A geometric formulation of the causal interpretation of quantum mechanics

Foundations o f Physies Letters, Vol. 6, No. 3, 1993

A G E O M E T R I C F O R M U L A T I O N O F T H E C A U S A L I N T E R P R E T A T I O N OF Q U A N T U M M E C H A N I C S

W . R. W o o d

Faculty of Natural and Applied Sciences Trinity Western University, 7600 Glover Road Langley , British Columbia, Canada V3A 6H 4

G. P a p i n i

D¢partm¢nt of Phy~ics, University of Rcgina Regina, Saskatchewan, Canada S~S OA2

Received September 23, 1992; revised February 8, 1993

The principles of the causal interpretation are embodied in a conformally invariant theory in Weyl spaee. The particle is represented by a spherieally symmetric thin-shell solution to Einstein's equations. Use of the Gauss-Mainardi-Codazzi formalism yields new insights into the issues of nonloeality, the quantum potential, and the guidance meehanism.

Key words: quantum mechanics, Weyl space, causal interpretation.

1. I N T R O D U C T I O N

The empirical evidence from a new generation of experiments (see, eg., [1]), motivated by the theoretical efforts of Einstein, Podol- sky and Rosen [2] and of Bell [3] to examine the tenets of quantum mechanics, indicates that a causal interpretation [4] is viable only if it contains an element of nonlocality. While this conclusion may, in the eyes of the sceptic, be tantamount to eondemning causal theories of the de Broglie-Bohm type, it would indeed be ironic if nonlocality were the message of nature and we insisted on ignoring it. In the authors ' opinion, it is a great inerit of the causal approach to have brought this to the lore. There ar G however, stringent limits on any manifestation of nonlocality within the context of a theory defined

207

0894-9875/93/0600-0207507.00/0 @ 1993 Plenum Publishing Corporation

208 Wood and Papini

in Minkowski space, which render nonlocality difficult to formulare mathematieally. Nonetheless, steps in this direetion have been taken [5]. The approach adopted here is to consider the issue of nonloeality within a geometrical context broader than that of Minkowski spaee. This approach is supported by the observation that, in general relativity, spaeetimes exist that eontain some nonlocal features. An interesting example is represented by anti-de Sitter spaee whose timelike geodesics can close on themselves [6,7].

The purpose of this Letter is to present a seheme whereby a bubble of anti-de Sitter spaee, for example, can be embedded in the space of Weyl such that the fundamental elements of the causal interpretation of quantum meehanics are contained within the geometric theory. The present analysis finds its correspondent in Vigier's re- eent solution of the problem of constructing solitonic solutions that are "piloted" by the solutions of the assoeiated linear Klein-Gordon and Schrödinger equations. In this view [8], an individuM particle is represented by a localized spherically symmetrie eonfiguration that is governed by a nonlinear ware equation and which is inseparable from a wave field that obeys a linear equation. The solitonic and linear ware aspects of the pargiele are related via de Broglie's guidance formula.

In a geometric approach to the eausal interpretation, ig is essential that a speeifie model of both the particle and ware aspeets of the theory be established. Ogherwise, it would be impossible to determine the precise mechanism (that is required by the Einsgein equations) by which energy and momentum are transferred from the wave phenomenon in the process of guiding the partiele. The absence of a model would also limit diseussion of the existence of nonlocal effects to mere conjecture. In the present geometric theory, a partiele is represented by a solution to the Einstein equations rather than a solitonic solution to a nonlinear wave equation. While this may appear to make the problem even more formidable, it actually becomes quige tractable if the particle is represented by a thin shell of marter. The Gauss-Mainardi-Codazzi (GMC) formalism ean then be used to facilitate the analysis of the mogion of the bubble, ineluding a geometric deseription of the guidance meehanism. In the classieal theory of general relativity, the guidanee meehanism is well known: The geometry~ which is determined by the distribution of matter, in turn, governs the motion of the marter itself. However, this classical guidance mechanism becomes insignifieant when applied to particles on the microscopic scale, where de Broglie's guidanee prineiple is required to explain quantum effects such as the ingerferenee pat tern in the two-slit experiment. Ig appears that a theory of gravitation whose domain of validity eneompasses the microscopie seale is required if the desired guidanee mechanism is to be given a geometric interpretation. A natural candidate is \¥eyFs eonformally invariant theory of gravitation and eleegromagnetism [9]. Weyl generalized

