Recovery of Local QFT and General Relativity

A pregeometric model succeeds only if one controlled phase reproduces both local quantum field theory and general relativity. Recovering a metric-like graph without local operators is insufficient; recovering relativistic quasiparticles on a fixed graph without universal dynamical gravity is also insufficient. The decisive consistency condition is that propagation, causal commutators, clocks, and the gravitational response identify the same effective geometry.

Required background. Quantum Information supplies operator algebras, channels, code subspaces, and state-dependent reconstruction.

Helpful background. Flat-Spacetime Quantum Field Theory supplies locality, particles, and scattering; Einstein Equations supplies the nonlinear infrared target; Quantum Gravity supplies comparison standards; and Causality and Global Structure supplies domains of dependence and global causal tests.

Approximate local quantum field theory

In continuum QFT, spacelike-separated gauge-invariant observables commute:

$$[\mathcal O(x),\mathcal O(y)]=0 \quad\text{when}\quad (x-y)^2\gt0$$

with the $(-,+,+,+)$ convention. A microscopic model need not possess points $x$ or exact commutativity, but its low-energy algebra should organize into approximately local subalgebras. A quantitative statement specifies an emergent separation $L$, a low-energy code subspace, and a bound such as

$$\lVert[\mathcal O_A,\mathcal O_B]\rVert \le C e^{-L/\xi}+\varepsilon_{\mathrm{UV}}.$$

One must then recover cluster decomposition, stable quasiparticles or conformal operators, and a vacuum or thermal state with the expected correlation structure. The bound should follow from microscopic dynamics, not from placing the degrees of freedom on the desired background lattice.

Gauge constraints complicate factorization. Physical operator algebras may have centers or require edge data, so “one tensor factor per region” can be the wrong locality criterion. Relationally dressed gravitational observables are never strictly compactly supported; the appropriate claim is controlled approximate locality in a specified dressing and regime.

Lorentz symmetry and one causal cone

A discrete substrate often selects a frame. Low-energy Lorentz symmetry requires more than a nearly linear dispersion relation for one field: limiting speeds, polarization structures, interaction vertices, and radiative corrections must approach a common invariant form. Microscopic Lorentz violation can generate low-energy operators without suppression by powers of energy over the microscopic scale. In a Yukawa-theory example, the correction to the relative coefficients of temporal and spatial kinetic terms scales with the squared interaction coupling even as the preferred-frame cutoff grows Collins et al. 2004, appendix A, eq. (A.2). This is a warning about radiative stability, not a theorem that every discrete model violates Lorentz symmetry: a proposed protection mechanism must be checked with interactions included.

The cone inferred from commutator support should agree with that inferred from wave-packet propagation and with null directions of the reconstructed metric. Matter species must share it to the required accuracy. If gravity propagates on a different cone, the construction has not recovered the equivalence principle even if each sector separately looks relativistic.

Corrections should be stated as an expansion in $E/M_*$, lattice spacing over wavelength, inverse bond dimension, or another controlled parameter. Merely observing numerical isotropy at one resolution is not a continuum limit.

Massless spin two and universal coupling

In a Lorentz-invariant S-matrix with a massless spin-2 particle, decoupling of unphysical polarizations constrains the soft emission factor. For external momenta $p_i$ and soft graviton polarization $\varepsilon_{\mu\nu}$,

$$\mathcal M_{n+1}\sim \frac{\kappa}{2}\sum_i\eta_i \frac{p_i^\mu p_i^\nu\varepsilon_{\mu\nu}} {p_i\cdot q}\,\mathcal M_n.$$

Here $\kappa=\sqrt{32\pi G}$, $q$ is the outgoing soft momentum, and $\eta_i=+1$ for outgoing hard particles and $-1$ for incoming ones, with their $p_i$ future directed. Symmetrization has weight one half. Gauge invariance under $\varepsilon_{\mu\nu}\mapsto\varepsilon_{\mu\nu}+q_{(\mu}\xi_{\nu)}$ requires $\sum_i\eta_i p_i^\mu=0$ with one common gravitational coupling. This is the soft-graviton route to universal coupling Weinberg 1964, pp. B1050–B1052.

Universal linear coupling is not yet nonlinear general relativity. The emergent theory must reproduce Hamiltonian and momentum constraints, two propagating tensor polarizations, diffeomorphism redundancy, and the self-coupling of gravitational energy. Extra scalar or vector modes must decouple or be phenomenologically bounded. A composite spin-2 excitation also faces assumptions behind no-go results—Lorentz covariance, a conserved local stress tensor, and particle interpretation—which must be checked rather than invoked as slogans.

