Waves, particles and fields: an explicitly covariant approach
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Waves, particles and fields: an explicitly covariant approach Alberto Strumia
Received: 29 April 2012 / Published online: 4 December 2012 © Università degli Studi di Napoli "Federico II" 2012
Abstract The idea of associating particle trajectories with wave propagation rays exploited in a previous paper in the context of general relativity with a synchronous gauge, here is examined with no assumptions on co-ordinate choice (no synchronous gauge condition on the metric). Identification of particle Hamilton–Jacobi equation with wave-sheet equation in a space–time with more than 4 dimensions, is performed in an explicitly covariant formulation, leading to a Kaluza–Klein type theory involving Klein–Gordon equation arising from dilaton field equations. De Broglie and EinsteinPlanck quantum relations are also deduced in a natural way. Adding suitable Yang– Mills fields provides unification of gravitational, electromagnetic, weak and strong interactions into a 16 dimensional space–time geometry. The electron mass gap is also avoided compactifying extra dimensional co-ordinates on fractalized closed paths. Keywords Wave propagation · Hamiltonian particle dynamics · Covariant Hamilton–Jacobi equation · Quantum mechanics Mathematics Subject Classification (2000)
35Q40 · 81R20 · 81R20 · 12E99
Communicated by prof. Salvatore Rionero. Work developed during a sabbatical year at the C.i.r.a.m. (http://www.ciram.unibo.it) of the University of Bologna (Italy). A. Strumia (B) Department of Mathematics, University of Bari, Via E. Orabona, 4, 70125 Bari, Italy e-mail: [email protected]
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1 Introduction In the present paper we offer a generalization of our previous work [20], in the sense that we generalize the results previously obtained under the restrictive assumption of synchronous gauge, here removing the latter co-ordinate condition on the metric choice and providing an explicit covariant formulation of the proposed theory. Furthermore, thanks to this more general approach, several achievements are also gained. It is shown that matter fields together with all the interaction fields involved (i.e. gravitational, electromagnetic, weak and strong ones) are unified into a Kaluza–Klein [15] type theory in a higher dimensional space–time manifold V n+1 (n > 3) [18], where the dilaton fields can be interpreted as matter fields and (differently than in [20]) no more additional scalar fields are required to govern matter waves. With such geometrical approach, all the interaction fields involved in the particle standard model (for a review article see, e.g. [13]) appear to be unified avoiding Higgs mechanism thanks to higher dimensionality of space–time. In [20]—following an intriguing suggestion by Boillat [4,7] (which he developed in the context of non linear wave propagation theory [5]) that the trajectories of motion of (stable) particles could be identified with the rays of propagation of (exceptional) waves across the physical space—we have exploited a more general assumption. Our generalized approach iden
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