# Γ5 and chiral symmetry

1. Sep 5, 2010

### lihurricane

recently i am reading chiral symmetry in QFT. Almost all textbooks define γ5 as a chiral
operator without saying some reasons. i am very confused why γ5 has something to do with
chiral symmetry, can somebody explain it more intuitively and physically? who first introduce γ5 as a chiral operator? and i also want to
know what is a chiral transformation? is it a transformation for example which can transform a left-handed to a right-handed?

Thanks!

2. Sep 6, 2010

### DrDu

The states projected out by the projectors $$(1 \pm \gamma_5)/2$$ can be seen to be interconverted into each other by the parity operator $$i \gamma_0$$ as it anticommutes with gamma_5. On the other hand time reversal as given by $$i\gamma_1\gamma_3$$ and taking the complex leaves these states invariant. Compare this to the properties of a chiral molecule under parity and time-reversal: under parity (space inversion), a D molecule will be transformed into an L molecule and vice versa, while time inversion leaves the chirality of the molecule the same.
The chirality transformation corresponds to a multiplication of these two components by phase factors $$\exp (\pm i \phi)$$.