13.2 Hypersurface Evolution and Its Limits
A natural starting point is the Tomonaga–Schwinger equation (Tomonaga 1946; Schwinger 1948),
i ħ δΨ_Σ / δΣ(x) = H_int(x) Ψ_Σ,
which describes how a state changes under local deformations of a spacelike hypersurface. Its formal
integrability condition is [H_int(x), H_int(y)] = 0 for spacelike-separated x and y. For interacting
fields this expression requires a treatment of operator-valued distributions and their domains; it
is not by itself an existence theorem. CTGP – explained proposes an open-system analogue, δρ_Σ /
δΣ(x) = ℒ_x(ρ_Σ), with conditional integrability (Proposition 17.16) and a conditional continuum
limit (Theorem 17.17). These are valuable starting points, but they are ensemble-level results: they
concern density operators. The next subsection shows why a single-outcome ontology needs more.
13.3 Ensemble Consistency Is Not Realized-History Consistency
Proposition 13.2 (Equal ensembles, different histories). There exist two instrument sequences whose
channels coincide at every stage, so that they assign the same density operator to every stage for
every initial state, but whose realized-history measures differ.
Proof. On a qubit, let the first instrument be either I_1 = { |0⟩⟨0|, |1⟩⟨1| } or I_2 = { 1/√2, Z/√2
}, where Z = |0⟩⟨0| − |1⟩⟨1|, and let the second be the measurement of X with outcomes ±, {|+⟩⟨+|,
|−⟩⟨−|}. Both first instruments implement the complete dephasing channel ρ ↦ |0⟩⟨0|ρ|0⟩⟨0| +
|1⟩⟨1|ρ|1⟩⟨1|, so the averaged states agree at every stage. For the initial state |+⟩, I_1 yields
outcomes 0 and 1 with probability 1/2 each and leaves |0⟩ or |1⟩, after which the X outcome is
uniformly random. I_2 yields its two outcomes with probability 1/2 each and leaves |+⟩ or |−⟩, after
which the X outcome is perfectly correlated with the first outcome. The joint laws of (first event,
second event) are uniform in the first case and perfectly correlated in the second: the mutual
information is 0 bits in one case and 1 bit in the other. □
Proposition 13.2 is an instance of the non-uniqueness of unravellings of a quantum channel (Wiseman
& Milburn 2010). Its consequence for CTGP is direct: a covariant or hypersurface-consistent master
equation does not by itself supply a covariant stochastic ontology. Two theories can agree on every
density operator on every hypersurface and still disagree about the probabilities of the realized
histories that, on CTGP, are what actually happened. Compatibility with a single-outcome ontology
therefore needs a consistency theorem at the level of realized histories, not only at the level of
states.
13.4 A Realized-History Consistency Theorem for Causally Local Instruments
The following theorem is a finite-dimensional analogue of the required relativistic result. Events
occur at a finite set of slots, ordered by a partial order intended to represent causal precedence;
a foliation is any way of advancing through the slots consistently with that order.
Definition 13.3 (Causally local event structure). A causally local event structure consists of a
finite partially ordered set (E, ≺) of event slots, a finite-dimensional Hilbert space H with
initial state Ψ_0, and for each slot x an instrument {K^x_a : a ∈ A_x} on H, such that for every
pair of incomparable (spacelike) slots x and y, every K^x_a commutes with every K^y_b. A foliation
is a linear extension λ = (x_1, …, x_N) of ≺; its successive initial segments {x_1, …, x_j} are the
regions to the past of its successive hypersurfaces. For an outcome assignment α = (a_x)_{x∈E}, set
v_λ(α) = K^{x_N}_{a_{x_N}} ⋯ K^{x_1}_{a_{x_1}} Ψ_0, P_λ(α) = ‖v_λ(α)‖².
Theorem 13.4 (Realized-history consistency). In a causally local event structure:
(a) Foliation independence. v_λ(α) does not depend on the foliation λ. Hence the probability of
every complete event history, P(α) = P_λ(α), and the corresponding final state are foliation-
independent, and Σ_α P(α) = 1.
(b) Consistency on every hypersurface. For every order ideal I ⊆ E (the past of some
hypersurface) and every assignment α_I of outcomes to the slots in I, the probability of α_I and
the conditional state on the hypersurface bounding I are the same for every foliation of I, and
they do not depend on the instruments at slots outside I.
(c) No dependence on instruments outside the designated predecessor set. For every set X ⊆ E,
the marginal law of the outcomes in X depends only on the instruments in its predecessor set
with respect to ≺, ↓X = { y : y ⪯ x for some x ∈ X }. Identifying ↓X with the operationally
relevant relativistic causal past of X requires additional physical assumptions, discussed after
Corollary 13.5.
