Thought it would be wise to recall some of the concepts before proceeding further.So,I started with revising the section on Four Vectors of sec 6 of Landau-Lifshitz(LL).
there is one thing i wanted to clarify.A quantity that is a product of a pseudoscalar and a scalar is a pseudoscalar; the product of a pseudoscalar with another pseudoscalar would be a scalar.I can extend the same argument to vectors and tensors; and then i would claim that (levi-civita_iklm)*(levi-civita_prst) is a true tensor.
Simple. Look at the action of parity (P) on a tensor: P S = S means that S is a (regular) tensor, P A = -A means that A is a pseudotensor in the sense that you use. The tensor SA clearly has the property P(SA)= -SA and hence is a pseudotensor. If S' is another tensor and A' another pseudotensor, then, in the same way, one can check that P(SS') = SS' and P(AA') = AA', so both these are tensors. (In all this I've left out tensor indices, because the argument does not depend on the ranks of the tensors/pseudotensors).
ReplyDeleteThe difference between a tensor and a pseudotensor lies only in the way that the discrete pieces of the full Lorentz group act on them. If you restrict your attention to the proper orthochronous Lorentz group, then they are the same. Therefore, the composition law is exactly the same as for the group Z(2).