arXiv subjects

Jürgen Struckmeier

Publications and source records attributed to Jürgen Struckmeier.

At least 19 recordsLinked to original sources

arXiv2026

Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians

In the extended Lagrange formalism of classical point dynamics, the system's dynamics is parametrized along a system evolution parameter $s$, and the physical time $t$ is treated as a \emph{dependent} variable $t(s)$ on equal footing with all other configuration space variables $q^{i}(s)$. In the action principle, the conventional classical action $L dt$ is then replaced by the generalized action $L_{e}ds$. A direct reparametrization of the conventional action yields an extended Lagrangian that is homogeneous of degree one in the velocities. A regular extended Lagrangian that is not homogeneous of degree one can also reproduce the conventional dynamics on a suitably selected invariant hypersurface. With this representation, the physical motion takes place on a hypersurface within a symplectic extended phase space of even dimension. The formal similarity of conventional and extended Lagrange formalisms provides a direct connection with the proper-time path integral approach to relativistic quantum physics. To provide an example, the non-homogeneous extended Lagrangian $L_{e}$ of a classical relativistic spinless point particle in an external electromagnetic field will be presented. This extended Lagrangian is a quadratic function in the velocities, and its selected zero-energy hypersurface is the classical mass shell. With this $L_{e}$, the generalized path integral first yields an evolution equation in $s$. Imposing the corresponding quantum mass-shell condition then yields the Klein-Gordon equation as the physical wave equation. This establishes the relation between the classical extended Lagrangian description and proper-time quantization for a charged spin-zero particle.

quant-ph↗

arXiv2026

Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems

We will present a constrained description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-independent Hamiltonian system on the conventional symplectic phase space. The energy constraint and the pertaining extended symplectic structure will be derived from a generalized formulation of Hamilton's variational principle. An extended Hamiltonian $H_{1}=k(H-e)$ generates the physical dynamics on the constraint surface, with $k$ specifying the parameterization of time. The extended canonical transformation theory naturally permits transformations that also map the time scales of original and destination system while preserving the form of the canonical equations. For generating functions without explicit dependence on the common evolution parameter, the extended Hamiltonian transforms as a scalar function. The Lorentz transformation and prescribed time scaling transformations will be shown to represent particular canonical transformations in the symplectic extended phase space. Coordinate-dependent time re-scalings and the constrained Kustaanheimo-Stiefel mapping will be distinguished from invertible canonical coordinate transformations. Furthermore, an extended generating function that maps explicitly time-dependent Hamiltonians into time-independent ones will be presented for the time-dependent damped harmonic oscillator and for an explicitly characterized class of time-dependent potentials. For a general potential, a related auxiliary system yields a trajectory-dependent reconstruction identity rather than a universal linear evolution law. In the appendix, the singular Legendre transformation of the extended Lagrangian $L_{1}$ will be shown to yield the energy constraint without uniquely fixing its Hamiltonian extension away from the constraint surface.

math-ph↗

arXiv2026

Generalized Noether theorem in the realm of Hamiltonian point dynamics

Noether's theorem in the realm of point dynamics establishes the correlation of a constant of motion of a Hamilton-Lagrange system with a particular symmetry transformation of its action functional. Although usually derived in the Lagrangian formalism, the Hamiltonian formalism provides a natural context for this correlation. Canonical transformations preserve the canonical form of the action up to a boundary term. Within the class considered here, a symmetry additionally preserves the functional form of the specified extended Hamiltonian. Every smooth phase-space first integral admits locally a corresponding canonical Noether symmetry when momentum-dependent transformations and boundary terms are allowed. In order to represent symmetry mappings that include a transformation of time, we use the extended Hamiltonian formalism. This formalism enables us to define generating functions of canonical transformations that also map time and energy in addition to the conventional mappings of canonical space and momentum variables. The physical energy hypersurface and the choice of the evolution parameter are distinguished from identities on the full extended phase space. As an example, a manifest representation of the symmetry transformation associated with the Runge-Lenz invariant of the Kepler system is derived. A suitable choice of evolution parameter makes its time-shifting representative a symmetry of the specified extended action.

