1,559 research outputs found
Mechanical analysis of finite idempotent relations
We use the technique of interactive theorem proving to develop the theory and anenumeration technique for finite idempotent relations. Starting from a short mathematical characterization of finite idempotents defined and proved in Isabelle/HOL, we derive first an iterative procedure to generate all instances of idempotents over a finite set. From there, we develop a more precise theo- retical characterization giving rise to an efficient predicate that can be executed in the programming language ML. Idempotent relations represent a very basic, general mathematical concept but the steps taken to develop their theory with the help of Isabelle/HOL are representative for developing algorithms from a mathematical specification
Algebraic Quantum Mechanics and Pregeometry
We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra
Kleene algebra with domain
We propose Kleene algebra with domain (KAD), an extension of Kleene algebra
with two equational axioms for a domain and a codomain operation, respectively.
KAD considerably augments the expressiveness of Kleene algebra, in particular
for the specification and analysis of state transition systems. We develop the
basic calculus, discuss some related theories and present the most important
models of KAD. We demonstrate applicability by two examples: First, an
algebraic reconstruction of Noethericity and well-foundedness; second, an
algebraic reconstruction of propositional Hoare logic.Comment: 40 page
The semigroup structure of Gaussian channels
We investigate the semigroup structure of bosonic Gaussian quantum channels.
Particular focus lies on the sets of channels which are divisible, idempotent
or Markovian (in the sense of either belonging to one-parameter semigroups or
being infinitesimal divisible). We show that the non-compactness of the set of
Gaussian channels allows for remarkable differences when comparing the
semigroup structure with that of finite dimensional quantum channels. For
instance, every irreversible Gaussian channel is shown to be divisible in spite
of the existence of Gaussian channels which are not infinitesimal divisible. A
simpler and known consequence of non-compactness is the lack of generators for
certain reversible channels. Along the way we provide new representations for
classes of Gaussian channels: as matrix semigroup, complex valued positive
matrices or in terms of a simple form describing almost all one-parameter
semigroups.Comment: 20 page
Aspects of Algebraic Quantum Theory: a Tribute to Hans Primas
This paper outlines the common ground between the motivations lying behind
Hans Primas' algebraic approach to quantum phenomena and those lying behind
David Bohm's approach which led to his notion of implicate/explicate order.
This connection has been made possible by the recent application of orthogonal
Clifford algebraic techniques to the de Broglie-Bohm approach for relativistic
systems with spin.Comment: 18 pages. No figure
Quantum mechanical virial theorem in systems with translational and rotational symmetry
Generalized virial theorem for quantum mechanical nonrelativistic and
relativistic systems with translational and rotational symmetry is derived in
the form of the commutator between the generator of dilations G and the
Hamiltonian H. If the conditions of translational and rotational symmetry
together with the additional conditions of the theorem are satisfied, the
matrix elements of the commutator [G, H] are equal to zero on the subspace of
the Hilbert space. Normalized simultaneous eigenvectors of the particular set
of commuting operators which contains H, J^{2}, J_{z} and additional operators
form an orthonormal basis in this subspace. It is expected that the theorem is
relevant for a large number of quantum mechanical N-particle systems with
translational and rotational symmetry.Comment: 24 pages, accepted for publication in International Journal of
Theoretical Physic
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