Locality of Dynamics vs. Globality of Fields
1 An Overlooked Contradiction
Classical physics gives us a fundamental intuition:
Interactions are local, propagating through neighboring forces or fields within finite time.
Causality is temporal: the past influences the present and future via continuous evolution.
Yet, standard quantum mechanics seems to tell us something different:
The Schrödinger equation governs a globally defined wavefunction across all space.
Quantum entanglement exhibits “instantaneous correlations at a distance.”
The Aharonov–Bohm (A–B) effect appears to show that “particles sense distant magnetic flux even in field-free regions.”
Bell experiments are interpreted as evidence that “the world is fundamentally nonlocal.”
Thus, a seemingly unbridgeable rift appears between local dynamics and global field structure:
on one side, special relativity demands local causality;
on the other, quantum formalism presents global nonlocality.
The goal of this chapter is, from the perspectives of Natural Quantum Theory (NQT) and the Global Approximation Interpretation (GAI), to systematically clarify:
How dynamical equations (e.g., Schrödinger, Hamilton–Jacobi) remain fundamentally local evolution laws;
Why global wavefunctions and “nonlocal” appearances naturally emerge when we perform global spectral analysis and approximations;
How phenomena like the A–B effect and quantum entanglement can be demystified within a unified framework of local interactions + global boundary/topological constraints;
And how, in NQT, “local dynamics + global field structure” converges into a clear physical picture:
Ontology: continuous fields and topological structures
Dynamics: strictly local
Globality: arises from field boundaries, resonances, and spectral projections—not from “spooky action.”
2 Locality in Classical Dynamics: From Newton to Hamilton–Jacobi
2.1 Local Equations and Causal Chains
In classical mechanics, motion is governed by local differential equations:
Newton’s second law:
m d²x/dt² = F(x, t)
tells us explicitly: acceleration is determined by the force at the current position.In field theory (e.g., electromagnetism), Maxwell’s equations are also local partial differential equations: the field’s change at any point depends only on nearby fields and sources.
Special relativity further enforces locality:
Interactions propagate via fields or particles at or below light speed.
Any causal influence must travel along timelike or lightlike paths in spacetime.
Whether in Lagrangian or Hamiltonian form, the essence of dynamics is:
Local equations + initial conditions → continuous time evolution → definite causal chain.
2.2 Hamilton–Jacobi Equation: The “Classical Mother Equation” Unifying Waves and Particles
The Hamilton–Jacobi equation (HJE) offers a deeper perspective.
For the action function S(q, t), it reads:
∂S/∂t + H(q, ∂S/∂q, t) = 0
In the geometric optics analogy:
Level surfaces of S act like wavefronts.
Particle trajectories follow the direction of ∇S.
The bundle of trajectories and the wavefronts are two descriptions of the same dynamics.
In the NQT framework, HJE is viewed as the classical mother equation unifying wave and particle:
On one hand, it yields deterministic trajectories.
On the other, the geometric structure of S inherently carries phase-like properties.
This implies:
Quantum “waviness” and classical “particle trajectories” are already unified in a single action field.
The Schrödinger equation is merely a spectral encoding—a complex-number representation—of this action field, not an introduction of a new, non-causal ontology.
3 Schrödinger Equation: Local Hamiltonian + Global Spectrum
3.1 Local Hamiltonian Density
The standard Schrödinger equation is:
iℏ ∂ψ(x,t)/∂t = Ĥ ψ(x,t)
where
Ĥ = −(ℏ²/2m) ∇² + V(x,t)
From the operator structure, Ĥ is a local differential operator:
The potential V(x,t) depends only on the local environment at x.
The kinetic term ∇² is a neighborhood operator, reflecting local curvature and diffusion.
At this level, the Schrödinger equation introduces no action-at-a-distance—it is simply another form of local dynamics.
