12.1.  The projected field theory: emergence of quantum fields

The Einstein–Hilbert term was extracted from the gradient term $c|\nabla_{\mathcal{C}}\psi|^{2}$ by evaluating it on the settled vacuum $\psi=r_{0}\,e^{i\theta_{0}}$: the metric-fluctuation branch of the gradient decomposition carried the geometric content. That decomposition has a second branch. The same gradient term also governs fluctuations of the order parameter about the vacuum, and it is this branch, not the vacuum value but the spectrum of excitations around it, that projects into quantum field theory. Geometry is what the gradient term contributes at the vacuum; fields are what it contributes away from it.

Writing the order parameter near a point on $\mathcal{V}$ as

$$\psi(\mathbf{x})=\bigl{(}r_{0}+h(\mathbf{x})\bigr{)}\,e^{i(\theta_{0}+\theta(\mathbf{x}))},$$ (121)

and retaining terms through quadratic order in the fluctuations $h$ and $\theta$, the gradient term gives $(\nabla h)^{2}+(r_{0}+h)^{2}(\nabla\theta)^{2}$, and the potential $-\varepsilon|\psi|^{2}+b|\psi|^{4}$ contributes a curvature in the radial direction and nothing in the phase direction (the flat direction along $\mathcal{V}$). The radial curvature at $r_{0}$, with $r_{0}^{2}=\varepsilon/(2b)$ from equation (78), is

$$m_{h}^{2}=\mathcal{F}^{\prime\prime}(r_{0})=4\varepsilon,$$ (122)

so the radial mode is massive with a mass set by the excess tension $\varepsilon=|T(C)|-\tau$, while the phase mode is exactly massless. This is the precise structural realisation of the qualitative claim that mass measures distance from the threshold: deeper instantiation ($\varepsilon$ larger) produces a heavier radial excitation, while the Goldstone phase mode sits at zero mass because motion along $\mathcal{V}$ costs nothing.

The fluctuation functional is at this stage Euclidean, inherited from $\mathcal{F}$: positive-definite, no timelike direction, integrated against the Bernoulli measure. It is not yet a field theory in the dynamical sense. The functor $\Phi$ supplies what is missing. As established by the F–S Dichotomy above, sublimation is the PST instance of Wick rotation: the moment the $(+,-,-,-)$ signature emerges from $\Phi$ is the moment the Euclidean quadratic functional becomes a Lorentzian quadratic action,

$$S_{2}[h,\theta]=\int_{M}\left[\tfrac{1}{2}(\partial h)^{2}-\tfrac{1}{2}m_{h}^{2}\,h^{2}+\tfrac{1}{2}r_{0}^{2}\,(\partial\theta)^{2}\right]\sqrt{-g}\;d^{4}x+\ldots,$$ (123)

where the ellipsis denotes interaction terms from $b|\psi|^{4}$. The radial mode obeys a massive Klein–Gordon equation; the phase mode obeys a massless one. Their propagators are the inverses of the respective quadratic operators, with the $i\varepsilon$ prescription not imposed by hand but fixed by the Lorentzian signature that $\Phi$ produces. Poincaré invariance of $S_{2}$ in the Minkowski limit is not a separate postulate: it is the flat-space restriction of the diffeomorphism invariance derived in Chapter 4. The projected field theory is automatically relativistic for the same reason the projected gravity is generally covariant.

Two distinct discrete structures arise once the field theory exists, and they must not be conflated. The first is topological. The vacuum manifold $\mathcal{V}\cong S^{1}$ has fundamental group $\pi_{1}(S^{1})=\mathbb{Z}$, so the phase $\theta$ can wind around $\mathcal{V}$ only an integer number of times: a configuration cannot return to itself across a fractional winding without discontinuity. This integer is the $\mathrm{U}(1)$ charge, and it is the origin of the discrete charge spectrum $L\in\hbar\mathbb{Z}$: the vacuum-manifold discussion above supplies the integer $n$ from $\pi_{1}(\mathcal{V}_{\hat{\eta}})$, while the unit $\hbar$ enters with canonical quantisation (§10.6) rather than being derived; the charge is angular momentum in the far-threshold regime and spin near it. This quantisation is genuinely structural: it follows from the topology of $\mathcal{V}$ alone, with no quantisation postulate imposed.