Causal Interpretation of Quantum Mechanics 209

the Riemannian geometry of general relativity by supposing that a vector parallel transported around a closed circuit would also ex- perience change in length aecording to the formula 5~ = ~t%Sz t'. The vector field ~~' together with the metric tensor gùù, which is defined modulo an equivalence class, comprise the fundamental fields of the new geometry. The choice of Weyl geometry is also strongly supported by Santamato's demonstration [10] of a "quantum force" associated with ~~, in bis stochastic theory of "geometric quantum mechanics" [11]. It should as weil be noted that London recognized long ago that Weyl's theory is susceptible to a quantum-mechanical interpretation [12]. Following a different approach, Weyl's geometry has just recently been shown [13] to support a probabilistic theory of measurement that is analogous to the path integral formulation of quantum mechanics.

2. T H E G A U S S - M A I N A R D I - C O D A Z Z I F O R M A L I S M

The GMC formalism in Weyl geometry, which is described in detail in Rel. 14, is briefly reviewed here. In Weyl geometry, one in- troduces a gauge-eovariant calculus [15] based on the gauge-covariant derivative FI and the semimetric connection F ~ , = F~~ + g~~,~~ - g~~ù - g~ùtc~, where an overbar is used to distinguish objeets from their Riemannian counterparts. The analysis of particles treated as thin shells is most naturally given in terms of the GMC formalism, where a timelike hypersurface E, which represents the history of the thin shell, divides spacetime into two four-dimensional regions (V x and VE), both of which have E as their boundary. The intrinsic metric on E is given by h#ù = g#ù - nt, n~, , where n # is well-defined as a unit spacelike (n~n ~ = 1) vector field normal to E, since the norm n , n ~ has a conformal weight of zero. The extrinsic curvature tensor in Weyl geometry is defined by/-t'#ù = - h ~ ~ n v = Kùù + h~,~,n~~~, where K~ù = - h ~ K ] ~ n ù . The development of the GMC formalism in Weyl geometry ultimately yields the equations [14]

h//~y/~//~~_~~~. = ~R~~ + f«~.~ ' / - f«~.f«ù, (1)

n h~ Rùù = D ~ K - D#K#~, (2)

and ~ % ~ h y h " S ~ ~ ~ ~ « = ~.,~~',; - zc~±c~, (3)

where Da is the operator of intrinsic, g_auße-covariant differentiation, L,~ is the Lie derivative and [( =_ I ( ~ . In applications to the

210 Wood and P a p i n i

problems involving thin shells of marter (defined in terms of a stress- energy tensor. . Tvù), it is useful to. re-express. . Eqs. (1)-(3) in terms of the Einstem tensor G»ù. The mtrmsm and normal components of the Einstein equations G#ù = Tvv are then identified ns the equations that govern the properties of the thin shell. From (1)-(3), the intrinsie and normal components of Gvù take the form [14]

1 nr, n" G~ = --~ (3R + Kt,ùI( ~'" - K 2) - Dt, t;# + 2h • ~# ~ù + 2Kn ~' ~~,

and

h~h;aù =3 a 5 + (K 5 _ hSK),n 1 «

- K Il 5 + ~h ~ (I/ùùK"" + IQ)

OL la ~ o - 2 (~<5 = I ' 5 1 < ) " % + 21, ehù ~ ,~ù

(6)

The intrinsic stress-energy tensor on E, which is defined by

S"ù - lim f r T~dn. (7) .--.o j_«

corresponds to the distributional part of T#ù. •n the Weyl-Dirae theory considered below, the existence of the surface stress-energy tensor St,ù is linked to the requirement that the theory be conformally invariant. The junction conditions for the gravitational field are obtained by integrating (4)-(6) in the normal direetion from -5 to +e across E and then taking the limit e -+ 0. One obtains the conditions

h ~ h ; S ~ = 7 ~ - h ~ 7 and n ù S ~ = O , (8)

where the jump in the extrinsic curvature is denoted by 3'~ -~ [K~], 7 = 7~, and g#ù and ~;t, are assumed to be eontinuous across E, but their normal derivatives are diseontinuous. It is also assumed that ~ù = 0 in the interior geometry V I, so that length integrability is established in the spaeetime region oeeupied by the particle. In this way, Weyl's geometrie interpretation of the exterior electromagnetic field may be reconciled with atomie standards of length without re- course to Dirac's two metric approach [16]. For the electromagnetic field, the junction conditions are given by [nùhJfùù] = h~ùj»


and [h;h•f%] = 0. Using (4) and (5), the jump in the equations nùG"~ = n~T% yields the intrinsic tensor equations

