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: Partition Functional and Effective Planck Constant

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 classical deterministic lattice dynamics are recovered by stationary-phase condition:

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

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

In the limit \(\hbar_{\text{eff}} \to 0\) (\(\gamma \to \infty\)), the path integral is dominated by the extremal classical path. Ordinary Mjolnir dynamics are therefore the leading-order behavior.

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 with [math]\displaystyle{ \hbar }[/math] replaced by [math]\displaystyle{ \hbar_{\text{eff}} }[/math]:

[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{ S_E[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{S_E[x{\text{bounce}}]}{\hbar_{\text{eff}}} \right) }[/math]

Here, \(x_{\text{bounce}}\) is the instanton minimizing \(S_E\). Since ħ_eff = ħ/γ, 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 \(e^{-S_E/\hbar_{\text{eff}}}\).
• Path ensemble approach: sample 50–200 classical trajectories and compute weighted averages.

Steps for Achieving the Numerical Prescription:

The numerical evaluation of the path-integral mapping proceeds sequentially as follows:

1. Discretization of time Choose the number of slices \(T\) and step size

[math]\displaystyle{ \Delta t = \frac{t_b - t_a}{T}. }[/math]

Fix the number of sampled paths \(P\).

2. Path generation

For each trial path, construct the sequence

[math]\displaystyle{ \{x_0, x_1, \dots, x_T\}, }[/math]

with endpoints \(x_0 = x_a\), \(x_T = x_b\).

3. Action evaluation

Compute the discrete action

[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]

4. Weighting Assign weights according to the regime:

[math]\displaystyle{ w = \exp\!\left( \frac{i}{\hbar_{\text{eff}}} S[x] \right), \qquad w_E = \exp\!\left( -\frac{1}{\hbar_{\text{eff}}} S_E[x] \right) }[/math]

for real-time and Euclidean (tunneling) calculations, respectively.

5. Observable averaging Compute expectation values as

[math]\displaystyle{ \langle O \rangle = \frac{\sum_x w[x]\, O[x]}{\sum_x w[x]}. }[/math]

6. Convergence check Verify stability of results with respect to both the number of paths \(P\) and the discretization parameter \(T\).

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).

9. Interpretation and Robustness

• Physical measurability of \(\gamma\):

The parameter

[math]\displaystyle{ \gamma = \frac{v_{\text{wave}}}{v_{m_b}} }[/math]

is directly measurable from lattice velocities and is not a free parameter. Sensitivity analysis should be performed to quantify how observables depend on variations in \(\gamma\).

• Ontological role of the path integral:

The path-integral mapping does not serve as a proof of quantum mechanics. Instead, it demonstrates how wave-like interference emerges naturally from deterministic lattice dynamics. The resulting deviations from standard quantum mechanics are falsifiable and provide testable predictions.

• Numerical considerations:
  • Oscillatory integrals in real time are computationally challenging.
  • Euclidean (Wick-rotated) sampling with weights \(\exp(-S_E/\hbar_{\text{eff}})\) is numerically stable for tunneling problems.
  • Stationary-phase guided sampling is effective for bound-state calculations, where fluctuations around the classical path dominate.

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.