Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 (Quantum Field Theory)

A continuous symmetry is a family of transformations depending continuously on parameters for which the action is unchanged, possibly up to a spacetime boundary term. For an infinitesimal field variation $\delta \varphi =ϵ\Delta \varphi$, a Noether current $j^{\mu }$ obeyswhen the Euler-Lagrange field equation holds. Its Noether chargeis conserved provided the flux ${\int }_{S_{\infty }}{j}^i{n}_{i}dS$ vanishes.

Noether theorem gives the implication from a differentiable global variational symmetry to an on-shell conserved current. A conserved current gives a conserved charge only under suitable boundary and convergence conditions. Conversely, a conserved charge generates a continuous symmetry through Poisson brackets classically or a commutator quantum mechanically when a regular Hamiltonian formulation exists. Currents can be changed by identically conserved improvement terms, and inverse Noether statements require regularity and the exclusion of such trivial currents, so the three notions are related but not literally in one-to-one correspondence.

For spacetime translations, the canonical stress-energy tensor isForraising the second index gives the symmetric tensorTranslation invariance gives four conserved currents $T^{\mu \nu }$, one for each fixed $\nu$, and the four conserved charges are the four-momentum$P^0$ is the energy and $\mathbf{P}$ is the spatial momentum.

The Lorentz transformation variation of a scalar is generated solely by its argument. The corresponding Lorentz current isUsing ${\partial }_{\mu }{T}^{\mu \nu }=0$ and symmetry of $T^{\rho \sigma }$,In particular, conservation of the boost chargeimpliesThe quantity denoted $V^i$ in the question is therefore the conserved momentum component $P^i$, assuming the same vanishing boundary flux.

The interaction $-g\varphi {\Psi }_i^{\ast }{\Psi }_i$ gives one cubic vertex $-ig$ joining a real-scalar line to a particle-antiparticle pair of either complex scalar species. The second-order term of the Dyson series isWick theorem contracts the incoming ${\Psi }_1{\overline{\Psi }}_1$ pair at one vertex, the outgoing ${\Psi }_2{\overline{\Psi }}_2$ pair at the other, and the two $\varphi$ fields with each other. The two assignments of $x,y$ cancel the factor $2!$. Thus there is one connected tree-level Feynman diagram, the $s$-channel exchangeWith standard relativistic external-state normalization,up to the physically irrelevant common sign convention for $\mathcal{M}$.

In the center-of-momentum frame, $q=(E_1,-\mathbf{p})$ and $p=(E_1,\mathbf{p})$, soPair creation is kinematically possible exactly when $s\ge 4M_2^2$. The threshold incident momentum is therefore

The Dirac spinor representation is the four-dimensional representation $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ of the connected Lorentz group. It is characterized byForone may takewith the signs of ${\Omega }_{\rho \sigma }$ fixed by the displayed vector-representation convention. This formula follows by differentiating the covariance relation and using the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$; moving both gamma indices upstairs changes the apparent parameter signs according to the metric.

Use the Weyl representation of the gamma matrices,

The displayed $R(\phi )$ rotates the coordinate components in the $xy$ plane by angle $-\phi$. Sinceits spin representative isAt a full turn,Thus a $2\pi$ spatial rotation changes the sign of a spin-one-half state, while a $4\pi$ rotation returns it to itself. This realizes the fact that the Spin group is a double cover of the proper orthochronous Lorentz group; observable spinor bilinears are unchanged by the sign.

For the displayed boost,At rest,For $p_2=(m\cosh \beta ,m\sinh \beta ,0,0)$, the Pauli matrix identity $e^{\beta {\sigma }^1}=\cosh \beta I_2+\sinh \beta {\sigma }^1$ givesTaking the positive matrix square roots,as required.

A gauge symmetry is a local redundancy in the fields used to describe one physical state. With metric signature $(+---)$, defineThe quantum electrodynamics Lagrangian isUnder the local U(1) gauge symmetryone has $D_{\mu }\psi \mapsto e^{-ie\chi }{D}_{\mu }\psi$ and $F_{\mu \nu }\mapsto F_{\mu \nu }$. Both terms in the Lagrangian are therefore invariant.

In Feynman gauge, the momentum-space QED Feynman rules are:

One integrates each undetermined loop momentum, includes a factor $-1$ for each closed fermion loop, and imposes overall momentum conservation. None of the following tree diagrams contains a loop.

LabelThis is Bhabha scattering. There are two diagrams:

With $s=(p_1+p_2{)}^2$ and $t=(p_1-p_3{)}^2$, a consistent external-fermion ordering givesThe displayed relative minus sign is the fermionic sign from putting the external fermion operators into the common chosen order.

LabelThere are two electron-exchange diagrams, distinguished by which labelled outgoing photon is emitted first along the fermion line. They are the $t$- and $u$-channel topologies. Their sum isThe two terms are added because the external photons are identical bosons. Replacing either polarization by its momentum makes their sum vanish by the Ward identity.

Label positron Compton scattering asThere are two positron-exchange diagrams: the $s$-channel ordering with internal momentum $-p-k$, and the $u$-channel ordering with internal momentum $-p+k^{\prime }$. Reading opposite to the fermion arrow givesIt is the crossed form of electron Compton scattering and again satisfies the Ward identity only after both orderings are included.