Representations of the Lorentz Group

[Yaelcita]

This is something I feel I should know by now, but I've always been very confused about. Specifically, how does one determine what each representation of the Lorentz group corresponds to? I mean, I know that the (1/2,0) and the (0,1/2) representations correspond to right and left handed spinors, and the (1/2,1/2) to a 4-vector, but why is that?. I have no idea how to determine this in general. What about the (0,1) and (1,0)? I know they form the electromagnetic tensor, but again I have no clue as to why.

All I can find is something about counting dimensions, or the possible values of j, but that doesn't seem to define things completely... I also found something about determining how those elements transform, but I don't get it. I don't understand what transformations I'm supposed to be looking at.

 

[dextercioby]

This is something I feel I should know by now, but I've always been very confused about. Specifically, how does one determine what each representation of the Lorentz group corresponds to? I mean, I know that the (1/2,0) and the (0,1/2) representations correspond to right and left handed spinors, and the (1/2,1/2) to a 4-vector, but why is that?

Because we DEFINE the spinors as <objects> which have particular behavior under the action of a group, in this case SL(2,C). So we DEFINE the LH spinor as the <object> which pertains to the (1/2,0) linear irreducible representation of SL(2,C).

I have no idea how to determine this in general.

Using the definitions and the Clebsch-Gordan theorem for SL(2,C).

What about the (0,1) and (1,0)? I know they form the electromagnetic tensor, but again I have no clue as to why.

This is a little involved. The e-m field tensor is the object transforming under the (1,0) direct sum with (0,1) representation when subject to the restricted homogenous Lorentz group.

All I can find is something about counting dimensions, or the possible values of j, but that doesn't seem to define things completely... I also found something about determining how those elements transform, but I don't get it. I don't understand what transformations I'm supposed to be looking at.

Well, linear algebra is a prerequisite for harmonic analysis (theory of representations of abstract algebraic structures on (topological) vector spaces), so, when talking about group representations, counting dimensions and paying attention to j's and j''s are a must. As for the bolded part (by me in the quote), well that's the essence of the whole topic. I'll give you some questions whose answers may help you find it:

1. What's a Lorentz transformation ?
2. What's the Lorentz group ?
3. In which theories does the Lorentz group matter the most ?
4. Why are representations of the Lorentz group important ?
5. In which theories do <objects> like A_mu, T_mu nu, psi_alpha, psi_alpha dot occur ?
6. In which theory do the representations of the Lorentz group occur ?
7. What is the classical field theory made up of ?

 

[Yaelcita]

I think my question about the way things transform was unclear. This is what I mean, with an example.

I know (1/2,1/2) is a Lorentz vector, so it's supposed to transform like, if A is my vector,
U(\Lambda)A\, U^\dagger (\Lambda)=\Lambda A
What I don't understand is how I can show that it transforms in that way simply by knowing that it lives in (1/2,1/2).

Of course, I mean this in general, not just for (1/2,1/2) but I thought the Lorentz vector example was the simplest to explain what I mean.

 

[dextercioby]

If it <lives> in the (1/2,1/2) representation, then it must have a particular <spinor index configuration>. The van der Waerden symbols (giving you the morphism between the restricted homogenous Lorentz group and SL(2,C)) will provide you with the vector index you need. Then it's just matrix multiplication and <index gymnastics> to show the expression that you wrote in LaTex.

 

[Yaelcita]

What are van der Waerden symbols? This is the first time I've heard that expression, and there's no Wikipedia article about it... Nor can I find it in Srednicki, Peskin and Schroeder or Weinberg...

 

[dextercioby]

The 3 Pauli matrices together with the unit matrix form a set of 4 matrices which is given a particular index structure:

\left(\sigma^{\mu}\right)_{\alpha\dot{\beta}}

That is the first van der Waerden symbol. The other one is an antisymmetric combination in vector indices of the first.

 

[Yaelcita]

Let me see if I understand this correctly.

What you are saying is that the only thing I know is its spinor index structure. In this case (the (1/2,1/2) case) I know my 'object' A must have one dotted and one undotted index, one from each SU(2). So I need to multiply it by another object, in this case the van der Waerden symbol, which also has one dotted and one undotted spinor index. Since the symbol also has a vector index, my new object ends up having a vector index, thus being a Lorentz vector (or thus satisfying the appropriate transformation, which makes it a vector). Is that right?

But this begs the question, what is the generalization of these van der Waerden symbols? I'm thinking of the F^{\mu \nu}[\tex] which you said was more involved, for example. Why is it more involved? I mean, I want to end up with two vector indices, so clearly something else needs to happen, but what? Is the difference that in this case I&#039;m talking about a direct sum of two representations? (1,0)+(0,1) Does that mean I&#039;ll just have two of these symbols? But that would mean that my object should have four spinor indices? Why? I think I&#039;m confusing myself.

 

[dextercioby]

Let me see if I understand this correctly.

What you are saying is that the only thing I know is its spinor index structure. In this case (the (1/2,1/2) case) I know my 'object' A must have one dotted and one undotted index, one from each SU(2). So I need to multiply it by another object, in this case the van der Waerden symbol, which also has one dotted and one undotted spinor index. Since the symbol also has a vector index, my new object ends up having a vector index, thus being a Lorentz vector (or thus satisfying the appropriate transformation, which makes it a vector). Is that right?

Perfect.

But this begs the question, what is the generalization of these van der Waerden symbols?

I told you, there's only one 'extension' of it which carries 2 vector indices and is a-symm between them.

I'm thinking of the F^{\mu \nu} which you said was more involved, for example. Why is it more involved? [...] Is the difference that in this case I'm talking about a direct sum of two representations? (1,0)+(0,1)

Yes, because normally you're looking for irreducible representations of the r.h. Lorentz group. However, you're forced into the reducible one, because you want your tensor to describe the e-m field, which is known to be parity-invariant.

Does that mean I'll just have two of these symbols? But that would mean that my object should have four spinor indices? Why? I think I'm confusing myself.

Not really, the e-m tensor is the sum between its self-dual and a-self-dual part. Each of the latter carry 2 vector indices and the spinor indices are contracted with the the second van der Waerden symbol like for the self-dual tensor:

F_{\mu\nu}^{(1,0)} = \psi^{\delta} \left(\sigma_{\mu\nu}\right)_{\delta}^{~\zeta} \chi_{\zeta}

EDIT: The name is actually: <Infeld-van der Waerden symbol(s)>.

 

[samalkhaiat]

This is something I feel I should know by now, but I've always been very confused about. Specifically, how does one determine what each representation of the Lorentz group corresponds to? I mean, I know that the (1/2,0) and the (0,1/2) representations correspond to right and left handed spinors, and the (1/2,1/2) to a 4-vector, but why is that?. I have no idea how to determine this in general. What about the (0,1) and (1,0)? I know they form the electromagnetic tensor, but again I have no clue as to why.

All I can find is something about counting dimensions, or the possible values of j, but that doesn't seem to define things completely... I also found something about determining how those elements transform, but I don't get it. I don't understand what transformations I'm supposed to be looking at.

See post #24,25 and 29 in
www.physicsforums.com/showthread.php?t=192572

 

[Yaelcita]

This has been most helpful. Thanks to both of you!