13.10 Randomly Located Collapses
Corollary 13.14 fixes the possible collapse regions in advance. The natural next step is to let the
law choose the locations, with a distribution that refers to no coordinates or foliation. This
subsection carries out that step on a fixed globally hyperbolic background, proves what survives
exactly, and identifies the two places where the construction meets obstructions.
Definition 13.20 (Poisson collapse network). Fix the free field of §13.6 on a globally hyperbolic
spacetime (M, g), in a quasi-free state such as the vacuum or another quasi-free Hadamard state. Fix
a Borel set D ⊆ M of finite four-volume, an intensity λ > 0, a width σ > 0, and a smearing rule x ↦
h_x that assigns to each x ∈ D a real test function whose compact support O_x contains x,
continuously in the topology of test functions. The collapse centres form a Poisson point process Ξ
on D with intensity measure λ dVol_g: the numbers of centres in disjoint Borel sets are independent
Poisson variables whose means are λ times the four-volumes of the sets (Last & Penrose 2017). A
realized configuration ξ = {x_1, …, x_n} is admissible if the regions O_{x_1}, …, O_{x_n} are
causally orderable in the sense of Definition 13.8. At each centre of an admissible configuration a
Gaussian collapse of φ(h_{x_i}) with width σ occurs (Proposition 13.12), in any order compatible
with the causal order of the regions. The realized history is the finite marked configuration {(x_i,
y_i)} of centres and outcomes.
The location of each collapse is fixed by the law, independently of the field state; only its
outcome is drawn by the Born rule. A separate hit probability is unnecessary, since thinning a
Poisson process independently with probability p yields a Poisson process of intensity pλ.
Theorem 13.21 (Poisson collapse networks: what holds exactly). The construction of Definition 13.20
defines a normalized collapse dynamics conditionally on the admissible sector 𝒜. It does not define
the dynamics on inadmissible configurations, so it is not an unconditional collapse law on the full
Poisson configuration space. Within that scope:
(a) Conditional foliation independence. For every admissible configuration ξ, the joint law Q_ξ
of the outcomes is the same for every linear extension of the causal order of its regions, and
it depends measurably on ξ.
(b) Averaging over locations. On the event 𝒜 of admissible configurations, the joint law of
centres and outcomes, Π(dξ) Q_ξ(dy) with Π the law of Ξ, is foliation independent, and so is the
law conditioned on 𝒜. Because the centres are drawn independently of the field, conditioning on
𝒜 does not bias the outcomes: given an admissible ξ, they have the Born-rule law Q_ξ.
(c) Conditional no-signaling. For an admissible ξ and any set Y of its collapses, the
unconditioned joint law of the outcomes in Y is unchanged by any operation j whose Kraus
operators commute with every W(g), g in the span of the h_y with y ∈ Y. This holds in particular
for every Weyl-diagonal operation localized in a region causally separated from all the regions
O_y, y ∈ Y.
(d) Covariance of the law. Let ψ be a time-orientation-preserving isometry of (M, g) with ψ(D) =
D, and suppose the smearing rule is ψ-covariant, h_{ψ(x)} = h_x ∘ ψ^{−1}. Then transforming the
field state by ψ transforms the joint law of centres and outcomes by ψ. This is covariance of
the law. A realized configuration is not ψ-invariant.
(e) Incremental sampling of the locations. For disjoint Borel sets D_1, D_2 ⊆ D, the
configurations Ξ ∩ D_1 and Ξ ∩ D_2 are independent, so the centres in each newly generated
region can be drawn from that region alone, without reference to later regions. This concerns
the locations only. It does not show that the quantum-state updates can be carried out
generatively along a succession of cosmic-time presents, particularly when collapse regions
straddle the level sets. That is a separate obligation, (E5) of §13.5, which §13.11 meets for
one candidate ordering rule.
Proof. (a) For a fixed admissible ξ, Theorem 13.13 applies, with the Gaussian collapses of
Proposition 13.12 as its Weyl-diagonal operations. In the GNS representation of a quasi-free state,
h ↦ W(h) is strongly continuous, so the Kraus operators, and with them the history probabilities,
depend continuously on the centres. (b) Integrate (a) against Π over 𝒜. The conditional law of the
outcomes given ξ is Q_ξ, because Ξ is independent of the field. (c) Apply Theorem 13.17 with 𝔐 the
von Neumann algebra generated by the W(g), g in the span of { h_y : y ∈ Y }. Condition (i) holds
because the Kraus operators of the target collapses lie in 𝔐, (ii) holds by Corollary 13.18, and
(iii) holds by the commutation assumption. (d) An isometry preserves dVol_g, so by the mapping
theorem for Poisson processes ψ(Ξ) has the law of Ξ. The free-field dynamics is covariant: the map
W(f) ↦ W(f ∘ ψ^{−1}) preserves the Weyl relations, because E is invariant under isometries, and it
carries the collapse at x to the collapse at ψ(x). (e) This is the complete independence property of
Poisson processes. □
Theorem 13.21 settles the conditional question. Two questions remain: whether admissible
configurations are typical, and whether the smearing rule can be chosen covariantly. The answer to
both is negative, in a precise sense.
