How parity exchanges right handed and left handed spinors

[victorvmotti]

Reading through David Tong lecture notes on QFT.On pages 94, he shows the action of parity on spinors. See below link: [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfIn (4.75) he confirms that parity exchanges right handed and left handed spinors.

Or for an arbitrary representation of the Clifford algebra:

$$ P: \psi_{+} \to \psi_{-} $$

Where

$$ \psi_{+}$$ and $$ \psi_{-} $$ are projections of the Dirac spinor $$\psi$$

But this is unclear to me.

To prove he says:

**Under parity, rotations don't change sign. But boosts do flip sign.**

Again this isn't clear. We are talking about a discrete Lorentz transformation, i.e. parity, on spacetime which is neither a rotation nor a boost. But why we are mixing them up? I mean a rotation and boost on the Dirac spinor?

 

[eljose79]

Thank you for bringing up this interesting question. I can understand your confusion about the statement made by David Tong regarding the action of parity on spinors.

Firstly, let's clarify the concept of parity. Parity is a discrete transformation that reflects a system about a chosen plane or point. In the context of quantum field theory, this transformation is often used to study the symmetry properties of particles and their interactions. When applied to a Dirac spinor, the parity transformation exchanges the right-handed and left-handed components, as you correctly pointed out.

Now, let's address the statement made by David Tong about rotations and boosts. In general, Lorentz transformations are a combination of rotations and boosts. However, in the context of parity, we are only interested in the transformations that do not involve any spatial rotations. This is because parity only affects the spatial coordinates and not the time coordinate. Therefore, when we consider the action of parity on a Dirac spinor, we only need to consider boosts and not rotations.

To understand why boosts flip the sign under parity, we need to look at the transformation properties of the Dirac spinor under boosts. It can be shown that under a boost, the left-handed component of the spinor transforms with a positive sign, while the right-handed component transforms with a negative sign. This is consistent with the statement made by David Tong, that boosts flip the sign of the spinor.

I hope this explanation helps clarify your doubts. If you have any further questions, please do not hesitate to ask. Happy learning!