Physics:Sean Collins' Path-Integral Mapping of the Mjolnir(A137)Lattice

Overview

The author (Sean Collins) has formulated and developed the conceptual and mathematical foundations of the Theory of Lattice (A137) Dynamics in previous works.

In this submission, the author, (Sean Collins), introduces further inroads into our understanding of Quantum Path-Integral Mapping of the Mjolnir (A137) Lattice via his Theory of Lattice (A137) Dynamics.

The Mjolnir (A137) lattice provides a deterministic substrate in which aggregated objects (e.g., nuclei, orbital aggregates, galactic branches) propagate. These aggregates experience both an effective kinetic term and a drag-derived potential. The path-integral formalism offers a compact mapping from lattice dynamics to quantum-like interference and tunneling phenomena.

Notation

• \(m\):mass of the aggregate (built from base mass\(m_b \))
• \(v_{\text{wave}} \): superluminal lattice wave speed (e.g. \( \kappa c \))
• \(v_{mb} \): characteristic base-particle cycle speed (lattice clock)
• \(\gamma = \dfrac{v_{\text{wave}}}{v_{m_b}} \): phase-synchronization parameter
• \(k \): drag coefficient (from \( F_{\text{drag}} = k \rho v_{\text{wave}} m v \))
• \(\rho(x) \): local lattice density (power-law or sigmoid form)
• \(V_{\text{drag}}(x) = k \rho(x) |x| \left(\dfrac{v_{\text{wave}}}{c}\right)^2 \): drag potential

The goal is to present a path-integral representation that:

1. Recovers deterministic lattice dynamics in the stationary-phase limit.

2. Introduces a control parameter \( \gamma \) governing interference vs. classicality.

3. Supplies a Euclidean action suitable for tunneling estimates.

1. Lattice to Continuum Action

1.1 Discrete Action

For a time interval \([t_a, t_b]\) split into \(N\) slices of size \(\Delta t\), a path is specified by \(\{x_0, x_1, \dots, x_N\}\) with fixed endpoints.

Discrete action increment:

[math]\displaystyle{ \Delta S_i = m \frac{(x_{i+1} - x_i)^2}{\Delta t} + V_{\text{drag}}(x_i)\,\Delta t }[/math]

Summing:

[math]\displaystyle{ S[x] \approx \sum_{i=0}^{N-1} \left[ m \frac{(x{i+1}-xi)^2}{\Delta t} + V{\text{drag}}(x_i)\,\Delta t \right] }[/math]

1.2 Continuum Limit

As \(\Delta t \to 0\):

[math]\displaystyle{ S[x(t)] = \int{ta}^{tb} \left[ m \dot{x}(t)^2 + V{\text{drag}}(x(t)) \right] dt }[/math]

This reflects the Mjolnir energy postulate \(E \approx m v^2\). The absence of the conventional \(1/2\) factor highlights the lattice-specific ontology.

2. Path Integral Definition

Define the partition functional with phase-sync parameter \(\gamma\):

[math]\displaystyle{ Z = \int \mathcal{D}[x(t)] \exp\!\left( \frac{i \gamma}{\hbar} S[x] \right), }[/math]

which is the standard Feynman Path Integral of Quantum Field Theory.

Introduce an effective Planck constant:

[math]\displaystyle{ \hbar_{\text{eff}} = \frac{\hbar}{\gamma}, }[/math]

which in principle essentially characterizes the Vuli-Ndlela Integral of the Theory of Entropicity(ToE).

So:

[math]\displaystyle{ Z = \int \mathcal{D}[x] \exp\!\left( \frac{i}{\hbar_{\text{eff}}} S[x] \right) }[/math]

Interpretation:

• Small \(\gamma \Rightarrow \hbar_{\text{eff}} \gg \hbar\) → strong interference (quantum-like).
• Large \(\gamma \Rightarrow \hbar_{\text{eff}} \ll \hbar\) → stationary-phase dominance (classical).