math-ph↗

arXiv2026

Tidal Forces in the Presence of Torsion and Nonmetricity

This work investigates how torsion and nonmetricity modify tidal accelerations in metric-affine gravity. We derive a projected deviation equation that generalizes the standard geodesic deviation equation to metric-affine geometry, and apply it to the relative acceleration of neighboring autoparallels in the weak-field, nonrelativistic limit. In this regime, the tidal acceleration separates into the usual Newtonian contribution and linear post-Riemannian corrections sourced by torsion and non-metricity. By decomposing torsion and nonmetricity into their irreducible Lorentz components, we identify the corresponding signatures in the tidal tensor and discuss to what extent these contributions can be distinguished. We then show how future direct tidal measurements could be translated into benchmark bounds on post-Riemannian tidal contributions, assuming probe dynamics sensitive to the affine connection. Our results suggest that the tidal acceleration may provide a systematic route toward probing post-Riemannian spacetime features in the future.

gr-qc↗

arXiv2026

Generalized U(N) gauge transformations in the realm of the extended covariant Hamilton formalism of field theory

The Lagrangians and Hamiltonians of classical field theory require compensating gauge fields in order to realize local internal gauge invariance with the standard transformation laws. This principle is developed in the extended covariant canonical transformation formalism, which also accommodates changes of spacetime coordinates. The construction yields locally U$(N)$-invariant matter--gauge-field actions and, in a separate application, diffeomorphism-covariant actions for ordinary vector fields and an independent affine connection. The latter application adopts a covariant-derivative coupling prescription whose antisymmetric vector-momentum sector contains a direct interaction with torsion. This interaction is not required by diffeomorphism covariance alone and does not apply to the internal gauge connection with its standard inhomogeneous transformation law. The compensating terms are distinguished from additional invariant interactions and from the gravitational dynamics, which must be specified independently.

nucl-th↗

arXiv2026

General U(N) gauge transformations in the realm of covariant Hamiltonian field theory

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. For regular Legendre transformations, the covariant canonical field equations are equivalent to the Euler-Lagrange field equations. The covariant canonical transformation theory provides a systematic means of defining admissible mappings that preserve the variational principle --- and hence the form of the field equations --- with the corresponding transformation of the Hamiltonian. Similar to the well-known canonical transformation theory of point dynamics, the canonical transformation rules for fields are derived from generating functions. As an interesting example, we work out the generating function of type F_{2} of a general local U(N) gauge transformation and thus derive a form-invariant Hamiltonian description under local U(N) gauge transformations, choosing the conventional quadratic gauge-field kinetic term. For a particular regular Dirac Lagrangian, the resulting gauge-invariant Lagrangian L_3 includes Pauli-coupling of an N-tuple of fermions with the set of bosonic gauge fields. Its coupling is controlled by an independent constant mass scale, and its relation to the Dirac-Pauli Lagrangian is a total divergence.

hep-th↗

arXiv2026

Covariant Hamiltonian representation of Noether's theorem and its application to SU(N) gauge theories

We present the covariant Hamiltonian representation of Noether's theorem and its application to Yang-Mills gauge theory. As the starting point, we re-formulate our previous presentation of the canonical Hamiltonian derivation of Noether's theorem, distinguishing admissible canonical transformations from symmetries of a specified Hamiltonian. The Noether currents of U(1) and U(N) gauge theories, and their restriction to SU(N), are derived from the respective infinitesimal generating functions of the pertinent symmetry transformations. The resulting conservation relations are distinguished from the complete field equations, which require the chosen kinetic Hamiltonian and a variational treatment of the antisymmetric gauge momenta.