Nonlinear and semiclassical checks

The effective action should reduce to

$$\Gamma_{\mathrm{IR}}= \frac{1}{16\pi G}\int d^4x\sqrt{-g}(R-2\Lambda) +\Gamma_{\mathrm{matter}}[g,\psi] +\Delta\Gamma,$$

with, for example,

$$\Delta\Gamma= \frac{1}{16\pi G}\int d^4x\sqrt{-g}\left[ \frac{a_1}{M_*^2}R^2 +\frac{a_2}{M_*^2}R_{\mu\nu}R^{\mu\nu} +\cdots\right] +\Delta\Gamma_{\mathrm{matter}}.$$

Here $a_i$ are dimensionless coefficients, and for $E\ll M_*$ these higher-dimension operators generate relative corrections of order $E^2/M_*^2$ in generic low-energy observables. The action must use one metric and yield conserved total stress energy. Recovering only linearized waves does not test black-hole solutions, cosmological backgrounds, constraint closure, or backreaction. Conversely, fitting a background equation does not establish the propagating quantum sector.

Jacobson’s equation-of-state derivation shows how universal local assumptions can force Einstein dynamics Jacobson 1995, pp. 1260–1263. For an emergent model, the stronger task is to derive those assumptions and $G$ from the microscopic state. Independent checks include the Newtonian potential, soft-graviton factor, gravitational-wave polarizations, Bianchi consistency, and matter free fall.

Results and open questions through 2026-08-20

This discussion reflects results available through 2026-08-20. Individual ingredients have convincing models, but no generic emergence mechanism is known to recover the entire package. Competing positions emphasize universality of massless spin-2 dynamics, the restrictive nature of no-go assumptions, or the possibility that locality and Lorentz symmetry are only approximate relational properties. The joint reconstruction remains unresolved and formulation-dependent.

Recovery target	Evidence or result	Scope and limitation
Approximate local algebras	Lattice systems, networks, and codes exhibit bounded influence and reconstructible regions	Background graph or factorization is often assumed
Lorentz-invariant excitations	Critical points and symmetry protection can generate relativistic dispersion	Common cones and radiative stability across all sectors are harder
Massless spin-2 universal coupling	Soft consistency strongly constrains a Lorentz-invariant S-matrix	Assumes an S-matrix regime and does not alone derive nonlinear dynamics
Einstein infrared action	Thermodynamic and effective arguments explain universality	Microscopic derivation of state, $G$, and controlled corrections is incomplete

The conclusions could change if one pregeometric model derives approximate local algebras, a protected common Lorentz cone, universal massless spin-2 coupling, and closed nonlinear constraints in one continuum limit, or if a no-go theorem excludes that package under its actual assumptions; the interpretation could also change with a controlled mismatch between the reconstructed cones.

Exercise

Suppose two emergent matter species have dispersions $E_a^2=c_a^2p^2+m_a^2$ with $c_1=1$ and $c_2=1+\delta$, while the spin-2 mode has speed one. Why is taking $p\to0$ not enough to recover one metric, and what must be shown?

Solution

For a massive species the rest term dominates as $p\to0$, hiding the limiting speed. At momenta large compared with the mass but below the microscopic cutoff, species 2 approaches speed $1+\delta$ and therefore defines a different cone when $\delta\ne0$. One must show that $\delta$ vanishes or is bounded and flows to zero under interactions, that radiative corrections do not regenerate species-dependent lower-dimension terms, and that operational commutators and the spin-2 propagation use the same cone. Matching one low-momentum energy value is not a Lorentz-symmetry test.

References

• Collins, John, Alejandro Perez, Daniel Sudarsky, Luis Urrutia, and Héctor Vucetich (2004), “Lorentz Invariance and Quantum Gravity: An Additional Fine-Tuning Problem?,” Physical Review Letters 93, 191301, DOI: 10.1103/PhysRevLett.93.191301. arXiv:gr-qc/0403053.
• Jacobson, Ted (1995), “Thermodynamics of Spacetime: The Einstein Equation of State,” Physical Review Letters 75, 1260–1263, arXiv:gr-qc/9504004.
• Weinberg, Steven (1964), “Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass,” Physical Review 135, B1049–B1056, DOI: 10.1103/PhysRev.135.B1049.