Polydimensional Relativity, a Classical Generalization of the Automorphism Invariance Principle (original) (raw)

Clifford-algebra based polydimensional relativity and relativistic dynamics

Foundations of Physics 31 (2001) 1185-1209, 2001

Starting from the geometric calculus based on Clifford algebra, the idea that physical quantities are Clifford aggregates (“polyvectors”) is explored. A generalized point particle action (“polyvector action”) is proposed. It is shown that the polyvector action, because of the presence of a scalar (more precisely a pseudoscalar) variable, can be reduced to the well known, unconstrained, Stueckelberg action which involves an invariant evolution parameter. It is pointed out that, starting from a different direction, DeWitt and Rovelli postulated the existence of a clock variable attached to particles which serve as a reference system for identification of spacetime points. The action they postulated is equivalent to the polyvector action. Relativistic dynamics (with an invariant evolution parameter) is thus shown to be based on even stronger theoretical and conceptual foundations than usually believed.

Relativity in Clifford's Geometric Algebras of Space and Spacetime

International Journal of Theoretical Physics, 2000

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cℓ 1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS) Cℓ 3 . STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA + of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA + are all that is needed to calculate physically measurable quantities (called measurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA + and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out timelike planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to the relative version of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra.

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.

Geometric Algebra for Special Relativity and Manifold Geometry

This thesis is a study of geometric algebra and its applications to relativistic physics. Geometric algebra (or real Clifford algebra) serves as an efficient language for describing rotations in vector spaces of arbitrary metric signature, including Lorentzian spacetime. By adopting the rotor formalism of geometric algebra, we derive an explicit BCHD formula for composing Lorentz transformations in terms of their generators — much more easily than with traditional matrix representations. This is used to straightforwardly derive the composition law for Lorentz boosts and the concomitant Wigner angle. Later, we include a gentle introduction to differential geometry, noting how the Lie derivative and covariant derivative assume compact forms when expressed with geometric algebra. Curvature is studied as an obstruction to the integrability of the parallel transport equations, and we present a surface-ordered Stokes’ theorem relating the ‘enclosed curvature’ in a surface to the holonomy ...

A classical derivation of spacetime

2015

The four dimensional spacetime continuum, as originally conceived by Minkowski, has become the default framework within which to describe physical laws. Due to its fundamental nature, there have been various attempts to derive this structure from more fundamental physical principles. In this paper, we show how the Minkowski spacetime structure arises directly from the geometrical properties of three dimensional space when modeled by Clifford geometric algebra of three dimensions Cℓ(ℜ 3). We find that a time-like dimension, as well as three spatial dimensions, arise naturally, as well as four additional degrees of freedom that we identify with spin. Within this expanded eightdimensional arena of spacetime, we find a generalisation of the invariant interval and the Lorentz transformations, with standard results returned as special cases. The power of this geometric approach is shown by the derivation of the fixed speed of light, the laws of special relativity and the form of Maxwell's equations, without any recourse to physical arguments. We also produce a unified treatment of energy-momentum and spin, as well as predicting a new class of physical effects and interactions.

Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime

Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension ict, with the unit imaginary producing the correct spacetime distance x 2 {c 2 t 2 , and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i~ffi ffiffiffiffiffiffi ffi {1 p , with the Clifford bivector i~e 1 e 2 for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e 1 and e 2 . We find that with this model of planar spacetime, using a twodimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.

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.

Dimensionally democratic calculus and principles of polydimensional physics

To be published in the proceedings of, 1999

A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the laws of physics should obey new polydimensional metaprinciples, for which Clifford algebra is the natural language of expression, just as tensors were for general relativity. Specifically, phenomena and physical laws should be invariant under local automorphism transformations which reshuffle the physical geometry. This leads to a new generalized unified basis for classical mechanics, which includes string theory, membrane theory and the hypergravity formulation of Crawford[J. Math. Phys., 35, 2701-2718 (1994)]. Most important is that the broad themes presented can be exploited by nearly everyone in the field as a framework to generalize both the Clifford calculus and multivector physics.

Generalized Relativistic Transformations in Clifford Spaces and their Physical Implications

A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in C-spaces. The most general transformations furnish a full mixing of the grades of the multivectorvalued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass M in C-spaces which differs from m. No longer the proper mass appearing in the relativistic dispersion relation E^2 −p^2 = m^2 remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.