J. Phys. Commun. 3 (2019) 065006 https://doi.org/10.1088/2399-6528/ab2820 PAPER Metric geometry and the determination of the Bohmian quantum OPEN ACCESS potential RECEIVED 6 February 2019 Paul Bracken REVISED 26 April 2019 Department of Mathematics, University of Texas, Edinburg, TX 78540, United States of America ACCEPTED FOR PUBLICATION E-mail: [email protected] 10 June 2019 Keywords: quantum theory, Finsler geometry, potential, curvature PUBLISHED 18 June 2019 Original content from this Abstract work may be used under A geometric approach to quantum mechanics which is formulated in terms of Finsler geometry is the terms of the Creative Commons Attribution 3.0 developed. It is shown that quantum mechanics can be formulated in terms of Finsler configuration licence. space trajectories which obey Newton-like evolution but in the presence of an additional kind of Any further distribution of this work must maintain potential. This additional quantum potential which was obtained first by Bohm has the consequence attribution to the author(s) and the title of of contributing to the forces driving the system. This geometric picture accounts for many aspects of the work, journal citation quantum dynamics and leads to a more natural interpretation. It is found for example that dynamics and DOI. can be accounted for by incorporating quantum effects into the geometry of space-time. 1. Introduction It is still a task to understand and interpret quantum mechanics. It has become clear that quantum mechanics can be looked at from a geometric point of view. In fact, geometry can be applied to the study of the structure and interpretation of quantum mechanics. This is due largely to the fact that geometrical considerations have resulted in advances in various other areas of physics, most notably, general relativity. Quantum mechanics is not so easy to formulate in terms of a dynamical, geometric description in terms of trajectories in some configuration space for a number of reasons. It will be seen here that a particular framework for such a task can be developed. A first reason for this is the important fact of quantum correlation and another is the complicated phenomenon of quantum entanglement. Both quantum coherence and decoherence effects, the state superposition principle and so forth have a natural description based on the Schrödinger equation. It has long been known that determinism is hard to reconcile with such a description of quantum dynamics. This problem has been discussed by both de Broglie and Bohm [1–3]. This view of things consists of material point trajectories carried along by a pilot wave that evolves according to the Schrödinger equation for the wavefunction of the system. Bohmian trajectories follow the flux lines of probability current which is a function of the position coordinate of the particle involved. This implies that there is a certain nonlocality inherent in the usual picture. Bohm realized that this kind of dynamics can be described by configuration space trajectories which follow a Newton’s law evolution, but under the influence of a quantum potential in addition to the external potential. This quantum potential can be calculated in closed form. This quantum potential is added to the classical potential function and it makes a contribution to the evolution of the system. It will be useful to have a brief description of Bohm’s theory at hand [4], to be able to refer to it further on. Bohm began from a series of basic postulates. An individual system comprises a wave propagating in space and time along with a point particle which moves under the guidance of the wave. The wave is described mathematically by Schrödinger’s equation i ¶=t yy()xx,,tH ( t), where H is the Hamiltonian. The motion of the particle is obtained as the solution to 1 xx˙ =St()∣,1.1xx= ()t () m © 2019 The Author(s). Published by IOP Publishing Ltd J. Phys. Commun. 3 (2019) 065006 P Bracken where S is the phase of y()x, t and m the mass. One initial condition for (1.1) suffices to solve it. These give a theory of motion which is consistent from a physical point of view. To get compatibility of the motion of an ensemble of particles with quantum mechanics, Bohm states that the probability that a particle in the ensemble resides between x and x+dx at time t is R23()x, tdxwhere R = ∣∣y 2. In Bohm’s approach, the wavefunction in polar form y = ReiS is substituted into Schrödinger’s equation. By isolating real and imaginary parts, he found the movement under the guidance of the wave happens in agreement with a law of motion of the form ¶S 1 2 ++S2 +=2 RV0.