When Landau (Classical Theory of Fields) introduces the Levi-Civita symbol, he writes

With respect to rotations of the coordinate system, the quantities $\epsilon^{iklm}$ behave like the components of a tensor; but if we change the sign of one or three of the coordinates the components of that tensor, being defined as the same in all coordinate systems, do not change, whereas some of the components of a tensor should change sign.

I am completely Ok with understanding that $\epsilon^{iklm}$ is a pseudotensor. What confuses me in this quote is the one/three choice, implying that if I reflect two coordinates none of the components of a (real) tensor should change sign. But as far as I can see, some of the components of a tensor will change sign whenever I reflect any number of coordinates. For example, if I have a two-rank tensor $A^{ij}$ and reflect two coordinates, say, $x'^0 = x^0$, $x'^1=-x^1$, $x'^2=-x^2$, $x'^3=x^3$, then, using the transformation rule for tensors, I get that, e.g., $A'^{01}=-A^{01}$, so some components of a tensor do change sign if I reflect two axes. If I am correct, why does Landau single out odd number of reflections?


That passage refers specifically to the 4th rank pseudotensor $\epsilon$. If you assume that $\epsilon$ is a tensor and calculate what happens when you flip an odd number of coordinates, you'll get a minus sign, which goes against the definition of what $\epsilon$ is.

$$\epsilon'^{0123}=\frac{\partial x'^0}{\partial x^a} \frac{\partial x'^1}{\partial x^b} \frac{\partial x'^2}{\partial x^c} \frac{\partial x'^3}{\partial x^d}\epsilon^{abcd} = \begin{cases}\epsilon^{0123} & \text{0, 2, or 4 flips} \\-\epsilon^{0123}& \text{1 or 3 flips} \end{cases}$$

Because $\epsilon$ does not transform in this way, it is a pseudotensor.

It's also worth noting that in general, flipping an even number coordinates is equivalent to a proper rotation, so the transformation properties of a general pseudotensor which distinguish it from a proper tensor do not manifest in those cases.

  • $\begingroup$ No, no. My answer is not about pseudotensor $\epsilon$ - as I indicated, I understand that. My answer is about behavior of an arbitrary regular tensor. Will some components of a tensor change sign, if we flip even number of coordinates? $\endgroup$ – Maximko Sep 15 '17 at 16:01
  • $\begingroup$ As I said, the passage refers to that specific pseudotensor, which is why the authors specifically designate an odd number of sign flips. In general, any number of sign flips will change at least some of the components of an arbitrary tensor. $\endgroup$ – J. Murray Sep 15 '17 at 16:07
  • $\begingroup$ Aha, so this is what confused me. From reading the passage I got the impression that Landau makes comparison to any "good old" tensor, whereas in fact he should have stated that the comparison is confined only to antisymmetric tensors (I guess). Thanks, I accept this as the answer. $\endgroup$ – Maximko Sep 15 '17 at 16:10
  • $\begingroup$ @Maximko Yes, that's right. Though I would point out that if you parse that passage carefully, he is saying that if you flip one or three coordinates, the components of $\epsilon$ would change sign if it were a tensor - implying that if you flip two coordinates, none of the components of $\epsilon$ should change. You extrapolated that to mean that none of the components of any tensor should change, which is not what the passage says. $\endgroup$ – J. Murray Sep 15 '17 at 16:15

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