3. Stationary-Phase and Classical Limit

The stationary-phase condition:

[math]\displaystyle{ \delta S[x] = 0 }[/math]

recovers the Euler–Lagrange equations for \(S\), i.e. deterministic lattice dynamics.

In the limit \(\hbar_{\text{eff}} \to 0\) (\(\gamma \to \infty\)), the path integral is dominated by the extremal classical path.

4. Short-Time Kernel and Schrödinger-like Reduction

The short-time propagator is then given by:

[math]\displaystyle{ K(x+\Delta x, \Delta t; x, 0) \approx \sqrt{\frac{m}{2\pi i \hbar_{\text{eff}} \Delta t}} \; \exp\!\left[ \frac{i m (\Delta x)^2}{2 \hbar{\text{eff}} \Delta t} - \frac{i \Delta t}{\hbar{\text{eff}}} V_{\text{drag}}(x) \right] }[/math]

Composition of kernels as \(\Delta t \to 0\) yields a Schrödinger-like equation:

[math]\displaystyle{ i \hbar{\text{eff}} \frac{\partial \Psi}{\partial t} = -\frac{\hbar{\text{eff}}^2}{4m} \nabla^2 \Psi + V_{\text{drag}}(x)\,\Psi }[/math]

5. Euclidean Action and Tunneling

Perform a Wick rotation \(t \to -i\tau\).

The Euclidean action becomes:

[math]\displaystyle{ SE[x(\tau)] = \int \left[ m \left(\frac{dx}{d\tau}\right)^2 + V{\text{drag}}(x(\tau)) \right] d\tau }[/math]

The Tunneling amplitude is:

[math]\displaystyle{ T \propto \exp\!\left( -\frac{SE[x{\text{bounce}}]}{\hbar_{\text{eff}}} \right) }[/math]

Here, \(x{\text{bounce}}\) is the instanton minimizing \(SE\). Larger \(\gamma\) reduces \(\hbar_{\text{eff}}\), and therefore suppresses tunneling.

6. Numerical Prescription

1. Discretization: Choose \(T\) slices, so that \(\Delta t = (t_b - t_a)/T\).

2. The Path action is:

[math]\displaystyle{ S[x] \approx \sum{i=0}^{T-1} \left[ m \frac{(x{i+1}-xi)^2}{\Delta t} + V{\text{drag}}(x_i)\,\Delta t \right] }[/math]

3. We employ the following sampling strategies:

• Direct Monte Carlo with oscillatory weights \(e^{iS/\hbar_{\text{eff}}}\).
• Complex Langevin or reweighting for oscillatory integrals.
• Euclidean sampling for tunneling: weights \(e^{-SE/\hbar{\text{eff}}}\).
• Path ensemble approach: sample 50–200 classical trajectories and compute weighted averages.

7. Physical Intuition for ToE Audience

• Economy of assumptions: Only lattice + action + phase-synch scaling \(\gamma\).
• Reduction: Stationary-phase → classical Mjolnir dynamics; path integral → Schrödinger dynamics.
• Control parameter: \(\gamma\) is physically meaningful, not a fudge factor.
• Falsifiability: Predicts deviations in line shapes, tunneling rates, and residuals compared to Standard Model expectations.

8. Suggested Figures and Tables

• Boxed derivation of \(S\), \(Z\), and \(\hbar_{\text{eff}}\).
• Stationary-phase vs. sampled path comparison.
• Convergence plots of observables vs. sampling size.
• Tunneling exponent \(SE/\hbar_{\text{eff}}\) vs. \(\kappa = v_{\text{wave}}/c\).
• Table: hydrogen binding and muon lifetime anchors (path-sum vs. observed).

Conclusion

The path-integral mapping provides a compact bridge from deterministic Mjolnir lattice dynamics to interference and tunneling phenomena. It introduces a physically measurable control parameter \(\gamma\) and an effective Planck constant \(\hbar_{\text{eff}}\), connecting lattice ontology to quantum-like behavior in a falsifiable framework.