Proof. (a) Any two linear extensions of a finite partial order are connected by a sequence of
adjacent transpositions of incomparable elements. To see this, let λ and λ′ first differ at position
j, where λ′ has x′. In λ, x′ occurs later; every element between position j and x′ in λ precedes x′
in λ, so it is not above x′, and follows x′ in λ′, so it is not below x′; hence it is incomparable
with x′. Moving x′ leftward in λ by adjacent transpositions with these elements yields a linear
extension that agrees with λ′ in one more position, and induction completes the argument. Each such
transposition exchanges two commuting operators and leaves v_λ(α) unchanged. Normalization follows
from Σ_a (K^x_a)† K^x_a = 1 applied slot by slot. (b) Because I is a down-set, any foliation of I
extends to a foliation of E that lists I first, and (a) applied to the sub-order I gives
independence of the foliation of I. Summing ‖v‖² over the outcomes of the slots after I and using
completeness of their instruments gives ‖v_I(α_I)‖², which involves only instruments in I. (c) Apply
(b) with I = ↓X, which is an order ideal containing X; the marginal on X is a marginal of the law of
α_I. □
Corollary 13.5 (Foliation-invisibility of the generative order in causally local finite models). In
a causally local event structure, suppose CTGP’s generative order is one particular foliation λ*.
Then every event proposition about any order ideal has the same probability whether computed along
λ* or along any other foliation. The realized assignment α is a single object, which different
foliations merely order differently; once the truth profile indexes events by slot rather than by
stage number, the complete profile after the whole region has been generated is the same set of
propositions whichever foliation is taken as generative.
Proof. Theorem 13.4(a)–(b). □
Corollary 13.5 is established for finite event structures under the stated locality assumption, not
for arbitrary relativistic quantum dynamics. It is the realized-history counterpart of CTGP’s
observational equivalence (CTGP – explained, §12.1): within causally local finite models, under the
commutation assumption of Definition 13.3, the choice of generative foliation has no empirical
signature, and a single realized history can be represented on any family of hypersurfaces without
multiplying the number of histories. If each K^x_a is invariant under a symmetry group in the sense
of §3, Lemma 3.7 applies slot by slot, and Theorem 13.4 holds in every frame.
Foliation-independent probabilities do not by themselves establish that any foliation is
ontologically privileged. What Corollary 13.5 shows is that privileging one foliation is empirically
invisible under the stated assumptions. That one foliation is in fact generative is CTGP’s
interpretive thesis, supplied separately, in the same way that Thesis 5.10 supplies the ontological
reading of description orbits.
The locality condition cannot be dropped. In the numerical check (Appendix B, item 14), a brickwork
arrangement of eight two-outcome instruments on six qubits has 136 foliations, and all of them
assign identical probabilities to all 256 outcome histories, to within 7 × 10⁻¹⁸. Changing the
instruments outside the predecessor set of a slot changes that slot’s marginal by at most 8 × 10⁻¹⁶.
But if two slots treated as spacelike act on overlapping qubits, so that their operators fail to
commute, the two orders assign history measures that differ by total-variation distance 0.165.
Theorem 13.4 must also be read with a caveat specific to quantum field theory. Part (c) is a
statement about the abstract order ≺: it says that outcomes do not depend on instruments outside a
designated predecessor set, not that relativistic operational no-signaling holds for arbitrary
spacetime regions or measurement procedures. The two coincide only if ≺ is the operationally
relevant causal order. For idealized point-like slots it is. For extended regions, however, the
transitive closure of “overlaps the causal future of” can relate two regions that are in fact
spacelike-separated, and naive local measurement rules then permit superluminal signaling (Sorkin
1993). A relativistic version of the theorem therefore needs a measurement framework in which
operations localized in regions respect relativistic causality; frameworks of this kind exist in
algebraic quantum field theory (Fewster & Verch 2020; Bostelmann, Fewster & Ruep 2021). The finite
theorem is an analogue of the relativistic result, not that result itself.
The relativistic target can now be stated precisely.
Target result 13.6 (Relativistic realized-history consistency). For a regulated relativistic single-
outcome model on a fixed globally hyperbolic background, given by a hypersurface-dependent family of
completely positive maps ℰ_{Σ_1→Σ_2} together with instruments assigning probabilities to realized
events between hypersurfaces, establish: (i) existence and normalization of the event-history
measure; (ii) composition consistency for nested regions; (iii) compatibility of operations in
spacelike-separated regions, in a framework free of Sorkin-type signaling; (iv) consistent
restriction of event histories to overlapping regions; and (v) a well-defined single-outcome
realization. Then prove that the probability of every common event-history proposition is
independent of the admissible intermediate hypersurface decomposition, and that the CTGP truth
profile updates consistently from the realized history.
Theorem 13.4 proves this for finitely many slots and finite-dimensional operators, Proposition 13.7
(§13.5) extends it to countably many slots with finite pasts, and Theorem 13.9 (§13.6) establishes
its measurement-model version for finite networks of gapless detectors on a free relativistic field.
The assumptions of the relativistic version must be made precise enough not to assume the
conclusion.