# Quantum Reference Frames and Single-Outcome Dynamics — A Compatibility Framework for Generative Presentism

Part 19 of 40 (1,299 words)



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.