43,157 research outputs found
A Physical Axiomatic Approach to Schrodinger's Equation
The Schrodinger equation for non-relativistic quantum systems is derived from
some classical physics axioms within an ensemble hamiltonian framework. Such an
approach enables one to understand the structure of the equation, in particular
its linearity, in intuitive terms. Furthermore it allows for a physically
motivated and systematic investigation of potential generalisations which are
briefly discussed.Comment: Extended version. 14 page
Common Axioms for Inferring Classical Ensemble Dynamics and Quantum Theory
The same set of physically motivated axioms can be used to construct both the
classical ensemble Hamilton-Jacobi equation and Schrodingers equation. Crucial
roles are played by the assumptions of universality and simplicity (Occam's
Razor) which restrict the number and type of of arbitrary constants that appear
in the equations of motion. In this approach, non-relativistic quantum theory
is seen as the unique single parameter extension of the classical ensemble
dynamics. The method is contrasted with other related constructions in the
literature and some consequences of relaxing the axioms are also discussed: for
example, the appearance of nonlinear higher-derivative corrections possibly
related to gravity and spacetime fluctuations. Finally, some open research
problems within this approach are highlighted.Comment: Final proceedings version. 6 pages. Presented at the 3rd QTRF
conference at Vaxjo, Sweden, June6-11 200
Algebraic Approach to Colombeau Theory
We present a differential algebra of generalized functions over a field of
generalized scalars by means of several axioms in terms of general algebra and
topology. Our differential algebra is of Colombeau type in the sense that it
contains a copy of the space of Schwartz distributions, and the set of regular
distributions with -kernels forms a differential subalgebra.
We discuss the uniqueness of the field of scalars as well as the consistency
and independence of our axioms. This article is written mostly to satisfy the
interest of mathematicians and scientists who do not necessarily belong to the
\emph{Colombeau community}; that is to say, those who do not necessarily work
in the \emph{non-linear theory of generalized functions}.Comment: 16 page
An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms
We offer an axiomatic definition of a differential algebra of generalized
functions over an algebraically closed non-Archimedean field. This algebra is
of Colombeau type in the sense that it contains a copy of the space of Schwartz
distributions. We study the uniqueness of the objects we define and the
consistency of our axioms. Next, we identify an inconsistency in the
conventional Laplace transform theory. As an application we offer a free of
contradictions alternative in the framework of our algebra of generalized
functions. The article is aimed at mathematicians, physicists and engineers who
are interested in the non-linear theory of generalized functions, but who are
not necessarily familiar with the original Colombeau theory. We assume,
however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page
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