Student Friendly Introduction to QFT · Problem 23

Vectors: Spin 1 Fields

Physics — QFT · dirac-field-hamiltonian

Problem

Prove that the last lines of (4-68) and (4-69) sum to zero. Hint: First use the Dirac equation with a single p\mathbf{p} eigensolution of spin ss for a dd type field to show that (p̸+m)vs(p)=0(\not{p} + m)v_s(\mathbf{p}) = 0. Then, use the RH relation of (4-25) to show uˉr(−p)γ0p0vs(p)=0\bar{u}_r(-\mathbf{p})\gamma^0 p_0 v_s(\mathbf{p}) = 0. Then, use that to re-write the last line of (4-68) in terms of p̸\not{p}. Finally, combine that line with the last line of (4-69) and show that together they equal zero.
Intuitive explanation

The key insight is recognizing that antiparticle and particle solutions are related through charge conjugation and momentum reversal—they're not independent. By applying the Dirac equation to the vv spinor and using the orthogonality relations between uu and vv spinors (encoded in the RH relation), you'll see that the problematic terms actually encode the same physical constraint twice, just in different forms. When you combine them using the p̸\not{p} notation, the redundancy cancels exactly, leaving zero. This is a consistency check that the theory respects particle-antiparticle symmetry.

Intuitive explanation

The key insight is recognizing that antiparticle and particle solutions are related through charge conjugation and momentum reversal—they're not independent. By applying the Dirac equation to the vv spinor and using the orthogonality relations between uu and vv spinors (encoded in the RH relation), you'll see that the problematic terms actually encode the same physical constraint twice, just in different forms. When you combine them using the p̸\not{p} notation, the redundancy cancels exactly, leaving zero. This is a consistency check that the theory respects particle-antiparticle symmetry.

Related theory

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