13.7 Toward Spontaneous Events: Weyl-Diagonal Collapse
The proof of Theorem 13.9(b) uses one property of the gapless detector: its nonselective update
multiplies every Weyl operator by a number. Inserting arbitrary stochastic collapses into the model
would risk reintroducing the signaling of Example 13.10. The natural route toward spontaneous events
is therefore to keep that property and ask which event-generating instruments have it.
Definition 13.11 (Weyl-diagonal instrument). Let O be a bounded spacetime region and H_O a finite-
dimensional space of real test functions supported in O. An instrument with finitely or continuously
many outcomes is a Weyl-diagonal instrument localized in O if (i) its Kraus operators belong to the
von Neumann algebra generated by the Weyl operators W(h), h ∈ H_O, and (ii) its nonselective update
satisfies N*(W(g)) = c(g) W(g) for every test function g, for some function c.
Examples are the gapless detector of Definition 13.8, with c(g) = cos(λ E(f, g)), and the unitary
kick, with c(g) = e^{iαE(f,g)}. The random-displacement channel ρ ↦ ∫ W(h) ρ W(h)* dμ(h), with μ a
probability measure on H_O, is also Weyl-diagonal, with c(g) = ∫ e^{iE(h,g)} dμ(h). The nonselective
gapless detector is exactly the random displacement by +λf or −λf, each with probability 1/2.
Proposition 13.12 (Gaussian collapse of a smeared field). Let h be a real test function supported in
O and σ > 0. The instrument with outcome y ∈ ℝ and Kraus operators
K_y = (2πσ²)^{−1/4} exp(−(φ(h) − y)² / (4σ²))
is normalized, ∫ K_y† K_y dy = 1, and Weyl-diagonal, with
N*(W(g)) = exp(−E(h, g)² / (8σ²)) W(g).
Its outcome y is distributed by the Born rule for φ(h) smoothed at width σ, and the selective update
multiplies the spectral amplitude of φ(h) by a Gaussian of width σ centred at y. It is a collapse of
the smeared field φ(h) around the realized value y. Combining it with a hit probability p, so that
with probability 1 − p nothing happens, gives the Weyl-diagonal instrument with c(g) = 1 − p + p
exp(−E(h, g)² / (8σ²)).
Proof. Normalization is the Gaussian integral ∫ (2πσ²)^{−1/2} exp(−(x − y)²/(2σ²)) dy = 1, applied
in the spectral representation of φ(h). By the second Weyl relation, W(g)* φ(h) W(g) = φ(h) + E(g,
h), so each Kraus operator satisfies K_y W(g) = W(g) K_y(φ(h) + e) with e = E(g, h). Hence
N*(W(g)) = W(g) ∫ (2πσ²)^{−1/2} exp(−[(φ(h) + e − y)² + (φ(h) − y)²] / (4σ²)) dy.
Writing u = φ(h) − y, the exponent is −[2(u + e/2)² + e²/2] / (4σ²), and the integral over y equals
exp(−e²/(8σ²)), independently of φ(h). Since E(g, h)² = E(h, g)², this is the stated multiplier. The
version with hit probability adds (1 − p) W(g). □
Theorem 13.13 (Weyl-diagonal networks). Theorem 13.9(a) and (b) hold for every finite, causally
orderable network on a free field in which each operation is a Weyl-diagonal instrument localized in
its region, with finitely or continuously many outcomes. For (b), j may be any operation whose space
H_j satisfies E(h, f) = 0 for all h ∈ H_j and all f ∈ H_x, x ∈ X; in particular any operation in a
region causally separated from every region of X.
Proof. (a) Weyl operators of causally separated regions commute, and hence so do the von Neumann
algebras they generate and the Kraus operators in them. The proof of Theorem 13.4 then applies, with
sums over continuous outcomes replaced by integrals. (b) Let V_X be the sum of the spaces H_x for x
∈ X, and M_X the von Neumann algebra generated by the W(g), g ∈ V_X. In the Heisenberg picture, the
selective updates of the operations in X preserve M_X, because their Kraus operators lie in M_X. The
nonselective update of any other operation multiplies each W(g) by a number; it therefore preserves
the linear span of { W(g) : g ∈ V_X }, and, being normal, it preserves the σ-weak closure of that
span, which is M_X. For the operation j, every W(g) with g ∈ V_X commutes with j’s Kraus operators,
so N_j*(W(g)) = W(g) Σ K† K = W(g). The update of j thus acts as the identity on M_X, and j drops
out of the joint law of X. □
Corollary 13.14 (A regulated spontaneous-collapse network). Fix a finite, causally orderable family
of bounded regions O_k, with smearings h_k supported in O_k, hit probabilities p_k, and widths σ_k.
Let the law specify that at each O_k a collapse event occurs with probability p_k, independently of
the state, and that its centre y is drawn by the Born rule with the update of Proposition 13.12. The
resulting single-outcome process is spontaneous in the sense of GRW: its events are prescribed by
the law, not by an experimenter. It has foliation-independent history probabilities, its event
statistics in any set of regions do not depend on operations in causally separated regions,
including in Sorkin’s configuration, and it is generable from the present field state together with
the realized events. By Proposition 13.7, whose Kolmogorov extension holds equally for standard
Borel outcome spaces such as ℝ, the history measure extends to countable past-finite families of
regions.
Proof. Theorem 13.13 with Proposition 13.12, and Proposition 13.7. □
In the two-mode realization of Sorkin’s configuration used in Example 13.10, replacing the
intermediate operation by a Gaussian collapse with σ = 0.6 gives Bob’s probability of reading 1 as
0.3449027185 when a collapse is certain and 0.2541731691 when it occurs with probability 0.4. Both
values agree with the formula ½ − ½ exp(−λ_B²)(1 − p + p exp(−λ_B²/(2σ²))). Neither Alice’s detector
nor her kick changes them by more than 10⁻¹³ (Appendix B, item 18).
Corollary 13.14 is a first construction of spontaneous relativistic events with the properties CTGP
requires, but its limitations are substantial and should be stated plainly.
(L1) Fixed regions. The family of collapse regions is fixed in advance. This is a regularization
and is not Lorentz-invariant. Section 13.10 lets a covariant Poisson law choose the centres,
proves what then holds exactly, and identifies the obstructions to a fully covariant version.
(L2) Linear field observables. The collapse localizes the value of a smeared field, which is
linear in φ. Collapse theories that localize matter act on mass density, which is quadratic in
the fields, and functions of quadratic observables are not Weyl-diagonal in general; Theorem
13.13 does not apply to them. A mathematically consistent stochastic update need not produce the
localization that an objective-collapse ontology requires, and whether field-linear collapse can
play that role is open.
(L3) Energy. As in collapse models generally, the collapses do not conserve the field’s energy,
because narrowing the distribution of φ(h) broadens that of its conjugate. Proposition 13.16
gives the exact, state-independent cost of each collapse. Any physical version must respect the
experimental bounds on such effects (Bassi et al. 2013).
(L4) Free fields and fixed background. Interacting fields, for which commutators of smeared
fields are not multiples of the identity, and dynamical geometry are not covered.
The construction also illustrates Proposition 13.2 once more. The same random-displacement channel
can be unravelled by unitary kicks, which localize nothing, by the cosine and sine operators of the
gapless detector, which localize φ(f) only modulo π/λ, or by Gaussian collapses, which localize it
around a single value. No-signaling depends only on the channel. Localization depends on the
unravelling, which is the part that a collapse ontology must justify. Proposition 13.15 shows that
the unread channel of a Gaussian collapse is itself a Gaussian mixture of unitary kicks.