12June 1995

Physics Letters A 202 (1995) 46-49

PHYSICS LETTERS A

A geometric solution to the de Broglie variable mass problem

G. Papini a***‘, W.R. Wood b*2 a Department of Physics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2

b Faculty of Natural and Applied Sciences, Trinity Western University 7600 Glover Road, Lmgley, British Columbia, Canada V3A 6H4

Received 13 March 1995; revised manuscript received 19 April 1995; accepted for publication 26 April 1995

Communicated by P.R. Holland

Abstract

It is shown that in the geometric formulation of the causal interpretation of quantum mechanics given by the authors, the de Broglie variable mass M is real and the Klein-Gordon current timelike.

An outstanding issue in the de Broglie-Vigiercausal

interpretation of quantum mechanics [ l] has been the problem associated with the reality of the vari- able mass M. Within the context of second-order wave equations, this problem manifests itself in the mathe-

matical existence of negative probability densities and negative energy solutions. When the usual probabilis- tic interpretation is applied, these solutions cannot, in general, be given a physically meaningful interpreta- tion. In contrast, when a particle follows a timelike

causal trajectory, the situation changes radically. In this case, positive energy solutions are necessarily cor- related with positive values of M and positive prob-

ability densities and the sign of the energy remains

fixed along the trajectory [ 21. However, general solu-

tions of the Klein-Gordon equation do not ensure the reality of M.

A solution to the variable mass problem is contained within a recent geometric formulation of the causal in: terpretation of quantum mechanics [ 31. In that formu- lation the Gauss-Mainardi-Codazzi (GMC) formal-

* Corresponding author. ’ E-mail: papini@pico.phys.uregina.ca.

*E-mail: wrwood@twu.ca.

ism was applied to the Weyl-Dirac theory [4] with

the complex scalar field p = peiP and the constraint

relation

~~ = -(lnp),, + E’P,~, (1)

where E = f 1 and p, (p and K~ are real fields. The field

ip is assumed to be multivalued to ensure a nonvanish-

ing fpv = Kv,/.L - Kw,v and leads to a flux quantization

condition [ 41, so that length integrability is restored along certain paths.

The field equations, given here in terms of the Rie-

mannian fields, are [4]

iJvf~V = 4(k - 3e)pZ4P.” E jr, (2)

G PLY = LE +Ip,+~hg,,p2+Hp, -T&, Q2 fiu (3)

3(Ek-3)9&+ = -@+2Ap2)+~0,O’p (4) P

and

(k - 3.5) 0, (p+L+“) = 0, (5)

where k and h are real arbitrary constants, Epy is the usual Maxwell tensor, and IPy and H,, are given in

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G. Papini, W.R. Wood/ Physics Letters A 202 (1995) 46-49 47

Ref. [3], Taking the trace of (3) one recovers (4), while (5) follows from the conservation equation as-

sociated with (2). The theory also contains the wave equation [ 51

(OA+iKA)(O~+iK~)~-_$AI~J2~-~R~ =O. (6)

In fact, if one writes I++ = p eiX and defines x+ accord- ing to ‘YP,~ z X+ + K~ with a2 = f (ck - 3), then

the imaginary part of (6) yields (5), while the real

part coincides with (4) which can be expressed as

(X,p+~p)(X~p+KCL) = -i(R+2hp2) +$O,Wp.

(7)

Since 9, x, and K~ are real fields, (Y must be a real

constant. In the causal interpretation of quantum me-

chanics, Eq. (7) is identified as the Hamilton-Jacobi equation for a system of momentum x+ + K~ = Mup,

so that

Mu, = “p,P,

From (7) one finds

(8)

M2 = $p2+ ( $- $lpo’p > ,

which is the square of the de Broglie mass in the

present model. Using the GMC formalism, the field equations (2)

and (3) have been shown [3] to support a charged

timelike thin-shell solution (see the Appendix) that provides (i) a geometric model with conformal in-

variance broken in the interior space; (ii) a new pos- sibility to consider nonlocal effects when the interior

curved spacetime can have acausal properties such as

closed timelike curves [ 31; (iii) a transfer mechanism

for energy-momentum between the thin shell and the Madelung fluid; (iv) a geometric guidance condition for the bubble at the microscopic scale; and (v) a Hamilton-Jacobi equation (7) that can be applied di- rectly to the thin shell guaranteeing that the bubble will move in step with the Madelung fluid. By basing

the theory on the field equations (2) - ( 5) (from which the wave equation (6) is then identified), one is not bound to demonstrate that all possible generic solu- tions to the wave equation must be physically mean- ingful as is the case when the causal interpretation is

based solely on a wave equation. It is the field equa- tions (2)-( 5) that determine the set of physically ac-

ceptable solutions in the geometric approach. The outstanding deficiency noted above regarding

the reality of M is overcome in the present geometric

model due to the existence of the timelike thin shell

solution to Einstein’s equations which can be embed- ded in the Madelung fluid according to the junction

conditions discussed in Ref. [ 31. While spacelike and

timelike directions are distinguished in any relativis-

tic theory, a geometric theory permits these directions

in spacetime to be related to the motion of matter via Einstein’s field equations. That is, for a given folia-

tion of spacetime, the GMC formalism requires the

various timelike and spacelike components of G,, to be equated to the corresponding components of Tpy . In this way, a link is established between the proper-

ties of spacetime and matter that allows one to address the issue of whether or not a given four-vector that is associated with matter is timelike. It is due to the ab-

sence of this geometric structure that the possibility of spacelike four-momenta in de Broglie’s guidance

formula Pp = Mu, cannot be excluded in previous formulations of the causal interpretation derived from a scalar wave equation. The suggestion advanced here

is that a resolution to the problem of the reality of

M should be sought within the context of a geomet-

ric theory. In the particular model developed in Ref. [ 31, the timelike nature of Mu, can be demonstrated

as follows. The constraint ( 1) can be written in the gauge K: =

K/A + (lnp)+ so that K: = EP,~. This is permissible

due to the gauge covariance of the theory and this par-

ticular gauge is viable since p must be greater than one in the model in order for the thin shell to be un-

der a surface tension that balances the Coulomb re-

pulsion [ 61. For the static solution obtained in Ref.

