Abstract
At first we derive a field equation by means of essentially geometry free demands, that is, by means of a Cauchy-problem, superposition principle, finite propagation speed and a conservation law. These requirements characterise a symmetric hyperbolic system of partial differential equations of first order.
In what follows, the field equation will define the geometry of space-time (in the same way as light rays and point particles define ‘their’ geometry, cf. Ehlers, Pirani and Schild (1972)). A general discussion of the derived field equation leads to a metrical structure of Finslerian type, to the propagation of helicity states, to a path structure and to the propagation of spin states. We thereby investigate the concept of a generalised Clifford-algebra.
Specialisation to two light cones, two helicity and spin states and the requirement that there is at least one time-like group velocity, then leads to the usual Dirac equation which defines as its geometry a Riemann-Cartan spacetime with axial torsion only and interaction with the Maxwell field.
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References
Andersson, S.I. (1987): Quantitative measures of geometric and topological structure as generated by dynamics, Physica Scripta 35 225
Audretsch, J. (1981): Trajectories and Spin Motion of Massive Spin 1/2 Particles in Gravitational Fields, J. Phys A Math. Gen. 14 41
Audretsch, J. (1981a): Dirac Electron in Space-Time with Torsion: Spinor Propagation, Spin Precession and Non-Geodesic Motion, Phys. Rev D 24 1470.
Audretsch, J. (1983): The Riemannian Structure of Space-Time as a Consequence of Quantum Mechanics, Phys. Rev. D 27, 2872.
Audretsch, J.; Gähler, F.; Straumann, N. (1984): Wave Fields in Weyl-Space and Conditions for the Existence of a Preferred Pseudo-Riemannian Structure, Comm. Math. Phys. 95 41
Audretsch, J.; Lämmerzahl, C. (1988): Constructive Axiomatic Approach to Space-Time Torsion, Class. Quantum Grav. 5 1285
Baekler, P.; Hehl, F.W.; Mielke, E. (1986): Nonmetricity and Torsion: Facts and Fancies in Gauge Approaches to Gravity, in Ruffini, R. (Hrsgb.): Proceedings of the 4th Marcel Grossmann Meeting on General Relativity, Elsèvier, Amsterdam.
Bargmann, V.; Michel, L.; Telegdi, V.L. (1959): Precession of the Polarisation of Particles Moving in a Homogeneous Electromagnetic Field, Phys. Rev. Lett. 2 435
Bleyer, U. (1988): Eine nicht-Lorentz-invariante Verallgemeinerung der Dirac-Gleichung: Begründungen und Konsequenzen, Dissertation B, Akad. d. Wiss. DDR, Potsdam.
Bleyer, U.; Liebscher, D.-E. (1986): Induced causality, Astron. Nachr. 307 267
Bleyer, U.; Liebscher, D.-E. (1987): Quantum mechanical consequences of pregeometry, Preprint. Potsdam (will appear in: Proceedings of the IVth Seminar on Quantum Gravity, Moscow 1987, World Scientific).
Bhabha, H.J. (1949): On the Postulational Basis of the Theory of Elemtary Particles, Rev. Mod. Phys. 21 451.
Carathéodory, C. (1956): Variationsrechnung und Partielle Differentialgleichungen erster Ordnung, B.G. Teubner, Leipzig
Castagnino, M. (1971): The Riemannian Structure of Space-Time as a Consequence of a Measurement Method, J. Math. Phys. 12 2203.
Childs, L. N. (1978): Linearising of n-ic Forms and Generalized Clifford Algebras, Linear and Multilinear Algebra 5 267.
Courant, R; Hilbert, D. (1962): Methods of Mathematical Phystcs, Vol. II, Interscience Publishers, New York.
Dencker, N. (1982): On the Propagation of Polarization Sets for Systems of Real Principal Type, J. Functional Analysis, 46 351.
Dieudonné, J. (1971): Èlements d’Analyse, Gauthier-Villars, Paris.
Duffin, R. J. (1938): On the Characteristic Matrices of Covariant Systems, Phys. Rev. 54 1114.
Ehlers, J.; Pirani, F.A.E.; Schild, A. (1972): The Geometry of Free Fall and Light Propagation, in: L. O’Raifeartaigh (ed.): General Relativity, Papers in Honour of J.L. Synge, Clarendon Press, Oxford.
