Abstract
A new theorem on random walks suggest some possible revisions of the foundations of Quantum Mechanics. This is presented below in the simplified framework of the description of the evolution of a material point in space. Grossly speaking, it is shown that the probabilities generated by normalizing the square modulus of a sum of probability amplitudes, in the setup of Quantum Mechanics, becomes asymptotically close (under the appropriate limiting conditions) to the probabilities generated by the usual causal processes of Classical Mechanics. This limiting coincidence has a series of interesting potential applications. In particular it allows us to reintroduce the concept of causality within the core of Quantum Mechanics. Moreover, it suggests, among other consequences, that gravitational interaction may not even exist. Even though the interpretations of Quantum Mechanics which follow from this mathematical result may seem to bring some unexpected innovations in the context of theoretical physics, there is an obvious necessity to study its theoretical impact on Quantum Mechanics. The first steps toward this aim are taken in the present article.
Highlights
An unexpected property of random walks [1] suggests a possible revision of the foundations of Quantum Mechanics
It is shown that the probabilities generated by normalizing the square modulus of a sum of probability amplitudes, in the setup of Quantum Mechanics, becomes asymptotically close to the probabilities generated by the usual causal processes of Classical Mechanics
Even though the interpretations of Quantum Mechanics which follow from this mathematical result may seem to bring some unexpected innovations in the context of theoretical physics, there is an obvious necessity to study its theoretical impact on Quantum Mechanics
Summary
The preceding arguments lead us to the following speculative suggestions, which are likely to compose the first steps towards a coherent proposal to revise the foundations of Quantum Mechanics. Our general idea is to give a physical meaning to the trajectory of A, simultaneously, and under the assumptions of Classical Mechanics This is rendered possible by a coupling argument, which turns out to be an established mathematical tool, see, e.g., [4], which allows us to embed the probability spaces (Ω, A, P) and (Ω', A', P') into a single joint space. A second step in this construction will come from the observation that, to render the system physically coherent, we must define at least two states for each isolated particle, depending upon whether they are stable or unstable The existence of these states underlies the possibility of a particle to interact with the outside world, and, in particular, to be detected or not. We are even led to question the existence of gravitational interaction
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