Proposition 13.22 (Orderability fails in large volume). Suppose D contains N pairwise disjoint Borel
cells C_1, …, C_N, each of four-volume at least v, such that O_x ∩ O_{x′} ≠ ∅ whenever x and x′ lie
in a common cell. Then
Pr(Ξ is admissible) ≤ [e^{−λv} (1 + λv)]^N.
In Minkowski space, if every h_x is a translate of one test function whose support has nonempty
interior, the cells can be taken as coordinate cubes of a fixed size. The probability that a
configuration in a large box is admissible then decays exponentially with the box’s four-volume.
Proof. Every region lies in its own causal future, so two overlapping regions each meet the causal
future of the other, and the relation of Definition 13.8 has a cycle. An admissible configuration
therefore has at most one centre in each cell. The cell counts are independent Poisson variables
with means μ_i ≥ λv, and Pr(count ≤ 1) = e^{−μ}(1 + μ) is decreasing in μ. For translates, suppose
the support contains a coordinate ball of radius r about some point c. If x and x′ lie in a cube of
side r/2, then |x − x′| < r, so c + x lies in both translated supports. □
Proposition 13.23 (The order of interfering collapses matters). Let h_1 and h_2 be real test
functions, σ_1, σ_2 > 0, and ρ a state in which φ(h_1) and φ(h_2) have finite variances V_1 and V_2.
Apply Gaussian collapses of φ(h_1) and φ(h_2) to ρ in either order. If collapse 1 is applied first,
the outcomes have variances V_1 + σ_1² and V_2 + σ_2² + E(h_1, h_2)²/(4σ_1²). If collapse 2 is
applied first, the outcomes have variances V_1 + σ_1² + E(h_1, h_2)²/(4σ_2²) and V_2 + σ_2². Hence,
whenever E(h_1, h_2) ≠ 0, the two orders give different joint laws, for every such state.
Proof. Each outcome is the Born-rule value of the collapsed observable in the state at its collapse,
plus independent Gaussian noise of variance σ_k² (Proposition 13.15(a)). If collapse 1 comes first,
its unread effect on the second outcome is the random kick of Proposition 13.15(b). Since
e^{itφ(h_1)} φ(h_2) e^{−itφ(h_1)} = φ(h_2) − t E(h_1, h_2), the distribution of φ(h_2) after the
kick is that of its value in ρ shifted by −t E(h_1, h_2), with t independent of the state, of mean
zero and of variance 1/(4σ_1²). The mean is unchanged and the variance grows by E(h_1,
h_2)²/(4σ_1²). The other order is symmetric. □
Proposition 13.24 (No Lorentz-invariant law for random smearings). On four-dimensional Minkowski
space, with a free field of mass m ≥ 0, or in any dimension with m > 0, let x be a point and μ a
Borel probability measure on the set of real test functions h with Eh ≠ 0. Then μ is not invariant
under the Lorentz transformations about x. In particular, no single test function with Eh ≠ 0 is
invariant under them.
Proof. By Proposition 13.16(c), and its analogue in other dimensions for m > 0, the unit vector u(h)
= ΔP(h)/(ΔP(h) · ΔP(h))^{1/2} is a continuous, Lorentz-equivariant map into the hyperboloid H³ of
future-directed unit timelike vectors, and it is unaffected by translating h. If μ were invariant,
its image under u would be a Lorentz-invariant probability measure ν on H³. Choose R with ν(K) >
1/2, where K is the closed hyperbolic ball of radius R about a unit vector e. A boost of rapidity
greater than 2R carries K onto the ball of radius R about a point at hyperbolic distance greater
than 2R from e, which is disjoint from K. By invariance it also has measure greater than 1/2, so the
total measure would exceed 1. □
The hypothesis on dimension and mass excludes one genuine exception. For a massless field in 1 + 1
dimensions, ΔP(h) is null whenever ĥ vanishes on one of the two null rays of the light cone, and the
map u is then undefined; the proof does not apply to such smearings. The proposition concerns the
class it states: point-centred smearings of a free field, drawn from a law attached to the centre.
It does not exclude every conceivable Lorentz-invariant stochastic localization mechanism.
Constructions with additional dynamical fields, different primitive ontologies, or event laws not of
this form lie outside it, and routes (iv) and (vi) of Remark 13.25 are of that kind.