Paper III — Relativistic Extension of Ordered Information Locality

DOI: 10.5281/zenodo.17925671 Relativistic Extension of Ordered Information Locality Paper III of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction 1 2 Operational Ingredients: Ordering, Clocks, Events, Signals 2 2.1 Ordering parameter and operational time 2 2.2 Clocks as phase accumulators 2 2.3 Events and coincidences 3 2.4 Signals and causal order 3 3 Inertial Frames and Lorentz-Covariant Structure 3 3.1 Operational inertial postulates 3 3.2 Linearity, composition, and transformation structure 4 3.3 Ignatowsky-type classification 4 4 Proper Time and the Clock Hypothesis 4 4.1 Clock hypothesis 4 4.2 Proper time as invariant operational parameter 4 5 Relativistic Operational Information Locality (ROIL) 5 5.1 Motivation: what replaces tin OIL? 5 5.2 Rest-frame formulation of spatial distinguishability 5 5.3 ROIL axiom 5 6 Lorentz Covariance and the Scalar Mass Shell 6 6.1 Translation-invariant free propagation and velocity operator 6 6.2 ROIL as a variance inequality for all states 6 6.3 From variance inequality to pointwise Lipschitz bound 7 6.4 Lorentz covariance and uniqueness of the dispersion 7 7 Wigner Classification and the Klein–Gordon Sector 8 7.1 Representation-theoretic context 8 7.2 Scalar identification and Klein–Gordon constraint 8 8 Gauge Structure as Operational Redundancy 8 8.1 Phase as temporal bookkeeping 8 8.2 Local phase freedom 9 8.3 Connection and minimal coupling 9 9 Scope and Limitations 9 9.1 What has been achieved 9 9.2 What has not been derived 10 9.3 Relationship to standard approaches 10 10 Conclusion 10 – i – A Example: Gaussian wavepacket and the nonrelativistic limit 11 A.1 Momentum-space evolution 11 A.2 Small-momentum expansion and Schr¨odinger limit 12 A.3 Spreading and ROIL scaling 12 A.4 Comparison with Dirac: why spin needs additional principles 12 DOI: 10.5281/zenodo.17925671 apers I and II of the Ordered-Dynamics Reconstruction Program show that quantum kinematics and free nonrelativistic Schr¨odinger dynamics can be constrained by operational consistency requirements relative to an abstract ordering parameter and bounded information propagation. The present paper extends that framework to the relativistic regime. We show that operational information locality admits a Lorentz-covariant generalization when inertial evolution is parametrized by proper time. This leads to a relativistic operational information locality bound (ROIL), a proper-time constraint on the growth of spatial distinguishability in the instantaneous rest frame of an inertial worldline. Requiring Lorentz covariance and universality of this bound singles out a unique invariant mass scale and constrains inertial kinematics to the standard scalar mass shell. Using Wigner’s classification, the corresponding sector is identified with the massive spin-0 representation, yielding Klein– Gordon dispersion as a representation of the kinematical constraint. Finally, local freedom in phase conventions for temporal bookkeeping forces the introduction of a connection and minimal coupling. The results are deliberately kinematical and single-particle in scope. No claim is made to derive special relativity, quantum field theory, or gauge dynamics from first principles. Rather, the contribution is to show that relativistic quantum kinematics and gauge coupling are compatible with—and constrained by—the same operational locality principles that underlie the nonrelativistic theory. Contents 1 Introduction Paper II introduced operational information locality (OIL) as a bound on how rapidly spatial distinguishability can grow under free evolution. Enforcing OIL constrains admissible generators and selects the Schr¨odinger class of nonrelativistic dynamics. The present paper asks a structurally parallel question in the relativistic setting: What is the Lorentz-covariant form of OIL, and what kinematical structure is forced by enforcing it universally across inertial frames? The goal is not to re-derive special relativity, nor to build interacting relativistic quantum field theory. Instead, we aim to exhibit a consistent extension of the locality-of-informationgrowth principle into the relativistic regime, and to identify the minimal kinematical consequences of doing so. The organizing idea is that the framework of Papers I–II distinguishes: (i) a structural ordering parameter used to speak about sequenced evolution, and (ii) operational time as – 1 – realized by physical clocks. In relativity, inertial observers agree that the appropriate operational parameter along an inertial worldline is proper