arXiv subjects
Publications and source records attributed to Andreas Aste.
A simple proof of Euler's formula which states that the sum of the reciprocals of all natural numbers squared equals $\pi^2/6$ is presented based on the distribution theory introduced by Laurent Schwartz. Additional identities are obtained as a byproduct of the derivation.
The concept of spontaneous symmetry breaking (SSB) generally lacks a simple and intuitive introduction in the literature. This gap is filled by defining SSB in a universal context beyond its usual applications in physics and by discussing some very simple, but stunning examples of the phenomenon.
This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space hints at the emergence of a complex structure. A widespread misconception concerning the measurement process in quantum mechanics and the hermiticity of observables is briefly discussed.
For decades, a lot of work has been devoted to the problem of constructing a non-trivial quantum field theory in four-dimensional space time. This letter addresses the attempts to construct an algebraic quantum field theory in the framework of non-standard theories like hyperfunction or ultra-hyperfunction quantum field theory. For this purpose model theories of formally interacting neutral scalar fields are constructed and some of their characteristic properties like two-point functions are discussed. The formal self-couplings are obtained from local normally-ordered analytic redefinitions of the free scalar quantum field, mimicking a non-trivial structure of the resulting Lagrangians and equations of motion.
A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory which are commonly not included in introductory courses when treating fermionic fields via the Dirac equation in the first place.
A concise point kinetic model of the explosion of a prompt supercritical sphere driven by a nuclear fission chain reaction is presented. The findings are in good agreement with the data available for Trinity, the first detonation of a nuclear weapon conducted by the United States Army as part of the Manhattan project. Results are presented for an implosion device containing pure plutonium-239, although the model can be easily applied to, e.g., uranium-235. The fizzle probability and corresponding yield of a fission bomb containing plutonium recovered from reactor fuel and therefore containing significant amounts of spontaneously fissioning plutonium-240 which can induce a predetonation of the device is illustrated by adding a corresponding source term in the presented model. Related questions whether a bomb could be made by developing countries or terrorist organizations can be tackled this way. Although the information needed to answer such questions is in the public domain, it is difficult to extract a consistent picture of the subject for members of organizations who are concerned about the proliferation of nuclear explosives.
The Gelfand pattern of the reduction of the N-fold tensor product of the fundamental representation of the special unitary group SU(2) by itself is studied in the framework of a finite Heisenberg model with infinite range, where N spins couple to each other with the same strength. The present findings are related to the microstatistics of non-rotating black holes for illustrative purposes.
Localization properties of scalar single particle states are analyzed by explicit calculational examples with a focus on the massless case. Problems arising from the non-existence of relativistic particle position operators respecting the causal structure of Minkowski spacetime are illustrated by exploring the conflicts arising from localization and causal properties commonly imposed on single particle states. These topics necessitate the introduction of quantum field theoretical localization concepts and are scarcely discussed and often misinterpreted in the literature.
A concise discussion of the basic principles of the controlled nuclear chain reaction is presented.
Fundamental problems of quantum field theory related to the representation problem of canonical commutation relations are discussed within a gauge field version of a van Hove-type model. The Coulomb field generated by a static charge distribution is described as a formal superposition of time-like pseudo-photons in Fock space with a Krein structure. In this context, a generalization of operator gauge transformations is introduced to generate coherent states of abelian gauge fields interacting with a charged background.
Symmetries are playing a very prominent role in natural sciences. In mathematics as the language of physics, symmetries are treated within the framework of group theory, which provides the tools to classify natural laws and physical objects like elementary particles. The present work discusses aspects of relativistic quantum field theory as the mathematical theory of particle physics which are relevant for the modern description of elementary particles and their associated fields hitherto considered as fundamental building blocks of the theory. Due to the abstract nature of quantum field theory, these aspects can only be touched by their exemplification within a review.
A concise discussion of a 3+1-dimensional derivative coupling model, in which a massive Dirac field couples to the four-gradient of a massless scalar field, is given in order to elucidate the role of different concepts in quantum field theory like the regularization of quantum fields as operator valued distributions, correlation distributions, locality, causality, and field operator gauge transformations.
A simple exposition of the rarely discussed fact that a set of free boson fields describing different, i.e. kinematically different particle types can be quantized with mutual anticommutation relations is given by the explicit construction of the Klein transformations changing anticommutation relations into commutation relations. The q-analog of the presented results is also treated. The analogous situation for two independent free fermion fields with mutual commutation or anticommutation relations is briefly investigated.
A concise discussion of the 3-dimensional irreducible (1,0) and (0,1) representations of the restricted Lorentz group and their application to the description of the electromagnetic field is given. It is shown that a mass term is in conflict with relativistic invariance of a formalism using electric and magnetic fields only, contrasting the case of the two-component Majorana field equations. An important difference between the Dirac equation and the Dirac form of Maxwell's equations is highlighted by considering the coupling of the electromagnetic field to the electric current.
The causal approach to perturbative quantum field theory is presented in detail, which goes back to a seminal work by Henri Epstein and Vladimir Jurko Glaser in 1973. Causal perturbation theory is a mathematically rigorous approach to renormalization theory, which makes it possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis. Epstein and Glaser solved this problem for a special class of distributions, the time-ordered products, that fulfill a causality condition, which itself is a basic requirement in axiomatic quantum field theory. In their original work, Epstein and Glaser studied only theories involving scalar particles. In this review, the extension of the method to theories with higher spin, including gravity, is presented. Furthermore, specific examples are presented in order to highlight the technical differences between the causal method and other regularization methods, like, e.g. dimensional regularization.
We analyze the electromagnetic pion and kaon form factor by including radiative and higher-twist effects within the framework of resummed pQCD in the space-like region. We focus on the transition from the perturbative to non-perturbative behavior in the phenomenological intermediate energy regime. Using a modified k_T factorization scheme with transverse degrees of freedom, we evaluate the non-perturbative soft contributions as distinct from the hard contributions, ensuring no double counting via the Ward identity at Q^2=0. The soft contributions are obtained via local quark-hadron duality while the hard contributions rest on the well known collinear factorization theorem using model wave functions with modified Brodsky-Huang-Lepage type ansatz and distribution amplitudes derived from light-cone QCD sum rules. Our analysis shows that the perturbative hard part prevails for large Q^2 beyond 50-100 GeV^2, while for low and moderate momentum transfers below 10-16 GeV^2 the soft contributions dominate over the hard part. Thus, we demonstrate the importance of including the soft contributions for explaining the experimental form-factor data.
A concise discussion of spin-1/2 field equations with a special focus on Majorana spinors is presented. The Majorana formalism which describes massive neutral fermions by the help of two-component or four-component spinors is of fundamental importance for the understanding of mathematical aspects of supersymmetric and other extensions of the Standard Model of particle physics, which may play an increasingly important role at the beginning of the LHC era. The interplay between the two-component and the four-component formalism is highlighted in an introductory way. Majorana particles are predicted both by grand unified theories, in which these particles are neutrinos, and by supersymmetric theories, in which they are photinos, gluinos and other states.
A condensed introduction to the basic concepts of causal perturbation theory is given. Causal perturbation theory is a mathematically rigorous approach to renormalization theory, which makes it possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis by avoiding infinities from the outset. It goes back to a seminal work by Henri Epstein and Vladimir Jurko Glaser published in 1973, where a specific causality condition was imposed at every order of perturbation theory in the case of scalar quantum field theory such that divergent integrals could be avoided in actual calculations of loop diagrams. In the meantime, the causal approach has been applied also to a wide range of gauge theories.