The second discrete structure is the Fock space [176], and it is a separate matter. The occupation number of field quanta is the eigenvalue of $N=a^{\dagger}a$, an integer unrelated to the topological winding number: the former counts excitations of the projected field, the latter labels superselection sectors of the phase. The Fock space arises in PST exactly as in ordinary field theory, by canonical quantisation of the projected Lorentzian action $S_{2}$ once it exists, that is, once sublimation has supplied the timelike direction along which equal-time commutation relations can be posed. PST therefore derives the arena in which canonical quantisation takes place and the charge spectrum the resulting states must carry. The canonical commutation relations $[\hat{x}_{\mu},\hat{p}_{\nu}]=i\hbar\,\delta_{\mu\nu}$ are derived in §10.6 from the Boolean gradient; the Born rule is derived in §10.2. The existence of the fermionic sector follows structurally from the Boolean–canonical-anticommutation-relations (CAR) isomorphism (§11.2). The functional-analytic treatment of the fermionic Fock space rests on the three closures of §11.3: CAR-to-continuum Mosco convergence, the Wightman axioms, and the substrate-derived Dirac equation. Two of those carried open residuals, the slice-wise CAR limit and microcausality W4; both now hold given A1 to A4 with A3 supplied by P3, the standing of equicoercivity A7, so the sector is closed on the Mosco chain’s terms (Appendix C). The statistics rule is a structural theorem from the Boolean exclusion of P1; the spin half is not derived. The narrower content of the present construction is: the projected field theory is a standard Lorentzian field theory carrying a topologically quantised $\mathrm{U}(1)$ charge, and its excitation spectrum is discrete for the ordinary reason, not because winding number and occupation number coincide.

The projection operator $\Pi$ carries the intrinsic scale $d_{0}$ derived in Chapter 15. Below $d_{0}$ the continuum description of $\Pi$ fails; above it the smooth field theory holds. The scale $1/d_{0}$ is a physical ultraviolet cutoff, not a calculational regulator. The divergences of standard quantum field theory (QFT), recovered as $d_{0}\to 0$, are artefacts of taking the continuum limit of a projection that was never continuous at arbitrarily short distances. Renormalisability is the statement that low-energy observables are insensitive to $d_{0}$, and it holds for separations $\gg d_{0}$ (bounded above at $\approx 2.3$ nm), precisely as the Casimir analysis of Chapter 15 requires.

The status of three items not directly delivered by this projected-field-theory construction is summarised explicitly (each is either derived in its own chapter or has a residual open piece, stated below):

The Born rule.

The identification of $|\psi|^{2}$ near threshold with measurement probability is derived from the Bernoulli measure and Gleason’s theorem [54] applied to $L^{2}(\mathcal{P}(D),\mu)$ in §10.2. The full measurement postulate (collapse rule) rests on three closures and one condition of §10.3 (explicit von Neumann measurement Hamiltonian, decoherence inheritance, threshold-crossing uniqueness + irreversibility theorem, and agreement with the Born rule). The fermionic-sector treatment rests on §11.3 (three closures: CAR-to-continuum convergence, Wightman axioms, substrate-derived Dirac equation), two of which hold given A1 to A4 with A3 supplied by P3 (Appendix C).

Spin and statistics.

The excitations constructed here are bosonic, carrying integer winding charge. The fermionic sector and the spin-statistics connection are not obtainable from the single complex order parameter alone; they are derived separately in Chapter 11 from the canonical anticommutation relations (CAR) algebra on the substrate’s Boolean distinction primitive, where the half-integer-spin excitations inherit the correct anticommutation relations as a structural consequence.

The full interacting theory.

The free-field Fock space and the $\phi^{4}$-type vertices are exhibited; the interacting vacuum of the scalar sector exists nonperturbatively in the regulated theory (Computation 200), and the gauge and fermion sector’s is inherited from the cutoff Standard Model.

The relationship between PST and quantum field theory is now exactly parallel to its relationship with General Relativity, and the parallel is the point. QFT is neither a competitor to PST nor identical to it. It is the Lorentzian projection, through $\Pi$, of the fluctuation spectrum of $\mathcal{F}$ about $\mathcal{V}$, just as General Relativity is the projection of the vacuum value of the same functional. PST is not itself a quantum field theory and could not be: $\mathcal{F}$ is Euclidean, atemporal, and precausal, and a quantum field theory requires the Lorentzian action $S$ that exists only after the threshold is crossed. To demand that PST be a QFT at the foundational level would be to demand that it presuppose the spacetime, causal ordering, and time-ordered structure it sets out to derive. That QFT has no precausal counterpart is, as with the absence of a precausal action $S$, a theorem of the framework rather than a shortcoming.