D# (h~hùZS~) + [nc~h:T%] = 0 (9)

and {KS} Sù" + [n:"T"ù] : O, (10)

where ( K ~ } denotes the average of K 5 across E. Equations (9) and (10) describe the balance of stress-energy-momentum between neighboring external fields and the thin shell. It is this balance that govern~ the dynamical behavior of the thin ~hell. Indeed, the requirement that the fields in the exterior Weyl space join at E in accordanee with the junction conditions places constraints on the motion of the bubble, sinee E represents the history of the thin shell. Within the context of the causal interpretation of quantum meehanics, it is par- ticularly significant that the interplay between the particle and wave aspects of the problem is an inherent feaßure of the present geometric formulation. In the following seetion, a particular Weyl-Dirac theory is shown to contain a sufficiently rich geometry to support descriptions of both a particle (represented by a thin shell) and its associated Madelung fluid.

3. A C O N F O R M A L L Y I N V A R I A N T T H E O R Y O F T H E W E Y L - D I R A C T Y P E

The GMC formalism is here applied to the Weyl-Dirac theory with complex scalar field/3 = pe i~, which was introduced in Ref. 15:

/ ~ = 1 #u f{-Tfùùf + I/fl~s~ + k Ir%/3r~"/~i + ),1/~1 ~

+ p< (~ùp - ~p:,ù) },/-~ d% (11)

where « = i l , k and A are real arbitrary constants, and p ,~ and fu- are real fields. Unlike Weyl's original action, (11) is linear in the scalar curvature/~ and as such is well-suited for application of the GMC formalism. The conformal invariance of the action (11) is ensured by the coupling between the scalar field 1/312 = p2 and /~ which, in turn, leads to a nonlinear stress-energy tensor for p. In (11), the Lagrange multiplier vector field 7# has been used to introduce the constraint which can be expressed as [15]

~ù = - ( l n p),ù + «:,ù (12)

212 Wood and Papi~

and leads to a flux quantization condition [15], so that length integrability is restored along certain paths. In the context of the causal interpretation . . . . of quantum mechanics, Eq. (12) is ~g~i nificant in that it estabhshes a geometmc interpretatmn for p and !P.

The constrained field equations, given here in terms of the Riemannian fields, are [15]

(13)

(14)

(16)

(17)

and

where E . . is the usual Maxwell tensor

and

(18)

Taking the trace of (14), one recovers (15), while (16) follows from the conservation of (13). The theory also contains the wave equation [17]

In fact, if one writes ~# = pe ix and defines X,v according to o~p,v -z X, . + ~ . with ~ = x / (6k - 3) /3 , the real part of (19) coincides with (15), while the imaginary part yields (16). In view of the close re- semblance of (11) to a relativistic Landau-Ginzburg action, the existence of Eq. (19) is not altogether surprising, nor is the existence of string solutions [18]. It should be noted, however, that the role of the nonlinear wave equation (19) in the present model is not to provide a solitonic representation of the particle but to establish a means by which, in the limit that the nonlinear terms are negligible, a connection with the probabilistic theory may be achieved} In the present geometric model, the particle and its associated wave (which are deemed inseparable in the causal interpretation of quantum mechanics) are shown below to be different aspects contained within the

(19)


full set of field equations (12) and (I3)-(16), ra ther than being identified with nonlinear and linear solutions of a scalar equation.

