(c) Explicit statistics. If detector B is preceded by detectors j_1, …, j_r whose outcomes are
not conditioned on, and the field is in a quasi-free state with vanishing one-point function,
then
P(B reads 1) = ½ − ½ exp(−2 λ_B² ω(φ(f_B)²))
· Π_{k=1}^{r} cos(2 λ_B λ_{j_k} E(f_{j_k}, f_B)).
Proof. (a) If two regions are incomparable, no causal curve joins them, so E(f_i, f_j) = 0, the Weyl
operators W(s f_i) and W(t f_j) commute for all s and t, and so do all bounded functions of φ(f_i)
and φ(f_j). The Kraus operators of incomparable detectors therefore commute. Theorem 13.4 applies,
in the form for bounded Kraus operators noted in the proof of Proposition 13.7, in the GNS
representation of the state.
(b) Compute the joint law of X in the Heisenberg picture, along any foliation. Every detector
outside X is marginalized, so it contributes its nonselective update N_k*. Every detector in X
contributes a selective update K† (·) K with K = cos(λφ(f_x)) or i sin(λφ(f_x)); each of these maps
a Weyl operator W(g) to a linear combination of Weyl operators W(g + c f_x) with real c. Starting
from the identity W(0), the operator under evaluation is therefore at every step a finite linear
combination of Weyl operators W(g) with g in the real span V_X of { f_x : x ∈ X }. Each nonselective
update of a detector k ∉ X multiplies such a W(g) by cos(λ_k E(f_k, g)). For k = j, E(f_j, g) = 0
for every g ∈ V_X by linearity, so the factor is 1, and the same holds for a kick. Detector j
therefore drops out of the joint law of X.
(c) Write sin²(λ_B φ(f_B)) = ½ − ¼ (W(2λ_B f_B) + W(−2λ_B f_B)). Each preceding detector,
marginalized, multiplies W(±2λ_B f_B) by cos(2 λ_B λ_{j_k} E(f_{j_k}, f_B)), and ω(W(±2λ_B f_B)) =
exp(−2 λ_B² ω(φ(f_B)²)). □
Part (b) is the statement that matters for CTGP, and its scope should be stated exactly. It is
state-independent; it holds for arbitrary smooth, compactly supported real smearings within finite,
causally orderable networks, under the free-field and gapless-detector assumptions; and it holds in
Sorkin’s configuration. There, Alice’s region A is causally separated from Bob’s region B, while an
intermediate region C overlaps the causal future of A and the causal past of B. A gapless detector
at C cannot carry Alice’s choice to Bob, because its nonselective update multiplies each Weyl
operator by a number instead of spreading it. Part (c) makes this explicit: Bob’s statistics depend
on a preceding detector only through the factor cos(2 λ_B λ_j E(f_j, f_B)), which equals 1 for every
detector causally separated from B. The essential property is that the nonselective update is
diagonal on Weyl operators. It holds for gapless linear couplings with any such smearing. It can
fail for detectors with internal dynamics during the interaction, and de Ramón, Papageorgiou and
Martín-Martínez (2021) find that spatially extended non-relativistic detector models can propagate a
field’s initial conditions faster than light. The conclusion agrees with detector-based measurement
theory, in which updates induced by local probe couplings avoid Sorkin-type signaling (Polo-Gómez,
Garay & Martín-Martínez 2022; compare Fewster & Verch 2020).
Three things should be distinguished in interpreting the theorem: the spacetime support O_j of the
detector–field interaction; the location and timing of the detector’s readout; and the effective
instrument on the field obtained by conditioning on that readout. Theorem 13.9 concerns the third.
It proves causal properties of the effective field instruments of the network, assuming that each
detector is read out after its interaction and treating the detector as an idealized probe. When a
coupling region is extended, a fully spacetime-local implementation of the detector together with
its readout would require further modeling of the probe itself. This qualification concerns physical
realization; it does not affect the algebraic proof.
Theorem 13.9 is exact within its model, and the numerical checks below only illustrate it. Exactness
within a model is not physical completeness. The model assumes a fixed spacetime, a free field,
externally specified couplings, and detectors with no internal Hamiltonian during the interaction,
and these assumptions are central to the proof. A nonzero energy gap makes the coupled detector
observable time-dependent, and the nonselective update then generally ceases to be diagonal on Weyl
operators. The theorem is therefore not a claim about arbitrary relativistic detector measurements,
realistic detector hardware, or interacting fields.
