From physical assumptions to classical and quantum Hamiltonian and Lagrangian particle mechanics

Abstract

The aim of this work is to show that particle mechanics, both classical and quantum, Hamiltonian and Lagrangian, can be derived from few simple physical assumptions. Assuming deterministic and reversible time evolution will give us a dynamical system whose set of states forms a topological space and whose law of evolution is a self-homeomorphism. Assuming the system is infinitesimally reducible-specifying the state and the dynamics of the whole system is equivalent to giving the state and the dynamics of its infinitesimal parts-will give us a classical Hamiltonian system. Assuming the system is irreducible-specifying the state and the dynamics of the whole system tells us nothing about the state and the dynamics of its substructure-will give us a quantum Hamiltonian system. Assuming kinematic equivalence, that studying trajectories is equivalent to studying state evolution, will give us Lagrangian mechanics and limit the form of the Hamiltonian/Lagrangian to the one with scalar and vector potential forces. i i instead of a vector in a complex inner product space. Those are taken as given. This means that if we have a system in front of us, we do not have a rigorous way to conceptually establish whether the system is Hamiltonian or not, whether it is classical or quantum. We have some heuristics we can apply, but, at the end the day, a system is Hamiltonian because it follows the Hamiltonian framework. 1

The fundamental connections between classical Hamiltonian mechanics, quantum mechanics and information entropy

“…To summarize the arguments laid out in [1], since the state of the system does not capture the state of the parts, we can imagine these continuously permutating without affecting the dynamics of the whole. The internal motion is then characterized by random variables, which still need to allow for the definition of an invariant density, and therefore they must, for a degree of freedom, form a pair (U, V ).…”

“…In a previous work [1] we identified a small set of physical assumptions from which classical and quantum particle mechanics can be rederived. By assumptions here we mean simplifying conditions that our system is supposed to satisfy under the processes under study.…”

“…For quantum mechanics, for the sake of brevity, we will cover only some of the general ideas. For details on both, we refer to the previous longer paper [1]. We then provide a classical analogue of the quantum uncertainty relationship and see how both can be understood in terms of information entropy conservation under the different initial assumptions.…”

Towards a general mathematical theory of experimental science

“…Developing a general mathematical theory for experimental science could be compared to what happened in mathematics during the first half of the last century, when it was reorganized using logic and set theory as its foundation, which had profound repercussions in the field. Our preliminary work in both physics [7] and math [8] convinced us that such a goal is possible and within reach. The aim of the present work is to lay down the beginning of a general mathematical theory of experimental science, limited to the part concerning experimental verification.…”

Space-time structure may be topological and not geometrical

“…One difference is that in our work both classical and quantum cases are derived on equal footing, identifying the key point of divergence between the two theories. [5] Another difference is that our approach aims to start with primitives that are necessary to do physics. Our basic building block is the notion of a verifiable statement: an assertion for which an experimental test is available that would confirm, in finite time, that the assertion is true.…”