I don't exactly understand how parity is related to chirality. On wikipedia it says that the parity transformation can be thought of as a test for chirality of a physical phenomenon. How those two concepts are related?


1 Answer 1


Chiralty comes from the greek word for hand and can be traduced as handedness in english.

Something is chiral if it is not invariant under a parity transformation. Prototypes of this appear in classical mechanics and field theory. An example of a chiral object in classical mechanics is the angular momentum.

I see particle physics mentioned as a topic, but even though the term chiral is most often used in particle physics I feel like as a term it is equally valid in any other fild of physics.

I'm gonna still use the Dirac equation as an example of chirality in particle physics because maybe it is the most common example. The Dirac equation reads \begin{equation} (i\gamma^\mu\partial_\mu - m)\psi = 0 \end{equation}

The chirality operator is $\gamma^5 = i \gamma^0\gamma^1\gamma^2\gamma^3$ and in an appropriate representation of the Clifford algebra $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}$ called the chiral representation this matrix is diagonal. A parity transformation of a spinor is represented by the matrix $\gamma^0 $ hence computing the transformed chirality operator \begin{equation} \gamma^0\gamma^5(\gamma^0)^{-1} = - \gamma^5 \end{equation}

In this sense a left handed fermion (i.e. eigenvector of $\gamma^5$ with eigenvalue 1) becomes a right handed fermion under parity.

  • $\begingroup$ Why is the parity transformation represented by gamma^0 in the dirac spinor space? $\endgroup$
    – Mathew
    Nov 30, 2021 at 18:14

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