13.2.  Path-integral quantisation of the projected theory

The Euclidean modal functional $\mathcal{F}[\psi,C]$, equipped with the Bernoulli product measure $\mu$ derived in §2.5, defines a substrate partition function:

$$Z_{\mathrm{sub}}\;=\;\int_{\mathcal{C}}\exp\!\bigl{(}-\mathcal{F}[\psi(C),C]\bigr{)}\,d\mu(C).$$ (153)

The functor $\Phi:\mathbf{S}\to\mathbf{G}$ maps each configuration to an instantiated spacetime geometry; the F–S dichotomy of Chapter 7 establishes that this is the precausal counterpart of Wick rotation. The instantiated partition function is therefore:

$$Z_{\mathrm{QFT}}\;=\;\int\mathcal{D}\psi\,\mathcal{D}g_{\mu\nu}\,\exp\!\bigl{(}iS[\psi,g]\bigr{)},$$ (154)

with $S=S_{\mathrm{EH}}+S_{\mathrm{matter}}$ the Einstein–Hilbert plus matter action of Chapter 7. This is the path integral of quantum gravity coupled to the modal field. The substrate grain $\ell_{*}=\hbar c/M_{*}\approx 1.5\times 10^{-19}$ m provides a physical ultraviolet cutoff $\Lambda_{\mathrm{UV}}=1/\ell_{*}$: not a calculational regularisation but a genuine breakdown scale of the continuum description. Renormalisability is the statement that low-energy observables are insensitive to $\ell_{*}$, which holds for separations $\gg\ell_{*}$ exactly as the Casimir analysis requires.

Expand around the vacuum $\psi=r_{0}e^{i\theta_{0}}$, and the quadratic fluctuation action $S_{2}$ of equation (123) generates Feynman rules for two modes: a massive scalar (mass $m_{h}^{2}=4\varepsilon$, equation (122)) and a massless pseudoscalar (the Goldstone mode [43]). The metric fluctuations $h_{\mu\nu}$ supply graviton propagators alongside them. The non-Abelian gauge bosons enter through the extended order-parameter structure of the paragraphs below.