time, as recorded by ideal clocks. This motivates formulating a relativistic information-growth bound in terms of proper time. After establishing the minimal operational ingredients (ordering parameter, clocks, signals, causal order), we state operational inertial postulates and adopt the Lorentz group as the unique inertial symmetry compatible with them (Ignatowsky-type classification). We then introduce ROIL as a proper-time bound on spatial distinguishability growth in instantaneous rest frames. The central technical result is that enforcing ROIL with a universal mass scale across all inertial observers constrains the admissible inertial sector to the standard mass shell. Wigner’s classification then identifies the corresponding kinematical representation as the massive scalar sector, and Klein–Gordon dispersion appears as a representation of the constraint. Finally, we show that allowing local freedom in phase conventions forces the introduction of a connection and minimal coupling as a matter of consistency. Scope. We restrict to flat spacetime, inertial motion, and the single-particle spin-0 sector. We do not attempt to derive gauge-group choice, gauge-field dynamics, interacting QFT, or spinor structure. Those require additional principles developed in subsequent papers. 2 Operational Ingredients: Ordering, Clocks, Events, Signals 2.1 Ordering parameter and operational time As in Paper I, we assume an abstract ordering parameter τthat provides a notion of sequenced evolution. The ordering parameter is not an observable coordinate time and is not tied to any particular inertial frame. Its role is structural: it supports statements of the form “process Aoccurs before process B” and provides a background against which operational time can be defined. Operational time arises only through physical clocks and their agreement properties. Different observers may use different conventions and coordinate assignments, but the empirical content of time measurement is encoded in the behavior and comparison of clocks. 2.2 Clocks as phase accumulators We adopt a minimal operational characterization of ideal clocks. Definition 1 (Ideal clock).An ideal clock is a physical system with a stable internal cyclic degree of freedom, represented by a phase variable θ, such that along inertial transport the accumulated phase difference between two events on the clock’s worldline determines an elapsed operational time: ∆τclock := ∆θ ω0 ,(2.1) where ω0is a calibration constant fixing units. The key operational feature is that the clock measures elapsed time by phase accumulation, and different realizations of ideal clocks agree when transported along the same inertial history (formalized in Section 4). – 2 – 2.3 Events and coincidences Definition 2 (Event).An event is an operational coincidence of physical processes, such as the emission or reception of a signal, an interaction between systems, or a clock reading registered at a particular coincidence. No background metric or coordinate structure is assumed at this stage; events are primitive and are related only by operational procedures (signals and clock comparisons). 2.4 Signals and causal order Signals mediate physical influence. We assume signals propagate forward with respect to the ordering parameter and at finite speed. Definition 3 (Signal reachability and causal precedence).For events E1and E2, we write E1≺E2if a signal can be emitted at E1and received at E2. We take ≺to be a strict partial order: •Irreflexive: E⊀E(no signal is received before it is emitted). •Transitive: E1≺E2and E2≺E3implies E1≺E3. •Antisymmetric (strict): E1≺E2implies not (E2≺E1). These properties encode causal consistency at the operational level, without invoking spacetime geometry. 3 Inertial Frames and Lorentz-Covariant Structure 3.1 Operational inertial postulates We now adopt standard inertial postulates, stated operationally: [Inertial postulates] 1. Relativity: The operational form of physical laws is the same in all inertial frames. 2. Homogeneity: No inertial frame can detect an absolute origin of space or time from local experiments. 3. Isotropy: No inertial frame can detect an absolute spatial direction from local experiments. 4. Invariant two-way signal speed: There exists a finite constant csuch that the two-way measured speed of a designated class of signals is the same for all inertial observers. 