Levi-Civita Symbol and contravariance vs covariance

Question

I have a question regarding the Levi-Civita symbol and contravariance vs covariance. Some of this was asked in a previous post, but I think I need more clarification.

Consider the magnetic field: \begin{equation} B^k = \tilde{\varepsilon}^{ijk}\partial_i A_j \end{equation}

Qn 1: In the formula I wrote it as the Levi-Civita symbol $ \tilde{\varepsilon}$, is it right or should it be the Levi-Civita tensor $\varepsilon$ instead? Or does that not make sense since the formula $\vec{B} = \nabla \times \vec{A} $ is true only in cartesian coordinates and so we cannot use a tensor? (hope that made sense)

Qn 2: I wrote $B^k$ with an upper index to indicate that it is a contravariant vector - so that means under coordinate transformations $\tilde{\varepsilon}$ transforms too. Except that I'm confused as to how it transforms? Carroll (in his book spacetime & geometry) says that the Levi-Civita symbol is defined not to change under coordinate transformations (so its entries remain +1 -1 0), yet goes on to derive a transformation law $\tilde{\varepsilon}_{i'j'k'\cdots} = |g| \tilde{\varepsilon}_{ijk} \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} \cdots$. So which is it?

Qn 3: However in some books they write \begin{align} B_k = \epsilon_{ijk}\partial_i A_j. \end{align} In this case it seems like they are ignoring the tensorial nature of the object and just treating $\tilde{\varepsilon}$ as simply a symbol that gives the values +1, -1 or 0 in the summation. This occurs too, for example, in writing the commutation relations down for the spin operators: \begin{align} [J_i, J_j] = i \tilde{\varepsilon}_{ijk}J_k, \end{align} where I am pretty sure in this case the symbol is just thought of as just a number. (although $J_i$ are vectors of the Lie algebra...?). So, am I attributing too much meaning to the Levi-Civita symbol?

Qn 4: In the course of my work I'm doing I have to contend with the angular momentum operator, which I wrote as \begin{align} L^k = \tilde{\varepsilon}^{ijk}g_{ia}r^ap_j. \end{align} In cartesian coordinates this reproduces $L^x, L^y, L^z$, since for example $L^z = x p_y - y p_x$. But is this formula right? Changing to spherical coordinates doesn't seem to give $L^z = \frac{\partial}{\partial \phi}$ and so on...

Qn 5: Just a general question about the Levi-Civita symbol. Does it make sense to raise/lower indices in the Levi-Civita symbol? I know that you can do it for the tensor because the Hodge star operator uses it.

Answer 1

This thread on physicsforums elaborates a bit on the difference between Levi-Civita symbols and tensors. Based on that, I conclude...

1) Your index notation formula for the magnetic field should use the Levi-Civita tensor, then. The "symbol" is a convenient thing, but this expression must be written with tensors.

2) Carroll likely made a mistake and meant to talk about the Levi-Civita tensor's transformation properties.

3) Any expression where the same index appears on the bottom twice (or the top twice) is just laziness on the part of the author. It's a common laziness, especially in contexts where one doesn't discriminate between covariant and contravariant components, however.

Actually, I'm not sure what your question is here.

4) Again, I would say that this expression should use the tensor, not the symbol.

5) I see no reason why you wouldn't.

Just some general remarks on the Levi-Civita tensor/symbol and what they represent: a flat space has a unique "volume form" or "pseudoscalar". There is a unit volume element that all other volumes are scalar multiples of. This volume has an orientation (think of a volume spanned by vectors according to the right-hand rule vs. a left-hand rule).

The Levi-Civita tensor and symbol are related to this notion. The tensor represents the components of the unit volume element with respect to volume elements built by combinations of basis vectors.

The volume element can be used to perform duality operations. This is the foundation of the Hodge star operator. Using the $N$-dimensional Levi-Civita tensor on a tensor object with $k$ free indices yields a new object with $N-k$ free indices. This can be described geometrically, too. In 3d, scalars would go to volumes, vectors to planes, planes to vectors, and volumes to scalars under this duality operation. It is the explicit mapping of planes to vectors that is so often performed with duality--it allows us to cheat in 3d by dealing with only scalars and vectors. Planes and volumes can then be mapped back to vectors and scalars by duality. This is exactly what is done with the magnetic field and angular momentum. You should see clearly that both of these vectors, if not for the use of the Levi-Civita tensor, would be expressions with 2 free indices, and antisymmetric on those indices. These objects are called bivectors.

The Levi-Civita tensor and symbol are often used in physics and math to treat expressions through duality rather than directly--even when, in my opinion, this obscures the real physics of the problem or covers up for a shortcoming of the notation. Just the other day around here we had a question about building up 4-volumes from a single plane. Geometrically, this is obvious--you can't build a 4-volume from a single plane. But in index notation, it was cumbersome at best, involving finding the dual plane through use of the Levi-Civita tensor and taking traces.

Overall, the Levi-Civita tensor and its many indices can be difficult to work with, especially in arbitrary coordinate systems. I once heard a professor bemoan that all the identities another professor had taught students with Levi-Civita had only used the symbol--i.e. the tensor in cartesian coordinates--and so they weren't valid in arbitrary coordinate systems. The solution suggested was to teach students about tensor densities, which was met with skepticism at best, since there were only three professors in the whole department that, in the other professor's view, either cared about or even knew about tensor densities. I think part of this view is why the Levi-Civita symbol is often used instead; it's just easier to prove some things in cartesian coordinates, even if the resulting expression is not really correct (not really a tensor because the metric determinant has been ignored, etc.).