# Lorentz transform of Levi-Civita Symbol

I was reading about Lorentz transformations and frequently I hear the notion of Lorentz transforming quantities like $$\epsilon^{\mu \nu \rho \sigma}$$. But I have never heard an explanation as to why that is even something that should be done.

My problem with this notion is that even $$\epsilon^{\mu \nu \rho \sigma}$$ has indices that suggest the transformation behaviour is not an observable at all. Rather just defined as being either 1,-1 or 0 in certain cases. The transformation behaviour of 4 vectors like $$x^{\mu}$$ or $$A^{\mu}$$ is a physical necessity because it corresponds to how those quantities will look like if you measure them in either frame.

But the Levi Civita tensor is not something that is measured, it's just a mathematical object. And it doesn't make sense to me that a purely abstract non physical object should at all be subject to change under Lorentz transformation which are a set of rule for transforming physical observations from one frame to another.

Just the explanation of: "Look it has indices!" doesn't cut it for me. Of course the Levi Civita Symbol is invariant anyway for certain Lorentz transformations and differs at most by a sign but that's beside the point.

Any thoughts on this?

What's "actually going on" is that if you have some tensor object $$A_{\alpha \beta \gamma \delta}$$, it transforms in the tensor representation $$\left(\frac{1}{2}, \frac{1}{2}\right) \otimes \left(\frac{1}{2}, \frac{1}{2}\right) \otimes \left(\frac{1}{2}, \frac{1}{2}\right) \otimes \left(\frac{1}{2}, \frac{1}{2}\right) = 4(0,0) + 6(1,0) + 6(0,1) + 9(1,1) + 2(0,2) + 2(2,0) + 3(1,2) + 3(2,1) + (2,2)$$ of the Lorentz group, which is $$SU(2)_L \otimes SU(2)_R$$. But, $$\frac{1}{2} \otimes \frac{1}{2} = 1 \oplus 0$$, so there is a singlet of the Lorentz group hiding in this tensor representation. You see that we have four singlets, three of these dimensions are spanned by contractions like: $${A^{\alpha \beta}}_{\alpha \beta}, {{{A^{\alpha}}_\alpha}^\beta}_\beta, {{A^{\alpha\beta}}_{\beta\alpha}}$$
The $$\epsilon^{\alpha \beta \gamma \delta}$$ is what pulls out the last singlet, so that you end up with a scalar. To me, its a bit of abuse of notation, I'm not sure its meaningful to think of the indices on the epsilon tensor as "actually being" lorentz indices.
• The Levi-Cita symbol $\epsilon^{ijkl}$ has only $\pm 1$ as its components. The Levi-Civita tensor $\varepsilon^{ijkl}= \epsilon^{ijkl}/\sqrt{-g}$ has different components in different frames because the $\sqrt{-g}$ that is necessary for it to transform as a tensor changes as you change frame. Jun 5, 2022 at 20:17