Ehlers, J. (1973): Survey of General Relativity Theory, in Israel, W. (ed.): Relativity, Astrophysics and Cosmology, D. Reidel, Dordrecht.
Fisher, A.E.; Marsden, J.F. (1979) The Initial Value Problem and the Dynamical Formulation of General Relativity, in S.W. Hawking, W. Israel (ed.): General Relativity, an Einstein centennary survey, Cambridge Univ. Press.
Gårding, L. (1985): Hyperbolic Differential Operators, in Jäger, W.; Moser, J.; Rem-mert, R. (Hrsgb.): Perspectives in Mathematics, Anniversary of Oberwolfach 1984, Birkhäuser Verlag, Basel.
Harish-Chandra (1947): On Relativistic Wave Equations, Phys. Rev. 71 793.
Hille, E.; Phillips, R.S. (1957): Functional Analysis and Semi-Groups, American Mathematical Society.
Hörmander, L. (1984): The Analysis of Linear Partial Differential Operators, Volume I–IV, Springer-Verlag, Berlin.
Kasper, U. (1986): On the importance of quantum mechanics for the axiomatic approach to the theory of gravitation, in Proceedings of the Conference on Differential Geometry and its Aopplications, in August 1986 in Brno, CSSR.
Kemmer, N. (1939): The particle aspect of meson theory, Proc. Roy. Soc. A173 91.
Komura, T. (1968): Semigroups of Operators in Locally Convex Spaces, J. Functional Analysis 2 258
Liebscher, D.-E. (1985a): The Geometry of the Dirac Equation, Ann. Physik 42 35
Loinger, A. (1985): Weylian geometry and first order wave equations, Nouvo Cim. 88B 9
Matsumoto, M. (1970): The Theory of Finsler Connections, Publications of the Study Group of Geometry.
Meyer, R.; Schroeter, G. (1981): The Application of Differential Geometry to Ray Acoustics in Inhomogeneous and Moving Media, Acustica 47 105.
Nono, T. (1971): Generalised Clifford algebras and linearisations of a partial differential equation. in Ramakrishnan, A. (ed.): Proceedings of the Conference on “Clifford-Algebra, its Generalisations and Applications”, Matscience, The Institute of Mathematical Sciences, Madras.
Rafanelli, K.; Schiller, R. (1964): Classical Motions of Spin 1/2 Particles, Phys. Rev. 135 B279.
Rauch, H. (1983): in S. Kamefuchi (ed.): Proceedings of the International Symposium on Foundations of Quantum Mechantcs, Tokyo.
Reichenbach, H. (1928): Philosophie der Raum-Zeit-Lehre; deGruyter, Berlin
Riemann, B. (1854): Ueber die Hypothesen, die der Geometrie zugrunde liegen, Habilitationsvortrag, in Weber, H. (Hersg.): Bernhard Riemanns gesammelte mathematische Werke und wissenschaftlicher Nachlass, B.G. Teubner, Leipzig 1876.
Roby, N. (1969): Algèbres de Clifford des formes polynomes, C.R. Acad. Sc. Paris A268 484.
Rubinow, S.J.; Keller, J.B. (1963): Asymptotic Solution of the Dirac Equation, Phys. Rev. 131 2789.
Rund, H. (1959): The Differential Geometry of Finsler Spaces, Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Springer-Verlag, Berlin
Svendsen, E.C. (1982): Unitary one-parameter groups with finite speed of propagation, Proc. Amer. Math. Soc. 84, 357.
Wightman, A.S. (1970): Relativistic Wave Equations as Singular Hyperbolic Systems, in Spencer, D.C. (ed.): Partial Differential Equations, Proceedings of the Symposium in Pure Mathematics, American Mathematical Society.
Yoshida, K. (1971): Functional Analysis, Springer-Verlag.
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Lämmerzahl, C. (1990). The Geometry of Matter Fields. In: Audretsch, J., de Sabbata, V. (eds) Quantum Mechanics in Curved Space-Time. NATO ASI Series, vol 230. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3814-1_2
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DOI: https://doi.org/10.1007/978-1-4615-3814-1_2
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