13.5 Toward the Relativistic Case: Countable Event Structures and the Remaining Assumptions
One step beyond Theorem 13.4 can be taken now: from finitely many event slots to countably many,
provided every slot has only finitely many predecessors.
Proposition 13.7 (Countable past-finite event structures). Let (E, ≺) be a countable partially
ordered set in which every slot x has a finite predecessor set ↓x. Let H be a Hilbert space with
initial state Ψ_0, and attach to each slot an instrument with finitely many outcomes and bounded
Kraus operators, such that the Kraus operators of incomparable slots commute. Then:
(a) There is a unique probability measure P on the product σ-algebra of the space ∏_{x∈E} A_x of
complete outcome assignments (the σ-algebra generated by cylinder sets) whose marginal on every
finite order ideal I is the law P_I(α_I) = ‖v_I(α_I)‖² of Theorem 13.4.
(b) E admits generative orders, that is, enumerations x_1, x_2, … in which every initial segment
is an order ideal. Sequential generation along any generative order reproduces P, so P does not
depend on which generative order is chosen.
(c) For every finite set X ⊆ E, the marginal of P on X depends only on the instruments in the
finite predecessor set ↓X.
Proof. Theorem 13.4 and its proof apply verbatim to bounded Kraus operators on an arbitrary Hilbert
space, since they use only finite products of operators and the completeness relation. (a) If I ⊆ J
are finite order ideals, I is an order ideal of the finite structure J, so by Theorem 13.4(b) the
marginal of P_J on I is P_I. Every finite set X lies in the finite order ideal ↓X, and finite order
ideals are closed under finite unions, so {P_I} is a consistent family indexed by a directed set
that exhausts E. Since every A_x is finite, the Kolmogorov extension theorem yields a unique
probability measure P on ∏_{x∈E} A_x with these marginals. (b) Enumerate E as e_1, e_2, …, and for k
= 1, 2, … append the elements of the finite set ↓e_k that have not yet been listed, in an order
compatible with ≺. Every predecessor of such an element lies in ↓e_k and is listed earlier, so every
initial segment is an order ideal, and every element is eventually listed. Sequential generation
along any generative order assigns the law P_I to each initial segment I (Theorem 13.4(b)), and
hence, by the uniqueness in (a), the measure P. (c) The marginal of P on X is the marginal of
P_{↓X}, which by Theorem 13.4(b) involves only the instruments in ↓X. □
Proposition 13.7 removes the finiteness of the event set while keeping discreteness: countably many
slots, finitely many outcomes per slot, and bounded operators. It constructs a probability measure
over infinite histories; it does not assert that the conditional quantum state converges to a limit
after infinitely many events. Past-finiteness is natural in regulated models, and it is the defining
property of causal sets grown from a beginning, whose generative orders are the natural labelings of
the growth process (§15.2(iii)). In the numerical check (Appendix B, item 14), the law on an order
ideal computed from that ideal’s instruments alone agrees with the marginal of the full law to
within 1.1 × 10⁻¹⁶, as part (a) requires.
A relativistic extension of Theorem 13.4 must supply, in addition, at least the following.
(E1) Localization. Instruments attached to spacetime regions rather than to abstract slots, with
continuous outcome spaces handled by measure-valued instruments.
(E2) Causal locality of operations. The Kraus operators by which events update the state must be
localized in the corresponding regions, so that, by microcausality, those attached to spacelike-
separated regions commute. This is a requirement on the operators themselves. It does not by
itself make the instruments causally admissible (Example 13.10).
(E3) Operational causal order. The predecessor relation must coincide with the causal precedence
that is relevant to signaling, and the actual instruments and their compositions must be shown
not to signal outside it. Commutativity of field observables and localization of Kraus operators
are not enough for this (Example 13.10). For extended regions it requires a measurement
framework that excludes Sorkin-type signaling, such as the probe-based schemes of Fewster and
Verch (2020) and Polo-Gómez, Garay and Martín-Martínez (2022); Theorem 13.9 verifies it for
gapless detector networks on a free field.
(E4) Continuum limit. A limit in which the countable, past-finite slot structure is replaced by
spacetime regions, with the history measure surviving the limit as a point process or measure-
valued construction.
(E5) Generative hypersurfaces. The level sets of cosmological time (§13.1) must define order
ideals of the event structure. This requires a convention for events whose regions straddle a
level set, and, for generability, a physically instantiated present state on each level set.
Proposition 13.7 does not derive a preferred cosmic-time slicing; it shows only that any
generative order of the events yields the same history measure.
