# Antisymmetric tensor and pseudotensor

I am reading Landau & Lifshitz's classical field theory book, and there is a paragraph which confused me:

"With respect to rotations of the coordinate system, the quantities $e^{iklm}$ behave like the components of a tensor; but if we change the sign of one or three of the coordinates the components $e^{iklm}$, being defined as the same in all coordinate systems, do not change, whereas some of the components of a tensor should change sign. Thus $e^{iklm}$ is, strictly speaking, not a tensor, but rather a pseudotensor."

$e^{iklm}$ is the antisymmetric tensor. I would like to ask

1. what does the passage means
2. how tensor changes under proper rotation
3. how tensor changes under improper rotation
4. how pseudotensor changes under proper rotation
5. how pseudotensor changes under improper rotation
6. how to decompose an improper rotation into the product of rotation and reflection
7. det(rotation x reflection) = -1. Does det R' = -1 implies R' is an improper rotation?

1. what does the passage means

I hope it will be clear from my answers to the other 6 questions.

2-3. how tensor changes under proper or improper rotation

By rotations we mean Lorentz transformations. Spatial rotations are just special cases of Lorentz transformations. An arbitrary Lorentz transformation is given by a $$4 \times 4$$ matrix $$\Lambda_{\mu}^{\;\nu}$$, which satisfies the following condition: $$\eta_{\mu \nu} = \eta_{\mu' \nu'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'}.$$ This is equivalent to saying that the matrix, when applied to a 4-vector, does not change its invariant interval.

Note that if we take $$\det$$ from both sides, we get $$(\det \Lambda)^2 = 1 \quad \Longrightarrow \quad \det \Lambda = \pm 1.$$

Lorentz transformations with positive (negative) determinants are called proper (improper) respectively.

Tensors transform under Lorentz transformations according to $$A_{\mu \nu \dots \sigma} \rightarrow A_{\mu' \nu' \dots \sigma'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'} \dots \Lambda_{\sigma}^{\;\sigma'}.$$

4-5. how pseudotensor changes under proper or improper rotation

Pseudotensors transform under Lorentz transformations according to $$B_{\mu \nu \dots \sigma} \rightarrow \det \Lambda \cdot B_{\mu' \nu' \dots \sigma'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'} \dots \Lambda_{\sigma}^{\;\sigma'}.$$

1. how to decompose an improper rotation into the product of rotation and reflection

Consider, for example, an (improper) spatial reflection, given by $$\Lambda_{\text{refl}} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{array}\right).$$

Any improper Lorentz transformation can be written as $$\Lambda_{\text{improper}} = \Lambda_{\text{proper}} \cdot \Lambda_{\text{refl}},$$

where $$\cdot$$ is matrix multiplication.

1. det(rotation x reflection) = -1. Does det R' = -1 implies R' is an improper rotation?

Exactly, it is the definition.

• Thanks so much. It is very clear, and I just want to make sure one thing: Is it $\Lambda = \Lambda_\mu^{\mu'}...\Lambda_\sigma^{\sigma'}$? – Kevin Kwok Oct 12 '16 at 15:08
• @KevinKwok no, by $\Lambda$ I mean just $\Lambda_{\mu}^{\;\nu}$ as a matrix. You are quite welcome. – Prof. Legolasov Oct 12 '16 at 23:29
• I meant the $\Lambda$ in the expression $\det \Lambda$. – Kevin Kwok Oct 13 '16 at 3:16
• @KevinKwok so did I. $\det \Lambda$ is the determinant of the matrix $\Lambda_{\alpha}^{\;\beta}$. – Prof. Legolasov Oct 13 '16 at 4:16
• In question 2-3, does $\Lambda = \Lambda_{\mu}^{\;\mu'}\Lambda_{\nu}^{\;\nu'}$? In question 4-5, does $\Lambda = \Lambda_{\mu}^{\;\mu'}...\Lambda_{\sigma}^{\;\sigma'}...$? – Kevin Kwok Oct 13 '16 at 15:48