How Can the Lorentz Group Be Represented on \(\mathbb{R}^{4^n}\)?

[the1ceman]

Homework Statement

i) Show that the Lorentz group has representations on any space $$\mathbb{R}^d$$
for
any d = 4n with n = 0, 1, 2, . . .. Show that those with n > 1 are not
irreducible. (Hint: here it might be useful to work with tensors in index
notation and to think of symmetry properties.)

Homework Equations

The Attempt at a Solution

I have no idea where to begin, i do however know how to show that they are not irreducible.
I mean if this was R^4 tensor itself n times than i could form rank (n,0) tensors, but this is R^4^n, which has vectors as elements, with 4^n components. Please help

 

[the1ceman]

Anyone?

 

[the1ceman]

Come on ppl someone must know!

 

[tim_lou]

Seriously, this problem is by no means trivial and you just have to crank it out or consult some Lie groups book.

Look at the lie algebra structure (tangent space at identity) first and then you can exponentiate the lie algebra representation. The irreducible representations are actions on spinors. You should be able to find those info in standard QFT books.

 

[the1ceman]

Look at the lie algebra structure (tangent space at identity) first and then you can exponentiate the lie algebra representation. The irreducible representations are actions on spinors. You should be able to find those info in standard QFT books.

It cannot be this long, because this question is part of question from a past exam paper. Either there is a mistake in the question and the R^d should actually mean
$$\mathbb{R}\otimes\mathbb{R}\otimes\cdots\otimes\mathbb{R}$$
d times(in which case the question is ok) or there are reps of the lorentz group on eg R^16 which i do not know about.

 

[tim_lou]

if it's from past exam paper, the professor probably expects you to know the results and derivations already.

Like I said, the answer is actions on spinors, given by
$$e^{-i\omega_{\mu\nu}S^{\mu\nu}}$$

where $S^{\mu\nu}=[J^\mu, J^\nu]$ are the spin operators (I might be missing a factor of 2 or i). The derivations are very nontrivial but the results are standard. You can start by looking at the lie algebra of the lorentz group. You'll get 3 spin operators and 3 boost operators. You take linearly combinations and find out that the lie algebra separates. The representation for the lie algebra is then a direction sum of two spin. You can then exponentiate this lie algebra to get the corresponding lie group.

There is no way I can derive all these without seeing how they were done first. I suggest you consult some textbooks. Srednicki's book on QFT has a good chapter on this matter and Peskin's QFT has a guided multi-part problem that leads you to this result.

 

[Hurkyl]

Has everyone forgotten about trivial actions and direct sums?

 

[tim_lou]

Oh, wait, I think I might have misread your problem. I thought the problem asks for finding all representations of the Lorentz group. I apologize.

Instead, it asks for showing that for real representations of d=4n, the representation is reducible. I can't think of a good strategy right away. I believe you need to work through the commutation relations of Lie algebra and make some clever observations.

 

[the1ceman]

Has everyone forgotten about trivial actions and direct sums?

Ah yes! I will try this thanks.

 

[the1ceman]

Woops i made a mistake! d=4^n not d=4n. So that the direct sum method of fundamental reps will not work. Instead i was thinking of this: Is, for example
$$\mathbb{R}^4\otimes\mathbb{R}^4\cong\mathbb{R^{16}} ?$$
Then the tensor product of the fund (or trivial) rep with itself is also a rep of R^16 ? Is this correct?