154 research outputs found
Geometrization of the Dirac theory of the electron
Using the concept of parallel displacement of a half vector, the Dirac equations are generally written in invariant form. The energy tensor is formed and both the macroscopic and quantum mechanic equations of motion are set up. The former have the usual form: divergence of the energy tensor equals the Lorentz force and the latter are essentially identical with those of the geodesic line
Leibniz Equivalence. On Leibniz's (Bad) Influence on the Logical Empiricist Interpretation of General Relativity
Einstein’s “point-coincidence argument'” as a response to the “hole argument” is usually considered as an expression of “Leibniz equivalence,” a restatement of indiscernibility in the sense of Leibniz. Through a historical-critical analysis of Logical Empiricists' interpretation of General Relativity, the paper attempts to show that this labeling is misleading. Logical Empiricists tried explicitly to understand the point-coincidence argument as an indiscernibility argument of the Leibnizian kind, such as those formulated in the 19th century debate about geometry, by authors such as Poincaré, Helmholtz or Hausdorff. However, they clearly failed to give a plausible account of General Relativity. Thus the point-coincidence/hole argument cannot be interpreted as Leibnizian indiscernibility argument, but must be considered as an indiscernibility argument of a new kind. Weyl's analysis of Leibniz's and Einstein's indiscernibility arguments is used to support this claim
Introducing Groups into Quantum Theory (1926 -- 1930)
In the second half of the 1920s, physicists and mathematicians introduced
group theoretic methods into the recently invented ``new'' quantum mechanics.
Group representations turned out to be a highly useful tool in spectroscopy and
in giving quantum mechanical explanations of chemical bonds. H. Weyl explored
the possibilities of a group theoretic approach towards quantization. In his
second version of a gauge theory for electromagnetism, he even started to build
a bridge between quantum theoretic symmetries and differential geometry. Until
the early 1930s, an active group of young quantum physicists and mathematicians
contributed to this new challenging field. But around the turn to the 1930s,
opposition against the new methods in physics grew. This article focusses on
the work of those physicists and mathematicians who introduced group theoretic
methods into quantum physics.Comment: Accepted by Historia Mathematica. The second version is considerably
changed in the section on Heisenberg, due to critical comments by an
anonymous referee. Other parts of the original manuscript have been improve
Distributions: The evolution of a mathematical theory
AbstractThe theory of distributions, or generalized functions, evolved from various concepts of generalized solutions of partial differential equations and generalized differentiation. Some of the principal steps in this evolution are described in this paper
Weyl laws for partially open quantum maps
We study a toy model for "partially open" wave-mechanical system, like for
instance a dielectric micro-cavity, in the semiclassical limit where ray
dynamics is applicable. Our model is a quantized map on the 2-dimensional
torus, with an additional damping at each time step, resulting in a subunitary
propagator, or "damped quantum map". We obtain analogues of Weyl's laws for
such maps in the semiclassical limit, and draw some more precise estimates when
the classical dynamic is chaotic.Comment: 35 pages, 5 figures. Corrected typos. Some proofs clarifie
General neutrino mass spectrum and mixing properties in seesaw mechanisms
Neutrinos stand out among the elementary particles because of their unusually small masses. Various
seesaw mechanisms attempt to explain this fact. In this work, applying insights from matrix theory, we are in a position
to treat variants of seesaw mechanisms in a general manner. Specifically, using Weyl's inequalities, we discuss
and rigorously prove under which conditions the seesaw framework leads to a mass spectrum with exactly three light
neutrinos. We find an estimate of the mass of heavy neutrinos to be the mass obtained by neglecting light neutrinos,
shifted at most by the maximal strength of the coupling to the light neutrino sector. We provide analytical conditions
allowing one to prescribe that precisely two out of five neutrinos are heavy. For higher-dimensional cases the inverse
eigenvalue methods are used. In particular, for the CP-invariant scenarios we show that if the neutrino sector
has a valid mass matrix after neglecting the light ones, i.e. if the respective mass submatrix is positive definite, then
large masses are provided by matrices with large elements accumulated on the diagonal. Finally, the Davis-Kahan
theorem is used to show how masses affect the rotation of light neutrino eigenvectors from the standard Euclidean
basis. This general observation concerning neutrino mixing, together with results on the mass spectrum properties,
opens directions for further neutrino physics studies using matrix analysis
- …