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 ?

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

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 ?

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, 20169 yr by ajb

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.

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.

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

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.

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.

Thanks all