Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 40 / 1

Use the Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$, $\hslash =c=1$, and $\square ={\partial }_a{\partial }^a=-{\partial }_t^2+{\nabla }^2$. The Proca field has nonzero mass $m$. Vary its action, using the fact that the gauge field strength is an antisymmetric second-rank tensor:The integration by parts discards a boundary term, with the variation fixed at the boundary. Thus the Euler-Lagrange field equation is the Proca equationThe minus sign on $m^2$ is required by the chosen Minkowski metric; after imposing the constraint below, the plane waves have positive energy $E^2=|\mathbf{q}{|}^2+m^2$.

Take the four-divergence of the Proca equation. Commuting partial derivatives and using that $F^{ab}$ is an antisymmetric second-rank tensor giveConsequently, for $m\ne 0$, ${\partial }_a{A}^a=0$. This is the Lorenz constraint in Proca theory: an equation of motion enforces it, rather than a choice of gauge fixing. It leaves $(\square -m^2)A^a=0$. At $m=0$ the divergence argument supplies no such constraint; the massless gauge symmetry requires a separate treatment.

Choose the Fourier transform conventionThen ${\partial }_a$ becomes $i{q}_a$. Applying this to the Lorenz constraint in Proca theory yieldsThus every physical Fourier transform mode is orthogonal to its four-momentum in the Minkowski metric.

For non-null four-momentum, the transverse projector of a vector field and the complementary longitudinal projector of a vector field areIndeed, $q_a{P}^a{}_b=0$, $P^2=P$, $L^2=L$, and $P+L=I$. Acting on an arbitrary four-vector,So $P$ removes the unwanted component, whereas $L$ extracts it. On the massive mass shell, $q^2=-m^2$ and $P^a{}_b={\delta }_b^a+q^a{q}_b/m^2$. This on-shell form should not be used as an off-shell linear projection: away from $q^2=-m^2$ it is not idempotent. The non-null hypothesis matters; this decomposition is undefined at $q^2=0$.

Put $q^a=(E,0,0,k)$ with $E=\sqrt{k^2+m^2}$. An orthonormal real basis of polarization vectors isEach polarization vector obeys $q_a{ϵ}_{\lambda }^a=0$. The Minkowski metric gives ${ϵ}_{\lambda }\cdot {ϵ}_{{\lambda }^{\prime }}={\delta }_{\lambda {\lambda }^{\prime }}$, since $E^2-k^2=m^2$. The first two are transverse to the spatial momentum; the longitudinal polarization of a massive vector boson is spatially longitudinal but still orthogonal to the full four-momentum. The polarization sum for a massive vector boson isIn the rest frame all three polarization vectors are spatial unit vectors. Their three independent positive-norm states are the physical spin states of a massive spin-one particle.

There are two closely related objects to distinguish. Inverting the quadratic Proca action gives the usual covariant Proca propagator. After integration by parts, its kernel isThe matrix inverse with the Feynman i-epsilon prescription givesMultiplication by $K^{ab}$ gives $i{\delta }_b^a$ in the distributional limit. The numerator agrees with the polarization sum for a massive vector boson at the poles, but is not a transverse linear projection at arbitrary four-momentum.

For the literal canonical time-ordered product of $A_a$ and $A_b$, the nondynamical component $A_0$ produces the Proca time-ordering contact term. In the chosen time coordinate the full answer isTo see the local term directly, the three physical polarization vectors give ${\sum }_{\lambda }{ϵ}_{\lambda 0}^2=|\mathbf{q}{|}^2/m^2$. The canonical $00$ two-point correlation function therefore has Fourier transformBy contrast, ${\eta }_{00}+q_0^2/m^2=|\mathbf{q}{|}^2/m^2-(q^2+m^2)/m^2$, so the covariant expression contains an additional $i/m^2$. The mixed and spatial components have no additional contact term. ThusIf $T$ is used to mean covariant time ordering, commonly denoted $T^{\ast }$, the conventional answer is instead just ${\Delta }^{\mathrm{cov}}$. Both conventions have the same propagating poles and agree away from coincidence; explicitly separating them respects the printed definition as an ordinary time-ordered product.