}$$

Now I know this isn't right, but I'm tempted nonetheless to write the Volume form in terms of the gamma 5 matrix as:

$$dV=\psi^{\dagger}\gamma^{5}\psi d^{4}x$$

I guess what's really bothering me is that when I see the axial current charge before second quantization:

$$Q_{axial}=\intop_{M}\psi^{\dagger}\gamma^{5}\psi d^{4}x$$

I feel like I'm looking at a volume integral (which may be locally deformed by the map $\psi$). In this case I don't feel very suprised that axial current isn't conserved after quantization.

We live in an expanding universe any spacelike volume is going to be nonconserved in time by some quantity related to the time derivative of our metric's scale parameter.

Now I get that quantization breaks conservation of the axial current (a choice made between that and the gauge fields), can we go the other way and say breaking axial current conservation (by letting space expand) quantizes the gauge fields? I honestly don't know.

I guess what I'm asking is what IS the correct way to relate the volume form of a spacetime to the $\gamma^5$ matrix?

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