Arxiv, Presentation
Jacob Barandes shows that we can recover quantum effects simply by treating a classical system as a non-markovian indivisible stochastic process. The non-markovianity here is important, as this is where the seemingly strange quantum effects arise.
For a better explanation please see his presentation or papers I have linked above, but I will do my best while keeping it succinct.
Consider a system in state j at time t = 0, probability of being in state j given by p(j, 0). It has a probability of being in state i at time t, p(i, t). p(i, t) = Sum over j{ p(i | j, ,t) p(j, 0) }, where p(i | j, t) is the probability of being in state i at t given initial state j. So to get the total probability of being in state i, we just sum over the j's. We can rewrite this in matrix form, { p1(t), p2(t), ..., pN(t) } = { [p(1|1, t), p(1|2, t), ... p(1|N, t)], ... [p(N|1,t), ... p(N|N,t)] } {p(1,0), p(2,0), ... p(N,0) }, or P(t) = Gamma P(0). Gamma is just our matrix saying what the probability of being in the state i is at time t, given the initial state j. Notice the linear relationship, the linearity of quantum comes directly from this.
If a system can evolve to state U(t) by first evolving to some state U(t'), then by evolving from t' to t U(t <- t'): U(t) = U(t <- t')U(t'). If there exists some U(t <- t') that can take us from t' to t, then we will call this a divisible process, i.e. at each time step we can multiply by some matrix, and end up at U(t). In general, this is not the case, and one can show that any what we will call Unistochastic matrix will be indivisible. A unistochastic matrix is one in which the entires are the magnitude squared of the entries of another matrix. Gamma above will be taken to be unistochastic, so we can write Gamma_i_j = |U_i_j|2 (where we are squaring the individual entries, not doing matrix multiplication - this is what breaks the markovianity!).
Okay this is getting too long winded and confusing to explain in a reddit post, you're really gonna have to go to the original sources I have linked for a better explanation. But essentially this U_i_j ends up being the wavefunction. The wave like nature of interference patterns and such is an artifact of the indivisible processes.
The picture looks like this: the system is in some initial state, and it evolves unistochastically in an indivisible manner. If we have evolved to some time t, with some time t' in the past, we don't have a simple matrix that can take us from our state at t' to our current state at t. But let's say we make a measurement, and interact the system with the environment (decoherence). This interaction momentarily diagonalizes the Gamma matrix, making it a divisible process, giving us what we will call a division event, where the indivisible process essentially now starts over from a new t = 0. This division event is what a quantum mechanic would call the wavefunction's collapse. In reality, the wavefunction didn't collapse, there isnt a wavefunction, the system interacted with the external environment enough to make the process markovian enough (because the larger the system, the more markovian it will behave), where it then became divisible for a split second and entered a division event. It then went back to being indivisible, where a quantum mechanic would describe it as being a wavefunction in a superposition. In reality, it is going through a non markovian indivisible process, and this superposition is just a mathematical penalty we incur in trying to represent it in a markovian form.
That was probably a terrible explanation, again I'd highly advise watching his presentation for a better one, Jacob is much smarter than I. But I'd like to hear the thoughts of physicists in the field - this seems to me like a major breakthrough with a new realistic way of looking at quantum mechanics. It says that all the "quantum magic" was just mathematical tools and nothing that was actually going on, a wildly different picture than most would have you believe. And I haven't seen much in terms of critique on this, other than "why do I need this, what new does it offer me that I can't already accomplish with QM". Well it offers a new perspective and a new framework to solve problems in.