Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

When physicists write expressions involving spinors $\psi \in S \otimes V$, where $S=S_+ \oplus S_-$ is a complex spinor representation of a spin group $Spin(2d)$ and $V$ is a complex representation of a non-Abelian Lie group $G$ preserving an inner product, they often continue to make use of explicit bases for the Clifford algebra. What is the standard mathematically invariant way of writing and defining expressions like $\bar \psi, \psi^\dagger, \psi^*, \psi^\dagger \gamma^0, \psi^\dagger \gamma^0 \gamma^5, etc... $ without using such explicit bases, but using only

  • The (presumably Hermitian) inner product on $S$.
  • The (presumably Hermitian) inner product on $V$.
  • Tensor products and direct sums of vector spaces and their elements.
  • Invariants inside the Clifford Algebra and other algebraic structures etc...
  • Clearly specified complex vector spaces like $V$, $\bar V$, $V^\star$, $\bar V^\star$.

In particular, what is the standard mathematically invariant definition of the spinorial source term $J(\psi)$ in the Yang-Mills Equation $$ d_A^* F_A = J(\psi) $$ where $V$ represents a non-trivial representation of a non-abelian $G$?

share|improve this question
add comment

1 Answer 1

I don't have the book in front of me but I think a good place to start for this question and similar ones is

Quantum Fields and Strings: A Course for Mathematicians by Pierre Deligne

http://www.amazon.com/Quantum-Fields-Strings-Course-Mathematicians/dp/0821820125

share|improve this answer
    
The relevant chapter is on the arXiv: Deligne, Freed, Supersolutions (arXiv:hep-th/9901094) –  Urs Schreiber Apr 23 '13 at 19:03
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.