13.3.  Local gauge symmetry: the photon as a connection

The global $\mathrm{U}(1)$ symmetry $\psi\mapsto e^{i\alpha}\psi$ is a symmetry of the action $S_{2}$ with $\alpha$ a single number. But the projection $\Pi$ maps the continuum of substrate configurations to the full manifold $(M,g)$: different loci of $M$ are images of different configurations $C$, each of which independently selects a vacuum phase $\theta(x)\in\mathcal{V}\cong S^{1}$. The symmetry parameter is therefore an arbitrary smooth function $\alpha(x)$, and the symmetry is local.

When $\alpha$ depends on $x$, the kinetic term $r_{0}^{2}\,\partial_{\mu}\theta\,\partial^{\mu}\theta$ is not invariant: under $\theta(x)\mapsto\theta(x)+\alpha(x)$ the derivative picks up $\partial_{\mu}\alpha(x)$. The geometric resolution is that $\theta(x)$ is a section of a $\mathrm{U}(1)$ principal bundle over $M$, and the correct derivative is the gauge-covariant one $D_{\mu}\theta=\partial_{\mu}\theta-A_{\mu}$, where $A_{\mu}(x)$ is the connection. Under $A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\alpha(x)$, the covariant derivative is invariant. The field strength $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the curvature of the connection, and the gauge kinetic term

$$S_{\mathrm{Maxwell}}\;=\;-\frac{1}{4e^{2}}\int_{M}F_{\mu\nu}F^{\mu\nu}\sqrt{-g}\;d^{4}x,$$ (155)

is supplied by the inner fluctuations of the spectral action (Computation 6), not by integrating out $\theta$.

What integrating out $\theta$ actually gives, and which boson this is. Expanding $r_{0}^{2}D_{\mu}\theta\,D^{\mu}\theta$ produces $r_{0}^{2}A_{\mu}A^{\mu}$, and that term is gauge invariant precisely because $\theta$ is there to be eaten: $(\partial\theta-A)$ is itself invariant, and in unitary gauge $\theta=0$ nothing is left over. This is the Stueckelberg form of the Abelian-Higgs mechanism, and it gives the connection a mass $m_{A}^{2}=2e^{2}r_{0}^{2}$ at the condensate scale. So the boson gauged here is not the photon. The premise of this section is $|\psi|=r_{0}\neq 0$, that is, that the $\mathrm{U}(1)$ acting on $\theta$ is broken, and a gauged broken $\mathrm{U}(1)$ gives a massive vector, which is a superconductor rather than electromagnetism (§9.2). Once the electroweak embedding of §13.4 is in place, this is the $Z$ (Computation 172).

The photon is the connection of the unbroken combination $Q=T_{3}+Y/2$, massless because $Q$ annihilates the vacuum ((186), §13.4). That is the Standard Model’s own unbroken-generator argument, which PST inherits; and there $Q$’s annihilation of the vacuum is what fixes $Y(H)=+1$, so masslessness is an input to that fit rather than an independent output of it. Minimal coupling of charged fields via $D_{\mu}$ follows from gauge covariance of the projected kinetic terms (Computation 166). The gauging is not an additional postulate: it is the consequence of $\Pi$ assigning an independent vacuum phase to each spacetime point.

13.3.1.  Minimal coupling is forced, and this is the algebra

“Follows from gauge covariance” is an assertion until someone does the variation, and the variation says something sharper than the phrase does. Computation 166 carries it out with explicit Dirac matrices and a completely general spinor.

The variation is one term, and it names the current. Under $\psi\mapsto e^{i\alpha(x)}\psi$ the free Dirac Lagrangian changes by

$$\delta\mathcal{L}\;=\;-\,\bigl{(}\bar{\psi}\gamma^{\mu}\psi\bigr{)}\,\partial_{\mu}\alpha,$$ (156)

exactly, with no other term. The object that appears is the vector current, so only a counterterm built on that same current can cancel it. That is already the uniqueness statement: the coupling is not chosen from a list of gauge-invariant candidates, it is read off the variation.

And the transformation law is an output. Writing the counterterm as $-e\,(\bar{\psi}\gamma^{\mu}\psi)A_{\mu}$ and asking what shift of $A_{\mu}$ cancels (156) returns $A_{\mu}\mapsto A_{\mu}-e^{-1}\partial_{\mu}\alpha$, and returns it uniquely. Neither the form of the vertex nor the gauge transformation rule is put in by hand; both fall out of requiring the phase to be local.

The one alternative fails twice. The other dimension-four candidate is the axial coupling $-e\,(\bar{\psi}\gamma^{\mu}\gamma^{5}\psi)A_{\mu}$. It cannot cancel (156), because that variation carries the vector current and this term carries a different one. And it could not source a gauge field in any case: $\partial_{\mu}\partial_{\nu}F^{\mu\nu}$ vanishes identically by antisymmetry, so $\partial_{\mu}F^{\mu\nu}=e\,j^{\nu}$ admits only a conserved source, and on shell the two currents behave differently,

$$\partial_{\mu}\bigl{(}\bar{\psi}\gamma^{\mu}\psi\bigr{)}=0,\qquad\partial_{\mu}\bigl{(}\bar{\psi}\gamma^{\mu}\gamma^{5}\psi\bigr{)}=2im\,\bar{\psi}\gamma^{5}\psi,$$ (157)

both verified against the explicit spinor. The axial current is not conserved for $m\neq 0$, and is therefore not available.

What this fixes is the form of the electromagnetic vertex, and that is all it fixes. The strength is not supplied. An earlier version wrote $e^{-2}\propto r_{0}^{2}$, which cannot hold: $1/e^{2}$ is dimensionless in four dimensions while $r_{0}^{2}$ carries $[\text{mass}]^{2}$, and in $D$ dimensions the two carry $[\text{mass}]^{D-4}$ and $[\text{mass}]^{D-2}$, so the proportionality is impossible in every dimension. The combination $e^{2}r_{0}^{2}$ that does carry $[\text{mass}]^{2}$ is the Stueckelberg mass of §13.3, not an inverse coupling. So the vertex’s form is forced (Computation 166) and its magnitude is an open input; the photoelectric cross-section computed from it (Computation 189) inherits that.