Example 13.10 (Localized Kraus operators are not enough). Replace the detector at C by the unsharp
Lüders instrument with Kraus operators √(g(φ(f_C))) and √(1 − g(φ(f_C))), where g(x) = (1 + tanh
2x)/2. Its Kraus operators are bounded functions of the smeared field in C, so it satisfies (E2),
and they commute with every operation localized in a region causally separated from C. But its
nonselective update is not diagonal on Weyl operators:
Σ_s √(g_s(φ(f_C))) W(h) √(g_s(φ(f_C))) = W(h) Σ_s √(g_s(φ(f_C) + E(h, f_C))) √(g_s(φ(f_C))),
with g_+ = g and g_− = 1 − g. The right-hand side is W(h) times a non-constant function of φ(f_C),
which fails to commute with φ(f_A) when E(f_A, f_C) ≠ 0. An operation by Alice can therefore change
Bob’s statistics through C even though A and B are causally separated. This is Sorkin’s problem. A
numerical realization reproduces the commutation pattern of Sorkin’s configuration (E(f_A, f_B) = 0,
E(f_A, f_C) = E(f_C, f_B) = 1) with two oscillator modes in the vacuum state, using λ_A = 0.9, λ_C =
0.8, λ_B = 0.7, and a kick with α = 1.5 (Appendix B, item 17). With a gapless detector at C, Bob’s
probability of reading 1 is 0.3665447170, equal to the formula of Theorem 13.9(c), and neither
Alice’s detector nor her kick changes it by more than 10⁻¹⁵. With the unsharp Lüders instrument at
C, Bob’s probability is 0.275038, and Alice’s detector shifts it by −0.021492 and her kick by
−0.046628. These values change by less than 10⁻⁷ when the Fock cutoff is raised from 40 to 50 quanta
per mode. Because Theorem 13.9(b) is state-independent and uses only the commutation relations, the
two-mode realization is an exact test of the algebraic mechanism, not a computation in a particular
spacetime.
Example 13.10 shows why (E2) and (E3) must be kept separate. Localizing the Kraus operators of each
operation is not enough. Causal admissibility is a property of the actual instruments and of their
composition, and it must be proved for the class of operations in use; Theorem 13.9 proves it for
gapless detector networks on a free field, and Theorem 13.13 for every network of Weyl-diagonal
instruments.
Connection to CTGP. In a gapless detector network the realized detector outcomes are Tier 1 events,
and the complete present state on an admissible hypersurface consists of the field state together
with the detectors’ records. For a specified network, the law of the next outcome depends only on
that present state, so the network is generable in the sense of Definition 12.1, (A2)–(A5). The
truth profile records the realized outcomes and is dynamically idle. Because cosmological time is in
general only locally Lipschitz (§13.1), the construction is formulated for arbitrary admissible
orderings. A generic globally hyperbolic spacetime also admits smooth Cauchy temporal functions
(Bernal & Sánchez 2003, 2005). Whenever the coupling regions can be separated by level sets of the
cosmological time T, ordering them by T is a linear extension of ≺, because T increases strictly
along future-directed causal curves. CTGP’s generative order is then one of the admissible
foliations, and by Theorem 13.9(a) and Corollary 13.5 the history law is the same for it as for any
other. Three steps must be kept apart here. First, the network’s history law is independent of the
admissible ordering of its regions; Theorem 13.9 establishes this. Second, the background spacetime
admits a regular cosmological time whose level sets separate the regions; this is an assumption
about the geometry. Third, those level sets are the ontological presents; this is CTGP’s
interpretation. Theorem 13.9 supports only the first step, which is what makes the other two
consistent with it. Within this class, then, the measurement-model version of Target 13.6 holds for
finite networks: (i)–(v) are satisfied, with (iii) secured by part (b). What remains open is the
spontaneous version, detectors with internal dynamics, interacting fields, and infinitely many or
continuously distributed events, that is, (E4) and (E6).