TLPL: Ter Law Particle Lattice
A Unified Quantum–Geometric Particle and Field Theory of Sector Dynamics

by D Law

copyright 2026

fair use only

Abstract

We present TLPL (Ter Law Particle Lattice), a unified toy model that couples a discrete sector decomposition of spacetime with quantum state dynamics and a rapidity‑derived scalar field. Sectors act as coarse‑grained spacetime cells; quantum states are superpositions over sector basis vectors; an effective curvature–source relation couples quantum expectation values back into geometry; and transition amplitudes between sectors are exponentially suppressed by geometry and field gradients. The result is a closed quantum–geometric feedback system that reproduces sector clustering, curvature‑dependent isolation, and rapidity‑aligned propagation. We derive the model formally, present example dynamics in 1+1 dimensions, provide numerical scenarios, and supply figures and dashboard modules for TLPL integration.

1 Introduction

The TLPL framework formalizes an intuition: spacetime can be usefully modeled as a lattice of sectors—finite regions with local geometric and field properties—over which quantum states propagate. Unlike continuum quantum field theory, TLPL emphasizes coarse‑grained connectivity and geometry‑dependent tunneling. The model is intended as a flexible platform for exploring emergent geometry, quantum back‑reaction, and transport on discrete sector graphs.

This document provides a full derivation, example dynamics, numerical illustrations, and dashboard‑ready modules for integration into TLPL documentation systems.

2 Conceptual Overview

Sectors Si: coarse spacetime cells associated with displacements Δxμ.


Sector Hilbert space: basis ∣ϕi⟩ with global state ∣ψ⟩=∑iai∣ϕi⟩.


Rapidity scalar ϕ(t,x)=12ln⁡ ⁣(t+xt−x): encodes local boost/kinematic structure.


Curvature–source coupling: Rμν=κS[ϕ,ψ].


Tunneling: Tij≈e−2κaij with aij a field/curvature dependent barrier.


Dynamics: Schrödinger‑type evolution on the sector graph with Hamiltonian entries set by Tij and local potentials.

These elements form a closed loop: ∣ψ⟩ → S → Rμν → Tij → evolution of ∣ψ⟩.

3 Formal Model
3.1 Sector decomposition

Let M be a spacetime manifold. Define a partition into sectors {Si}i∈I such that each sector Si corresponds to a finite region with representative coordinate xiμ and characteristic displacement Δxiμ. Adjacency is defined by a symmetric relation Aij∈{0,1} indicating whether sectors Si and Sj are neighbors.
3.2 Hilbert space and quantum state

Associate to each sector Si a normalized basis vector ∣ϕi⟩. The global state is
∣ψ(t)⟩=∑i∈Iai(t) ∣ϕi⟩,∑i∣ai(t)∣2=1.

Probability of sector occupancy is Pi(t)=∣ai(t)∣2.
3.3 Rapidity scalar field

On a 1+1 slice with coordinates (t,x), define the scalar
ϕ(t,x)=12ln⁡ ⁣(t+xt−x).

For each sector Si we define ϕi≡ϕ(ti,xi). Differences ∣ϕi−ϕj∣ measure kinematic separation between sectors.
3.4 Curvature–source relation

Introduce an effective scalar source S and coupling κ. Postulate
Rμν=κ S[ϕ,ψ],

with a simple functional choice
S[ϕ,ψ]=γ1∑i∣ai∣2 F(ϕi)+γ2 G({ϕi},{ai}),

where F and G are real functions chosen to reflect field gradients and sector correlations, and γ1,2 are constants. For example,
F(ϕi)=∣∇ϕ∣Si2,G=∑i,j∣ai∣2∣aj∣2 hij(ϕi,ϕj),

with hij a symmetric kernel.

This choice makes S an expectation value of field‑dependent quantities in the state ∣ψ⟩.
3.5 Barrier parameter and tunneling

Define an effective barrier between sectors:
aij=α ∣ϕi−ϕj∣+β R‾ij+δ dij,

where:

∣ϕi−ϕj∣ is rapidity separation,


R‾ij is curvature averaged between sectors,


dij is a geometric distance measure,


α,β,δ>0 are constants.

Then the tunneling coefficient is
Tij=exp⁡ ⁣(−2κaij).

For nonadjacent sectors we may set Tij to be negligible or include long‑range suppressed terms.
3.6 Hamiltonian and evolution

Construct an effective Hamiltonian H on the sector basis:
Hij={Vi,i=j, \[4pt]− J Tij Aij,i≠j,

with Vi a local potential and J a scale factor. The Schrödinger equation for amplitudes is
iℏdaidt=∑jHij aj.

Because Tij depends on {ak} via curvature Rμν and S[ϕ,ψ], the system is nonlinear and self‑consistent.

4 Self‑Consistency and Closure

To close the model we specify:

Field evaluation: compute ϕi at sector centers.


Source functional: choose F,G and constants γ1,2.


Curvature mapping: map scalar S to an effective curvature measure Ri per sector (e.g., Ri=κS or a normalized variant).


Barrier update: compute aij from ϕi,Ri.


