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


*

*what does the passage means

*how tensor changes under proper rotation

*how tensor changes under improper rotation

*how pseudotensor changes under proper rotation

*how pseudotensor changes under improper rotation

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

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

 A: 
  
*
  
*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
$$ B_{\mu \nu \dots \sigma} \rightarrow \det \Lambda \cdot B_{\mu' \nu' \dots \sigma'} \; \Lambda_{\mu}^{\;\mu'} \Lambda_{\nu}^{\;\nu'} \dots \Lambda_{\sigma}^{\;\sigma'}. $$


  
*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.


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

Exactly, it is the definition.
