Griffiths Introduction to Elementary Particles · Problem 7.9

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · charge-conjugation

Problem

The charge conjugation operator (

C

) takes a Dirac spinor into the ``charge-conjugate'' spinor

\psi_c

, given by

\psi_c = i\gamma^2\psi^*

[See Halzen and Martin, ref. 7, Sect. 5.4.] Find the charge-conjugates of

u^{(1)}

and

u^{(2)}

, and compare them with

v^{(1)}

and

v^{(2)}

Intuitive explanation

Charge conjugation reveals a deep symmetry between particles and antiparticles in the Dirac equation. The key insight is that applying $C$ to a particle spinor (like $u$ ) should yield an antiparticle spinor (like $v$ ), up to phase and normalization—this is the physical content of $\psi_c = i\gamma^2\psi^*$ . You'll need to use the explicit forms of $u^{(1,2)}$ and $v^{(1,2)}$ , apply complex conjugation and the $\gamma^2$ matrix, and then compare the results to see how the two spinor types are related by this symmetry operation.

Intuitive explanation

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