T h e M a d e l u n g F lu id . The ene rgy-momentum tensor H#ù tha t appears in the Einstein equations (14) ean be identified with the Madelung fluid that is associated with the wave equation(19) as follows. From Eq. (15) one obtains

1 1 (R + 2Ap 2) + p[:]#[:]#p (x, , + ~ù) (x'" + ~ ' ) = - ~ (20)

which, in the causal interpretat ion of quan tum mechanics, is the Hamil ton-Jacobi equation for a system of m o m e n t u m X ,# + ~~ = Mu". From (20) one finds

1 (21)

Since M is a scalar field of conformal weight - ! , the product Mul~ is a gauge-invariant quantity, and its gauge-covariant derivative with respect to the parameter s along the curve is given by

[3 [3 (Ma») = o~-3-:(¢p#, ) - o~u~E]ù(q%,) = o~u"[:]ù[Y]~,c2. ( I S

(22)

--Ol Noting that F ~~ is symmetr ic in # and u, one can derive the rela- t ionship

[:3ù[23ùp - E]ù@ù~ = ~o,,ù - cp,ù» = ¢fùù, (23)

where the last equality follows as a consequence of (12) for q0 multi- valued. Subst i tu t ing (23) into (22) and sett ing [~t,~~"T = -M2/a2, one obtains the equation of mot ion

Q --eafùùu ~ QùM. ~(Muù) = (24)

The dynamical propert ies of the Madelung fluid are usually assoeiated with the Hamilton-Jaeobi and eontinuity equations. How- ever, by limiting the dynamieal analysis to sealar ra ther than ten- sorial equations, a well-known diffieulty [8] arises in the de Broglie- Bohre theory when one a t tempts to identify the souree of the energy- m o m e n t u m tha t is responsible for the the guidance of the partiele embedded in the fluid. This problem ia reaolved in the present geo- m¢tric formulation due to the correspondence between the HamiIton- Jacobi equation and the Einstein equations. In fact, the equat ion of

214 Wood and Papini

motion (24) also follows from the Einstein equations when the con- traeted Bianchi identities are employed. Following a straightforward ealeulation, the divergenee of T#ù in (14) ean be expressed as

ù 2 ,~ 6c~ 2

P P - - - ¢ , ù ~ ù ~ , " . (25)

For a~o#, = Muv, the tensor H#ù is seen to represent a perfect (irrotational) fluid with equal pressure and energy density

1 Htù, =-6M2 (u»u,, H- -~guù) • (26)

The conservation equation that follows from (25) then becomes

u [:]ùuù+uù[-lùu +~M,~, u#u" + ~gù - :a fù ,u +-p,~,u~,u = O. P

(27) Equation (27) eontains the dynmnieal information for the fields eom- prising T#ù. Multiplying (27) by (g% + u~'u«), one obtains the equation of motion

Murrtùuù = - « ù A ù u ~ - Mù(~ù ~ + ~ ù « ) , (28)

which, when re-expressed in terms of the gauge-covariant operators, is seen to be identical to the equation of motion (24) that follows from the Hamilton-Jaeobi equation. When (27) is contracted with u ~, one reeovers the equation of continuity (16). Henee, the dynamieal eontent of the Einstein equations is equivalent to that of the ware equation (19) when H~ù takes the form of the Madelung fluid tensor (26). The eontribution due to p in Eq. (28) enters through the field M. Having thus identified the Madelung fluid part of the geometrie theory, it remains to demonstrate that the nonlinear terms in I#ù ean support a partiele model. This is aecomplished by obtaining a statie, spherieally symmetrie thin shell solution to the Einstein equations when the Madelung fluid tensor Huv is negleeted.

T h e Par t lc le . Applieation of the GMC formalism requires the determination of the interior and exterior line elements

,~,~,E=--«"~dt~,~+~""~«, 2 +,'2(g02 + sin2 0~ó~), (29)

as weil as the intrinsic stress-energy tensor Steù. In V I it is assumed that n , = 0 and that the sealar field acqmres the constant value p = po whieh breaks the interior eonformal invarianee and fixes the


scale of the particle. reduces to

The interior stress-energy tensor from (14)

T L = ~~~ù~p0 ~ , (30)

so that the interior metric is given by

1 . 2 ~ (31) e -»" = l + - ~ a p o r = e v l .

