Representation Theory and Particle Theory

[fscman]

I am familiar with the representation theory of finite groups and Lie groups/algebra from the mathematical perspective, and I am wondering how quantum mechanics/quantum field theory uses concepts from representation theory. I have seen the theory of angular momentum in quantum mechanics, and I realized that L_x, L_y, and L_z, the components of angular momentum, are elements of the Lie algebra SO(3). I also heard of the notion that irreducible representations correspond with elementary particles and that the Casimir element can measure scalar quantities such as mass. Unfortunately, my knowledge in this area is merely a bunch of scattered facts. Could anyone explain the foundations of the relationship between representation theory and quantum physics, or provide a resource (book or website) that explains this connection? Thanks.

 

[vanhees71]

You find this in my manuscript on quantum field theory

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

[IRobot]

I would recommend the second chapter of Weinberg's Quantum theory of Fields, he does construct the particle states from the irreducible representations of the Lorentz Group.

 

[Fredrik]

The best place I know to read about particles and irreducible representations is chapter 2 of Weinberg's QFT book. I'm not saying that it's great, only that it's not bad for a physics book, and that I don't know a better place. Link

Edit: D'oh. Too slow again.

 

[IRobot]

Yep, burned ;)

 

[lugita15]

Ballentine has some discussion of irreducible representations (although he doesn't use that term) and their role in QM.

BTW, I have the opposite problem. I have a general idea about the role of Lie groups in QM, but I'd like to learn some math: representation of finite groups, semisimplicity, irreps, and then once I've mastered that stuff Lie groups and Lie algebras. Does anyone know of a good intro to that kind of material, something accessible to me having only taken a standard undergrad abstract algebra course?

 

[Markus Hanke]

Ballentine has some discussion of irreducible representations (although he doesn't use that term) and their role in QM.

BTW, I have the opposite problem. I have a general idea about the role of Lie groups in QM, but I'd like to learn some math: representation of finite groups, semisimplicity, irreps, and then once I've mastered that stuff Lie groups and Lie algebras. Does anyone know of a good intro to that kind of material, something accessible to me having only taken a standard undergrad abstract algebra course?

That's pretty interesting...how did you manage to understand its role within - and connection to - QM if you didn't know the mathematics of group & representation theory ? I am trying to self-study this area at the moment ( I'm not a scientist by trade ), but am finding it hard going.

 

[IRobot]

Ballentine has some discussion of irreducible representations (although he doesn't use that term) and their role in QM.

BTW, I have the opposite problem. I have a general idea about the role of Lie groups in QM, but I'd like to learn some math: representation of finite groups, semisimplicity, irreps, and then once I've mastered that stuff Lie groups and Lie algebras. Does anyone know of a good intro to that kind of material, something accessible to me having only taken a standard undergrad abstract algebra course?

Shouldn't be hard to find what you are looking for, a lot of group theory books start by introducing the representation theory and its concepts before going to the special case of Lie Groups, one example:
https://www.amazon.com/dp/0387788654/?tag=pfamazon01-20