3.2 Nonrelativistic Approximation and Emergent Globality
The key lies in two often-overlooked facts:
(a) Nonrelativistic approximation
Deriving the Schrödinger equation implicitly assumes infinite propagation speed of interactions.
In reality, electromagnetic influences propagate at finite speed (c), but this temporal structure is compressed into instantaneous Hamiltonian action.
This erases the time needed to establish coherence and global patterns.
(b) Globality of spectral analysis
Solving the Schrödinger equation typically involves finding eigenstates:
Ĥ φₙ = Eₙ φₙThen expanding any state as:
ψ(x,t) = Σₙ cₙ e^(−iEₙt/ℏ) φₙ(x)
This is fundamentally a global spectral decomposition: eigenmodes are selected over the entire space and boundary conditions.
Thus:
The equation remains a local PDE,
but its solutions are global eigenmodes—inherently nonlocal in spatial support.
This is the core of the Global Approximation Interpretation (GAI):
The Schrödinger equation is an effective global equation arising from two approximations:
(1) nonrelativistic (infinite signal speed)
(2) global spectral representation
Its apparent “nonlocality” is not ontological nonlocality, but rather:
In spectral representation, local causal processes are projected onto global modes, and the local causal chain is “smeared out” in the representation.
4 Boundaries, Resonance, and Natural Quantization: Where Globality Comes From
4.1 Boundary Conditions and Eigenmodes
In any wave system, boundaries lead to discrete eigenmodes:
A guitar string with fixed ends → only certain resonant frequencies.
Microwave or optical cavities → discrete electromagnetic modes.
Electrons in a Coulomb potential → discrete atomic energy levels.
NQT emphasizes:
Quantization = spectral discreteness of confined waves, not an added postulate.
Boundary conditions and global geometry determine allowed eigenmodes.
“Energy quantization” in the Schrödinger equation is just global spectral analysis under constraints.
Therefore, the so-called “global wavefunction” is not the fundamental entity of reality, but rather:
A mathematical encoding of all possible resonant modes allowed by the system’s boundaries and topology.
4.2 How Global Constraints Reshape Local Dynamics
In the physical world, local interactions—through feedback, resonance, and dissipation over time—spontaneously establish stable eigenmodes:
Interactions themselves are local.
But when countless local processes occur within a confined geometry, the system converges to global modes satisfying boundary conditions.
In this process:
Coherencing (establishing phase coherence) is a time-consuming global process.
Once eigenmodes are established, the system appears as “discrete energy levels” or “stable quantum states.”
The Schrödinger equation ignores this formation process and directly uses the final global modes as an approximation.
Thus, at the deepest level, dynamics remains governed by local field equations;
what we see in quantum theory is their global spectral image under topological and boundary constraints.
5 Aharonov–Bohm Effect: Local Fields + Global Topology
The A–B effect is often hailed as the pinnacle of “nonlocal phase”:
Electrons travel in regions where B = 0, yet their interference phase depends on the enclosed magnetic flux Φ:
Δϕ = (q/ℏ) Φ
Superficially, electrons experience no local magnetic force but “know” the distant flux—seemingly violating locality.
From the NQT/GAI perspective, this is naturally understood:
Locality is preserved:
The physical entities are the electromagnetic fields E, B and the vector potential structure A_μ.
Although B = 0 along the path, A_μ has a nontrivial circulating structure around the flux region.
The electron’s field couples locally to A_μ at every point, accumulating phase along the path.Global topology matters:
The total phase is the line integral of A_μ along the path.
This integral depends only on the topological class of the loop (how many times it winds around the flux).
In other words:
The A–B effect is not action-at-a-distance, but the natural result of
local field coupling + global topological structure
expressed in spectral (wave) representation.
This aligns perfectly with our earlier view:
Local dynamics records the relationship between path and background field via phase transport.
At the wavefunction level, this appears as a global phase difference, read out via interference.
