How Do You Add Angular Momentum in Fock Space?

[lugita15]

When adding the angular momenta of two particles, you use Clebsch-Gordan coefficients, which allow you, in fancy language, to decompose the tensor product of two irreducible representations of the rotation group into a direct sum of irreducible representations. (I'm not exactly clear on what this means, so on a side note can someone suggest a good book on representation theory of Lie groups?)

If we want to add the angular momenta of even more particles, then we have to use other coefficients, like the Wigner 3j, 6j, or in general 3nj symbol. But what if you have an unknown number of particles, like you do in quantum field theory? In this case quantum states live in a so-called Fock space made of a direct sum of infinitely many tensor products of Hilbert spaces (which is another thing I'm unclear on). So then how do you add angular momentum in Fock space? To put it another way, how do you decompose a tensor product of irreducible representations into a direct sum if you have a variable number of terms in the tensor product?

Any help would be greatly appreciated.
Thank You in Advance.

 

[eljose79]

Thank you for your question about adding angular momenta in Fock space. You are correct that in quantum field theory, the quantum states exist in a Fock space, which is a direct sum of infinitely many tensor products of Hilbert spaces.

In order to add angular momenta in Fock space, we use a similar approach as in the case of two particles. We decompose the tensor product of irreducible representations into a direct sum of irreducible representations using a generalization of the Clebsch-Gordan coefficients, known as the Racah-Wigner coefficients. These coefficients take into account the variable number of terms in the tensor product and allow us to add angular momenta for an arbitrary number of particles.

As for a good book on representation theory of Lie groups, I would recommend "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction" by Brian C. Hall. It provides a clear and accessible introduction to the subject, including the Clebsch-Gordan coefficients and their generalizations.

I hope this helps answer your question. If you have any further inquiries, please don't hesitate to ask.