arXiv subjects
Publications and source records attributed to C. Baumgarten.
Many papers have been published over the years that either conjecture or even (claim to) prove the universality of the form of Maxwell's equations. We present yet another derivation of Maxwell's equations and discuss the conclusions suggested by Maxwell universality, namely the logical inevitability of the Lorentz transformations and the mathematical inconsistency of Newtonian physics.
It is demonstrated that energy conservation allows for a straight derivation of Newtonian mechanics without an apriori definition of the concept of work. Furthermore it is shown that energy must be depicted as a function of position and momentum in order to obtain the correct relativistic equations. Accordingly it is argued that not only quantum theory but also special relativity is intrinsically a Hamiltonian theory which requires a description of the dynamics using coordinate and momentum instead of velocity. Furthermore it is argued that the usual historical order of the ``formulations'' of mechanics, from Newtonian via Lagrangian to Hamiltonian mechanics, is illogical and misleading. We suggest that it should be reversed.
We give an exceptionally short derivation of Schroedinger's equation by replacing the idealization of a point particle by a density distribution.
We report the simplest possible form to compute rotations around arbitrary axis and boosts in arbitrary directions for 4-vectors (space-time points, energy-momentum) and bi-vectors (electric and magnetic field vectors) by symplectic similarity transformations. The Lorentz transformations are based exclusively on real $4\times 4$-matrices and require neither complex numbers nor special implementations of abstract entities like quaternions or Clifford numbers. No raising or lowering of indices is necessary. It is explained how the Lorentz transformations can be derived from the most simple second order Hamiltonian of general significance. Since this approach exclusively uses the real Clifford algebra $Cl(3,1)$, all calculations are based on real $4\times 4$ matrix algebra.
We report about the methods used in, and the performance of, the new fast and light-weight linear beam transport program MinT. MinT provides, beyond the usual linear ion optics, methods to compute the effects of beam degradation, multiple scattering and beam collimation. This is specifically important in facilities where the ion beam passes matter, for instance in proton therapy beamlines with an energy degrader as in the Proscan facility at PSI, but also for modelling the beam traversing the Muon- and Pion-production targets of the Paul Scherrer Institut's high intensity proton accelerator (HIPA). MinT is intended to be useful as a support tool for the HIPA and Proscan control rooms. This requires to have useful results within a few seconds. Hence simplicity and speed of calculation is favoured against higher accuracy. MinT has been designed not only to replace the FORTRAN 77 codes TRANSPORT and TURTLE, but to combine and extent their capabilities. MinT is a byte-code-compiler which translates an input language, described by syntactic rules. This allows for control structures like ``if-then-else'' or ``while''-loops, thus providing a high flexibility and readability.
We discuss the key role that Hamiltonian notions play in physics. Five examples are given that illustrate the versatility and generality of Hamiltonian notions. The given examples concern the interconnection between quantum mechanics, special relativity and electromagnetism. We demonstrate that a derivation of these core concepts of modern physics requires little more than a proper formulation in terms of classical Hamiltonian theory.
When compared to quantum mechanics, classical mechanics is often depicted in a specific metaphysical flavour: spatio-temporal realism or a Newtonian "background" is presented as an intrinsic fundamental classical presumption. However, the Hamiltonian formulation of classical analytical mechanics is based on abstract generalized coordinates and momenta: It is a mathematical rather than a philosophical framework. If the metaphysical assumptions ascribed to classical mechanics are dropped, then there exists a presentation in which little of the purported difference between quantum and classical mechanics remains. This presentation allows to derive the mathematics of relativistic quantum mechanics on the basis of a purely classical Hamiltonian phase space picture. It is shown that a spatio-temporal description is not a condition for but a consequence of objectivity. It requires no postulates. This is achieved by evading spatial notions and assuming nothing but time translation invariance.
