# Is there a relationship between axial $U(1)$ symmetry and parity transformations?

Some books, such as Maggiore, write that the axial $U(1)$ transformation is the chiral transformation. A Dirac field $\Psi \in (\frac12, \frac12)$ undergoes a chiral transformation in the following manner.

$$\Psi \mapsto e^{i\beta\gamma_5}\Psi$$

In a chiral basis, we can write $\Psi \equiv \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix}$ where $\psi_L \in (\frac12,0)$ and $\xi_R \in (0,\frac12)$. Then,

$$\psi_L \mapsto e^{-i\beta}\psi_L \,,\qquad \xi_R \mapsto e^{i\beta}\xi_R \,.$$

Maggiore defines chiral fields to be ones that violate the axial $U(1)$ symmetry.

Whereas books such as Srednicki define chiral fields to be ones that violate parity. A parity transformation does the following.

$$\begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix} \mapsto \begin{pmatrix} \xi_R\\\psi_L \end{pmatrix}$$

Srednicki discusses the axial $U(1)$ symmetry in a different chapter as a global symmetry in consistent gauge theories.

So, which of the definitions of chirality is true? And if they are equivalent, how so?

• Both P and an axial rotation do not leave $\psi_L, \xi_R$ invariant. Jul 31 '17 at 15:51
• @CosmasZachos Yes, that's true. But are they necessary and sufficient for each other: parity violation $\Leftrightarrow$ axial asymmetry? In other words, are the two definitions equivalent? Jul 31 '17 at 15:57

They are both manifestly true. Any confusion is traceable to notational overkill. In your 2d Weyl basis where superfluous spinor indices are ignored, an axial rotation is but $$e^{-i\beta \sigma_3} =1\!\!1 \cos \beta -i \sigma_3 \sin \beta ,$$ where, of course, the finite rotation $\beta=\pi/2$ amounts to the above reducing to $-i\sigma_3= -\tfrac{i}{2}(1\!\!1+i\sigma_2)\sigma_1 (1\!\!1-i\sigma_2)$, so unitarily equivalent to $-i\sigma_1$.
Likewise, $$P=\sigma_1$$ which has the same eigenvalues as the π/2 rotation above, and provides its conjugation matrix, $$P e^{-i\beta \sigma_3} P= e^{i\beta \sigma_3},$$ since $\sigma_1 \sigma_3\sigma_1=-\sigma_3$.
We are talking about fermions here, but, parity-odd spinless states like pseudoscalars (which might be used to violate parity in an action) can be associated with fermion spinless bilinears which comport with this axial rotation scheme: so you might as well consider the two bases connected by $(1\!\!1-i\sigma_2)/\sqrt2$ equivalent.