Hamiltonian update: set Hij and integrate Schrödinger equation for ai(t).


Iterate until convergence or for time evolution.

This iterative loop implements the TLPL feedback.

5 Example Dynamics in 1+1 Minimal Lattice
5.1 Setup

Consider three sectors S1,S2,S3 in a line with centers at x1=−ℓ,x2=0,x3=+ℓ. Choose time slice t=t0>ℓ so ϕ is defined. Let
ϕ1=0,ϕ2=η,ϕ3=2η.

Assume initial amplitudes a1(0)=1,a2(0)=a3(0)=0. Choose uniform local potentials Vi=V and small uniform curvature baseline R.
5.2 Tunneling coefficients

Compute
T12=e−2κ(αη+βR+δℓ),T23=e−2κ(αη+βR+δℓ),
T13=e−2κ(2αη+βR+2δℓ).

Nearest neighbors dominate.
5.3 Evolution

With Hamiltonian
H=(V−JT12−JT13−JT12V−JT23−JT13−JT23V),

the amplitudes oscillate between sectors with frequencies set by Tij. Increasing R or η reduces Tij, slowing mixing and producing sector localization.

6 Numerical Scenario and Plots
6.1 Parameter choices

Example parameters for simulation:

ℏ=1, J=1, V=0


α=1.0, β=0.5, δ=0.2


κ=0.8, η=0.5, ℓ=1.0


Baseline curvature R=0.1
6.2 Expected behavior

Low curvature R→0: rapid mixing across sectors.


High curvature: exponential suppression of Tij, localization.


Large rapidity spacing η: directional suppression, preferential propagation along aligned sectors.
6.3 Example plots (to be generated in TLPL dashboard)

Probability vs time for P1,P2,P3.


Tunneling map Tij heatmap.


Sector graph with edge widths ∝Tij.

(ASCII figure placeholders provided in Section 8 for conversion.)

7 Discussion

TLPL is a flexible toy model bridging discrete sector graphs and continuum geometric intuition. It captures:

Back‑reaction: quantum state influences curvature via S[ϕ,ψ].


Geometry‑controlled transport: tunneling depends on curvature and rapidity.


Emergent localization: high curvature isolates sectors, resembling gravitational trapping.


Directional propagation: rapidity differences bias transitions.

Limitations include the simplified curvature relation Rμν=κS, the ad hoc choice of barrier functional, and the absence of a full relativistic field action. These are deliberate: TLPL is intended as a modular platform for exploring phenomenology and numerical experiments.

8 Figures and Diagrams

Below are ASCII‑safe diagrams and figure descriptions for conversion to SVG/Canvas in the TLPL dashboard.
Figure 1 Sector Lattice Schematic
Code

Figure 1: TLPL Sector Lattice (linear example) S1 --- S2 --- S3 --- S4 | | | phi1 phi2 phi3 Edges weighted by T_ij; nodes labeled by phi_i and local R_i.

Figure 2 Rapidity Field Geometry
Code

Figure 2: Rapidity Field along 1+1 slice t ^ | / rapidity contours | / | / phi(t,x) = 0.5 ln((t+x)/(t-x)) |/_________________ x

Figure 3 Curvature Feedback Loop
Code

Figure 3: Feedback Loop |-- |psi> amplitudes a_i | --(expectation)--> S[phi,psi] --(coupling kappa)--> R_mu_nu ^ | | v <--------------------------- T_ij(a,phi,R) -------------------------------

Figure 4 Sector Graph with Edge Weights
Code

Figure 4: Sector Graph (example) [S1]--(w12)--[S2]--(w23)--[S3] | | (w13) (w34) | | [S4]-----------------------[S5] Edge width ~ T_ij; node color ~ local curvature R_i.

Figure 5 Example Probability Flow
Code

Figure 5: P_i(t) oscillations (schematic) P | | /\ /\ | / \ /\ / \ |____/____\/__\/____\____ t P1 P2 P3 P1 P2 P3


9 Conclusion

TLPL provides a compact, modular framework for studying quantum dynamics on a geometry that is both discrete and responsive to quantum state properties. It is suitable for numerical experiments, visualization, and conceptual exploration of emergent geometry and transport phenomena.

10 Appendices
Appendix A Notation

Si: sector index.


∣ϕi⟩: sector basis state.


ai: amplitude for sector Si.


ϕ(t,x): rapidity scalar.


S[ϕ,ψ]: scalar source functional.


Rμν: Ricci curvature.


Tij: tunneling coefficient.


Hij: Hamiltonian matrix.
Appendix B Example derivation of tunneling factor

Starting from a WKB‑like ansatz for a one‑dimensional effective barrier of height U and width a, transmission scales as exp⁡(−2∫2m(U−E) dx). Replacing the integral by an effective barrier parameter proportional to geometric and field measures yields the phenomenological form T≈e−2κa used in TLPL.
Appendix C Suggested parameter regimes for simulation

Exploratory: κ∈[0.1,1.0], α∈[0.5,2.0], β∈[0.1,1.0].


Localization: καη≳1.


Delocalization: καη≪1.