Tensor Rank and Spin

[Markus Hanke]

As we all know there is a correspondence between the tensor rank of a field representation, and the spin of the associated particle - so for example, a spin-½ is represented by a spinor field, a spin-0 by a scalar field, a spin-1 by a vector field, and so on. This seems to just be taken for granted in all of the texts I have seen, but is never really explained. So my question is, what is the deeper reason for this correspondence ? Is there a mathematical reason for it, specifically in terms of group theory and/or differential geometry ?

 

Any ideas, anyone ?

[ajb]

You need to look at the representaion theory of the Lorentz group...

[Markus Hanke]

You need to look at the representaion theory of the Lorentz group...

 

 

Ok, thanks for the pointer, I haven't studied that specific topic yet. Goes on my reading list now

https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group

 

Holy crap...it seems that this is quite a complex topic. This is going to need more than just a fleeting read before bedtime

Have you any recommendations for a math textbook that treats this topic in some detail, at undergrad level ?

[ajb]

Ok, thanks for the pointer, I haven't studied that specific topic yet. Goes on my reading list now

I can't recall much from the top of my head - but you are looking for irreducible representations of the (complete) Lorentz group. It is not just tensors you are interested but also spinor reps.

Edited July 22, 2016 by ajb

[Mordred]

I can dig up some pdfs for you. The subject is quite intensive. I have to go through my archives to find the articles I found most useful.

[ajb]

Have you any recommendations for a math textbook that treats this topic in some detail, at undergrad level ?

Kaku sketches some things in his book on quantum field theory. It is not the best, but it will give you something to start with.

[Mordred]

I believe Sean Caroll also has some papers in arxiv on the Lorentz group.

[Markus Hanke]

Thank you all

My point is precisely that I want to learn about the various possible representations of that group ( and I can see even from the Wiki article that there is more going on that just simple tensors and spinors ), and how they relate to spacetime physics and particle physics. I know only very general group theory, but haven't dived into the details of that particular group yet. At least now I know where to look for further research.

[Mordred]

Well I can give some preliminary reading material. You will need the basic particle physics aspects to understand the groups.

 

for example Baryon octect and meson nonet groups.

 

http://arxiv.org/abs/0810.3328 A Simple Introduction to Particle Physics

 

http://arxiv.org/abs/0908.1395 part 2

 

part 2 gets into the Relativity aspects.

[Markus Hanke]

Thanks all