Dynamic equation --> Phase–path integral

Can the dynamic equation be rewritten as a phase–path integral?

If every phase trajectory that starts at cell i and ends at cell j in Qaether theory is weighted by the energy associated with that trajectory, the usual lattice path-integral kernel

$$K\bigl[\phi_i\!\to\!\phi_j;\,T\bigr] \;=\; \!\! \int_{\phi(t_0)=\phi_i}^{\;\;\phi(t_0+T)=\phi_j}\! \mathcal D\phi\;\exp\!\Bigl\{\,i\,S_{\text{eff}}[\phi]\Bigr\}$$

is recovered. The derivation goes in three steps:

 

1. Identify the energy (Lagrangian) of a single time-slice

• For a small time step \(\Delta t\) write the finite–difference velocities $$\dot\phi_i \;\longrightarrow\; \frac{\phi_i^{n+1}-\phi_i^{n}}{\Delta t}$$
• The energy stored in that slice is $$E^{(n)} \;=\;\sum_i \frac{I_i(m_i)}{2}\Bigl(\dot\phi_i^{(n)}\Bigr)^2 \;+\; \sum_{\langle ij\rangle} \Bigl[ -K_{ij}\,\Im\chi_{ij} +U_0\,\Im(\chi_{ij}^6) +q_e\,A_{ij}\,\Im\chi_{ij} +g\,\vec C_i\!\cdot\!\vec A_{ij}\,\Im\chi_{ij} \Bigr] \;$$ $$+\; \sum_i P_i(m_i)\,V_s\,\ell_p\bigl(1-\cos\phi_i^{(n)}\bigr)$$ (Up to an irrelevant constant, the potential is the negative of the bracketed force terms.)
• Define the discrete Lagrangian $$L^{(n)} \;=\; -E^{(n)}$$

 

2. Build the discrete-time action

$$S_{\text{lat}} \;=\; \sum_{n}\! \mathcal L^{(n)}\,\Delta t \;\;=\; \sum_{n}\!\Bigl[ \sum_i \tfrac12 I_i(m_i) \bigl(\dot\phi_i^{(n)}\bigr)^2 - \sum_{\langle ij\rangle}V_{\langle ij\rangle}^{(n)} - \sum_i V_{{\rm Void},i}^{(n)} \Bigr]\Delta t$$

• \(V_{\langle ij\rangle}^{(n)}\) collects the four interaction terms and
  $$V_{{\rm Void},i}^{(n)} = P_iV_s\ell_p(1-\cos\phi_i)$$

 

3 Take the continuum limit → path integral

• Let \(\Delta t\to0\) and \(a=\ell_p\to0\) while \(T=N\Delta t\) stays finite. One obtains the continuous action $$S_{\text{eff}}[\phi] \;=\;\int_{t_0}^{t_0+T}\!dt \biggl\{ \sum_i \frac{I_i(m_i)}{2}\,\dot\phi_i^2 \;-\; \sum_{\langle ij\rangle} \Bigl[ -K_{ij}\,\Im\chi_{ij} +U_0\,\Im(\chi_{ij}^6) +q_e\,A_{ij}\,\Im\chi_{ij} +g\,\vec C_i\!\cdot\!\vec A_{ij}\,\Im\chi_{ij} \Bigr] \;$$ $$-\; \sum_i P_i(m_i)V_s\ell_p \bigl(1-\cos\phi_i\bigr) \biggr\}$$
• Because the weight of every trajectory is $$\exp\!\bigl\{iS_{\text{eff}}[\phi]\bigr\}$$
• summing (integrating) over all trajectories between the fixed endpoints produces the desired path-integral kernel $$K(i\!\to\!j;T)\;=\; \int_{\phi(t_0)=\phi_i}^{\;\;\phi(t_0+T)=\phi_j} \!\!\!\!\!\!\!\!\!\! \mathcal D\phi\; e^{\,iS_{\text{eff}}[\phi]}$$
• The dynamic equation of Qaether is recovered by taking the stationary-phase (variational) condition \(\delta S_{\text{eff}}/\delta\phi_i = 0\)
• Hence the Qaether dynamics does admit a standard phase-path-integral formulation once one integrates over every discrete phase route from cell i to cell j.
• It tells that Qaether Theory is not just a set of discrete, cell-by-cell dynamical rules, but in fact admits exactly the same kind of path-integral quantization that underlies continuum quantum field theory:
  1. Global Sum over Histories
     • Every way of “stepping” your phase field from cell i at time t₀ to cell j at time t₀+T contributes, weighted by
       $$\exp\bigl(i\,S_{\rm eff}[\phi]\bigr)$$
     • Nothing mystical is required—just as in standard QFT you sum over all field configurations, in Qaether you sum over all discrete phase trajectories.
  2. Stationary-Phase → Classical EOM
     • Taking \(\delta S_{\rm eff}=0\) reproduces your lattice equations of motion exactly.
     • This guarantees that the classical limit of the path integral is the very same phase-dynamics you wrote down.
  3. Quantization of Both Gauge and “Gravity” Sectors
     • All of your interaction terms—phase-coupling \(K_{ij}\), quantization potential \(U_0\), \(U(1)\) and \(SU(3)\) gauge couplings, Void pressure—enter on equal footing in the action.
     • You can therefore derive Feynman rules, compute loop corrections, and analyse quantum fluctuations of both the gauge fields and the emergent gravitational (Void/Regge) sector in one unified formalism.
  4. Consistency Checks & Renormalization
     • If the continuum limit is well-defined, you can study whether those higher-derivative or non-linear pieces spoil unitarity or causality, or whether they flow to safe infrared fixed points.
     • In principle, we can ask whether Qaether quantize to a renormalizable or asymptotically safe theory, or it require a UV completion (e.g. stringy, asymptotic safety scenarios)
  5. Connection to Standard QFT and Gravity
     • By casting Qaether into a path integral, you can directly compare it to lattice gauge theory, Regge calculus, CDT, and continuum EFTs of gravity + YM.
     • You can match couplings, identify ghost modes, test emergent Lorentz invariance at low energy, and see exactly how the discrete Void-phase picture dovetails with known quantum-gravity approaches.

 

Results:

Qaether is fully quantizable. Its discrete phase dynamics can be reformulated as a conventional path integral with action \(S_{\rm eff}[\phi]\), ensuring that:

• Its classical lattice EOM emerge in the stationary-phase limit,
• Quantum fluctuations of both gauge and gravitational degrees of freedom are systematically computable,
• Standard tools (Feynman diagrams, renormalization group, unitarity checks) become available to test its consistency and make experimental predictions.