Related papers: Dimensioned Algebra: the mathematics of physical q…
The aim of the paper is to discuss the relations between the three kinds of objects named in the title. In a sense, this is a survey of such relations; however, some new directions are also considered. This relates, especially, to sections…
In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and…
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real,…
Pure gravity and gauge theories in two dimensions are shown to be special cases of a much more general class of field theories each of which is characterized by a Poisson structure on a finite dimensional target space. A general scheme for…
The theory of measurement is employed to elucidate the physical basis of general relativity. For measurements involving phenomena with intrinsic length or time scales, such scales must in general be negligible compared to the (translational…
The problem of quantizing theories defined over configuration spaces described by non-commuting parameters is considered. In this paper we describe the first step in this direction, that is the definition of an integral over a general…
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same…
Phase-space realisations of an infinite parameter family of quantum deformations of the boson algebra in which the $q$-- and the $qp$--deformed algebras arise as special cases are studied. Quantum and classical models for the corresponding…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…
Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
In this thesis quadratic and cubic algebras, which are extensions of SU(1,1) and SU(2) are studied in detail, with particular attention being given to their construction, their finite and infinite dimensional irreducible representations and…
Particle physics has evolved in the past decade through evaluating the consequences of experimental measurements as well as exploiting theoretical tools that permit exploration of new model building and cosmological possibilities.…
Classical and quantum measurement theories are usually held to be different because the algebra of classical measurements is commutative, however the Poisson bracket allows noncommutativity to be added naturally. After we introduce…
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
In this paper we study associative algebras with a Poisson algebra structure on the center acting by derivations on the rest of the algebra. These structures, which we call Poisson fibred algebras, appear in the study of quantum groups at…
If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime.…
The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to…
A 3d generally covariant field theory having some unusual properties is described. The theory has a degenerate 3-metric which effectively makes it a 2d field theory in disguise. For 2-manifolds without boundary, it has an infinite number of…
We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized…