Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 2

The normalized Fock vacuum obeys $a_{\mathbf{p}}|0⟩=0$ for every momentum and $⟨0|0⟩=1$. Taking adjoints gives $⟨0|a_{\mathbf{p}}^{†}=0$. Of the four products in the mode expansion of $\varphi (x)\varphi (y)$, only $a_{\mathbf{p}}{a}_{\mathbf{q}}^{†}$ has a nonzero vacuum expectation value. The commutator impliesConsequentlyThis unordered Wightman function differs from the time-ordered Feynman propagator when the temporal ordering is reversed.

Write $t=x^0-y^0$ and $\mathbf{r}=\mathbf{x}-\mathbf{y}$. By time ordering for a bosonic field,Reverse $\mathbf{p}$ in the second integral. At fixed spatial momentum considerThe Feynman i-epsilon prescription places the poles at $+E_p-i0$ and $-E_p+i0$. For $t>0$, close in the lower half-plane, clockwise. The positive-energy residue and the orientation give $e^{-i{E}_{p}t}/(2E_p)$. For $t<0$, close counterclockwise in the upper half-plane; the negative pole has derivative denominator $-2E_p$, and its residue gives $e^{i{E}_{p}t}/(2E_p)$. Thus $I(t)=e^{-i{E}_p|t|}/(2E_p)$, reproducing the time-ordered mode expression. We obtainThis is a distributional boundary value, not an absolutely convergent Fourier integral. As an independent normalization check, $I^{\prime }(0^{+})-I^{\prime }(0^{-})=-i$, so $(\square +{\mu }^2){\Delta }_F=-i{\delta }^{(4)}$; the numerator is $i$ with the question's definition of the propagator.

The Yukawa interaction has one real-scalar leg and an oriented fermion line at each vertex. At lowest connected order, two vertices exchange one scalar, so the diagrams are of order ${\lambda }^2$.

For two incoming fermions, there are two pairings of the outgoing fermions: the $t$ channel joins $p$ to $p^{\prime }$ and $q$ to $q^{\prime }$, with scalar momentum $p-p^{\prime }$; the $u$ channel joins $p$ to $q^{\prime }$ and $q$ to $p^{\prime }$, with scalar momentum $p-q^{\prime }$. Fermion number conservation excludes an annihilation channel for two particles carrying the same charge.

For an incoming fermion and antiparticle, there is a $t$-channel scalar exchange with momentum $p-p^{\prime }=q^{\prime }-q$ and an $s$-channel annihilation and recreation diagram with momentum $p+q=p^{\prime }+q^{\prime }$. Arrows on antiparticle lines run opposite to the direction of physical propagation. These are the four tree scattering in Yukawa theory diagrams shown below; all external momenta are labelled.

The crossing in the $u$-channel drawing is not a vertex. Dashed internal lines are the real scalar field; solid arrows show fermion-number flow.

Use the Feynman vertex $-i\lambda$ for the scalar Yukawa coupling, the internal scalar propagator $i/(k^2-{\mu }^2+i0)$, incoming factors $u_r(p)$ and outgoing Dirac adjoints ${\bar{u}}_{r^{\prime }}(p^{\prime })$. Define $i\mathcal{M}$ as the coefficient in the connected S-matrix element after removing its overall momentum-conserving Dirac delta function. A direct pairing contributesThe crossed pairing interchanges the identical outgoing fermions and therefore has a relative fermionic sign of minus. With $t=(p-p^{\prime }{)}^2$ and $u=(p-q^{\prime }{)}^2$, the result isIt changes sign under exchange of the two outgoing momenta and spin labels, as required for identical fermions. No fermion spin sum or identical-particle phase-space factor belongs in this amplitude; those enter a relativistic scattering cross-section calculation.