# Formal definition of gauge field and spinors in QFT

I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where

• $i$ is a color index in the fundamental representation of the color group $\mathrm{SU}(N_\mathrm c)$, taking 3 different values;
• $\alpha$ is a Dirac spinor index from the Dirac representation of the Lorentz group, taking 4 different values;
• $f$ is a flavor index in the fundamental representation of the flavor group $\mathrm{SU}(N_\mathrm f)$.

Then it is a complex number and also a Grassmann number. And then of course it is a field because it depends on spacetime $(\vec x, t)$. So what I got is the following formal definition: $$\psi \colon \mathbb R^{1,3} \mapsto \mathfrak{su}(N_\mathrm c) \otimes \Lambda^1\,\mathrm{SU}(N_\mathrm f) \otimes\Lambda \mathbb C \otimes \Lambda^1 \mathbb R^{1,3} \,.$$ where the left side is supposed to be the spacetime and on the left it is the color structure, flavor structure, Grassman structure and the Lorentz structure.

How do I represent the statements “color vector”, “flavor vector”, “Grassmann field” and “Lorentz vector” with this type of tensor product space? Similarly, how does it work for the gauge field $A$ and the gluon field strength tensor $G$?

• Just a little remark: $SU(N_f)$ is only an approximate symmetry. If you are talking about quarks (which I assume, since you mentioned color symmetry), rather use the $SU(2)_L$ left-handed weak isospin symmetry instead and write three copies of the resulting theory to have the three generations of fermions. – Photon Aug 4 '17 at 22:04
• A spinor field on a manifold can be viewed as a section of the Clifford bundle of the tangent bundle. Where the Clifford bundle is the bundle whose fiber at a point p is the Clifford algebra of the tangent space at p. In the case of a G-gauge theory, you would consider the pullback of a spinor field on a principal G-bundle to the base spacetime manifold. – user73352 Aug 4 '17 at 22:45