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

Codex  0 2026-10-25

Put

z = x - y

and

d Π_{p} = d^{3} p / [(2 π)^{3} 2 E_{p}]

. The Dirac adjoint of the printed mode expansion of a Dirac field is

\overset{ˉ}{ψ}_{a} (x) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} \sum_{s} [e^{i p \cdot x} a_{p}^{s †} \overset{u}{ˉ}_{a}^{s} (p) + e^{- i p \cdot x} b_{p}^{s} \overset{v}{ˉ}_{a}^{s} (p)] .

The Fock vacuum is annihilated by

a, b

. The canonical anticommutation relations therefore give the two non-time-ordered vacuum expectation values

⟨ 0∣ \overset{ˉ}{ψ}_{a} (x) ψ_{b} (y) ∣0 ⟩ ⟨ 0∣ ψ_{b} (y) \overset{ˉ}{ψ}_{a} (x) ∣0 ⟩ = \int d Π_{p} e^{- i p \cdot z} s \sum \overset{v}{ˉ}_{a}^{s} (p) v_{b}^{s} (p), = \int d Π_{p} e^{i p \cdot z} s \sum \overset{u}{ˉ}_{a}^{s} (p) u_{b}^{s} (p) .

In 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:

T \overset{ˉ}{ψ}_{a} (x) ψ_{b} (y) = θ (z^{0}) \overset{ˉ}{ψ}_{a} (x) ψ_{b} (y) - θ (- z^{0}) ψ_{b} (y) \overset{ˉ}{ψ}_{a} (x) .

Thus the result following from the stated canonical anticommutation relations and standard time ordering is

C_{ab} (x, y) \equiv ⟨ 0∣ T \overset{ˉ}{ψ}_{a} (x) ψ_{b} (y) ∣0 ⟩ = \int d Π_{p} [θ (z^{0}) e^{- i p \cdot z} s \sum \overset{v}{ˉ}_{a}^{s} v_{b}^{s} - θ (- z^{0}) e^{i p \cdot z} s \sum \overset{u}{ˉ}_{a}^{s} u_{b}^{s}] .

The 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 ψ_{b} (w) \overset{ˉ}{ψ}_{a} (0) ∣0 ⟩

and let

μ

denote this free Dirac field's mass. Then

C_{ab} (x, y) = - S_{F} (y - x)_{ba} = - \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i ( \neq p + μ ) _{ba}}{p ^{2} - μ ^{2} + i 0} e^{- i p \cdot (y - x)} .

The exchange of the spinor indices is essential:

\overset{v}{ˉ}_{a} v_{b}

is the

(b, a)

component of

v \overset{v}{ˉ}

, not its

(a, b)

component.

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