Paper IV — Local Interactions and Potentials from Operational Locality

Local Interactions and Potentials from Operational Locality Paper IV of the Ordered-Dynamics Reconstruction Program Paul Cooney DOI: 10.5281/zenodo.17925675 Abstract Papers I–III of the Ordered-Dynamics Reconstruction Program derive quantum kinematics, free nonrelativistic dynamics (Schr¨odinger class), and a Lorentz-covariant extension (scalar mass shell) from operational principles based on ordered reversible evolution and bounded information growth. The present paper addresses interaction structure. In standard nonrelativistic quantum mechanics, interactions enter through Hamiltonians of the form H=P2/(2m)+V(X), but the restriction to multiplicative potentials is usually assumed rather than derived. We introduce an Operational Locality principle (OL) stating that a localized intervention cannot instantaneously influence the initial-time rate of change of observables supported in spatially disjoint regions. We show that OL implies a sharp no-coupling condition on the interaction generator: matrix elements between disjoint spatial sectors must vanish. Using standard von Neumann algebra arguments, we prove a rigidity theorem: any self-adjoint operator satisfying this no-coupling condition must be a multiplication operator V(X) for some real-valued measurable function V. The result is then elevated operationally via the Trotter product formula: e−i(T+V)t/ℏ= limn→∞(e−iT t/(nℏ)e−iV t/(nℏ))n, which shows that admissible dynamics decomposes, at arbitrarily fine resolution of the ordering parameter, into alternating phases of transport (constrained by Operational Information Locality, Paper II) and local query (constrained by OL). Momentum-dependent and derivative-coupling terms are excluded as violations of this separation of operational resources. We clarify the relationship to gauge connections, which are treated separately in Paper VIII as compensators of local descriptive redundancy rather than as matter–matter interaction terms. Finally, we note a forward consequence for Paper IX: if interactions are forced into V(X) form, then generic environments monitor position, naturally pre-structuring spatial record formation. 1 1 Introduction Nonrelativistic quantum mechanics is commonly presented with Hamiltonians of the form H=P2 2m+V(X),(1) motivated by analogy with classical kinetic and potential energy. While (1) is empirically successful, its structural restriction raises a foundational question: Why must fundamental interactions in nonrelativistic quantum mechanics be multiplicative in position? In Papers I–III, the reconstruction program treats mathematical structure as a consequence of operational consistency constraints relative to an abstract ordering parameter. Paper I establishes the minimal convex/linear framework and reversible dynamics needed to speak operationally about systems and statistics. Paper II introduces Operational Information Locality (OIL), a bound on the growth of spatial distinguishability under free evolution, and shows that it selects the Schr¨odinger class of generators. Paper III shows that a Lorentz-covariant analog of OIL selects the scalar mass shell and compatible relativistic kinematics. 1.1 What Paper IV adds: “change beyond motion” Paper II constrains transport: how distinguishability can spread under free evolution. Paper IV addresses a different aspect of change: local influence, i.e. the structure of additional generators that represent interventions, forces, couplings, or external control. The paper introduces an operational locality postulate (OL) stating that a localized intervention cannot instantaneously influence the initial-time rate of change of observables supported in spatially disjoint regions. Translated into operator language, OL becomes a sharp no-coupling condition between disjoint spatial sectors. We prove that this condition uniquely forces the interaction generator to be a multiplication operator V(X). 