5. Causal consistency: The causal precedence relation ≺is preserved under inertial transformations. Remark 1 (What is input vs. output).Axiom 3.1 is an empirical input characterizing inertial physics. The Lorentz group is not assumed as a geometric postulate; rather, it is the unique transformation group compatible with these operational requirements (up to the Galilean limit). – 3 – 3.2 Linearity, composition, and transformation structure Homogeneity implies that inertial transformations map uniform motion to uniform motion. Under mild regularity assumptions, this forces transformations between inertial coordinate descriptions to be affine. If origins are chosen to coincide, they become linear: x′µ= Λµνxν.(3.1) The relativity principle and consistency of changing frames further require closure under composition, giving a group structure for admissible Λ. 3.3 Ignatowsky-type classification The classic Ignatowsky approach and its refinements show that Axiom 3.1 restricts admissible inertial transformation groups to two cases. [Ignatowsky-type classification] Assume Axiom 3.1 and linearity of inertial transformations between coincident origins. Then the admissible inertial transformation group is either: •the Galilean group (corresponding to c→ ∞), or •the Lorentz group, preserving a quadratic form of signature (+,−,−,−). Remark 2 (References and proof status).Complete derivations appear in Ignatowsky [1] and later treatments such as L´evy-Leblond [3] and Berzi–Gorini [2]. We adopt the Lorentzian case, consistent with a finite invariant signal speed. 4 Proper Time and the Clock Hypothesis 4.1 Clock hypothesis We now state the universality assumption required to identify the operational inertial time parameter. [Clock hypothesis] The elapsed time recorded by an ideal clock between two events on its inertial worldline depends only on the worldline segment connecting those events, and not on the clock’s internal constitution. Remark 3 (Operational content).Axiom 4.1 asserts agreement between distinct physical clocks when transported along the same inertial history, and is supported by extensive experimental evidence (relativistic time dilation in unstable particle lifetimes, clock comparison experiments, and precision timing applications). 4.2 Proper time as invariant operational parameter Lorentz covariance implies there exists (up to units) a unique scalar functional of timelike inertial displacement that is invariant under inertial transformations. In coordinates adapted to an inertial frame, it takes the form dτ =1 cpc2dt2−dx2.(4.1) [Selection of proper time] Assuming Axiom 4.1 and Lorentz covariance (Theorem 3.3 in the Lorentzian case), the operational time recorded by ideal inertial clocks is proportional to the Lorentz-invariant scalar dτ defined by Eq. (4.1). – 4 – Proof sketch. Ideal clocks are characterized by phase accumulation. Universality demands that the elapsed operational time be an invariant scalar under inertial transformations. In the Lorentzian case, the only invariant scalar constructed from the infinitesimal inertial displacement is the square root of the invariant quadratic form; thus the clock time is proportional to dτ. Remark 4 (Role in the reconstruction program).Proper time is not introduced as a primitive geometric entity, but as the unique operational parameter selected by clock universality under Lorentz covariance. It is the natural parameter with respect to which relativistic informationlocality constraints can be stated without frame ambiguity. 5 Relativistic Operational Information Locality (ROIL) 5.1 Motivation: what replaces tin OIL? In Paper II, OIL bounds the growth of spatial distinguishability under free nonrelativistic evolution: Varψ(X(t)) ≤Varψ(X(0)) + t2 m2Varψ(P)+O(|t|),(5.1) for an appropriate operational notion of position and momentum. The key issue in the relativistic setting is that coordinate time tis frame dependent. A Lorentz-covariant bound must be formulated using an invariant evolution parameter. Section 4 identifies proper time τas that parameter along inertial worldlines. 5.2 Rest-frame formulation of spatial distinguishability A fully covariant four-variance ∆Xµ∆Xµis not operationally well-behaved due to Lorentz signature, and because time components are not measured in the same way as spatial components. Instead, we formulate ROIL in the instantaneous rest frame of an inertial worldline, where spatial position and momentum have direct operational meaning. Let X(τ) and Pdenote spatial position and momentum operators in the instantaneous rest frame at proper time τ. Define ∆X(τ)2:= 3 X i=1 Varψ(Xi(τ)),∆P2:= 3 X i=1 Varψ(Pi).