Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 3

For an infinitesimal spacetime-dependent parameter, the action variation has the Noether current formIn canonical quantization, apply the corresponding transformation inside an equal-time correlation function. Integrating the resulting Ward-Takahashi identity over a thin time interval around an insertion givesComparison with $\delta {\phi }_m=i{\alpha }^a{T}_{mn}^a{\phi }_n$ yieldsin the conventions of the question. Current conservation makes $Q^a$ time independent when the surface flux vanishes, so the same charge generates the symmetry at every time.

A symmetry generated by $Q^a$ is spontaneously broken when the vacuum is not invariant:A local order parameter is obtained by finding a field ${\phi }_m$ for whichwith the sign depending on the commutator convention. This criterion is useful in infinite volume because it can remain finite even when the spatially integrated charge itself is not a well-defined normalizable operator.

For an unbroken internal symmetry, $[H,Q^a]=0$ and $Q^a|0⟩=0$. If $H|n⟩=E_n|n⟩$, then $Q^a|n⟩$ is either zero or another state of the same energy, so particles fill degenerate symmetry multiplets.

In a spontaneously broken phase, applying the finite symmetry changes the vacuum and moves to a distinct infinite-volume superselection sector. The charge is not a well-defined finite-norm operator within one such sector, so the usual argument that $Q^a|n⟩$ is a physical partner state can fail; particle masses need not occur in multiplets of the broken group. Formally, however,Thus the broken directions are degenerate with the vacuum. In a local relativistic theory these zero-energy, long-wavelength excitations become the Goldstone bosons.

Goldstone's theorem states that every spontaneously broken generator of a continuous global internal symmetry gives a massless scalar excitation in a Lorentz-invariant quantum field theory, subject to the standard locality and positivity assumptions.

Choose a local field $\phi$ with $⟨0|[Q^a,\phi (0)]|0⟩\ne 0$ and write $Q^a=\int d^{3}x,J_0^a$. Translation invariance givesLorentz covariance forces the contribution of a spin-zero intermediate state to have ${\rho }_{\mu }(p)=p_{\mu }\rho (p^2)$. Current conservation then implies $p^2\rho (p^2)=0$. The nonzero equal-time integral demanded by the order parameter rules out $\rho =0$, so the spectral density must contain support at $p^2=0$. Hence there is a massless one-particle pole withwhich is the required Goldstone boson. Independent broken generators give independent massless modes in the ordinary relativistic case.

Up to a vacuum-energy constant and the topological Abelian $F\tilde{F}$ term, the most general renormalizable Abelian Higgs model isIt has three local dynamical parameters, $g,m^2,\lambda$. Stability requires $\lambda >0$, and spontaneous symmetry breaking occurs for $m^2<0$. Writingand choosing unitary gauge removes $\pi$. The physical masses areThe interactions include $g^{2}vh{A}_{\mu }{A}^{\mu }$, $g^2{h}^2{A}_{\mu }{A}^{\mu }/2$, $-\lambda v{h}^3$, and $-\lambda h^4/4$. The original massless vector's two polarizations and the complex scalar's two real components become a massive vector with three polarizations and one massive scalar. This rearrangement, in which the gauge field absorbs the would-be Goldstone mode and acquires mass without explicitly breaking gauge invariance, is the Higgs mechanism.