Clifford Algebra: A Case for Geometric and Ontological Unification (original) (raw)
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Title of Document: CLIFFORD ALGEBRA: A CASE FOR GEOMETRIC AND ONTOLOGICAL UNIFICATION
2016
abhors singularities. “So should we, ” responds the physicist. However, the epistemic assessments of Batterman concerning the matter prove to be less clear, for in the same vein he write that singularities play an essential role in certain classes of physical theories referring to certain types of critical phenomena. I devise a procedure (“methodological fundamentalism”) which exhibits how singularities, at least in principle, may be avoided within the same classes of formalisms discussed by Batterman. I show that we need not accept some divergence between explanation and reduction (Batterman 2002), or between epistemological and ontological fundamentalism (Batterman 2004, 2005). Though I remain sympathetic to the ‘principle of charity ’ (Frisch (2005)), which appears to favor a pluralist outlook, I nevertheless call into question some of the forms such pluralist implications take in Robert Batterman’s conclusions. It is difficult to reconcile some of the pluralist assessments that ...
Clifford Algebra, Geometry and Physics
The Nature of Time: Geometry, Physics and Perception, 2003
The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold (C-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. Within C-space we can perform the so called polydimensional rotations which reshuffle the multivectors, e.g., a bivector into a vector, etc.. A consequence of such a polydimensional rotation is that the signature can change: it is relative to a chosen set of basis vectors. Another important consequence is that the well known unconstrained Stueckelberg theory is embedded within the constrained theory based on C-space. The essence of the Stueckelberg theory is the existence of an evolution parameter which is invariant under the Lorentz transformations. The latter parameter is interpreted as being the true time -associated with our perception of the passage of time.
Clifford space as the arena for physics
Foundations of Physics 33 (2003) 1277-1306, 2003
A new theory is considered according to which extended objects in nnn-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of centre of mass coordinates. While the usual centre of mass is a point, by generalizing the latter concept, we associate with every extended object a set of rrr-loops, r=0,1,...,n−1r=0,1,..., n-1r=0,1,...,n−1, enclosing oriented (r+1)(r+1)(r+1)-dimensional surfaces represented by Clifford numbers called (r+1)(r+1)(r+1)-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called CCC-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but CCC-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in CCC-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of CCC-space.
Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism
New Structures for Physics, 2010
In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical light-cone structure of space-time, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics through the Bohm formalism which emerges quite naturally from the connection, showing that it is a structural feature of the mathematics. We then generalise the approach to include the symplectic Clifford algebra, which leads us to a non-commutative geometry with projections onto shadow manifolds. These shadow manifolds are none other than examples of the phase space constructed by Bohm. We also argue that this provides us with a mathematical structure that fits the implicateexplicate order proposed by Bohm.
Fundamental Physical Theories: Mathematical Structures grounded on a Primitive Ontology
In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a clear three-dimensional primitive ontology that evolves according to a law determined by the wave function. The primitive ontology is what matter is made of while the wave function tells the matter how to move. One might think that what is important in the notion of primitive ontology is their three-dimensionality. If so, in a theory like classical electrodynamics electromagnetic fields would be part of the primitive ontology. I argue that, reflecting on what the purpose of a fundamental physical theory is, namely to explain the behavior of objects in three-dimensional space, one can recognize that a fundamental physical theory has a particular architecture. If so, electromagnetic fields play a different role in the theory than the particles and therefore should be considered, like the wave function, as part of the law. Therefore, we can characterize the general ii structure of a fundamental physical theory as a mathematical structure grounded on a primitive ontology. I explore this idea to better understand theories like classical mechanics and relativity, emphasizing that primitive ontology is crucial in the process of building new theories, being fundamental in identifying the symmetries. Finally, I analyze what it means to explain the word around us in terms of the notion of primitive ontology in the case of regularities of statistical character. Here is where the notion of typicality comes into play: we have explained a phenomenon if the typical histories of the primitive ontology give rise to the statistical regularities we observe.
The alternative formalism based on the generalized Clifford algebra
The article presents the unification of two Maxwell's systems equations (homogeneous and inhomogeneous) within the generalized Clifford algebra. In this new formalism, an electromagnetic current and certain gauges acquire a geometric meaning associated with the properties of space.
2007
I summarize Silberstein, et. al's (2006) discussion of the derivation of the Heisenberg commutators, whose work is based on and . I argue that Bohr and Kaiser's treatment is not geometric enough, as it still relies on some unexplained residual notions concerning the unitary representation of transformations in a Hilbert space. This calls for a more consistent characterization of the role of i than standard QM can offer. I summarize David Hestenes ' (1985,1986) major claims concerning the essential role Clifford algebras play in such a fundamental characterization of i, and I present a Cliffordalgebraic derivation of the Heisenberg commutation relations (taken from ). I argue that their derivation exhibits a more fundamentally geometrical approach, which unifies geometric and ontological content. I also point out how some of Finkelstein's ontological notions of "chronon dynamics" can give a plausible explanatory account of RBW's "geometric relations."
Adversus Singularitates: The Ontology of Space–Time Singularities
I argue that there are no physical singularities in space–time. Singular space– time models do not belong to the ontology of the world, because of a simple reason: they are concepts, defective solutions of Einstein’s field equations. I discuss the actual implication of the so-called singularity theorems. In remarking the confusion and fog that emerge from the reification of singularities I hope to contribute to a better understanding of the possibilities and limits of the theory of general relativity.
Applications of Clifford Algebras in Physics
Lectures on Clifford (Geometric) Algebras and Applications, 2004
Clifford's geometric algebra is a powerful language for physics that clearly describes the geometric symmetries of both physical space and spacetime. Some of the power of the algebra arises from its natural spinorial formulation of rotations and Lorentz transformations in classical physics. This formulation brings important quantum-like tools to classical physics and helps break down the classical/quantum interface. It also unites Newtonian mechanics, relativity, quantum theory, and other areas of physics in a single formalism and language. This lecture is an introduction and sampling of a few of the important applications in physics.
2010
In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.