NDLI: The Complex Algebra of Physical Space: A Framework for Relativity

Content Provider	Springer Nature Link
Author	Baylis, W. E. Keselica, J. D.
Copyright Year	2012
Abstract	A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the natural origin of the Minkowski space-time metric. The CAPS formalism is Lorentz covariant and gives expression to persistent vectors in physical space as time-like planes in space-time. Commuting projectors $${P_{\pm} = \frac{1}{2} (1 \pm h)}$$ project CAPS onto two-sided ideals, one of which is APS. CAPS has the same dimension as the space-time algebra (STA) if both are considered real algebras, and it distinguishes covariant roles of elements, as does STA. Its structure, however, is closer to APS, with a volume element that belongs to the center of the algebra and a simple relation between space-times of opposite signature. Furthermore, CAPS, unlike STA, distinguishes point-like space-time inversion of a Dirac spinor from a physical rotation. To illustrate its use, CAPS is applied to the Dirac equation and to the fundamental symmetry transformations of the equation and Dirac spinors. The physical interpretations of both the equation and the spinor are clarified, and it is seen that the space-time frame $${\{\gamma_{\mu}\}}$$ arises fully from relative vectors and does not imply the existence of an absolute space-time frame.
Ending Page	561
Page Count	25
Starting Page	537
File Format	PDF
ISSN	01887009
e-ISSN	16614909
Journal	Advances in Applied Clifford Algebras
Issue Number	3
Volume Number	22
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-07-17
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	algebra of physical space Dirac spinor ideal Theoretical, Mathematical and Computational Physics hyperbolic complex numbers Complex APS Physics Mathematical Methods in Physics relativity Special relativity Clifford algebras, spinors Dirac equation Spinor and twistor methods Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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