The optimization of paths in the $R^{3,1}$ space time by Markov Chain Monte Carlos

Abstract: We propose a method to obtain the optimal weight function of 9 paths in (3+1)D space-time whose length is less than or equal to $2\times (6+2)$ lattice units. The factor 2 comes from inclusion of opposite direction path or time reversed paths. There are $2\times 2$ time shifts, which we assume that they can be regarded as stochastic Markov processes. We prepare the input 9D vector ${\bf X}$ and a $9\times 9$ matrix ${\bf W}$ and a bias vector ${\bf b}$, and consider affine transformations ${\bf Z}^{(h)}={\bf X}^{(in)}{\bf W}^{(h)T}+{\bf b}^{(h)}$ and ${\bf A}^{(h)}=\sigma({\bf Z}^{(h)})$ from an input layer to a hidden layer, the hidden layer to another hidden layer and from the hidden layer to an output layer, using the transformation ${\bf Z}^{(x)}={\bf A}^{(h)}{\bf W}^{(out) T} +{\bf b}^{(x)}$ and ${\bf A}^{(x)}=\sigma({\bf Z}^{(x)})$. By choosing the matrix ${\bf W}$ a diagonal matrix, and introducing the information of action of the 9 paths, a simple Monte Carlo simulation yields actions on a 2D plane spanned by $e_1, e_2$ for a fixed $u_2=j_2 e_2$ as a function of $u_1=j_1 e_1$. The action at high momentum region has small fluctuations, but at small momentum region, has large fluctuation. Generalizing $\bf W$ including mixing of paths, we search the optimal weight function using the Machine Learning (ML) techniques. For fixed point actions, actions of the output layer are defined by the output of final hidden layer