math-ph↗

arXiv2026

Induced Couplings and Causal Bounds from Nondegenerate Dirac Lagrangians

The standard Dirac Lagrangian is linear in the field derivatives and therefore has a vanishing Hessian. We identify the minimal null deformations of this Lagrangian that make the covariant Legendre map locally invertible. Imposing global phase invariance, reality, proper Poincaré covariance, and absence of external background tensors leaves the two-parameter spinorial bivector $\mathcal E^{μν}=\ellσ^{μν} +\ell_5{}^\starσ^{μν}$, where the star denotes the Hodge dual and $\ell^2+\ell_5^2\neq0$. This extends the analysis of \cite{struckmeier2024pauli} by a parity-odd term. After minimal $U(1)$ gauging, this free null term is no longer variationally trivial and induces magnetic- and electric-dipole Pauli operators, together with an identically conserved dipole current. These dipole terms make the species-dependent regularization lengths directly constrainable by precision moment measurements. Metric-affine gauging in a Lorentz-spinor prescription then produces spin-curvature, torsion, and nonmetricity corrections to the Dirac operator. Applying the Velo-Zwanziger criterion, we exclude all nonzero pure axial, pure trace, and mixed axial-trace torsion backgrounds, as well as nonzero pure Weyl and second-trace nonmetricity backgrounds. The combined trace-vector nonmetricity sector is not excluded only when the effective trace vector vanishes. Tensor torsion and general tracefree nonmetricity remain unclassified without further algebraic assumptions, while the Levi-Civita limit preserves the metric light cone and leaves only lower-order curvature-dependent mass terms. Thus, gauging a Legendre-regular Dirac representative turns a free variational ambiguity into observable couplings, while non-Riemannian sectors are sharply constrained by causality bounds.

physics.gen-ph↗

arXiv2024

Bounding the Cosmological Constant using Galactic Rotation Curves from the SPARC Dataset

Dark energy (and its simplest model, the Cosmological Constant or $Λ$) acts as a repulsive force that opposes gravitational attraction. Assuming galaxies maintain a steady state over extended periods, the estimated upper limit on $Λ$ studies its pushback to the attractive gravitational force of dark matter. From the SPARC dataset, we select galaxies that are best fitted by the Navarro-Frenk-White (NFW) and Hernquist density models. Introducing the presence of $Λ$ in these galaxies helps to establish the upper limit on its repulsive force. This upper limit on $Λ$ is around $ρ_{\left(<Λ\right)} \sim 10^{-25}$~kg/m$^3$, only two orders of magnitude higher than the one measured by Planck. {We show that for galaxies with detectable velocities far from the galaxy core, the upper limit on $Λ$ is lower. Furthermore, we show that galaxies and other systems follow the same principle: for larger orbital periods the upper limit on $Λ$ is lower. Consequently, we address the implications for future measurements on the upper limit and the condition for detecting the impact of $Λ$ on galactic scales.

astro-ph.CO↗

arXiv2024

The problem of self-consistent particle phase space distributions for periodic focusing channels

Charged particle beams that remain stationary while passing through a transport channel are represented by ``self-consistent'' phase space distributions. As the starting point, we assume the external focusing forces to act continuously on the beam. If Liouville's theorem applies, an infinite variety of self-consistent particle phase space distributions exists then. The method is reviewed how to determine the Hamiltonian of the focusing system for a given phase space density function. Subsequently, this Hamiltonian is transformed canonically to yield the appropriate Hamiltonian that pertains to a beam passing through a non-continuous transport system. It is shown that the total transverse beam energy is a conserved quantity, if the beam stays rotationally symmetric along the channel. It can be concluded that charged particle beams can be transmitted through periodic solenoid channels without loss of quality. Our computer simulations, presented in the second part of the paper, confirm this result. In contrast, the simulation for a periodic quadrupole channel yields a small but constant growth rate of the rms-emittance.

physics.acc-ph↗

arXiv2024

On the stability and emittance growth of different particle phase-space distributions in a long magnetic quadrupole channel