1.2 Operational upgrade: transport vs. local query (Trotter “tick–tock”) The result will be framed not merely as “deriving the textbook form,” but as a rigidity statement about the operational logic of time evolution. Using the Trotter product formula, we interpret the sum H=T+Vas the unique way to interleave two primitive phases at arbitrarily fine resolution of the ordering parameter: 1. Transport phase (OIL): information spreads under T. 2 2. Local query phase (OL): local influence is encoded by V(X). Mixed generators that entangle these resources (e.g. derivative couplings) are excluded as violations of operational locality. 1.3 Roadmap Section 2fixes notation and assumptions. Section 3distinguishes OIL (Paper II) from OL (this paper) and explains why both are needed. Section 4 develops the “tick–tock” operational reading via Trotterization. Section 5 states OL precisely and derives the no-coupling condition. Section 6proves the multiplication-operator characterization and gives the main uniqueness theorem. Section 7treats momentum-dependent and derivative coupling explicitly. Section 8gives concrete examples and non-examples. Section 9 clarifies the relationship to gauge connections (Paper VIII). Section 10 notes the pointer-structure consequence and links forward to Paper IX (records and operational irreversibility). 2 Framework and notation We work on the Hilbert space H=L2(Rd). Position operators X= (X1, . . . , Xd) act as multiplication by x, and momentum operators P= (P1, . . . , Pd) act as Pj=−iℏ∂jon an appropriate domain, with canonical commutation relations [Xi, Pj]=iℏδij. The free Hamiltonian is taken to be H0=P2 2m,(2) whose Schr¨odinger-class form is justified in Paper II as the generator compatible with OIL for nonrelativistic transport. Interactions are introduced additively: H=H0+Hint,(3) with Hint assumed self-adjoint on a suitable domain. For a measurable region Ω ⊂Rd, let ΠΩ:= M 1 Ω(4) denote the orthogonal projection given by multiplication by the indicator function 1 Ω. We say that a bounded operator Bis supported in Ω if B= ΠΩBΠΩ.(5) 3 3 Two locality principles: OIL versus OL The reconstruction program uses two distinct locality constraints, addressing different operational questions. 3.1 Operational Information Locality (OIL): bounded transport OIL (Paper II) is a bound on how rapidly spatial distinguishability can grow under free evolution. In its simplest variance form, it constrains the growth of position variance in time in terms of momentum variance and a mass scale. Operationally, OIL limits information transport: distant regions cannot become distinguishable “too quickly” under unperturbed evolution. 3.2 Operational Locality (OL): no instantaneous influence OL (this paper) is a constraint on influence at the initial instant: a localized intervention in region Ω1cannot affect the initial-time rate of change of observables supported in a disjoint region Ω2. 3.3 Why both are needed (and why H0is not treated as an OL interaction) At first glance, the kinetic term H0=P2/(2m) has nonlocal kernels in the position representation and thus does not satisfy the no-coupling condition we derive for Hint. This is not a contradiction: H0represents transport rather than intervention. Its nonlocality is precisely the mechanism of propagation, and its rate is controlled by OIL over finite time intervals. By contrast, Hint is intended to encode additional influence beyond free transport. OL constrains this additional structure so that it does not introduce instantaneous cross-region sensitivity beyond what is already present in transport and already bounded by OIL. Remark 3.1 (Prospect of unification).In relativistic field-theoretic settings, microcausality (commutativity at spacelike separation) unifies “no instantaneous influence” and “finite propagation speed” into a single causal locality principle. The present paper is deliberately nonrelativistic and singleparticle: we therefore treat OIL and OL as complementary operational inputs that are mutually compatible and jointly restrict the allowed structure. 4 The “tick–tock” structure of ordered evolution The theorem proved in this paper has an algebraic content (OL implies Hint =V(X)) and an operational content: admissible evolution splits into transport and local query in the ordered-time limit. 