() 1.2 ¶tm22mR Equation (1.2) is equal to the classical Hamilton-Jacobi equation except for the appearance of the term 2 Q =2R.1.3() 2mR It is referred to as the quantum potential. The equation of motion can be expressed as well in the form, d2x m =-()QV + ,1.4 () dt 2 where x=x (t) is the trajectory of the particle associated with its wavefunction. One approach to obtain a geometric formulation and interpretation of quantum mechanics is to introduce a particular geometric structure on which to base everything. Finsler geometry is a metric geometry which is well suited to this purpose [5]. It is described by a metric tensor which is a function of both the position and momentum variables. The metric in Finsler geometry is in fact a generalization of a Riemannian metric, and the case in which the metric is developed from the quantum potential will be studied. The approach of Chern to Finsler geometry will be followed [6, 7]. The theory that results by proceeding in this way accounts for quantum effects as a consequence of the geometry of space-time [8, 9]. The curvature is self-induced as well as generated by the collection of particles in a nonlocal fashion resulting in a curved configuration space. The metric tensor and as a consequence the curvature of the manifold depends on the coordinates of the particle, as well as all the other particles of the system. The wavelike nature of quantum mechanics is then confined to accounting for how space is described by the particular geometry. The main result of proceeding in this fashion is the conclusion that quantum mechanical properties are a net result of the metric used for the underlying space itself [10–15]. 2. Geometry of the projectivized tangent bundle and the Hilbert form A mathematical presentation of the geometric setting will be given so that later the physical application can be adapted in a straightforward manner. Let M be an m-dimensional manifold. This manifold is said to be a Finsler manifold if the length s of any curve described by t ¼(()ut1 ,, um ()) t in coordinates atb is given by the integral b ⎛ du1 dum ⎞ sFuu=¼¼⎜ 1,,,m ,,⎟ dt .() 2.1 òa ⎝ dt dt ⎠ In (2.1), F is a smooth non-negative function in 2m variables and the function F (x, y) vanishes only when y=0. It is required to be symmetrically homogeneous of degree one in the y variables Fx()∣∣()11,,;¼¼=¼¼ xmmll y ,, y l Fx 11 ,,;,, x mm y y , l Î . () 2.2 A Finsler manifold M has a tangent bundle p: TM M and a cotangent bundle p**: TM M. From TM the projectivized tangent bundle of M is obtained and denoted PTM, by identifying the non-zero vectors differing from each other by a real factor. Geometrically, PTM is the space of line elements on M. Let u i, 1im be local coordinates on M. Then a non-zero tangent vector can be represented in the form ¶ XX= i ,2.3() ¶ui with the X i not all zero. The uXii, are local coordinates on TM. They are also local coordinates on the space PTM with the X i homogeneous coordinates which are determined up to a real factor. A fundamental idea is to regard PTM as the base manifold of the vector bundle pT* M which is pulled back by means of the canonical projection map pPTM: M defined by pu()()ii,. X= u i () 2.4 2 J. Phys. Commun. 3 (2019) 065006 P Bracken Since the function F (uXii, ) is homogeneous of degree one in the X i variables, Euler’s theorem for homogeneous functions implies that ¶F X i = F.2.5() ¶X i Differentiating (2.5) with respect to X j, we obtain ¶F ¶2F ¶F + X i = .2.6() ¶X j ¶¶XXij¶X j Clearly (2.6) simplifies to ¶2 F X i = 0.() 2.7 ¶¶XXij Result (2.7) implies that the first derivatives of F with respect to the X i are homogeneous functions of degree zero in the X i coordinates. Such functions are functions on PTM. Suppose (vYii, ) is another local coordinate system on PTM, then it is the case that ¶vi Y i = X j,2.8() ¶X j hence ¶F ¶F ¶v j = .2.9() ¶X ij¶Y ¶ui Now an important one-form ω can be defined as ¶F ¶F w = dui = dvi.2.10() ¶X i ¶Y i It follows that ω in (2.10) is independent of the choice of local coordinates, hence it is defined intrinsically on PTM. The form (2.10) is usually referred to as the Hilbert form. By Euler’s theorem, the arclength integral (2.1) with respect to M can be formulated as b s = w.2.11() òa The integral (2.11) is called Hilbert’s invariant integral. On exterior differentiation, the Hilbert form yields a connection with some remarkable properties.