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 in- terpreted, when p(r) is present to ensure eonformal invarianee, as a manifestation of the exterior Weyl spaee with a stress-energy tensor

2p 2 Et, u + I#ù + Ag#up 2 • (32)

The exterior metric, expressed in terms of the arbitrary function p(r), is given by [14]

e -~~ = l + r 1 - p--+=+lAp2r26 (33)

and e~~ (/0p)_2 [ 1 2 m «2 ~ ]

= - - - + + Ap2r 2 (34) flr ~

where m,q and 60 are integration constants and a prime denotes differentiation with respect to r. When Tùù is substituted into (7), the surface stress-energy tensor is found to take the form [14]

h ù1~;sù~ = - 2 e h 5 , (3õ)

where a _= [nt'(ln p),#]; that is, the surface stress-energy tensor is induced when the normal derivative of In p across E is diseontinuous. The intrinsic stress-energy tensor (35) is characteristic of a domain wall of surface energy 2er. For cr > 0, the bubble is under a surface tension thafl opposes the Coulomb repulsion due to the surface charge. In ghis way, the particle finds its origin in the field p that (i) fixes the scale in V z, (ii) ensures eonformal invariance in V E, and iii) induces the smffaee tension needed for stability, a gension thag is equently. introdueed by hand in the literagure. For the static bub-

ble, one can show from (9) ghag « = constant, and from (10) that the normal forces acting on the respective sides of E are not equal.

216 Woodand Papini

To determine the dynamical properties of a spherical shell, one usually considers the intrinsic metric

d~ 2 = - d r 2 + R2(~-)(dO 2 + sin 2 0d¢2), (36)

where the equation r = R(T) describes the motion of the shell in the comoving frame of the particle. In general relativity, a charged, spherically symmetric shell has an exterior geometry given by the static Reissner-Nordström metric, even when the bubble is pulsating [20]. In the particle model presented in [14] it turns out that, because « is not an arbitrary input parameter, the only exact solution corresponding to the static metric (33) and (34) is the equilib- r ium solution R~q = q2/(8porn), although an approximate solution with small radial oscillations also exists [14]. The generalization of Birkhoff's theorem fails in the present model, because the scalar field p establishes a link between the thin shell and the exterior metric; a pulsating bubble with a scalar-field-induced surface stress-energy tensor will affect the exterior eonformally invariant metric. This type of intimate relationship between the particle and its surrounding geometry is anticipated in the causal interpretation. In the static, spherically symmetric case considered, where p(r) is an arbitrary function in V E and the Madelung fluid is neglected, the divergence of T~v vanishes identically and the tangential motion remains inde- terminate. However, if external influences such as other particles were present, the resulting boundary conditions imposed on p could influence the particle in the manner required in the causal interpretation. Also, a complete dynamical description requires that the particle and Madelung fluid be combined into a single picture.

T h e Causa l I n t e r p r e t a t i o n . An important element of the causal interpretation of quantum mechanics is the manner in which the partiele is viewed as being inseparable from the Madelung fluid. The task of uniting the particle and wave patts is easily accom- plished in the present geometrie model. Taking ~,ù = 0 in V I and h2~0,v discontinuous across E allows us to embed the bubble in the Madelung ituid in accordance with (9) and (10), while the sùrface stress-energy tensor (35) remains unchanged. For nt'~, t, = 0, the normal component of the exterior fluid momentum at E takes the form nùH% = -3M2nù . From this it follows that n u h ~ H % = 0 and n~n"H% = - 3 M 2. As a consequence, the Madelung fluid tensor H#ù does not contribute to (9), the time component of which, in the rest frame of the thin shell, governs the transfer of energy between the particle and its neighboring fields [21]. While the particle does not draw energy from H#ù as one might expect [8], it can be shown that, if [p,,] varies on E, then Iuù will transfer energy to the thin shell such that « ~ constant. Although the Madelung fluid


doesn't serve as an energy source for the particle, it does influence the motion of the thin shell through (10), which represents Newton's second law [21]. In this manner, the bubble acquires a new dynamieal nature as it is guided in its motion by the fluid in V E. The realization of this guidance proeess ean be seen by eonsidering the dynamieal behavior of a fluid element at a point P on the exterior surface of the bubble, where the four-veloeity of the fluid element is denoted @(P) = dzt'(P)/dsE. By construetion, the metric tensor

is eontinuous aeross E at P and consequently, ds~E(P) = ds~(P). Hence, at any point P on E, the four-veloeity of the thin shell is given by

u~(P) = dzù P dzù = u } ( p ) = ~ ~ , ù ( p ) , ~S E = d8 E p

(37)

which is recognized as the guidance formula advanced by de Broglie. In the present approach, the guidance condition can be extended beyond being valid only at a given point by noting that the motion of the fluid along its worldline from P to a subsequent point P ' , at whieh the fluid element and the bubble are still in contact, can be viewed as the result of a conformal transformation induced by the faetor [22]

«2 = 1 - M2 ([~]M) -2 ([~3uv) 2 (38)

The metric tensor at P~ in V E can therefore be obtMned from its corresponding value at P by applying a conformal transformation with the same factor ~2 and, by continuity, the intrinsic metric at P ' is also determined. The resulting identity, hu~(P' ) = ~2ht, u(P), leads to the conclusion that the bubble and fluid taust move in step. CõnsequentIy, the tIamiIton-Jacobi equation (2) rnay be applied to the particle itself, as required in the causal interpretation of quantum meehanics.