4 4.1 Trotter product formula as an operational decomposition Let T:= H0denote transport (fixed by OIL), and let V:= Hint be an additional self-adjoint generator. Whenever T+Vis self-adjoint on a suitable domain, the Trotter product formula states e−i(T+V)t/ℏ= lim n→∞ e−iTt/(nℏ)e−iV t/(nℏ)n,(6) with convergence in the strong operator topology under standard hypotheses. Equation (6) yields an operational reading of the sum H=T+V: evolution over a macroscopic interval tcan be approximated at increasingly fine resolution by alternating primitive steps: 1. Transport step (OIL): e−iT δt/ℏpropagates distinguishability according to the free law. This step is responsible for information transmission and is constrained by OIL. 2. Local query step (OL): e−iV δt/ℏmodifies phases locally in the position basis and represents localized influence that does not instantaneously couple disjoint regions. Thus H=T+V(X) is the unique way to interleave bounded transport with strictly local influence at the infinitesimal ordered-time scale. 4.2 Why mixed generators are operationally unstable A generator that mixes transport and query at the interaction level (e.g. g(X)·Por P a(X)Pas an added interaction) does not simply “add a force”. It couples the local strength of influence to the instantaneous generator of translation, thereby producing immediate cross-region sensitivity to localized perturbations. This is precisely what OL forbids. Section 7makes this exclusion explicit. 5 Operational Locality 5.1 Operational locality postulate Let Ω1,Ω2⊂Rdbe measurable regions with Ω1∩Ω2=∅and dist(Ω1,Ω2)> 0. Axiom 5.1 (Operational Locality (OL)).Let UΩ1(ϵ) = e−iϵAΩ1be a oneparameter family of local unitaries supported in Ω1, i.e. AΩ1= ΠΩ1AΩ1ΠΩ1 is self-adjoint. Let BΩ2be any bounded observable supported in Ω2, i.e. BΩ2= ΠΩ2BΩ2ΠΩ2. Then for any state |ψ⟩supported in Ω2, the localized perturbation in Ω1does not affect the initial-time rate of change of ⟨BΩ2⟩: ∂ ∂ϵ d dtϵ=0,t=0 ⟨ψ|UΩ1(ϵ)†eiHt/ℏBΩ2e−iHt/ℏUΩ1(ϵ)|ψ⟩= 0.(7) 5 Remark 5.1 (Operational content).OL is a minimal “no instantaneous influence” requirement. It does not forbid propagation effects at later times; those are bounded by OIL. It forbids only a dependence of the initial derivative at Ω2on interventions supported in a disjoint region Ω1. 5.2 From OL to a no-coupling condition We now derive the operator constraint corresponding to Axiom 5.1. Proposition 5.1 (No-coupling condition).Assuming Axiom 5.1, the interaction term satisfies ΠΩ2Hint ΠΩ1= 0 for all disjoint measurable Ω1,Ω2.(8) Proof. Fix disjoint Ω1,Ω2and choose a state |ψ⟩supported in Ω2, i.e. |ψ⟩= ΠΩ2|ψ⟩. Let UΩ1(ϵ) = e−iϵAΩ1with AΩ1= ΠΩ1AΩ1ΠΩ1, and let BΩ2be supported in Ω2. Define F(ϵ, t) := ⟨ψ|UΩ1(ϵ)†eiHt/ℏBΩ2e−iHt/ℏUΩ1(ϵ)|ψ⟩.(9) Expand to first order in both ϵand t(retaining the mixed term): F(ϵ, t)=⟨ψ|BΩ2|ψ⟩+it ℏ⟨ψ|[H, BΩ2]|ψ⟩−iϵ ⟨ψ|[AΩ1, BΩ2]|ψ⟩ −ϵt ℏ⟨ψ|[AΩ1,[H, BΩ2]] |ψ⟩+O(t2)+O(ϵ2)+O(ϵt2).(10) Taking ∂ϵ∂tat (ϵ, t) = (0,0) isolates the mixed term: ∂ ∂ϵ d dtϵ=0,t=0 F(ϵ, t) = −1 ℏ⟨ψ|[AΩ1,[H, BΩ2]] |ψ⟩.(11) By OL, this must vanish for all choices of AΩ1,BΩ2, and |ψ⟩supported in Ω2: ⟨ψ|[AΩ1,[H, BΩ2]] |ψ⟩= 0.(12) Now choose BΩ2= ΠΩ2and choose AΩ1= ΠΩ1. Since Ω1,Ω2are disjoint, ΠΩ1ΠΩ2= 0 and hence [ΠΩ1,ΠΩ2] = 0. Equation (12) becomes ⟨ψ|[ΠΩ1,[H, ΠΩ2]] |ψ⟩= 0.(13) Using H=H0+Hint and the fact that |ψ⟩is supported in Ω2, the contribution from H0to (13) is independent of the perturbation in Ω1and represents transport effects already controlled by OIL (Section 3). The operative OL constraint is therefore imposed on Hint: ⟨ψ|[ΠΩ1,[Hint,ΠΩ2]] |ψ⟩= 0.(14) 6 Expanding the commutators gives [ΠΩ1,[Hint,ΠΩ2]] = ΠΩ1HintΠΩ2−ΠΩ2HintΠΩ1(15) because ΠΩ1ΠΩ2= 0. Sandwiching between |ψ⟩= ΠΩ2|ψ⟩yields 0=⟨ψ|ΠΩ2HintΠΩ1|ψ⟩.(16) Since |ψ⟩supported in Ω2is arbitrary, this implies ΠΩ2HintΠΩ1= 0, proving (8). Remark 5.2 (Interpretation).Equation (8)states that Hint cannot map a state supported in Ω1into Ω2at the generator level whenever the regions are disjoint. This is the operator-theoretic content of “no instantaneous influence” for the additional interaction structure. 6 Characterization of local interaction operators 6.1 Multiplication operator characterization We now show that the no-coupling condition forces Hint to be a multiplication operator. Theorem 6.1 (Multiplication operator characterization).Let Abe a selfadjoint operator on L2(Rd)such that ΠΩ2AΠΩ1= 0 for all disjoint measurable Ω1,Ω2.