General relativity as an extended canonical gauge theory

It is widely accepted that the fundamental geometrical law of nature should follow from an action principle. The particular subset of transformations of a system's dynamical variables that maintain the form of the action principle comprises the group of canonical transformations. In the context of canonical field theory, the adjective "extended" signifies that not only the fields but also the space-time geometry is subject to transformation. Thus, in order to be physical, the transition to another, possibly noninertial frame of reference must necessarily constitute an extended canonical transformation that defines the general mapping of the connection coefficients, hence the quantities that determine the space-time curvature and torsion of the respective reference frame. The canonical transformation formalism defines simultaneously the transformation rules for the conjugates of the connection coefficients and for the Hamiltonian. As will be shown, this yields unambiguously a particular Hamiltonian that is form-invariant under the canonical transformation of the connection coefficients and thus satisfies the general principle of relativity. This Hamiltonian turns out to be a quadratic function of the curvature tensor. Its Legendre-transformed counterpart then establishes a unique Lagrangian description of the dynamics of space-time that is not postulated but derived from basic principles, namely the action principle and the general principle of relativity. Moreover, the resulting theory satisfies the principle of scale invariance and is renormalizable.Comment: 14 page

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

We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The extended Hamiltonian $H_{1}$ and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional Hamiltonian $H$ will be derived from a generalized formulation of Hamilton's variational principle. The extended canonical transformation theory then naturally permits transformations that also map the time scales of original and destination system, while preserving the extended Hamiltonian $H_{1}$, and hence the form of the canonical equations derived from $H_{1}$. The Lorentz transformation, as well as time scaling transformations in celestial mechanics, will be shown to represent particular canonical transformations in the symplectic extended phase space. Furthermore, the generalized canonical transformation approach allows to directly map explicitly time-dependent Hamiltonians into time-independent ones. An ``extended'' generating function that defines transformations of this kind will be presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials. In the appendix, we will reestablish the proper form of the extended Hamiltonian $H_{1}$ by means of a Legendre transformation of the extended Lagrangian $L_{1}$.Comment: 24 page

Stochastic effects in real and simulated charged particle beams

The Vlasov equation embodies the smooth field approximation of the self-consistent equation of motion for charged particle beams. This framework is fundamentally altered if we include the fluctuating forces that originate from the actual charge granularity. We thereby perform the transition from a reversible description to a statistical mechanics' description covering also the irreversible aspects of beam dynamics. Taking into account contributions from fluctuating forces is mandatory if we want to describe effects like intrabeam scattering or temperature balancing within beams. Furthermore, the appearance of ``discreteness errors'' in computer simulations of beams can be modeled as ``exact'' beam dynamics that is being modified by fluctuating ``error forces''. It will be shown that the related emittance increase depends on two distinct quantities: the magnitude of the fluctuating forces embodied in a friction coefficient $\gamma$, and the correlation time dependent average temperature anisotropy. These analytical results are verified by various computer simulations.Comment: 11 pages, 9 figure

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 to comprise gauge fields in order to be form-invariant under local gauge transformations. These gauge fields have turned out to correctly describe pertaining elementary particle interactions. In this paper, this principle is extended to require additionly the form-invariance of a classical field theory Hamiltonian under variations of the space-time curvature emerging from the gauge fields. This approach is devised on the basis of the extended canonical transformation formalism of classical field theory which allows for transformations of the space-time metric in addition to transformations of the fields. Working out the Hamiltonian that is form-invariant under extended local gauge transformations, we can dismiss the conventional requirement for gauge bosons to be massless in order for them to preserve the local gauge invariance.The emerging equation of motion for the curvature scalar turns out to be compatible with the Einstein equation in the case of a static gauge field. The emerging equation of motion for the curvature scalar R turns out to be compatible with that from a Proca system in the case of a static gauge field.Comment: 27 page

Generic Theory of Geometrodynamics from Noether's theorem for the Diff(M) symmetry group

We work out the most general theory for the interaction of spacetime geometry and matter fields---commonly referred to as geometrodynamics---for spin-$0$ and spin-$1$ particles. The minimum set of postulates to be introduced is that (i) the action principle should apply and that(ii) the total action should by form-invariant under the (local) diffeomorphism group. The second postulate thus implements the Principle of General Relativity. According to Noether's theorem, this physical symmetry gives rise to a conserved Noether current, from which the complete set of theories compatible with both postulates can be deduced. This finally results in a new generic Einstein-type equation, which can be interpreted as an energy-momentum balance equation emerging from the Lagrangian $L_{R}$ for the source-free dynamics of gravitation and the energy-momentum tensor of the source system $L_{0}$. Provided that the system has no other symmetries---such as SU$(N)$---the canonical energy-momentum tensor turns out to be the correct source term of gravitation. For the case of massive spin particles, this entails an increased weighting of the kinetic energy over the mass in their roles as the source of gravity as compared to the metric energy momentum tensor, which constitutes the source of gravity in Einstein's General Relativity. We furthermore confirm that a massive vector field necessarily acts as a source for torsion of spacetime. Thus, from the viewpoint of our generic Einstein-type equation, Einstein's General Relativity constitutes the particular case for spin-$0$ and massless spin particle fields, and the Hilbert Lagrangian $L_{R,H}$ as the model for the source-free dynamics of gravitation.Comment: 33 page

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.Comment: 24 pages, 3 figure

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.Comment: 31 pages, 12 figure

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.Comment: 9 pages, 7 figure