Understanding Lorentz Representations and Their Corresponding Identities

[Sombrero]

Hello everyone,

In wikipedia when searching Lorentz representations, there is given that (1/2,0)*(0,1/2) corresponds to Dirac spinor representation and (1/2,1/2) is vector representation, but in P.Ramond's book "Field Theory - A Modern Primer" I read (1/2,0)*(0,1/2)=(1/2,1/2), obviously I suffer from leak of knowledge.

Please advise me how do I understand this line (1/2,0)*(0,1/2)=(1/2,1/2) properly and not to think that Dirac representation = Vector representation?

Also how do I identify that (1,0) representation from (1/2,0)*(1/2,0)=(0,0)*(1,0) corresponds to ""self-dual 2 form representation", could you tell me where do I find these rules or some books to read about this?

Thank you in advance

 

[dextercioby]

There's a difference between the direct sum and direct product representations. The direct sum of Weyl representations (1/2,0) and (0,1/2) is a Dirac representation (it is reducible) and the direct product of Weyl representations (1/2,0) and (0,1/2) is a vector representation (it is irreducible).

Ramond is correct and the last time i checked the Wiki page http://en.wikipedia.org/wiki/Representations_of_the_Lorentz_Group
was correct as well.

Daniel.

 

[Sombrero]

Dear dextercioby thanks for your reply,
Could you advise me some books about this issue?

Thank you in advance

 

[dextercioby]

Yes, i think Moshe Carmeli's "Group Theory and General Relativity" should be an easy useful read. Wu Ki Tung's "Group Theory in Physics" is also easy to read.

Especially for the Lorentz group the best by far treatment is in Wiedemann-Kirsten et al. "Introduction to supersummetry". It will help you se what's with the (anti)selfdual 2-forms.

Daniel.

 

[Sombrero]

Dear PF,
Just recently I got book (Group Theory of Physics by Wu-Ki Tung), I had a glance through it.
I have a question:

SU(2)*SU(2) is isomorphic to Lorentz group (Let symbol “*” be a direct product), so to know everything on Lorentz group it is sufficient to be familiar with SU(2). Let us consider the direct product of fundamental representations of SU(2) - SU(2)*SU(2) when j=1/2 for each and from these two SU(2) with J=1/2 each we get (1/2,1/2) correct? Correct. (1/2 , 1/2) is vector representation so does all this mean that from two sets of generators of fundamental representation of SU(2) (from Pauli Matrices) somehow we get Lorentz generators for vector transformation?


What I say wrong here?
Thank you

 

[dextercioby]

Nothing is wrong up to the fact that $SU(2)\times SU(2)$ is only locally isomorphic to $\mathcal{L}_{+}^{\uparrow}$.

 

[Sombrero]

Dexter I really appreciate ur quick responses that's really very helpful. Thanks

So can I from two sets of generators of fundamental representation of SU(2) (from Pauli Matrices) somehow get Lorentz generators for vector transformation?

The generators of a direct product representation are the sums of the corresponding generators of its constituent representations.

Just reformulating my question: Do sums of generators of fundamental SU(2) representations give corresponding generators of vector resentation?

Transformations of Weyl representation are represented by Pauli matrices, and this direct product of Weyl representations (1/2,0) and (0,1/2) is a vector representation. Can I get (4*4) Generator for vector transformation (vector representation) from Pauli matrices?

Thank you