[ 61, K:K~’ < 0 and as a consequence (D,~ is timelike. It then follows from (8) that M must be real since (Y is real. This result, obtained in the static case, also holds true in a frame comoving with the thin shell, and is therefore quite general. Indeed, due to the co- variant nature of M under conformal transformations

that preserve the sign of the line element, M2 must be positive in general.

The condition for ‘p+ to be timelike can also be expressed within the context of the theorem of Frobe- nius. In terms of differential forms, Eq. (8) is given

48 G. Papini, W.R. Wood/Physics Letters A 202 (1995) 46-49

by Appendix

(10)

The condition for UP to be orthogonal to hypersurfaces of constant pp. and hence for ~p,~ to be timelike, is given by du A u = 0. Recognizing that, due to the multivalued nature of p, dp is not closed even though it is an exact one-form, the condition for timelike (P,~ becomes

In the geometric model presented in Ref. [ 31, the tensor HPy that appears in (3) is identified with the Madelung fluid in the causal interpretation. As such, this fluid tensor is viewed as the mediator of external influences on a particle and as a consequence is ini- tially neglected in the determination of the spherically symmetric solution to the Einstein equations that pro- vides a particle model.

d24pAdp=0. (11)

Eq. ( 11) is satisfied in the static case considered above, where d24p N dx” A dx’ and dqp N dx”.

Application of the GMC formalism requires the de- termination of the interior and exterior line elements

For timelike p+, it follows that the Maxwell cur- rent jp in (2) is also timelike without having to im- pose this as an auxiliary condition. In the present the- ory, the Maxwell current is proportional to the Klein- Gordon current [ 51 associated with the wave equa- tion, j& = cup*@, and is therefore also timelike and as such does not suffer from the difficulties normally associated with the current for a second order wave equation. It should be noted, however, that + in the present theory is a physical field and not immediately identifiable with a probabilistic wave function. In ad- dition, the time component of the current can be made positive by choosing the positive sign of the radical in the definition of LX. It then follows that positive (nega- tive) energy particles will correspond to positive (neg- ative) values of M and positive (negative) values of j[o. In this regard, it is interesting to observe that the equation of motion

ds:,, = - evlE dt:,, + ePLI.E dr*

+ r*(d0* + sin*ed+*), (13)

as well as the intrinsic stress-energy tensor S,, on the timelike hypersurface S. In the interior space it is assumed that K~ = 0 and that the scalar field acquires the constant value p = po which breaks the interior conformal invariance and fixes the scale of the particle. The interior stress-energy tensor from (3) reduces to

so that the interior metric is given by

-

e-pi = 1 + ihp$* = eQ . (15)

The interior space is then de Sitter (A < 0), Minkowski (A = 0) or anti-de Sitter (A > 0). The Coulomb potential of the charged shell gives rise to an exterior electromagnetic field that can be interpreted, when p(r) is present to ensure conformal invariance, as a manifestation of the exterior Weyl space with a stress-energy tensor

$(Mu,) = -&(Yfp,UV- 0, M (12)

is invariant under C and T transformations as dis- cussed by Dirac [ 71. In addition, Eq. ( 12) is invari- ant under M --+ -M together with time reversal. This indicates that, in the present theory, negative energy particles may be interpreted as positive energy parti- cles moving backward in time.

In summary, the problem of the de Broglie variable mass can be resolved when it is addressed within the context of a particular geometric theory which pro- vides the required distinction between timelike and spacelike motions.

(14)

T;“=LE +I,,+$Ag,,p*. 2p* w

(16)

The exterior metric, expressed in terms of the arbitrary function p(r) , is given by [ 61

and

e*=(&p)-* I-$+- (

4* 4p*r*

(18)

G. Papini, W.R. Wood/Physics Letters A 202 (1995) 46-49

where m, q and & are integration constants and a prime denotes differentiation with respect to r. The surface stress-energy tensor is found to take the form [ 61

h” h “9 = -2ah” PLP v B’ (19)

where o = [nJ‘( In p),,] ; that is, the surface stress- energy tensor is induced when the normal derivative of In p across Z is discontinuous. The intrinsic stress- energy tensor (19) is characteristic of a domain wall of surface energy 2u, where h,, is the intrinsic metric on X. For L+ > 0, the bubble is under a surface ten- sion that opposes the Coulomb repulsion due to the surface charge. In this way, the particle finds its ori- gin in the field p that (i) fixes the scale in V’, (ii) ensures conformal invariance in VE, and (iii) induces the surface tension needed for stability, a tension that is frequently introduced by hand in the literature.

This work was supported in part by the Natural Sci- ences and Engineering Research Council of Canada. One of the authors (G.P.) wishes to thank Dr. K. Den- ford, Dean of Science, University of Regina, for con- tinued research support.

49

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[51

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