Casper Ablij’s Post

Electromagnetism and Fluid Dynamics Classical electrodynamics has a continuity equation that has exactly the same form. In electromagnetism, ρ is the electrical charge density, and J is the electrical current density. The continuity equation expresses mathematically the fact that electrical charge is locally conserved. There is also a similar equation in fluid dynamics. Specifically, for an Eulerian inviscid fluid, local conservation of the fluid is expressed by ∂t ρm+∇⋅(ρm v)=0, where ρm is the mass density of the fluid, and v is the fluid velocity. (Inviscid means zero viscosity, and Eulerian means watching a fixed point in space as the fluid flows past.) Interpretation in Quantum Mechanics Since quantum mechanics has a continuity equation for the probability density, it is often said that the probability “flows like a fluid.” However, this analogy takes a bit more work to justify. The fact that a continuity equation exists for some scalar quantity does not necessarily mean that quantity behaves like a fluid. In electromagnetism, we do not say that electrical charge “flows like a fluid.” The dynamics of electric charge is governed by Maxwell’s equations, which are not the same as Euler’s equations for fluid flow. The existence of a continuity equation just means that the quantity in question is locally conserved. In quantum mechanics, probability is locally conserved. Otherwise the theory would not make much sense. It is possible to make the analogy between the time evolution of probability in quantum mechanics and fluid flow more precise. https://lnkd.in/gKvQZpVG

My new introduction to quantum mathematical physics is lastly available: Lie, Poisson, Jacobi structures and quantisation Abstract: Rigorous mathematical formulation of the problem of quantisation of dynamical systems on graded and supermanifolds using the theory of Lie algebroids, Poisson-Jacobi geometries and formal deformation of Lie algebras. https://lnkd.in/dpSMWSfC

Understanding Diffusion Models by Feynman's Path Integral https://lnkd.in/eXPa7NiK The paper introduces a novel approach to understanding score-based diffusion models by employing Feynman's path integral formalism, traditionally used in quantum physics. It provides a comprehensive description of generative models and demonstrates the derivation of backward stochastic differential equations and loss functions. The innovation lies in the introduction of an interpolating parameter, h, bridging stochastic generation (h=1) and deterministic probability flow ODEs (h=0), analogous to Planck's constant in quantum physics. Employing this analogy, the authors apply the WKB expansion to evaluate negative log-likelihood (NLL), offering a theoretical tool to explain the performance gap between stochastic and deterministic sampling. This work not only reveals a deeper connection between diffusion models and physics but also proposes a new method for explicit NLL calculation in noisy sampling processes.

PDF Advances In Density Functional Theory Per-Olov Löwdin (Eds.) digsell https://lnkd.in/eXDGZVYX Quantum mechanics can describe the detailed structure and behavior of matter, from electrons, atoms, and molecules, to the whole universe. It is one of the fields of knowledge that yield extraordinary precessions, limited only by the computational resources available. Among these methods is density functional theory (DFT), which permits one to solve the equations of … Read More » https://lnkd.in/eAcnZaZD

Why did such serious people take so seriously axioms which now seem so arbitrary? I suspect that they were misled by the pernicious misuse of the word "measurement" in contemporary theory. This word very strongly suggests the ascertaining of some preexisting property of some thing, any instrument involved playing a purely passive role. Quantum experiments are just not like that, as we learned especially from Bohr. The results have to be regarded as the joint product of "system" and "apparatus," the complete experimental set-up. But the misuse of the word "measurement" makes it easy to forget this and then to expect that the "results of measurement" should obey some simple logic in which the apparatus is not mentioned. The resulting difficulties soon show that any such logic is not ordinary logic. It is my impression that the whole vast subject of "Quantum Logic" has arisen in this way from the misuse of a word. I am convinced that the word "measurement" has now been so abused that the field would be significantly advanced by banning its use altogether, in favor for example of the word "experiment." –John Stewart Bell 🍁 On the Impossible Pilot Wave, Foundations of Physics 12, 989–99 (1982).