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Hamiltonian Dynamics in Fundamental Physics
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Hamiltonian Dynamics in Fundamental Physics

A Special Issue of Entropy (ISSN 1099-4300) belonging to the section "Quantum Information".

Deadline for manuscript submissions: 15 February 2027 | Viewed by 3091

Editor

Prof. Dr. Lawrence Horwitz

E-Mail Website1 Website2
Guest Editor
1. Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
2. Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
3. Department of Physics, Ariel University, Ariel 40700, Israel
Interests: relativistic quantum mechanics and quantum field theory; theory of classical and quantum unstable systems and chaos; quantum theory on hypercomplex Hilbert modules; complex projective spaces in quantum dynamics; relativistic statistical mechanics and thermodynamics; high-energy nuclear structure and particle physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Hamiltonian dynamics provides a profound and unifying framework for describing the evolution of physical systems, from classical point particles to relativistic fields and quantum states. Its principles, centered on phase space structure, symplectic geometry, and conservation laws, are the bedrock of modern theoretical physics. This formalism is not only essential for analyzing integrability, chaos, and stability but also serves as the critical pathway towards quantization, bridging the classical and quantum worlds through canonical methods.

In contemporary fundamental physics, the application of Hamiltonian dynamics is pushing the boundaries of our understanding in diverse and profound areas. A particularly rich avenue of research concerns Hamiltonians of the form pgp (with tensor indices), which naturally generate, via the Hamilton equations, a dynamical manifold with g as the metric. The evolution of such geometric structures—whether driven by intrinsic curvature-driven flows such as the Ricci flow or by other physical perturbations—offers fertile ground for exploration. The study of these evolving manifolds, including questions of their convergence, stability, and singularity formation, is of significant mathematical and physical interest, linking fundamental physics to geometric analysis.

This Special Issue aims to highlight recent advancements and foster new research at the intersection of Hamiltonian dynamics and the core problems of fundamental physics. We seek contributions that explore both the formal development of the Hamiltonian framework and its cutting-edge applications. We also welcome reviews offering a synthesis of recent progress.

Topics of interest include, but are not limited to, the following:

  • Canonical General Relativity and Quantum Gravity: Constraint analysis, ADM formalism, Ashtekar variables, and applications in loop quantum gravity or spin foams.
  • Hamiltonian Formulations of Gauge Theories: Canonical quantization, BRST formalism, and topological field theories.
  • Gravitational Waves and Binary Dynamics: Hamiltonian methods for post-Newtonian and post-Minkowskian calculations, effective-one-body formalism, and chaos in N-body systems.
  • Geometric Flows and Hamiltonian Dynamics: Ricci flow and other curvature flows as generated by Hamiltonian systems; stability and convergence properties of geometrically evolving manifolds.
  • Cosmological Perturbation Theory: Hamiltonian approaches to the evolution of scalar and tensor perturbations and quantum-to-classical transition.
  • Integrability and Chaos: Classical and quantum chaos in gravitational systems, out-of-time-order correlators (OTOCs), and the Kolmogorov–Arnold–Moser (KAM) theory in fundamental contexts.
  • Symplectic Geometry and Geometric Quantization: Advanced mathematical foundations and their physical implications.
  • Statistical Mechanics and Thermodynamics of Gravity: Black hole thermodynamics, the concept of entropy in isolated gravitational systems, and the connection to information theory.

This Special Issue of Entropy aims to compile original research and review articles that explore the central role of Hamiltonian dynamics in addressing the most pressing questions in fundamental physics, gravitation, and cosmology.

Prof. Dr. Lawrence Horwitz
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Hamiltonian dynamics
  • canonical quantization
  • general relativity
  • symplectic geometry
  • quantum gravity

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Published Papers (4 papers)