(E6) Interactions and renormalization. The instruments must arise from an interacting,
renormalized field dynamics rather than being posited.
Proposition 13.7 settles none of (E1)–(E6). It shows that the step from finitely many to countably
many events, with a single history measure independent of the generative order, raises no difficulty
of its own. Section 13.6 shows that a finite network of gapless detectors on a free relativistic
field satisfies (E1)–(E3), and (E5) when its coupling regions are separated by level sets of
cosmological time, and §13.7 extends this to spontaneous Gaussian collapses of smeared fields on a
fixed family of regions. Section 13.10 extends it further to collapses at Poisson-distributed
locations, conditionally on causal orderability, and §13.9 gives a sufficient condition for (E3)
that does not depend on free fields. (E4) and (E6) remain open.
13.6 A Relativistic Construction: Gapless Detectors on a Free Field
The requirements (E1)–(E3) and (E5) can be met exactly in one relativistic setting: finite networks
of gapless two-level detectors coupled linearly to a free scalar field on a fixed globally
hyperbolic spacetime. In this measurement model the realized events are detector outcomes triggered
by specified couplings, not spontaneous flashes; the spontaneous version is discussed in §13.7.
Setting. Let φ be a free real scalar field on a globally hyperbolic spacetime (M, g), obeying a
linear hyperbolic field equation. For real test functions f and h, the smeared fields satisfy [φ(f),
φ(h)] = i E(f, h), where E is the causal propagator, the difference of the advanced and retarded
fundamental solutions (sign conventions aside). Its key property is that E(f, h) = 0 whenever no
causal curve joins the supports of f and h (Wald 1994; Hollands & Wald 2015). Write W(f) = exp(i
φ(f)) for the Weyl operators, which satisfy
W(f) W(h) = e^{−(i/2) E(f,h)} W(f + h), W(−h) W(g) W(h) = e^{i E(h,g)} W(g).
The field may be in any state ω; where a formula needs one, ω is a quasi-free state with vanishing
one-point function, such as the vacuum, for which ω(W(g)) = exp(−½ ω(φ(g)²)).
Definition 13.8 (Gapless detector network). A gapless detector network consists of finitely many
two-level detectors j = 1, …, n. Detector j has a coupling region O_j, a real test function f_j
(smooth and compactly supported) with support in O_j, and a coupling constant λ_j. It starts in the
state |0⟩, interacts through the unitary U_j = exp(i λ_j σ_x^{(j)} ⊗ φ(f_j)), and is then read out
by a projective measurement of σ_z^{(j)}. The regions are causally orderable: the transitive closure
≺ of the relation “O_i meets the causal future of O_j”, for i ≠ j, is a strict partial order.
Gapless means that the detector has no internal Hamiltonian while it interacts. The time-ordered
evolution generated by a coupling λ_j χ_j(t) σ_x ∫ F_j φ then differs from U_j only by a phase,
because the commutators of the interaction at different times are multiples of σ_x² = 1. Since U_j =
cos(λ_j φ(f_j)) + i σ_x sin(λ_j φ(f_j)), the instrument induced on the field by detector j has Kraus
operators
K^j_0 = cos(λ_j φ(f_j)), K^j_1 = i sin(λ_j φ(f_j)),
which are bounded functions of the smeared field in O_j and satisfy (K^j_0)† K^j_0 + (K^j_1)† K^j_1
= 1. The nonselective update of detector j acts on Weyl operators by multiplication with a number:
N_j*(W(g)) = ⟨0| U_j† (W(g) ⊗ 1) U_j |0⟩ = cos(λ_j E(f_j, g)) W(g).
This follows from the second Weyl relation, applied separately to the two eigenvalues s = ±1 of σ_x,
each occurring with probability 1/2 in |0⟩. A unitary kick W(α f_j) acts in the same diagonal way:
W(−α f_j) W(g) W(α f_j) = e^{i α E(f_j,g)} W(g).
Theorem 13.9 (Gapless detector networks). In a gapless detector network on a free field, for any
state of the field:
(a) Realized-history consistency. The probability of every complete outcome history, and the
corresponding conditional state, are the same for every foliation, that is, for every linear
extension of ≺.
(b) Operational no-signaling, including Sorkin configurations. Let X be a set of detectors, and
let j ∉ X be a detector with E(f_j, f_x) = 0 for every x ∈ X; this holds in particular when no
causal curve joins O_j to any O_x. Then the joint law of the outcomes of the detectors in X is
the same whether detector j is operated with any coupling λ_j, replaced by a kick W(α f_j), or
not operated at all, whatever other detectors are operated before, between, or after them.