The behavior of K-V, waterbag, parabolic, conical and Gaussian distributions in periodic quadrupole channels is studied by particle simulations. It is found that all these different distributions exhibit the known K-V instabilities. But the action of the K-V type modes becomes more and more damped in the order of the types of distributions quoted above. This damping is so strong for the Gaussian distribution that the emittance growth factor after a large number of periods is considerably lower than in the case of an equivalent K-V distribution. In addition, the non K-V distributions experience in only one period of the channel a rapid initial emittance growth, which becomes very significant at high beam intensities. This growth is attributed to the homogenization of the space-charge density, resulting in a conversion of electric-field energy into transverse kinetic and potential energy. Two simple analytical formulae are derived to estimate the upper and lower boundary values for this effect and are compared with the results obtained from particle simulations.

physics.plasm-ph↗

arXiv2024

Theoretical studies of envelope oscillations and instabilities of mismatched intense charged-particle beams in periodic focusing channels

The behavior of mismatched intense charged-particle beams in periodic transport channels of the solenoid and quadrupole type is studied theoretically. The envelope-oscillation frequencies of the mismatched beam are obtained by the smooth-approximation method and by numerical evaluation of the linearly perturbed K-V envelope equations. Phase shifts of the envelope oscillations and growth rates in the case of instability are calculated for a solenoid and a magnetic quadrupole (FODO) channel using the parameters of the Maryland and GSI beam transport experiments. For comparison and the purpose of illustration, the K-V equations are integrated numerically, and envelope curves as well as single-particle trajectories for for mismatched beams are shown in graphical form. In addition, computer simulation studies with the PARMILA code were performed, and results are presented both for K-V and a Gaussian distribution in transverse phase space.

physics.acc-ph↗

arXiv2024

Improved envelope and emittance description of particle beams using the Fokker-Planck approach

Beam dynamics calculations that are based on the Vlasov equation do not permit the the treatment of stochastic phenomena such as intra-beam scattering. If the nature of the stochastic process can be regarded as a Markov process, we are allowed to use the Fokker-Planck equation to describe the change of the phase space volume the beam occupies. From the Fokker-Planck equation we derive equations of motion for the beam envelopes and for the rms-emittances. Compared to previous approaches based on Liouville's theorem, these equations contain additional terms that describe the temperature balancing within the beam. Our formalism is applied to the effect of intra-beam scattering relevant for beams circulating in storage rings near thermodynamical equilibrium. In this case, the Fokker-Planck coefficients can be treated as adiabatic constants of motion. Due to the simplified analysis based on `beam moments', we obtain fairly simple equations that allow us to estimate the growth rate of the beam emittance.

physics.acc-ph↗

arXiv2024

Energy-second-moment map analysis as an approach to quantify the irregularity of Hamiltonian systems

A different approach will be presented that aims to scrutinize the phase-space trajectories of a general class of Hamiltonian systems with regard to their regular or irregular behavior. The approach is based on the `energy-second-moment map' that can be constructed for all Hamiltonian systems of the generic form $H=p^{2}/2+V(q,t)$. With a three-component vector $s$ consisting of the system's energy $H$ and second moments $qp$, $q^{2}$, this map linearly relates the vector $s(t)$ at time $t$ with the vector's initial state $s(0)$ at $t=0$. It will turn out that this map is directly obtained from the solution of a linear third-order equation that establishes an extension of the set of canonical equations. The Lyapunov functions of the energy-second-moment map will be shown to have simple analytical representations in terms of the solutions of this linear third-order equation. Applying Lyapunov's regularity analysis for linear systems, we will show that the Lyapunov functions of the energy-second-moment map yields information on the irregularity of the particular phase-space trajectory. Our results will be illustrated by means of numerical examples.

physics.comp-ph↗

arXiv2023

Rigorous derivation of dark energy and inflation as geometry effects in Covariant Canonical Gauge Gravity

The cosmological implications of the Covariant Canonical Gauge Theory of Gravity (CCGG) are investigated. CCGG is a Palatini theory derived from first principles using the canonical transformation formalism in the covariant Hamiltonian formulation. The Einstein-Hilbert theory is thereby extended by a quadratic Riemann-Cartan term in the Lagrangian. Moreover, the requirement of covariant conservation of the stress-energy tensor leads to necessary presence of torsion. In the Friedman universe that promotes the cosmological constant to a time-dependent function, and gives rise to a geometrical correction with the EOS of dark radiation. The resulting cosmology, compatible with the $Λ$CDM parameter set, encompasses bounce and bang scenarios with graceful exits into the late dark energy era. Testing those scenarios against low-z observations shows that CCGG is a viable theory.

