Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB

• URL: http://arxiv.org/abs/2406.07829v1
• Date: Wed, 12 Jun 2024 02:50:43 GMT
• Title: Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB
• Authors: Mustafa Türe, Mithat Ünsal,
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond the correspondence principle. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton's characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ-formalism is energy, not time. Thus, we are led to consider Fourier transform of path integral, spectral path integral, $\tilde Z(E)$. This admits a representation in terms of a quantum Hamilton's characteristic functions for perturbative and non-perturbative periodic orbits, generalizing Gutzwiller's sum. This results in a path integral derivation of exact quantization conditions, complementary to the exact WKB analysis of differential equations. We apply these to generic $\mathbb Z_2$ symmetric multi-well potential problems and point out some new instanton effects, e.g., the level splitting is generically a multi-instanton effect, unlike double-well.

Related papers

• Scaled quantum theory. The bouncing ball problem [0.0]
  The standard bouncing ball problem is analyzed under the presence of a gravitational field and harmonic potential. The quantum-classical transition of the density matrix is described by the linear scaled von Neumann equation for mixed states.
  arXiv  Detail & Related papers  (2024-10-14T10:09:48Z)
• Quantifying non-Hermiticity using single- and many-particle quantum properties [14.37149160708975]
  The non-Hermitian paradigm of quantum systems displays salient features drastically different from Hermitian counterparts. We propose a formalism that quantifies the (dis-)similarity of these right and left ensembles, for single- as well as many-particle quantum properties. Our findings can be instrumental in unveiling new exotic quantum phases of non-Hermitian quantum many-body systems.
  arXiv  Detail & Related papers  (2024-06-19T13:04:47Z)
• From reasonable postulates to generalised Hamiltonian systems [0.0]
  Hamiltonian mechanics describes the evolution of a system through its Hamiltonian. In both quantum and classical mechanics, Hamiltonian mechanics demands a precise relationship between time evolution and observable energy.
  arXiv  Detail & Related papers  (2024-02-29T07:50:51Z)
• Generalized quantum measurement in spin-correlated hyperon-antihyperon decays [11.594851987280764]
  We introduce a generalized quantum measurement description for decay processes of spin-1/2 hyperons. We validate this approach by aligning it with established theoretical calculations. We employ quantum simulation to observe the violation of CHSH inequalities in hyperon decays.
  arXiv  Detail & Related papers  (2024-02-26T13:54:20Z)
• A dynamic programming interpretation of quantum mechanics [0.0]
  We introduce a transformation of the quantum phase $S'=S+frachbar2logrho$, which converts the deterministic equations of quantum mechanics into the Lagrangian reference frame of particles. We show that the quantum potential can be removed from the transformed quantum Hamilton-Jacobi equations if they are solved as Hamilton-Jacobi-Bellman equations.
  arXiv  Detail & Related papers  (2024-01-08T18:43:40Z)
• Quantum generalized Calogero-Moser systems from free Hamiltonian reduction [0.0]
  The one-dimensional system of particles with a $1/x2$ repulsive potential is known as the Calogero-Moser system. We present a detailed and rigorous derivation of the generalized quantum Calogero-Moser Hamiltonian.
  arXiv  Detail & Related papers  (2022-11-10T18:34:17Z)
• Dispersion chain of quantum mechanics equations [0.0]
  The paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation.
  arXiv  Detail & Related papers  (2022-09-28T12:58:19Z)
• Geometric relative entropies and barycentric Rényi divergences [16.385815610837167]
  monotone quantum relative entropies define monotone R'enyi quantities whenever $P$ is a probability measure. We show that monotone quantum relative entropies define monotone R'enyi quantities whenever $P$ is a probability measure.
  arXiv  Detail & Related papers  (2022-07-28T17:58:59Z)
• Hamiltonian singular value transformation and inverse block encoding [12.386348820609626]
  We show how to perform the quantum singular value transformation for a matrix embedded as a block of a Hamiltonian. We also show how to use the Hamiltonian quantum singular value transformation to perform inverse block encoding to implement a unitary of which a given Hamiltonian is a block.
  arXiv  Detail & Related papers  (2021-04-03T13:58:27Z)
• Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
  We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow. First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric. We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
  arXiv  Detail & Related papers  (2020-10-29T06:15:30Z)
• Hamiltonian operator approximation for energy measurement and ground state preparation [23.87373187143897]
  We show how to approximate the Hamiltonian operator as a sum of propagators using a differential representation. The proposed approach, named Hamiltonian operator approximation (HOA), is designed to benefit analog quantum simulators.
  arXiv  Detail & Related papers  (2020-09-07T18:11:00Z)

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.