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Research

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39 pages, 497 KB  
Article
Invariant Boltzmann-Shannon Entropy for Black-Holes: A Manifestly-Covariant Canonical Quantum-Gravity Approach
by Claudio Cremaschini, Ramesh Radhakrishnan and Gerald Cleaver
Entropy 2026, 28(8), 932; https://doi.org/10.3390/e28080932 - 20 Aug 2026
Viewed by 602
Abstract
A novel theoretical study of Boltzmann-Shannon entropy arising in information-statistic theory applied to black-hole physics is proposed. The invariant setting implemented is represented by the manifestly-covariant quantum-gravity theory expressed in canonical Hamiltonian form. In such a framework the appropriate statistical interpretation relies on [...] Read more.
A novel theoretical study of Boltzmann-Shannon entropy arising in information-statistic theory applied to black-hole physics is proposed. The invariant setting implemented is represented by the manifestly-covariant quantum-gravity theory expressed in canonical Hamiltonian form. In such a framework the appropriate statistical interpretation relies on the configuration-space quantum expectation value of physical observables over the 4− scalar quantum-gravity probability density function (PDF). A representation for the black-hole Boltzmann-Shannon entropy is obtained for a Gaussian PDF profile and by establishing simultaneously a relationship between the black-hole invariant energy-content and the mean value of the quantum-gravity nonlinear Bohm potential. This yields a non-trivial functional dependence of the Boltzmann-Shannon entropy on the black-hole surface area, to be interpreted as a quantum statistical entropy counting black-hole bulk quantum-gravity states. The mathematical setting is shown to preserve manifest covariance and be self-contained within quantum-gravity realm. Comparisons with literature treatments dealing with thermodynamic or kinetic-statistical entropies that lead to the Bekenstein-Hawking black-hole surface entropy linear relation or its proposed quantum modifications are discussed. Full article
(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)
39 pages, 468 KB  
Article
4+1 Gravitation in the SHP Formalism
by Martin Land
Entropy 2026, 28(4), 417; https://doi.org/10.3390/e28040417 - 8 Apr 2026
Viewed by 618
Abstract
The Stueckelberg–Horwitz–Piron (SHP) formalism describes particles and fields traced out as spacetime events functionally dependent on an external evolution parameter τ. This approach addresses a number of difficulties associated with the problem of time. In SHP general relativity, the state of the [...] Read more.
The Stueckelberg–Horwitz–Piron (SHP) formalism describes particles and fields traced out as spacetime events functionally dependent on an external evolution parameter τ. This approach addresses a number of difficulties associated with the problem of time. In SHP general relativity, the state of the unconstrained phase space variables {xμ(τ),pν(τ)} specifies a 4D block spacetime M(τ) that evolves to an infinitesimally close 4D block spacetime M(τ+δτ) under a scalar Hamiltonian. As the configuration of matter and energy evolves with τ it induces changes in the spacetime metric γμν(x,τ), leading to τ-dependent geodesic equations for the phase space variables. The 4+1 approach in gravitation generalizes the 3+1 formalism of Arnowitt, Deser, and Misner (ADM) to construct τ-dependent Einstein field equations, a canonical Hamiltonian formalism, and an initial value problem for γμν(x,τ). To conform to known gravitational phenomenology, we must respect the 5D symmetries associated with the free fields—the geometrical constructs relevant to M(τ) as an embedded hypersurface—and the O(3,1) symmetries of 4D matter. The 4+1 formalism has been discussed in a series of publications. The goal of this paper is to provide a systematic review of the subject, make a few corrections and some significant additions, and present the theory in a concise and orderly fashion. Full article
(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)
28 pages, 394 KB  
Article
A Geometry of Hamiltonian Mechanics
by Gil Elgressy and Lawrence Horwitz
Entropy 2026, 28(4), 379; https://doi.org/10.3390/e28040379 - 27 Mar 2026
Viewed by 967
Abstract
We develop a local, patchwise geometric framework that embeds a broad class of potential Hamiltonian dynamical systems into a family of Riemannian Hamilton patches built over an underlying Gutzwiller manifold. We adopt a conformal (Jacobi) ansatz and a frame-adapted reconstruction procedure, through which [...] Read more.
We develop a local, patchwise geometric framework that embeds a broad class of potential Hamiltonian dynamical systems into a family of Riemannian Hamilton patches built over an underlying Gutzwiller manifold. We adopt a conformal (Jacobi) ansatz and a frame-adapted reconstruction procedure, through which we construct, on each patch, a pulled-back metric, along with a reduced (truncated) connection (not a metric-compatible connection) and a corresponding dynamical curvature tensor governing geodesic deviation in the Hamilton coordinates. Then, using the Poisson–Hodge reconstruction, we reconstruct coordinate potentials, enforcing harmonic obstructions, and along with exactness and Jacobian nondegeneracy conditions, we obtain explicit elliptic bounds that control the connection and curvature residuals. On the basis of this construction, we formalize the notion of a Hamilton manifold such that reparametrized geodesics approximate Newton trajectories with controlled acceleration and tolerances. As a generalized structural framework, to promote the local Jacobi reconstructions to a coherent dynamical evolution and provide a dynamical closure, we introduce a patchwise hyperbolic geometric flow for the pullback metric coupled to a kinetic (Vlasov) closure that controls reconstruction and curvature residuals. Under natural regularity, ellipticity, and overlap-tolerance assumptions, together with precise estimates that control the reconstruction and curvature errors, we establish short-time well-posedness of the coupled Vlasov–hyperbolic geometric flow that defines the patchwise Hamilton manifold. Motivated by this construction of the Hamilton manifold with atlas-dependent time, we propose convergence and stability conjectures for dissipative and conservative (non-dissipative) hyperbolic geometric flows. On a single patch, these conjectures characterize local orbital stability (in the sense of coercivity modulo symmetry) and identify local linear instability when unstable linear modes are present. On a finite atlas (the Hamilton manifold with atlas-dependent time), we state conjectures under which local stability propagates to global stability, provided that overlap residuals remain uniformly sufficiently small. The framework identifies the geometric origin of local instability diagnostics used in Hamiltonian mechanics and outlines a practical strategy for verifying stability or instability, numerically or analytically, on finite coverings of configuration space (the Hamilton manifold). Full article
(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)

