The dimension matters for the last statement of (c). In 1 + 1 dimensions the same formula holds with
d³k/(2π)³ replaced by dk/(2π), and ΔP(h) is timelike if m > 0. For m = 0, however, the light cone
consists of two null rays, ĥ can vanish on one of them, and ΔP(h) is then null.
Three consequences follow. First, the exact cost agrees with the standard collapse-model result.
With σ² = r_C²/2, K_y is the GRW localization operator of width r_C, and part (a) gives 3ħ²/(4m
r_C²) per hit, hence 3λħ²/(4m r_C²) per unit time at hit rate λ (compare Bassi et al. 2013). Second,
part (b) is the exact case of the weak-collapse expansion N* = exp(−ad_A²/(8σ²)), with ad_A(B) = [A,
B], which gives N*(H) − H = −(1/(8σ²)) [A, [A, H]] + O(σ^{−4}) whenever the series makes sense. The
expansion terminates when [A, [A, H]] is a multiple of the identity. Third, the field cost grows as
the collapse region shrinks. For a massless field and h_ℓ(x) = ℓ^{−4} χ(x/ℓ), substituting k → k/ℓ
gives ΔP(h_ℓ) = ℓ^{−3} ΔP(χ) at fixed σ, while the vacuum variance of φ(h_ℓ) scales as ℓ^{−2}. At a
fixed ratio ε of σ to the vacuum spread of φ(h_ℓ), the energy injected per collapse is therefore
proportional to ħc/(ε² ℓ). Part (d) gives the corresponding local statement on a curved background.
In the numerical checks (Appendix C.1, items C1 and C2), the Kraus integral and the Gaussian mixture
of kicks both reproduce the multiplier exp(−(a − a′)²/(8σ²)), to 2.4 × 10⁻¹⁴ and 7.8 × 10⁻¹⁶. For a
particle with an anharmonic potential, the energy increment is ħ²/(8mσ²) to twelve digits, in the
ground state and in a random state alike. On a lattice field, the double-commutator and Fourier
expressions agree to 5 × 10⁻¹⁶. For a massive field in 1 + 1 dimensions, under a boost of rapidity
0.5, ΔP transforms as a four-vector to 1.3 × 10⁻¹⁶.
13.9 Algebra Preservation: A Sufficient Condition for No-Signaling
The proofs of Theorems 13.9(b) and 13.13(b) use the free field only to show that a certain algebra
of observables is carried into itself by the operations between the one being tested and the
targets. That structure can be isolated, and it does not depend on central commutators.
Theorem 13.17 (No-signaling by algebra preservation). Let ω be a normal state on B(H), and let o_1,
…, o_n be operations performed in this order, each a normal instrument I_m with a standard Borel
outcome space (Ω_m, 𝔉_m). Write I_m*(Δ) for the Heisenberg dual of I_m(Δ) and N_m* = I_m*(Ω_m) for
the dual of its unread channel. For a set X of target operations and Borel sets Δ_x, x ∈ X, the
unconditioned joint probability of the target outcomes is
P(Δ_x : x ∈ X) = ω(T_1*(T_2*(⋯ T_n*(1) ⋯))),
where T_m* = I_m*(Δ_m) if m ∈ X and T_m* = N_m* otherwise, so that the outcomes of all non-target
operations are summed over rather than conditioned on. Fix a non-target operation j, and suppose
there is a linear subspace 𝔐 ⊆ B(H) containing 1 such that:
(i) I_x*(Δ)(𝔐) ⊆ 𝔐 for every target operation x performed after j and every Δ;
(ii) N_k*(𝔐) ⊆ 𝔐 for every non-target operation k ≠ j performed after j;
(iii) N_j*(B) = B for every B ∈ 𝔐.
Then the joint law of the target outcomes is the same whether j is performed, omitted, or replaced
by any other operation satisfying (iii). Condition (iii) holds in particular whenever j has Kraus
operators in the commutant 𝔐′, and operations performed before j are unrestricted.