An outstanding deficiency in the causal interpretation of quan- turn mechanics has been the lack of a condition that requires the de Broglie "variable mass" M to be real, so that unphysical solutions of the wave equation do not arise. It has been shown [19] that the reality of M is ensured if, on some initial spacelike hypersurface, the motion is timelike. The inability of previous formulations to guar- antee that this condition is satisfied sterns from the fact that, when the causal interpretation is based soMy on a scalar equation, the for- mal introduction of de Broglie's guidance formula does not exclude the possibility that the initial four-momentum be spacelike. The latter may be required to be timelike, but this places restrictions on allowed solutions to the ware equation that one cannot justify

218 Wood and Papini

independently. In the geometric model presented above, the condition that p,~ be timelike is supported by the Maxwell equation (13),

• c = 4ea2p29L# oc uù. Hence, the restriction to physically since 3t, meaningful solutions of the wave equation emerges as a consequence of self-consistency of the complete set of field equations in the present

eometric model. It is interesting to note that the guidance formula 7) requires M / a to be real. As it is this combination that also

occurs in the equation of motion (24), it is evident that M / a is a more appropriate definition of the "variable mass" in this model.

Another important element in the causal interpretation is the quantum potential. The terms in the brackets of (21) provide a generalization of the quantmn potential in Minkowski spacetime that has been identified as the vehicle by which quantmn effects arise in the causal interpretation. Holland and Kyprianidis have argued [23] that "the universal sufficient condition for the classical limit of quantum mechanical motions is the vanishing of the quantum potential" rather than the usual limit h ~ O/n -+ c~. A sußcient condition for the quantum potential to vanish is p --- constant. Under this condition, the conformally invariant theory (11) essentially tal<es the form of the classical Einstein-Maxwell theory. For R ~ 0 in V E, the de Broglie mass (21) then reduces to a real constant when A > 0. For A E = A I, this suggests taking the interior space to be anti-de Sitter. This particular choice for the interior space will be of significance in the discussion of nonlocal effects in the following section. The implications that the generalized quantum potential in (21) has for the nonlocal character ascribed to the quantum potential in the de Broglie-Bohm interpretation will also be considered below.

4. T H E I S S U E OF N O N L O C A L I T Y

Perhaps the most compelling feature of the bubble model presented above is the manner in which the interior space V I is made distinct from the exterior space V E. This property not only makes it possible to break the conformal invariance in the interior space, whereby standards of length can be introduced into the theory while Weyl's geometric interpretation of the exterior electromagnetic field is preserved, but the locality rsquirernents of the exterior space need not be irnposed in the interior space. The suggestion advanced here is that the curved interior space, which is that of anti-de Sitter for A > 0 in the present model, is responsible for the observed nonlocal effects.

The properties of anti-de Sitter space have made it popular in theories of confinement [24] and supergravity [25]. One such property that follows from consideration of radial null geodesics in anti-de Sitter space is that "information" can propagate an infinite distance


