Physical Applications of a Generalized Clifford Calculus

Presenter: William M. Pezzaglia Jr. (wpezzag@clifford.org) Department of Physics, Santa Clara University, Santa Clara, CA 95053, USA

Presentation at section on "Analysis of Dirac Operators", at the first annual ISAAC Conference (International Society for Analysis, its Applications and Computation,

http://www.math.udel.edu/isaac/conferen/congr97.htm) at the University of Delaware, June 2-6, 1997.

Abstract A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of physical matter (e.g. quantum spin). The Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physical interpretation is natural coupling of the classical spin to linear motion replacing the usual ad-hoc derivation of the Papapetrou equation. A generalized curvature is proposed for the Clifford manifold in which the connection does not preserve the rank of a multivector under parallel transport.

The concepts presented in this talk were later summarized in the conference proceedings, available as preprint gr-qc/9710027.

Index to Color Transparencies.

1. I. Title Page

2. II. The Clifford Manifold o A. Polygeometric Space o B. Geometric Algebra o C. Polygeometric coordinates in Physics

3. III. Multivector Calculus o A. Polydimensional Differentials o B. Differential Multiforms o C. Spinning Particles in Curved Space

4. IV. Pan-Dimensional Curvature o A. Panvector Formulation of Physics

1. Automorphism Invariance o B. PolyAffine Space o C. Pan-Dimensional Geometrodynamics

5. V. Epagogy

6. VI. References

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References The concepts presented in this talk were later summarized in the conference proceedings,

available as preprint gr-qc/9710027.

• J. Crawford, Local automorphism invariance: Gauge boson mass without a Higgs particle, J. Math. Phys. 35, 2701 (1994); Hypergravity in Clifford Geometric Algebras, with Applications in Physics, Mathematics and Engineering, W. Baylis ed. (Birkhauser 1996).

• J.S.R. Chisholm and R.S. Farwell, Unified Spin Gauge Theory of Electroweak

and Gravitational Interactions, J. Phys. A22 1059 (1989); Spin gauge theories: principles and predictions, in Clifford Algebras and Their Applications in Mathematical Physics, proceedings of 3rd Conference, Belgium 1993, R. Delanghe et al (Kluwer 1993) pp. 367-374).

• W. Pezzaglia, Polydimensional Relativity, a Classical Generalization of the

Automorphism Invariance Principle, in the Proceedings of the 4th Conference on Clifford Algebras and their Applications in Mathematical Physics, Aachen, Germany 1996, K. Habetah ed., Kluwer (1998), pp. 305-317. Preprint gr-qc/9608052.

• W. Pezzaglia and A. Differ, A Clifford Dyadic Superfield from Bilateral

Interactions of Geometric Multipsin Dirac Theory, in Proceedings of XXIIth International Conf on Differential Geometric Methods in Theoretical Physics, Ixtapa-Zihuatanejo, Mexico 1993, J. Keller ed., Advances in Applied Clifford Algebras (Proc. Suppl.) 4 (S1) (1994) pp. 437-446. Preprint gr-qc/9311015

• I.B. Khriplovich, Particle with internal angular momentum in a gravitational

field, Sov. Phys. JETP 69, 217 (1989)

• index of other talks at: http://www.clifford.org/~wpezzag/talks.html • This URL: http://www.clifford.org/wpezzag/talk/97santabarb/

Updated: 2005May20