How does uncertainty propagate in Hamiltonian mechanics?

[@Cjeska192]

You might want to look into continues value QKD and specifically two-mode squeezed states. This topic essentially revolve around Gaussian states purely determined by the covariance and how to evolve and manipulate these. They are also only interested in quadratic Hamiltonians because the state stays Gaussian.

[@AssumptionsofPhysicsResearch]

But that would be for quantum mechanics (I assume QKD = Quantum key distribution)

[@Cjeska192]

@@AssumptionsofPhysicsResearch Ah okay so you are interested in the classical case. Then what is the probability distribution for your covariance? What ensemble/statistics/distribution are you interested in? Is the covariance you define substracing the average? Is 1/2 indistinguishable or can i just build a 2D oscillator out of it? Is your covariance equal time?

[@ThomasEmilioVilla]

Sorry Gabriele, I had missed this most interesting video! Well, in order to find out how a 1 degree of freedom manifold and momentum commute I guess the perfect theoretical framework is the Jackiw-Teitelboim gravity, a toy model in 2 D spacetime (1 timelike and 1 spacelike degree of freedom). This could help you formalize the unitary hamiltonian dynamics. Off diagonal terms vanishes quickly because, as a decoherentist would tell you (I'm not one, sorry...) the diagonal terms are "pointer states" and resist until macroscopic phenomenological scale...I guess that for a 1 degree of freedom system is easier to decohere than a cat :)

[@AssumptionsofPhysicsResearch]

@ThomasEmilioVilla This is just classical mechanics: nothing to decohere! :-D

[@ThomasEmilioVilla]

@@AssumptionsofPhysicsResearch Oh sorry! I tried to kill a mosquito with a cannon! :) I understood "covariant metrics", not matrix! :) I think you could rearrange the elements of the matrix in 4!=24 different ways, since p1, p2, q1, q2 are four variables. The diagonal would still yeld the same result, isn't it? Then you have at the beginning two "block matrix" isolated with one degree of freedom and they get in contact, time flows and they mix up and start swapping the values of the covariant matrices. If you do not introduce some sort of damping mechanism then I guess what you obtain is some sort of harmonic oscillator, going on forever and ever, a sort of oscillatory behaviour... :)

[@AssumptionsofPhysicsResearch]

@@ThomasEmilioVilla " tried to kill a mosquito with a cannon! :)" 🤣🤣🤣 It's the same intuition that I have... the uncertainty is going to "rotate around". I need to understand how exactly it works.

[@wwkk4964]

Just leaving a comment here so that the algorithm can propogate the video! (Not withstanding any uncertainties!)

[@santiagomartinez3417]

This is what chatGPT says: 1 Cramer-Rao Bound: The Cramer-Rao bound is a fundamental result in statistics that provides a lower bound on the variance of an estimator. In the context of Hamiltonian mechanics, the Cramer-Rao bound has been used to study the fundamental limits of uncertainty propagation. 2 Liouville's Theorem: This fundamental theorem in Hamiltonian mechanics states that the phase space volume of a closed system is conserved over time. In other words, the flow of a Hamiltonian system is incompressible. Liouville's Theorem has far-reaching implications for uncertainty propagation, as it implies that the probability density of the system's state is conserved along the flow. 3Poincaré's Integral Invariant: This theorem, also known as Poincaré's Lemma, states that the integral of the symplectic form (a fundamental geometric object in Hamiltonian mechanics) over a closed curve in phase space is invariant under the Hamiltonian flow. This result has important implications for the study of periodic orbits and the behavior of uncertainty in Hamiltonian systems. 4 Kolmogorov-Arnold-Moser (KAM) Theory: KAM theory is a collection of results that describe the behavior of quasi-periodic orbits in Hamiltonian systems. In particular, KAM theory provides a framework for understanding the stability of orbits in the presence of small perturbations. This theory has important implications for the study of uncertainty propagation in higher-dimensional systems. 5 Symplectic Geometry: Symplectic geometry is a branch of mathematics that studies the geometric properties of Hamiltonian systems. Symplectic geometry provides a powerful framework for understanding the behavior of uncertainty in Hamiltonian systems, particularly in higher-dimensional systems. Key concepts in symplectic geometry include symplectic forms, symplectic manifolds, and symplectic transformations.

[@AssumptionsofPhysicsResearch]

I did ask chatGPT the actual question: it didn't seem to know any relationship between the covariance matrix and the symplectic group...