Past exam of the quantum field theory course of the University of Oxford / 2025 / 3 / d / Solution

Put $z=x-y$ and $d{\Pi }_p=d^3\mathbf{p}/[(2\pi {)}^{3}2E_{\mathbf{p}}]$. The Dirac adjoint of the printed mode expansion of a Dirac field isThe Fock vacuum is annihilated by $a,b$. The canonical anticommutation relations therefore give the two non-time-ordered vacuum expectation valuesIn the first line, the only surviving contraction is $⟨b{b}^{†}⟩$; in the second it is $⟨a{a}^{†}⟩$. Both of these unordered contractions have positive coefficients.

However, fermionic time ordering includes a minus sign when it interchanges two odd fields:Thus the result following from the stated canonical anticommutation relations and standard time ordering isThe second term printed with a plus sign is not correct for standard fermionic time ordering. Its plus-plus expression would be an ungraded chronological product. The missing minus cannot be absorbed by a phase choice for $u$ or $v$, since each spin sum contains a spinor and its adjoint.

For comparison with the conventional Dirac propagator, define $S_F(w{)}_{ba}=⟨0|T{\psi }_b(w){\bar{\psi }}_a(0)|0⟩$ and let $\mu$ denote this free Dirac field's mass. ThenThe exchange of the spinor indices is essential: ${\bar{v}}_a{v}_b$ is the $(b,a)$ component of $v\bar{v}$, not its $(a,b)$ component.