PART V. A Compatibility Program: Quantum Fields, Quantum Clocks, and Quantum Gravity
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12. The CTGP Admissibility Criterion
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Parts II–IV establish compatibility for a finite, non-relativistic model with spatial quantum
reference frames. Part V asks how the same commitments can be carried toward relativistic quantum
field theory (RQFT), temporal quantum reference frames, and quantum gravity. It does not redesign
CTGP. It states the commitments to be preserved, turns them into a single admissibility criterion,
proves the results that can be proved now in finite form, and identifies precisely what would have
to be established, and what would count as an obstruction, in each of the three directions. Results
in this Part are labeled by status: theorems and propositions are proved; definitions marked as
proposals are offered for adoption rather than derived; conjectures and target results are open.
12.1 The Central Principle and the Commitments to Preserve
The organizing principle of the program is this:
CTGP requires one objectively generated physical history. It does not require every observer,
clock, quantum representation, or mathematical formulation to describe that history in the same
way.
The principle is already at work in Parts II–IV: frame descriptions differ, sometimes as product
versus entangled states (§6.3), while determinacy, truth, and the history measure are frame-
independent (Proposition 5.2, Theorems 5.12 and 5.16). It leaves room for relativistic covariance,
for quantum clocks, and potentially for quantum geometry, without abandoning the central ontology.
It does not guarantee success: some quantum-gravity proposals could be fundamentally incompatible
with CTGP, and no rigorous relativistic single-outcome model with CTGP’s structure yet exists.
The commitments to be preserved are those of §2.3, grouped as follows.
(P1) Presentism (C-a): only the currently generated physical configuration exists.
(P2) A single realized history (C-d): there is one factual sequence of physical outcomes.
(P3) Generability (C-f): the complete present physical state and the laws fix the probability
distribution of successors, without indispensable independent future boundary inputs.
(P4) Causal inheritance (C-g): whatever part of history can still affect future physics is
instantiated in present physical variables.
(P5) Relational descriptive freedom (C-j): different valid reference frames may describe the
same physical history differently.
(P6) Emergent classical cosmic time (C-k): in the general-relativistic regime where the required
structure exists, the objective present corresponds to the level sets of a regular cosmological
time function.
Commitment (P6) is conditional on the classical spacetime admitting the required structure, and it
should not be imposed unchanged on a quantum regime in which classical geometry is undefined (§15).
CTGP’s negative result on density-weighted generation (C-l) constrains every realization: the
extension must not make different regions become real at rates set by local matter density. Local
matter, fields, clocks, and stochastic events may evolve differently while belonging to one
underlying generative ordering.
12.2 The Criterion
Write the ontological state at a generative stage σ as Ω_σ = (S_σ, L_σ), with S_σ the complete
physically instantiated state and L_σ its primitive past-tensed truth profile. A physical extension
must supply a forward transition law K_{σ,σ′}(S_σ, dS′ de) for σ′ > σ, where e denotes the realized
event data of the transition.
Definition 12.1 (CTGP-admissible theory; a proposal). A physical theory is CTGP-admissible on a
specified domain if it has a formulation with the following features.
(A1) Present state. A complete, physically instantiated present state S_σ on each stage of a
generative ordering.
(A2) Forward law. Transition kernels K_{σ,σ′} that are normalized and satisfy the composition
relation K_{σ,σ″} = K_{σ′,σ″} ∘ K_{σ,σ′} for σ < σ′ < σ″, with integration over intermediate
states and concatenation of event data.
(A3) One realized history. A single realized sample path of events, with the truth profile
updated from it, L_{σ′} = Update(L_σ, S_σ, e, S_{σ′}).
(A4) No indispensable future input. No independent future boundary condition is required, in the
sense of CTGP – explained, §12.2: future dependence disqualifies only if no empirically
equivalent forward formulation exists (compare §11).
(A5) Causal idleness of the past. Pr(S_{σ′}, e | S_σ, L_σ) = Pr(S_{σ′}, e | S_σ).
(A6) Frame and gauge consistency. Admissible reference-frame and gauge transformations preserve
the probabilities and the identities of realized physical events.
(A7) Classical realization. Wherever the domain includes the classical general-relativistic
regime in which CTGP claims (P6), the generative ordering is realized by the level sets of
regular cosmological time.
Primitive past-tensed truths may be included in the ontology, but by (A5) they supply no causal
input absent from the complete physical state.
Definition 12.1 is a proposal, not a theorem about any interacting relativistic or quantum-
gravitational theory. Its value is that it turns the broad question whether CTGP can be compatible
with quantum physics into a list of precise mathematical obligations.
