Physics Letters A 170 (1992) 409—412 PHYSICS LETTERS A North-Holland Maximal acceleration of thin shells in Weyl space G. Papini and W.R. Wood Department ofPhysics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Received 9 June 1992; accepted for publication 21 September 1992 Communicatedby J.P. Vigier It is shown that maximal acceleration, which arises in various theories when aspects of quantum mechanics are taken into account, also arises in a geometriccausal approach to quantum mechanics in which a particle is represented by a bubble in Weyl space. This result supports the hypothesis that quantum effects can be given a spacetime description. When a vector in Weyl space [1] is transported In this way, a bubble solution is obtained in Weyl along an infinitesimal path, its length / changes ac- space that obeys a relativistic wave equation. cording to the law 61=liç~6x~, where ic, 4 represents The purpose of this Letter is to show that this same the additional degree of freedom that Weyl’s ge- model leads to maximal acceleration which, in the ometry has relative to Riemannian geometry. A di- work of several authors [6—8], is associated with rect consequence of this transport law is that the quantum mechanics ~‘. The present work in fact ex- equations of Weyl’s theory must remain invariant tends the concept of a particle-dependent maximal under the extended group of conformal transfor- acceleration beyond the flat eight-dimensional phase mations [2]. These transformations do not leave the space of Caianiello, for example, and confirms that usual four-acceleration invariant in magnitude and Weyl space provides an appropriate curved space in the resulting motion of particles in Weyl’s theory is which quantum phenomena may be studied. stochastic in nature due to the arbitrariness in the The analysis of particles treated as thin shells is local standards of length. Indeed, as suggested by most naturally given in terms of the Gauss— Santamato [3], one may assume Weyl’s transport Mainardi—Codazzi (GMC) formalism where a ti- law as the geometrical expression of an intrinsically melike hypersurface E, which represents the history random particle motion. The fact that Santamato’s of the thin shell, divides spacetime into two four- theory allows quantum phenomena to be given a dimensional regions (V 1 and YE), both of which have purely geometrical description suggests that, in prim- X as their boundary. The GMC formalism can be ciple, Weyl space may be used to gain further in- generalized [11] to Weyl geometry by employing the sights into the nature of quantum theory. Attempts in this direction have recently led us to the construc- tion of a model [4] in Weyl space that provides a ~‘ It has at times been argued that the causal structure of space- time in classical relativity also requires a limit on the proper geometrical analogue of Vigier s stochastic theory [51 acceleration of an extended object [9]: a ~ —‘, where A rep- where a solitonic particle is required to beat in phase resents the proper extension of the object. When considering with its associated linear wave so that it follows the the point-like limit 2-+0, classical general relativity can be used mean flow lines of the Madelung fluid. In our case [10] to set a lower limit on A at the Schwarzschild radius in the particle is represented by a spherically symmetric order to avoid v olations of causality in the presence ofa na- ked singulanty. While these classical considerations imply a thin shell solution to the tensorial Einstein equations maximal acceleration, the limiting value actually occurs at the in a particular Weyl—Dirac theory that provides for microscopic scale which lies in the domain of quantum the Madelung fluid and the guidance principle as well, mechanics. 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 409