Review

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29 pages, 8782 KB  
Review
Hamiltonian Dynamics and Fundamental Phenomena in Biophysics: A Review
by Matteo Gori, Roberto Franzosi, Giulio Pettini and Marco Pettini
Entropy 2026, 28(8), 928; https://doi.org/10.3390/e28080928 - 19 Aug 2026
Viewed by 308
Abstract
We review a theoretical and experimental programme with the aim of understanding two intimately related fundamental phenomena in biophysics: (i) the classical analogue of Fröhlich phonon condensation in macromolecules driven out of thermal equilibrium and (ii) the consequent activation of long-range resonant electrodynamic [...] Read more.
We review a theoretical and experimental programme with the aim of understanding two intimately related fundamental phenomena in biophysics: (i) the classical analogue of Fröhlich phonon condensation in macromolecules driven out of thermal equilibrium and (ii) the consequent activation of long-range resonant electrodynamic intermolecular forces. Both phenomena are underpinned by explicit Hamiltonian models. The first is derived by applying the time-dependent variational principle (TDVP) to the quantum Wu–Austin model, producing a fully classical Hamiltonian in action-angle variables whose nonlinear rate equations exhibit a nonequilibrium phase transition: the channelling of supplied energy into the lowest-frequency collective mode. The second is grounded in a classical electrodynamic Hamiltonian for two coupled oscillating dipoles whose normal-mode structure predicts long-range (∼1/r3) resonant interactions, absent at thermal equilibrium but activated by out-of-equilibrium collective oscillations. We also discuss a complementary Hamiltonian approach that connects Fröhlich’s rate equations directly to Hamilton’s equations of motion, clarifying the role of bath-mediated nonlinear coupling and the conditions for strong condensation at room temperature. In addition, the TDVP is applied to a Davydov–Holstein–Fröhlich Hamiltonian describing electron–phonon motion along the backbone of a specific DNA sequence and its cognate restriction enzyme, EcoRI: the time-domain Fourier cross-spectrum of the resulting electron currents exhibits a sharp co-resonance peak for the canonical recognition sequence that disappears upon randomisation, providing a sequence-specific electrodynamic signature of DNA–protein recognition. Experimental evidence from THz near-field spectroscopy, fluorescence correlation spectroscopy, and direct observation of protein clustering is reviewed in relation to these theoretical predictions. The results establish a coherent physical picture suggesting that metabolic energy supply can play a role in driving macromolecules into coherently oscillating states that activate selective, distance-reaching electrodynamic forces capable of contributing to the organisation of biochemical reactions in living matter. Full article
(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)
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