# quantum mechanics from disretized classical mechanics

Suppose you have a simple 1D harmonic oscillator given by the Lagrangian, say, $$L(x,\dot x) = \dot x^2 - \frac{1}{10}x^2.$$

Semi-discretizations of the Euler-Lagrange equations, where spatial variables are treated as continuous and time is discretized into finite-sized steps, is extremely well studied and forms the basis of scientific computing. I was playing around with purely discrete formulations of Hamilton's principle: where $x$ also is allowed to attain only values on the integers.

In this setting a trajectory $\gamma$ is described by a sequence of $\{x_0, x_1, \ldots, x_T\}\in\mathbb{Z}^T$ and the action can be discretized as $$S(\gamma) = \sum_{i=1}^{T-1}\left[(x_{i+1}-x_i)^2 - \frac{x_{i+1}^2}{10}\right].$$

Now Hamilton's principle would look like the following: a trajectory $\gamma$ is physical if $$S(\{x_0, \ldots, x_i, \ldots, x_T\}) \leq S(\{x_0, \ldots, x_i\pm 1, \ldots, x_T\})\quad \forall i\in \{1,\ldots,T-1\},$$ i.e, if you cannot improve the action of the trajectory by moving any position along the trajectory to either of its neighbors on $\mathbb{Z}$.

Euler-Lagrange equations don't really exist since nothing in sight is differentiable, but one can still turn the above boundary-value problem into an initial value problem by declaring the physical continuation of the trajectory $\{x_0, \ldots, x_i\}$ to be the point $x_{i+1}$ so that the augmented trajectory $\{x_0, \ldots, x_{i+1}\}$ still satisfies Hamilton's principle.

There may be multiple such $x_{i+1}$. In that case, we choose one valid $x_{i+1}$ uniformly at random.

If I simulate this process for the harmonic oscillator (with the initial conditions $x_0=x_1=-100$) I get the following, showing the PDF of $x$ animating over time:

movie

This doesn't quite look like time evolution of Schroedinger's equation, but it's certainly suggestive.

Is there some way to formalize all of this? Can quantum mechanics be formulated as arising from discretization of classic mechanics onto a lattice?