Investigación

 

Geometry of spin ½ particles

 

G. Sobczyk*

 

Universidad de las Américas-Puebla, Departamento de Físico-Matemáticas, 72820 Puebla, Pue., México.

 

Received 18 August 2014;
accepted 18 March 2015

 

Abstract

The geometric algebras of space and spacetime are derived by sucessively extending the real number system to include new mutually anticommuting square roots of ±1. The quantum mechanics of spin 1/2 particles are then expressed in these geometric algebras. Classical 2 and 4 component spinors are represented by geometric numbers which have parity, providing new insight into the familiar bra-ket formalism of Dirac. The classical Dirac Equation is shown to be equivalent to the Dirac-Hestenes equation, so long as the issue of parity is not taken into consideration, the latter quantity being constructed in such a way that it is parity invarient.

Keywords: Bra-ket formalism; geometric algebra; spacetime algebra; Schrödinger-Pauli equation; Dirac equation; Dirac-Hestenes equation; spinor; spinor operator.

 

PACS: 02.10.Xm; 03.65.Ta; 03.65.Ud

 

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References

* http://www.garretstar.com

1. L. Susskind, 9 YouTube Lectures: Quantum Entanglements, Part 1, (Stanford University 2008).         [ Links ]

2. T. Dantzig, NUMBER: The Language of Science, Fourth Edition, (Free Press, 1967).         [ Links ]

3. W.K. Clifford, On the Classification of Geometric Algebras, in Mathematical Papers by William Kingdon Clifford, edited by Robert Tucker, London, Macmillan and Co., 1882.         [ Links ]

4. D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed. (Kluwer 1999).         [ Links ]

5. G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number, Birkhauser, (New York 2013). http://www.garretstar.com/        [ Links ]

6. G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal 26, No. 4 (1995) 268-280.         [ Links ]

7. G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its Applications, 429 (2008) 1163-1173.         [ Links ]

8. E. Cartan, The Theory of Spinors, Dover Publications, (New York 1981).         [ Links ]

9. D. Hestenes, Spacetime Algebra, (Gordon and Breach 1966).         [ Links ]

10. P. Lounesto, Clifford Algebras and Spinors, 2nd Edition. Cambridge University Press, Cambridge, 2001.         [ Links ]

11. D. Hestenes, Clifford Algebra and the Interpretation of Quantum Mechanics, in Clifford Algebras and Their Applications in Mathematical Physics, edited by J.S.R. Chisholm and A.K. Common, NATO ASI Series C: Mathematical and Physical Sciences Vol. 183, D. Reidel (Publishing Company 1985).         [ Links ]

12. G. Sobczyk, Vector Analysis of Spinors (to appear), http://www.garretstar.com/nyuvas3-10-15.pdf        [ Links ]

13. G. Sobczyk, Spacetime Algebra of Dirac Spinors, (2015) (to appear), http://www.garretstar.com/diracspin03-24-15.pdf        [ Links ]

14. T.F. Havel, J.L. Doran, Geometric Algebra in Quantum Information Processing, Contemporary Mathematics, ISBN-10: 08218-2140-7, Vol. 305 (2002).         [ Links ]

15. C. Doran, A. Lasenby, Geometric Algebra for Physicists, (Cambridge 2007).         [ Links ]

16. D.J. Griffiths, Introduction to Quantum Mechanics, (Prentice Hall, Inc. 1995).         [ Links ]

17. D. Hestenes and R. Gurtler, Local Observables in Quantum Theory, Am. J. Phys. 39 (1971) 1028-1038.         [ Links ]

18. D. Hestenes, Zitterbewegung in Quantum Mechanics, Found Physics 40 (2010) 1-54. http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf http://geocalc.clas.asu.edu/pdf-preAdobe8/LocObsinQT.pdf        [ Links ]

19. D. Hestenes, "GEOMETRY OF THE DIRAC THEORY", in: A Symposium on the Mathematics of Physical Space-Time, Facultad de Quimica, Universidad Nacional Autonoma de Mexico, Mexico City, 67-96, (1981). http://geocalc.clas.asu.edu/pdf/Geom_Dirac.pdf        [ Links ]