Unitary inference: introduction

Mathematical inconsistencies of Statistical Field Theory and its consequences for Quantum Mechanics and Machine Learning

Let's assume for a moment that we live in a world governed by the deterministic Newton's laws of physics. We are studying the motion of a dust particle on the surface of a lake. We know that the position of the dust particle in the lake when measured at large enough time intervals is well approximated by a Wiener process (Brownian motion). However, the probability that the trajectory of the particle is differentiable at all times is null, according to the same Wiener process. This contradicts Newton's laws. We may argue that the Wiener process is just an approximation, that a more elaborated generalization of a Wiener process could work. But it is not known how, in fact reference bertoin94 shows that all Levy processes (a continuous-time analog of a random walk) with strictly positive Gaussianity predict a trajectory which is non-differentiable at all times.

The above implies that we need to give up on Levy processes or on Gaussianity, if we want to predict trajectories of the dust particle consistent with Newton's laws at all times. In other words, mainstream stochastic processes are incompatible with Newton's laws at all times, these stochastic processes only work at isolated points in time. When merging randomness with continuous-times, it's every man for himself.

The above implies that there is no solid mathematical ground where we can build any interpretation of Quantum Mechanics. Because an interpretation of Quantum Mechanics is mostly about what happens between measurements, when we consider continuous-time. Note that Quantum Mechanics makes predictions for weak measurements: measurements that do not change the result of a final measurement and that happen at isolated times, see tamir13. The ambiguity only arises when we consider continuous-time, where there is no solid mathematical ground anyway.

Since an inconsistent set of mathematical axioms allows us to prove anything, it should not be a surprise that Quantum Mechanics is full of fantastical phenomena, such as particles that seem to be in two different places at the same time between measurements. No one ever mentioned the obvious fact that all statements about such fantastical phenomena have no solid mathematical ground, because we are all just trying to make a living.

As shocking as the above discussion may be, we still have to consider the fact that since merging randomness with continuous-times is such a general and old mathematical problem (that is relevant already to Newton's physics), it was studied well before the Planck's black body (the most famous origin of Quantum Mechanics). And it is not surprising that a viable solution to such problem, proposed decades before Planck's black body in reference mehler1866, already includes the most part of what would later become known as the Quantum Field Theory of a free particle. In particular, it requires that the dynamics of an isolated system be defined by an Hamiltonian operator on a Fock Hilbert space, even when the dynamics respects Newton's physics.

It is not surprising because since Quantum Mechanics works well for isolated times, then a first guess for a generalization of it for continuous-times is to use Quantum Mechanics for free fields (since a trajectory is a field). And since merging randomness with continuous-times is a very old problem, there should be solutions to it older than Planck's black body, some of these solutions should include the most part of the best solution to it we know of today, which includes the Quantum Field Theory of a free particle. In other words, a strong case can be made that Quantum Mechanics and Quantum Field Theory were invented decades before Planck's black body.

Note that we are not talking about a few similarities between the theories. The title of reference mehler1866, clearly proposes to solve the exact same old mathematical problem, that everyone that studies Quantum Mechanics knows that it is crucial if we want to study random fields (including trajectories in continuous times). Moreover, the author is well known and credible. We are all just trying to make a living.