# What are the invariant definitions of spinorial quantities from mathematical physics?

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 $G$?

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