Proposition 12.2 (The finite model is CTGP-admissible on its domain). The discrete-stage and
continuous-time models of Part III satisfy (A1)–(A6), with (A6) restricted to the spatial QRFs of
§3; (A7) is vacuous on their domain, which contains no classical spacetime.
Proof. (A1): the present state is the ray [Ψ_σ] in H_phys. (A2): the kernels are Markov; in discrete
time, composition is the product of one-step kernels; in continuous time it follows from the
factorization of record densities (Proposition 9.9). (A3): Definition 4.6 and Theorem 4.8;
Definition 9.5 and Theorem 9.7. (A4): §10.2, with §11 for the sense of future dependence. (A5):
Corollary 4.10 and Proposition 9.9. (A6): Theorems 5.12, 5.16, and 9.10. □
12.3 Three Levels of Compatibility
“Compatibility” is used in three distinct senses, and keeping them apart is essential to every claim
in this Part.
Level | What has been established
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Conceptual | No identified logical contradiction under specified assumptions
Mathematical | A well-defined model satisfies CTGP’s stated compatibility conditions
Physical | The model reproduces the required observations and survives empirical tests
Table 3. Three levels of compatibility.
The finite model establishes mathematical compatibility on its own domain (Proposition 12.2). It
does not establish physical compatibility, and a mathematically consistent model can be physically
inadequate. The results below for the relativistic, temporal, and gravitational directions are of
three kinds: finite-dimensional theorems that are analogues of the required results, conditional
propositions, and open targets.
12.4 Ontology Versus Implementation
CTGP’s foundational ontology is what (P1)–(P6) assert. A relativistic flash model, a Bohmian field
theory, or a quantum-gravitational causal structure might each satisfy Definition 12.1 by quite
different physical mechanisms. They are therefore candidate realizations to be investigated, not
components of CTGP itself. The same holds for the ring model of Part III: it is one realization,
chosen because every claim about it can be proved.
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13. Relativistic Quantum Fields: Realized-History Consistency
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13.1 Background Structure
The first construction should not quantize gravity. It should assume a fixed classical spacetime (M,
g) with a regular cosmological time function T(p) = sup { L(γ) : γ a past-directed causal curve from
p }, and take the proposed CTGP presents to be the level sets Σ_t = T^{−1}(t).
The relevant regularity results are those of Andersson, Galloway and Howard (1998), and they must be
stated with care. If T is regular, meaning finite everywhere and tending to zero along every past-
inextendible causal curve, then (M, g) is globally hyperbolic; T is a time function, continuous and
strictly increasing along future-directed causal curves; every point p lies at the future end of a
maximal timelike geodesic of length T(p) issuing from the initial singularity; and T is locally
Lipschitz, with first and second derivatives existing almost everywhere. The nonempty level sets are
closed, acausal, edgeless, and future Cauchy surfaces: their future Cauchy horizons are empty
(Andersson, Galloway & Howard 1998, Corollary 2.6). They need not be Cauchy surfaces in the full,
two-sided sense, and the paper says so explicitly. The following proposition states exactly when
they are.
Proposition 13.1 (When a level set is a Cauchy surface). Let T be regular and let S_a = T^{−1}(a) be
nonempty. Then S_a is a Cauchy surface if and only if every future-inextendible causal curve
contains points at which T > a.
Proof. (⇐) Let γ be an inextendible causal curve. By regularity, T tends to zero toward the past end
of γ, and by hypothesis T exceeds a somewhere on γ. Since T is continuous and strictly increasing
along γ, γ meets S_a exactly once. Every inextendible causal curve therefore meets S_a exactly once,
so S_a is a Cauchy surface. (⇒) Suppose some future-inextendible causal curve γ has T ≤ a at all of
its points, and extend it to the past to an inextendible causal curve. Because T is strictly
increasing, T < a along the whole extended curve (were T = a at some point, T would exceed a beyond
it). The curve never meets S_a, so S_a is not a Cauchy surface. □
In FLRW models with a big-bang singularity, whether they expand forever or recollapse, cosmological
time is the FLRW time and every level set is a Cauchy surface. But a causal curve that ends at a
singularity inside a black hole typically has bounded T, and level sets beyond that bound are not
met by it. In universes containing black holes, later level sets can therefore be future Cauchy
surfaces without being Cauchy surfaces. Any CTGP claim that the present is a Cauchy surface should
accordingly be restricted to the level sets satisfying Proposition 13.1, with the future Cauchy
property as the general statement.
For quantum fields on such a background, a state is assigned to each admissible hypersurface, as a
density operator ρ_Σ or a vector state Ψ_Σ in a suitable representation. In curved-spacetime quantum
field theory there is in general no single preferred Fock representation, and the algebraic
formulation, in which states are positive normalized functionals on a net of local observable
algebras, is the more appropriate setting (Wald 1994; Hollands & Wald 2015).