In a cyclotron-based proton therapy facility, the energy changes are performed by means of a degrader of variable thickness. The interaction of the proton beam with the degrader creates energy tails and increases the beam emittance. A precise model of the degraded beam properties is important not only to better understand the performance of a facility already in operation, but also to support the development of new proton therapy concepts. The exact knowledge of the degraded beam properties, in terms of energy spectrum and transverse phase space, depends on the model used to describe the proton interaction with the degrader material. In this work the model of a graphite degrader has been developed with four Monte Carlo codes: three conventional Monte Carlo codes (FLUKA, GEANT4 and MCNPX) and the multi-purpose particle tracking code OPAL equipped with a simplified Monte Carlo routine. From the comparison between the different codes, we can deduce how the accuracy of the degrader model influences the precision of the beam dynamics model of a possible transport line downstream of the degrader.
Hamiltonian matrices appear in a variety or problems in physics and engineering, mostly related to the time evolution of linear dynamical systems as for instance in ion beam optics. The time evolution is given by symplectic transfer matrices which are the exponentials of the corresponding Hamiltonian matrices. We describe a method to compute analytic formulas for the matrix exponentials of Hamiltonian matrices of dimensions $4\times 4$ and $6\times 6$. The method is based on the Cayley-Hamilton theorem and the Faddeev-LeVerrier method to compute the coefficients of the characteristic polynomial. The presented method is extended to the solutions of $2\,n\times 2\,n$-matrices when the roots of the characteristic polynomials are computed numerically. The main advantage of this method is a speedup for cases in which the exponential has to be computed for a number of different points in time or positions along the beamline.
Despite the fact that the first-order beam dynamics models allow an approximated evaluation of the beam properties, their contribution is essential during the conceptual design of an accelerator or beamline. However, during the commissioning some of their limitations appear in the comparison against measurements. The extension of the linear model to higher order effects is, therefore, demanded. In this paper, the effects of particle-matter interaction have been included in the model of the transport lines in the proton therapy facility at the Paul Scherrer Institut (PSI) in Switzerland. To improve the performance of the facility, a more precise model was required and has been developed with the multi-particle open source beam dynamics code called OPAL (Object oriented Particle Accelerator Library). In OPAL, the Monte Carlo simulations of Coulomb scattering and energy loss are performed seamless with the particle tracking. Beside the linear optics, the influence of the passive elements (e.g. degrader, collimators, scattering foils and air gaps) on the beam emittance and energy spread can be analysed in the new model. This allows for a significantly improved precision in the prediction of beam transmission and beam properties. The accuracy of the OPAL model has been confirmed by numerous measurements.
Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the necessary number of postulates. We uncover the intrinsic connection of these areas of physics and describe them using a common symplectic Hamiltonian formalism. Our approach is based on a proper distinction between variables and constants, i.e. on a basic but rigorous ontology of time. We link these concept with the obvious conditions for the possibility of measurements. The derived consequences put the measurement problem of quantum mechanics and the Copenhagen interpretation of the quantum mechanical wavefunction into perspective. According to our (onto-) logic we find that spacetime can not be fundamental. We argue that a geometric interpretation of symplectic dynamics emerges from the isomorphism between the corresponding Lie algebra and the representation of a Clifford algebra. Within this conceptional framework we derive the dimensionality of spacetime, the form of Lorentz transformations and of the Lorentz force and fundamental laws of physics as the Planck-Einstein relation, the Maxwell equations and finally the Dirac equation.
We acuminate the idea of a final theory of physics in order to analyze its logical implications and consequences. It is argued that the rationale of a final theory is the principle of sufficient reason. This implies that a final theory of physics, presumed such a theory is possible, does not allow to incorporate substantial (non-trivial) propositions unless they are logically or mathematically deduced. Differences between physics and mathematics are discussed with emphasis on the role of physical constants. It is shown that it is logically impossible to introduce constants on the fundamental level of a final theory. The most fundamental constants emerging within a final theory are constants of motion. It is argued that the only possibility to formulate a final theory is necessarily a tautology: A final theory of physics can only be derived from those presumptions about reality that are inherent in the idea and practice of physics itself. It is argued that a final theory is based on the notion of objectivity, but it is logically impossible that an ideal final theory supports realism.