6 Entanglement and “Nonlocality”: Global Modes, Not Superluminal Causality
6.1 Entanglement as a Shared Global Coherent Mode
In GAI and NQT, entanglement is viewed as:
A shared vibrational mode of a multi-body system under specific boundary conditions—
analogous to normal modes of coupled strings: each string’s amplitude and phase are correlated,
but the correlation stems from the common mode, not from distant forces.
In this picture:
Entanglement requires time and local interaction:
Two particles must interact locally over finite time (e.g., via EM fields) to establish a shared global mode.
This process is fully consistent with local dynamics.Once the mode is established, the system is described in spectral space by a global state, e.g.:
|Ψ⟩ = (1/√2)(|↑↓⟩ − |↓↑⟩)
This is a compact encoding of the shared mode.Remote measurements simply read different projections of the same pre-existing mode:
Measuring one end selects an eigenvalue—
but this is just local access to a globally correlated structure, not “instantaneous change” of the distant particle.
6.2 Bell Experiments and the Assumption of Preparation Independence
Bell’s inequality relies on strong assumptions of preparation independence and causal separation.
But in NQT/GAI:
The states of two subsystems are not independent tensor products,
but projections of a single global field mode onto two spatial regions.This global mode intrinsically carries correlations, without requiring “spooky signals.”
Thus:
Entanglement correlations are the statistical manifestation of
“multi-body systems sharing global resonant modes,”
not “instantaneous state collapse at a distance.”
Locality holds at the ontological level (field equations, finite propagation speed).
Apparent “nonlocality” arises only because we use global wavefunctions to encode the system and interpret measurements statistically.
7 Unified View: Local Dynamics, Global Fields, Spectral Representation
Synthesizing the above, NQT unifies locality and globality as follows:
Ontological Layer
Physical reality consists of continuous fields and finite-scale topological structures (e.g., EM fields, extended charges, field vortices).
Field equations (Maxwell, Einstein in classical limit) are local PDEs.
Interactions propagate locally at finite speed, preserving causality.
Spectral Representation Layer (Quantum Layer)
Under given boundaries and topology, we perform global eigenmode decomposition of local field solutions.
The Schrödinger equation, state vectors, and operator algebra are tools for this global spectral analysis.
Quantization, entanglement, geometric phases—all reflect the structure of the eigen-spectrum and topology.
Source of Globality
Arises from boundaries, geometry, and topology—not from superluminal influence.
Results from mathematically representing the whole system as a sum of global eigenmodes, not from ontological nonlocality.
Duality of Local–Global
In time domain / field-equation view: all processes are local and causal.
In frequency domain / spectral view: the same processes appear as global modes and phase correlations.
These are complementary descriptions: the former is “process language,” the latter is “mode language.”
8 Summary
This chapter, from the perspectives of Natural Quantum Theory and the Global Approximation Interpretation, presents a unified picture of “locality of dynamics vs. globality of fields”:
Classically: Newton’s laws, Hamilton–Jacobi, and Maxwell’s equations are all local dynamical laws, ensuring causality and finite propagation.
In Schrödinger theory: the local Hamiltonian density, under nonrelativistic + global spectral approximations, yields global wavefunctions and eigenmodes—making “local dynamics → global modes” a mathematical reality.
Boundaries and topology: natural quantization and geometric phases arise from confined waves and topological structure; A–B and spin geometric phases are natural outcomes of “local coupling + global topology.”
Entanglement and nonlocality: entanglement is reduced to shared global vibrational modes of multi-body systems—no superluminal causality needed; Bell experiments reveal global mode structure, not instantaneous action.
NQT unification:
Ontology = continuous fields + finite topological structures
Dynamics = strictly local
Quantum mechanics = global spectral representation
In this view, “locality of dynamics” and “globality of fields” are not in conflict, but two complementary manifestations of the same physical reality in different time-frequency representations.
What truly needs revision is not the mathematics of quantum theory, but our physical interpretation of its formalism.