# QFT $\sigma$-model - Commutator of Pauli and Gamma matrices

To give some contest, I'm working on a $$\sigma$$-model given by the Lagrangian

$$L = \bar\psi i \partial_\mu\gamma^\mu \psi - g\bar\psi (\sigma + i\vec\pi \cdot \vec\tau \gamma^5)\psi + \frac{1}2(\partial_\mu \sigma \partial^\mu \sigma + \partial_\mu\vec\pi \cdot\partial^\mu\vec\pi) + V(\sigma^2 + \pi^2)$$

I omit the structure of the potential since it's irrelevant to my question; the point is that $$\psi$$ is a Dirac field, $$\sigma$$, $$\vec \pi$$ are respectively a scalar field and a 3-vector field, while $$\tau ^a$$ is the $$a$$-th Pauli matrix. What I understand is that $$\tau^a$$ acts as a 2x2 matrix on $$\psi$$, i.e. mixing the left and the right component of the Dirac spinor, but it leaves untouched the components of the left/right Weyl spinor. To give and example, I'm thinking that

$$\tau^1 \psi = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \psi_R \\ \psi_L \end{bmatrix}$$

and that (here I'm writing only $$\psi_L$$ but the same structure clearly holds for $$\psi_R$$) $$\psi_{L} = \begin{bmatrix} \xi_{L_1} \\ \xi_{L_2} \end{bmatrix}$$

My objective is computing the vectorial 4-current, associated to a symmetry (which is more explicit if I re-define the Lagrangian through the fields $$\Sigma = \sigma + i\vec\pi \cdot \vec \tau$$ and $$\psi_{L/R} = \frac{1}2 (1 \mp \gamma^5) \psi$$); but to rearrange the current I bump into a controversy. Let me show it. One of the contribution to the current is

$$\bar\psi_{L} i \gamma^\mu \frac{i}2 \tau^a \psi_L + \bar\psi_{R} i \gamma^\mu \frac{i}2 \tau^a \psi_R = \\ \frac{1}2 \psi^\dagger(1-\gamma^5)\gamma^0 i \gamma^\mu \frac{i}2 \tau^a \frac{1}2 (1 - \gamma^5) \psi + \frac{1}2 \psi^\dagger(1+\gamma^5)\gamma^0 i \gamma^\mu \frac{i}2 \tau^a \frac{1}2 (1 + \gamma^5) \psi = \\ -\frac{1}8 \bar\psi \gamma^\mu [(1-\gamma^5) \tau^a (1- \gamma^5) + (1+\gamma^5) \tau^a (1+ \gamma^5)]\psi = \\ -\frac{1}4 \bar\psi \gamma^\mu [\tau^a + \gamma^5 \tau^a \gamma^5] \psi$$

Now, the expected result (given for example in the problem 5.17 in Radovanovich - Problem Book Quantum Field Theory) would be $$-\frac{1}2 \bar\psi \gamma^\mu \tau^a \psi$$, which could be obtained if $$[\tau^a, \gamma^5] = 0$$.

My question: how could these matrices commute? Aren't they acting on the same space, more explicitly aren't they matrices on the Dirac spinor components?

• $\tau^a$ and $\gamma^\mu$ act on different vector spaces so they commute. Commented Jan 26, 2022 at 15:02
• @AccidentalFourierTransform Let me rephrase my question: it is clear that they have to act on different spaces, but I cannot see why. If the Pauli matrices mix the components, and so do the Gamma matrices, why am I allowed to consider them as acting on different spaces? Commented Jan 26, 2022 at 15:07

You are dramatically misreading the formalism setup, probably victimized by the use of Pauli matrices in explaining the structure of the Weyl representation of γ matrices. You may use Pauli matrices in a direct product structure when confusion can be prevented.

In particular, you are working in an 8-dimensional direct product space, not a 4-dimensional one. Let me first supplant the implicit indices which you skipped to get into this trouble.

The ψs are 8-vectors, $$\psi^i_\alpha$$, where the isospin indices i take values 1,2, and are acted upon the $$\tau^a_{ij}$$ matrices; while the α s are the four Dirac spinor indices, and are acted upon by all $$\gamma^\mu_{\alpha \beta}$$ matrices. So the two types of indices live in distinct Cartesian factor vector spaces, yielding 8-dimensional vectors, $$2\times 4$$, and acted upon by the 8×8 matrices $$\tau^a \otimes \gamma^\mu$$. (I hope you appreciate the bizarre bogus actions of the Pauli matrices you wrote are worse than meaningless.)

Crucially, note the as acting on the pseudoscalar pions have nothing to do with spacelike spacetime indices, possibly the source of your conflation of Pauli matrices used for two purposes here.

Moreover, the issue of "ordering" τs and γs doesn't even arise, since they are in different factor spaces of your direct product: τs act on isospin apples and γs on spinor oranges.

Write the explicit indices in your spinors and see that you may reorder transpose the apples and the oranges without ambiguity or obstruction.

• I think you couldn't be clearer. My problem was that, even if I had all the indices explicited, I couldn't understand why those indices were substancially different (basically I was still trying matrix product). That's because isospin didn't come across my mind at all. And of course this has nothing to do with R/L spin decomposition (which makes my post so embarassing) but indeed acts on a different space. A sincere thank you. Commented Jan 26, 2022 at 17:50