Proof. Let j be the m-th operation and set B = T_{m+1}*(⋯ T_n*(1) ⋯). Starting from 1 ∈ 𝔐 and
applying the maps T_n*, …, T_{m+1}* in turn, conditions (i) and (ii) keep every intermediate
operator in 𝔐, so B ∈ 𝔐. By (iii), T_j*(B) = N_j*(B) = B, which is the value obtained when j is
omitted, and the same holds for any replacement satisfying (iii). The remaining maps T_{m−1}*, …,
T_1* and the state ω are unchanged. If K_y ∈ 𝔐′ for almost every outcome y of j, then N_j*(B) = ∫
K_y† B K_y dν(y) = B ∫ K_y† K_y dν(y) = B. □
Corollary 13.18 (Random local automorphisms). Suppose that every unread non-target operation k
performed after j is an average of automorphisms, N_k*(B) = ∫ U_k(t)* B U_k(t) dμ_k(t) with
unitaries U_k(t) and a probability measure μ_k, and that 𝔐 is σ-weakly closed and mapped into itself
by B ↦ U_k(t)* B U_k(t) for μ_k-almost every t. Then (ii) holds. For a free field, Gaussian
collapses and random displacements are of this form with U_k(t) = e^{−itφ(h_k)} or W(h), each of
which multiplies every Weyl operator by a phase. They therefore preserve the von Neumann algebra 𝔐_X
generated by { W(g) : g ∈ V_X }, for every target set X, and Theorem 13.17 gives Theorem 13.13(b)
for these operations.
Proof. A weak integral of a bounded measurable family in a σ-weakly closed subspace lies in that
subspace, since a normal functional that vanishes on the subspace vanishes on every member of the
family and hence on the integral. □
The theorem locates the role of relativistic causality exactly. Microcausality supplies (iii): if 𝔐
is generated by observables localized in the target regions and j is localized in a region causally
separated from all of them, the Kraus operators of j commute with 𝔐. It does not supply (ii), which
is a condition on the composed dynamics of the intervening operations. Example 13.10 is a case in
which (ii) fails. The unread Lüders instrument at C maps W(s f_B) to W(s f_B) G(φ(f_C)) for a non-
constant function G, which lies outside the algebra generated by W(s f_B), and Alice can then signal
to Bob. Microcausality alone therefore does not exclude Sorkin-type signaling. What must be
controlled is closure of a target algebra under the composed intervening operations. The theorem
also concerns unconditioned statistics only: conditioning on the outcome of an intervening operation
is not covered, and in general it can reveal what was done at j. Its scope is that of a sufficient
condition. It shows that one particular unread operation cannot influence the unconditioned target
statistics; it does not cover protocols in which later operations are chosen according to earlier
outcomes, and it is not a theorem about every relativistic signaling protocol. Its operational
reading depends on identifying correctly where each operation is localized and which regions are
causally separated, and for interacting fields its hypotheses must be verified for physically
realizable instruments before it yields any conclusion.
Nothing in Theorem 13.17 refers to free fields. For interacting fields, conjugation by e^{itA}, with
A an interacting local observable, does not in general map a target algebra into itself, and the
problem is to find nontrivial instruments for which (i)–(iii) hold. Instruments induced by coupling
a probe to the field in a compact region are natural candidates, because their causal properties are
controlled by the coupled dynamics (Fewster & Verch 2020). An operationally causal measurement
scheme is not, however, a spontaneous collapse law, and the latter needs an additional physical
mechanism.
Target result 13.19 (Interacting probe instruments). For a scalar field with a λφ⁴/4! self-
interaction, coupled to a probe in a compact region, compute the induced instrument to the lowest
nontrivial order in λ. Check that the free limit reproduces §13.6. Then determine whether conditions
(i)–(iii), or the causal factorization of the probe framework, hold at that order for composed
operations, including an intermediate operation in Sorkin’s configuration, with a deliberately
inadmissible instrument as a negative control. Such a calculation establishes a result to the order
computed. It is not an existence theorem for interacting quantum field theory in four dimensions.