>[!warning] >This content has not been peer reviewed. # Classical motion emerges from a sum over paths ## What it is (established result) In the **Feynman path integral** (sum-over-histories) formulation of quantum mechanics, the transition amplitude from an initial configuration to a final configuration is obtained by summing contributions from **all possible paths** connecting them. Each path contributes a complex phase determined by the classical action \(S[x(t)]\): \[ \langle x_f, t_f \mid x_i, t_i \rangle \;=\; \int \mathcal{D}x(t)\; \exp\!\left(\frac{i}{\hbar} S[x(t)]\right). \] In the semiclassical (small-\(\hbar\)) regime, the dominant contribution comes from paths near those for which the action is stationary (\(\delta S = 0\)), recovering the **classical equations of motion** via the principle of least action \([1,2]\). --- ## How RRT / RST uses it RST uses the path-integral idea as an **engineering pattern**: - The substrate does not “know” a single correct future. It can be modelled as evaluating many candidate micro-updates (paths) and selecting the realised update under a finite budget. - The classical trajectory corresponds to the update history that maximises delivered identity / fidelity for minimal workload cost (least relational friction under A5-like constraints). In code this becomes a **parallel-search update rule**: sample many candidate micro-updates, score each by a fidelity-per-cost functional, and pick the realised event. (See the `Reality Engine` application for the runnable toy implementation.) --- ## Links | Role | Link | |:---|:---| | Simulator application | [[further applications/Reality Engine/Reality Engine (RST)]] | | Simulator code | [[further applications/Reality Engine/Reality Engine - Code]] | --- ## References [1] R. P. Feynman, "Space-Time Approach to Non-Relativistic Quantum Mechanics," *Reviews of Modern Physics* **20**, 367–387 (1948), `https://doi.org/10.1103/RevModPhys.20.367`. [2] R. P. Feynman and A. R. Hibbs, *Quantum Mechanics and Path Integrals*, McGraw–Hill, 1965.