Past exam of the quantum field theory course of the University of Oxford / 2018 / 4 / a / Solution

The displayed oscillator expansion and the displayed action require two separate corrections to describe one conventional positive-energy theory. First compute what the oscillator algebra literally implies. Assume the two oscillator species commute with each other and annihilate the vacuum. Same-charge contractions vanish, soFor the two mixed contractions the printed phases give the same kernel, with $z=x-y$:Thus $⟨T\varphi (x){\varphi }^{\ast }(y){⟩}_0=⟨T{\varphi }^{\ast }(x)\varphi (y){⟩}_0=C_{\mathrm{lit}}(z)$. In $D$ dimensions replace $d^{3}p$ and $d^{4}p$ by $d^{D-1}p$ and $d^{D}p$; the source's explicit three-dimensional spatial measure is the $D=4$ specialization. The covariant integral is manifestly Lorentz invariant. Withwe have $C_{\mathrm{lit}}(z)=D_F(z{)}^{\ast }$. This reversed-frequency scalar contraction has the opposite pole prescription to the usual positive-energy Feynman propagator. For a single zero-momentum mode at positive time separation, the printed contraction is $e^{+imt}/(2m)$, while $D_F$ is $e^{-imt}/(2m)$: this is a concrete phase counterexample, rather than an immaterial change of integration variable.

There is also a normalization of a complex scalar propagator conflict independent of those phases. The displayed Lagrangian density gives the canonical momentum ${\pi }_{\varphi }={\dot{\varphi }}^{\ast }/2$. Substituting the printed modes yieldsinstead of the canonical $+i{\delta }^{(3)}$. Reversing the mode phases repairs the sign but leaves $+i{\delta }^{(3)}/2$, so it does not repair the normalization. The complex scalar field convention with unit mixed propagator normally has no $1/2$ in its quadratic action.^[ref]

To retain the actual action in the paper, use $\varphi ={\varphi }_1+i{\varphi }_2$ with canonically normalized real components, or equivalently replace its mode expansion byThe corresponding path integral with the usual Feynman i-epsilon prescription haswith the two same-charge propagators still zero. This action-consistent kernel is used in b and c. Keeping conventional unit modes instead would require removing the quadratic factors $1/2$ from the action as well as reversing the phases. Those are distinct theories unless the field and its coupling are rescaled together.