(5.2) 5.3 ROIL axiom [Relativistic Operational Information Locality (ROIL)] Fix an inertial worldline and consider its instantaneous rest frame at proper time τ. There exists a universal constant m > 0 such that for any normalized state ψwith finite variances, ∆X(τ)2≤∆X(0)2+c2τ2 m2∆P2+O(|τ|),(5.3) and the bound must hold for all inertial observers (i.e. in all inertial frames, with the corresponding rest-frame operators). Remark 5 (Dimensional necessity of c2τ2/m2).Since [∆X2]=L2and [∆P2]=(ML/T)2, the coefficient multiplying ∆P2must have units T2/M2. Lorentz covariance singles out τ as the invariant time and cas the invariant speed scale, yielding the natural combination c2τ2/m2. – 5 – Remark 6 (Operational meaning).ROIL states that inertial evolution cannot generate spatial distinguishability faster than a universal rate controlled by a single scale m, measured per unit proper time. This is a locality-of-information-growth constraint, not a dynamical equation. 6 Lorentz Covariance and the Scalar Mass Shell 6.1 Translation-invariant free propagation and velocity operator To extract kinematical consequences from ROIL, we assume (as in Paper II for the free case) that inertial propagation is generated by a translation-invariant self-adjoint operator Hwhich is a function of spatial momentum: H=E(P),(6.1) with E:R3→Rsufficiently regular (at least C1almost everywhere on the support considered). In the Heisenberg picture, d dτ Xi(τ) = i ℏ[H, Xi(τ)] = i ℏ[E(P), Xi(τ)].(6.2) Using [f(P), Xi] = iℏ∂Pif(P) for suitable functional calculus, we obtain the velocity operator Vi:= d dτ Xi(τ)=∂PiE(P),(6.3) and hence X(τ)=X(0) + τV,(6.4) for inertial (free) propagation. 6.2 ROIL as a variance inequality for all states Let ψbe any normalized state with finite variances. From (6.4), and using Var(A+B)≤ 2Var(A) + 2Var(B), we obtain an exact identity ∆X(τ)2= ∆X(0)2+τ2∆V2+ 2τ 3 X i=1 Cov(Xi(0), V i),(6.5) where ∆V2:= PiVar(Vi) and Cov is the symmetric covariance. ROIL (Axiom 5.3) requires that ∆X(τ)2≤∆X(0)2+c2τ2 m2∆P2+O(|τ|).(6.6) Comparing with (6.5), the O(|τ|) term is consistent with the covariance term; the nontrivial content is that for all states, ∆V2≤c2 m2∆P2.(6.7) – 6 – 6.3 From variance inequality to pointwise Lipschitz bound We now convert (6.7) into a pointwise bound on the function v(p) := ∇pE(p). Work in the momentum representation, where Pacts by multiplication. Then V= ∇pE(p) is also multiplication by v(p). For a normalized momentum wavefunction ˜ ψ(p), define the probability density µ(d3p)=|˜ ψ(p)|2d3p. Then ∆P2=Z|p−¯ p|2µ(d3p),∆V2=Z|v(p)−¯ v|2µ(d3p),(6.8) with ¯ p=Rpµand ¯ v=Rvµ. The inequality (6.7) holding for all µis a rigidity condition. In particular, choose µ supported on two small disjoint balls centered at p1and p2with equal weight. Letting the ball radii shrink to 0 yields the two-point specialization 1 4|v(p1)−v(p2)|2≤c2 m2 1 4|p1−p2|2,(6.9) hence the global Lipschitz condition |v(p1)−v(p2)| ≤ c m|p1−p2|for all p1,p2.(6.10) Equivalently, vis globally (c/m)-Lipschitz and therefore ∇pvexists almost everywhere with operator norm bounded by c/m. 6.4 Lorentz covariance and uniqueness of the dispersion Equation (6.10) is the relativistic analog of Paper II’s rigidity step: a variance inequality holding for all states forces a Lipschitz constraint on the velocity function. To identify the dispersion, we now incorporate Lorentz covariance. In the Lorentzian inertial sector, the conserved quantities combine into a four-momentum Pµ= (E/c, P), and the only intrinsic scalar available to label an irreducible inertial sector is the quadratic Casimir PµPµ. Universality of the ROIL constant mmeans that the same inertial sector must be recognized by all observers. The only covariant way to encode a single universal scale is to fix the Casimir: PµPµ=m2c2.(6.11) In terms of E(p), (6.11) is E(p)2=m2c4+c2|p|2,(6.12) with the positive-energy branch selected by stability. [ROIL rigidity ⇒massive mass shell] Assume (i) translation-invariant inertial propagation H=E(P), (ii) ROIL with universal constant m(Axiom 5.3), and (iii) Lorentz covariance of the inertial sector (Theorem 3.3 in the Lorentzian case). Then ROIL implies the global Lipschitz bound (6.10) on v=∇E, and Lorentz covariance fixes the inertial sector to the massive shell (6.11), equivalently the dispersion (6.12). Remark 7 (Connection to Paper II’s rigidity lemma).The step from (6.7) to the pointwise Lipschitz bound (6.10) is structurally identical to Paper II’s variance–Lipschitz equivalence: a variance inequality holding for all states forces a Lipschitz constraint on the corresponding multiplier function (group velocity). Paper II then classifies the allowed C1,1dispersions in the nonrelativistic setting; here, Lorentz covariance further selects the unique dispersion with fixed invariant mass parameter, namely (6.12). – 7 –