(17) Then there exists a real-valued measurable function V:Rd→Rsuch that A=V(X)(multiplication by V). Proof. Condition (17) implies that Apreserves supports: if ψis supported in Ω, then Aψ is supported in Ω as well. Equivalently, Acommutes with all projections ΠΩ: [A, ΠΩ] = 0 for all measurable Ω.(18) Since finite linear combinations of such projections generate the algebra of bounded multiplication operators Mf(with f∈L∞), it follows that A commutes with all bounded multiplication operators. The bounded multiplication operators form a maximal abelian von Neumann algebra on L2(Rd). By the commutant characterization (see, e.g., [1,2]), any (possibly unbounded) self-adjoint operator commuting with this algebra is itself a multiplication operator. Self-adjointness forces the multiplier to be real-valued. 7 6.2 Main uniqueness theorem Combining Proposition 5.1 with Theorem 6.1 yields: Theorem 6.2 (Unique interaction structure under OL).Assume the Paper II transport generator H0=P2/(2m)and Axiom 5.1. Then any admissible interaction term Hint must be of the form Hint =V(X) (19) for some real-valued measurable V:Rd→R. Hence the most general Hamiltonian compatible with OIL (transport) and OL (no instantaneous influence for additional couplings) is H=P2 2m+V(X).(20) Remark 6.1 (Domain conditions).Standard sufficient conditions for essential self-adjointness of P2/(2m)+V(X)on C∞ 0(Rd)include Kato–Rellichtype hypotheses such as Vbeing relatively form-bounded with respect to P2. We do not attempt maximal generality here; the theorem is structural and holds at the level of operator form. 7 Excluded alternatives: momentum dependence and derivative coupling 7.1 Momentum-dependent interactions are nonlocal in position Consider Hint =f(P) for a real Borel function f. In Fourier space, f(P) acts as multiplication by f(p): ^ (f(P)ψ)(p)=f(p)˜ ψ(p).(21) Transforming back, (f(P)ψ)(x) = ZRdb f(x−x′)ψ(x′)dx′.(22) Unless fis constant, b fis not supported at the origin and the kernel has nontrivial off-diagonal support. Consequently, there exist disjoint Ω1,Ω2such that ΠΩ2f(P)ΠΩ1= 0, violating (8). Thus purely momentum-dependent added interactions are excluded by OL. 8 7.2 Derivative coupling as forbidden dependence of local influence on transmission The mathematical reason derivative couplings violate OL is that they fail the no-coupling condition between disjoint spatial regions. It is also useful to state the operational reason in the language of the series. The momentum operator Pis the generator of translations; it is the primitive resource that implements transport of distinguishability. A term such as g(X)·Pmakes the instantaneous strength of the coupling depend on the generator of translation. Operationally, this means: the local influence exerted at a point depends on how fast information is being transmitted away from that point. This conflates two roles treated as distinct throughout the program: 1. Transmission (transport): constrained by OIL as a bound on information growth under free evolution; 2. Query (local influence): constrained by OL as a prohibition of instantaneous cross-region sensitivity. OL enforces that interaction strength is a local state property (encoded by a multiplication operator in the position representation), not a transport property. In this sense, the exclusion of derivative coupling is a rigidity constraint on the operational separation of resources: local influence cannot be parametrically controlled by instantaneous translation. 7.3 Explicit kernel obstruction for linear derivative terms For completeness, note that in the position representation Pj=−iℏ∂j, so a term g(X)·Phas distributional kernel involving derivatives of δ(x−x′): ⟨x|g(X)·P|x′⟩=−iℏg(x)· ∇x′δ(x−x′).(23) Such kernels do not preserve disjoint supports in the sense of (8), and therefore violate OL when treated as added interaction generators. 8 Examples and applications Theorem 6.2 does not determine which potential is realized; it determines the allowed structural form of interactions consistent with OL. 8.1 Allowed examples Harmonic oscillator. V(X) = 1 2mω2|X|2is multiplication by a real function. 9