We take a new look at the details of symplectic motion in solenoid and bending magnets and rederive known (but not always well-known) facts. We start with a comparison of the general Lagrangian and Hamiltonian formalism of the harmonic oscillator and analyze the relation between the canonical momenta and the velocities (i.e. the first derivatives of the canonical coordinates). We show that the seemingly non-symplectic transfer maps at entrance and exit of solenoid magnets can be re-interpreted as transformations between the canonical and the mechanical momentum, which differ by the vector potential. In a second step we rederive the transfer matrix for charged particle motion in bending magnets from the Lorentz force equation in cartesic coordinates. We rediscover the geometrical and physical meaning of the local curvilinear coordinate system. We show that analog to the case of solenoids - also the transfer matrix of bending magnets can be interpreted as a symplectic product of 3 non-symplectic matrices, where the entrance and exit matrices are transformations between local cartesic and curvilinear coordinate systems. We show that these matrices are required to compare the second moment matrices of distributions obtained by numerical tracking in cartesic coordinates with those that are derived by the transfer a matrix method.
We discuss the principle possibility of stripping extraction in combination with reverse bends in isochronous separate sector cyclotrons (and/or FFAGs). If one uses reverse bends between the sectors (instead of drifts) and places stripper foils at the sector exit edges, the stripped beam has a reduced bending radius and it should be able to leave the cyclotron within the range of the reverse bend - even if the beam is stripped at less than full energy. We are especially interested in $H_2^+$-cyclotrons, which allow to double the charge to mass ratio by stripping. However the principle could be applied to other ions or ionized molecules as well. For the production of proton beams by stripping extraction of an $H_2^+$-beam, we discuss possible designs for three types of machines: First a low-energy cyclotron for the simultaneous production of several beams at multiple energies - for instance 15 MeV, 30 MeV and 70 MeV - thus allowing to have beam on several isotope production targets. In this case it is desired to have a strong energy dependence of the direction of the extracted beam thus allowing to run multiple target stations simultaneously. Second we consider a fast variable energy proton machine for cancer therapy that should allow extraction (of the complete beam) at all energies in the range of about 70 MeV to about 250 MeV into the same beam line. And third, we consider a high intensity high energy machine, where the main design goals are extraction with low losses, low activation of components and high reliability. The price that has to be paid for these advantages is an increase in size and/or in field strength compared to proton machines with standard extraction at the final energy.
The Courant-Snyder theory for two-dimensional coupled linear optics is presented, based on the systematic use of the real representation of the Dirac matrices. Since any real $4\times 4$-matrix can be expressed as a linear combination of these matrices, the presented Ansatz allows for a comprehensive and complete treatment of two-dim. linear coupling. A survey of symplectic transformations in two dimensions is presented. A subset of these transformations is shown to be identical to rotations and Lorentz boosts in Minkowski space-time. The transformation properties of the classical state vector are formulated and found to be analog to those of a Dirac spinor. The equations of motion for a relativistic charged particle - the Lorentz force equations - are shown to be isomorph to envelope equations of two-dimensional linear coupled optics. A universal and straightforward method to decouple two-dimensional harmonical oscillators with constant coefficients by symplectic transformations is presented, which is based on this isomorphism. The method yields the eigenvalues (i.e. tunes) and eigenvectors and can be applied to a one-turn transfer matrix or directly to the coefficient matrix of the linear differential equation.
A method is presented that enables to compute the parameters of matched beams with space charge in cyclotrons with emphasis on the effect of the transverse-longitudinal coupling. Equations describing the transverse-longitudinal coupling and corresponding tune-shifts in first order are derived for the model of an azimuthally symmetric cyclotron. The eigenellipsoid of the beam is calculated and the transfer matrix is transformed into block-diagonal form. The influence of the slope of the phase curve on the transverse-longitudinal coupling is accounted for. The results are generalized and numerical procedures for the case of an AVF cyclotron are presented. The algorithm is applied to the PSI Injector II and Ring cyclotron and the results are compared to TRANSPORT.