gr-qc↗

arXiv2023

Covariant Canonical Gauge Theory of Classical Gravitation for Scalar, Vector, and Spin-1/2 Particle Fields

The framework of the Covariant Canonical Gauge theory of Gravity (CCGG) is described in detail. CCGG emerges naturally in the Palatini formulation, where the vierbein and the spin connection are independent fields. Neither torsion nor non-metricity are excluded. The manifestly covariant gauge process is based on canonical transformations in the De Donder-Weyl Hamiltonian formalism, starting from a small number of basic postulates. Thereby, the original system of matter fields in flat spacetime, represented by non-degenerate Hamiltonian densities, is amended by spacetime fields. The coupling of matter and spacetime fields leaves the action integral of the combined system invariant under active local Lorentz transformations and passive diffeomorphisms, aka Principle of General Relativity. We consider the Klein-Gordon, Maxwell-Proca, and Dirac fields and derive the corresponding equations of motion. Albeit the coupling of the given matter fields to the gauge fields are unambiguously determined by CCGG, the dynamics of the free gauge fields must be postulated based on physical reasoning. Our choice allows to derive Poisson-like equations of motion also for curvature and torsion. The latter is proven to be totally anti-symmetric. The affine connection is a function of the spin connection and vierbein fields. Requesting the spin connection to be anti-symmetric gives naturally metric compatibility. The canonical equations combine to an extension of the Einstein-Hilbert action with a quadratic Riemann-Cartan concomitant that endows spacetime with inertia. Moreover, a non-degenerate, quadratic version of the free Dirac Lagrangian is deployed. When coupled to gravity, the Dirac equation is endowed with an emergent mass parameter, a curvature-dependent mass correction, and novel interactions between particle spin and spacetime torsion.

gr-qc↗

arXiv2023

Exact invariants for a class of three-dimensional time-dependent classical Hamiltonians

An exact invariant is derived for three-dimensional Hamiltonian systems of $N$ particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function $f_{2}(t)$, embodying a solution of a linear third-order differential equation whose coefficients depend on the explicitly known trajectories of the particle ensemble. Our result is applied to a one-dimensional time-dependent non-linear oscillator, and to a system of Coulomb interacting particles in a time-dependent quadratic external potential.

physics.class-ph↗

arXiv2023

Extended Hamilton-Lagrange formalism and its application to Feynman's path integral for relativistic quantum physics

With this paper, a consistent and comprehensive treatise on the foundations of the extended Hamilton-Lagrange formalism will be presented. In this formalism, the system's dynamics is parametrized along a system evolution parameter $s$, and the physical time $t$ is treated as a dependent variable $t(s)$ on equal footing with all other configuration space variables $q^{i}(s)$. In the action principle, the conventional classical action $L dt$ is then replaced by the generalized action $L_{e}ds$, with $L$ and $L_{e}$ denoting the conventional and the extended Lagrangian, respectively. It is shown that a class of extended Lagrangians $L_{e}$ exists that are correlated to corresponding conventional Lagrangians $L$ without being homogeneous functions in the velocities. Then the Legendre transformation of $L_{e}$ to an extended Hamiltonian $H_{e}$ exists. With this class of extended Hamiltonians, an extended canonical formalism is presented that is completely analogous to the conventional Hamiltonian formalism. The physical time $t$ and the negative value of the conventional Hamiltonian then constitute and an additional pair of conjugate canonical variables. The extended formalism also includes a theory of extended canonical transformations, where the time variable $t(s)$ is also subject to transformation. In the extended formalism, the system's dynamics is described as a motion on a hypersurface within an extended phase space of even dimension. With the extended Lagrangian $L_{e}$, it is shown that the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description. Moreover, the space-time propagator for a free relativistic particle will be derived. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.

quant-ph↗