On computing quantum waves and spin from classical and relativistic action

• URL: http://arxiv.org/abs/2405.06328v4
• Date: Tue, 05 Nov 2024 19:48:13 GMT
• Title: On computing quantum waves and spin from classical and relativistic action
• Authors: Winfried Lohmiller, Jean-Jacques Slotine,
• Abstract summary: We show that the Schroedinger equation of quantum physics can be solved using a generalized form of the classical Hamilton-Jacobi least action equation. The results, which extend to the relativistic setting, build on two developments. They suggest a smooth transition between physics across scales, with the Hamilton-Jacobi formalism extending to general relativity.
• Score: 0.0
• License:
• Abstract: We show that the Schroedinger equation of quantum physics can be solved using a generalized form of the classical Hamilton-Jacobi least action equation, extending a key result of Feynman applicable only to quadratic actions. The results, which extend to the relativistic setting, build on two developments. The first is incorporating geometric constraints directly in the classical least action problem. This leads to multi-valued least action solutions where each local action is its own set element. The multiple solutions replace in part the probabilistic setting by the non-uniqueness of solutions of the constrained problem. For instance, in the double slit experiment or for a particle in a box, spatial inequality constraints create impulsive constraint forces, which lead to multiple path solutions. Second, an approximate mapping $ \ \Psi \approx e^{\frac{i }{\hbar} \Phi } \ $ between action $\Phi$ and wave function $\Psi$ has been known since Dirac and even Schroedinger. We show that this mapping can be made exact by introducing a compression ratio of the proposed multi-valued action, which can in turn be interpreted as a probability density on classical trajectories of a fluid flow field. Branch points of the multi-valued least action imply a quantum wave collapse. These developments leave the results of associated Feynman path integrals unchanged, but the computation can be greatly simplified as only multi-valued least actions are used, avoiding time-slicing and zig-zag trajectories altogether. They also suggest a smooth transition between physics across scales, with the Hamilton-Jacobi formalism extending to general relativity, in a coordinate-invariant framework. In particular, the Klein-Gordon equation may have a natural extension to general relativity.

Related papers

• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB [0.0]
  Hamilton-Jacobi theory is a powerful formalism, but its utility is not explored in quantum theory beyond the correspondence principle. We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi theory.
  arXiv  Detail & Related papers  (2024-06-12T02:50:43Z)
• Efficient Quantum Simulation Algorithms in the Path Integral Formulation [0.5729426778193399]
  We provide two novel quantum algorithms based on Hamiltonian versions of the path integral formulation and another for Lagrangians of the form $fracm2dotx2 - V(x)$. We show that our Lagrangian simulation algorithm requires a number of queries to an oracle that computes the discrete Lagrangian that scales for a system with $eta$ particles in $D+1$ dimensions, in the continuum limit, as $widetildeO(eta D t2/epsilon)$ if $V(x)$ is bounded
  arXiv  Detail & Related papers  (2024-05-11T15:48:04Z)
• Revealing quantum effects in bosonic Josephson junctions: a multi-configuration atomic coherent states approach [1.450405446885067]
  We show that quantum effects beyond the mean-field approximation are easily uncovered. The number of variational trajectories needed for good agreement with full quantum results is orders of magnitude smaller than in the semiclassical case.
  arXiv  Detail & Related papers  (2023-02-10T16:10:20Z)
• Correspondence between open bosonic systems and stochastic differential equations [77.34726150561087]
  We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment. A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
  arXiv  Detail & Related papers  (2023-02-03T19:17:37Z)
• Path integral in position-deformed Heisenberg algebra with strong quantum gravitational measurement [0.0]
  We show that quantum gravity bends the paths of particles, allowing them to travel quickly from one point to another. It is numerically observed by the decrease in values of classical actions as one increases the quantum gravitational effects.
  arXiv  Detail & Related papers  (2022-04-29T14:21:30Z)
• Quantum dynamics corresponding to chaotic BKL scenario [62.997667081978825]
  Quantization smears the gravitational singularity avoiding its localization in the configuration space. Results suggest that the generic singularity of general relativity can be avoided at quantum level.
  arXiv  Detail & Related papers  (2022-04-24T13:32:45Z)
• Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions [0.0]
  We inspect the quantum problem with the help of the conventional Schr"odinger's equation. Surprisingly, the quantum and classical solutions exhibit striking similarities.
  arXiv  Detail & Related papers  (2022-02-25T11:04:39Z)
• A shortcut to adiabaticity in a cavity with a moving mirror [58.720142291102135]
  We describe for the first time how to implement shortcuts to adiabaticity in quantum field theory. The shortcuts take place whenever there is no dynamical Casimir effect. We obtain a fundamental limit for the efficiency of an Otto cycle with the quantum field as a working system.
  arXiv  Detail & Related papers  (2022-02-01T20:40:57Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Quantum dynamics and relaxation in comb turbulent diffusion [91.3755431537592]
  Continuous time quantum walks in the form of quantum counterparts of turbulent diffusion in comb geometry are considered. Operators of the form $hatcal H=hatA+ihatB$ are described. Rigorous analytical analysis is performed for both wave and Green's functions.
  arXiv  Detail & Related papers  (2020-10-13T15:50:49Z)
• External and internal wave functions: de Broglie's double-solution theory? [77.34726150561087]
  We propose an interpretative framework for quantum mechanics corresponding to the specifications of Louis de Broglie's double-solution theory. The principle is to decompose the evolution of a quantum system into two wave functions. For Schr"odinger, the particles are extended and the square of the module of the (internal) wave function of an electron corresponds to the density of its charge in space.
  arXiv  Detail & Related papers  (2020-01-13T13:41:24Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.