in a finite time [26]. Also, anti-de Sitter spaee permits the existence of closed timelike curves (CTC's). The traditional approach has been to avoid dealing with such curves that "would seem to lead to the possibility of local paradoxes" ([7], p. 189) by working with the universal covering space of anti-de Sitter space which does not contain any CTC's. More recently, however, the question whether the laws of physics actually forbid particles (of observers) from travelling along CTC's has been approached with a new open-mindedness in studies dealing with wormholes [27], cosmic strings [28], and the "quantum theory of computation" [29]. In contrast to these studies, the anti-de Sitter space that permits CTC's in the present model exists only in the spacetime that comprises the interior of the world tubes of extended elementary particles and consequently cannot be associated with a "time machine" in the usual sense. Rather than being the source of counterintuitive effects, CTC's are viewed hefe as providing a means by which certain effects that appear to be counterintuitive, at least classically, may find an explanation. In particular, it is argued that the nonlocal influences observed in experiments employing correlated particles may simply be a consequence of the fact that, while the world tubes of the correlated particles diverge after the disintegration process, they actually share a common anti-de Sitter geometry (Fig. 1). The anti-de Sitter spacetime in the interior region of the world tubes makes it possible for the two spatially separated particles to be linked by a CTC as follows. When a measurement is performed on one of the particles, a CTC connecting the measurement event P1 and the event of separation P2 would permit the correlated particle at Pa to assume its appropriate state in a manner that would appear, to an external observer, to be nonlocal. In this way then, the nonlocal effects may be explained without deny- ing the objective reality of elementary particles or comproraising the principles of relativity (in curved spacetime). The presence of CTC's does not lead to any logieal paradoxes in the present context; in fact, the notion of "free will" is necessarily violated when the particles are correlated. An interesting possibility that remains to be investigated is that a strut-like link [30] (or, equivalently, a string of negative energy density [31]) may form between two anti-de Sitter bubbles (with T{0 < 0) that are involved in a disintegration process.

5. C O N C L U D I N G R E M A R K S

Out suggestion that the separation of V r and V E plays an essential role in seeking to understand the intriguing nonlocal EPR- type correlations does not rêquire any nonlocal effects to occur in the exterior Weyl space. This situation is clearly not in keeping with the idea advanced in the de Broglie-Bohm theory that it is the quantum

220 Wood and Papini

I I I I I I I I

Fig. 1. The world tubes of correlated particles after a disintegration proeess with a CTC connecting the events P1, P2, and P3.

potential (that exists in V E in the present model) that is responsible for nonlocal phenomena. Bohm et al. [32] have argued that the invariance of the quantum potential under a scaling of ~b(x) by an arbitrary constant plays a fundamental role in the nonloeal nature of the theory. However, the significance of this invarianee property is brought into perspective when it is eompared to the global phase invariance of pre-gauge field theories that also "contradiets the ler- ter and spirit of relativity" [33] and as a eonsequence is replaced by local phase invariance. In the present geometric model, the generalized quantum potential in (21) is invariant under the eonformal transformation

~,ù = a2gt,ù, /5 = c - i p (39)

for the arbitrary func t ion o 2 (x) > 0. The generalization to invariance under the local scaling (39) indieates that the information regard- ing the partiele's environment contained in the quantum potential is propagated via the tensor field I#~ in a local manner. The suggestion


advanced here that the external guiding mechanism is local in nature does not contradict any experimental evidenee of nonlocal phenomena, since these results have only been obtained in experiments that involve eorrelated particles.

Although the geomëtric formulation of the causal theory presented above is fairly rudimentary, it nevertheless demonstrates that it is possible to inject the principles of the causal interpretation of quantum mechanics into a geometric theory in Weyl space. In the authors' opinion, this is an essential step towards obtaining a sat- isfactory causal theory of quantum phenomena. Besides the diffi- culty associated with nonlocal effeets in Minkowski space discussed above, the guidance mechanism, which is central to the causal approach, appears difficult to realise in a theory based on a scalar equation describing solitonie particles in Minkowski spaee. I-Iowever, by formulating the problem within the context of a theory of gravi- ration, whereby the description of the transfer of energy-momentum becomes an inherent feature, it becomes possible to demonstrate that the guidance principle is dictated by physics rather than by the physieist. Possible refinements to the particle model adopted in the present work are desirable in order that the issues assoeiated with a causal interpretation of quantum mechanics may be better addressed. Such refinements are currently under investigation.

A C K N O W L E D G E M E N T S

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

R E F E R E N C E S

1. A. Aspect, J. Dalibard and G. Roger Phys. Rer. Lett. 49 (1982) 1804.

2. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rer. 47 (1935) 777.

3. J. S. Bell, Phy~ics 1 (1964) 195. 4. L. de Broglie, NonIinear Ware Mechanics (Elsevier, Amsterdam,

1960). 5. A. Kyprianidis and J. P. Vigier, in Quantum Mechanics Versus

Local Realisrn, F. Selleri, ed. (Plenum, New York, 1988); D. Bohre and B. J. Hiley, Phys. Rep. 172 (1989) 93.

6. J. L. Synge, Relativity: The General Theory (North-Holland, Am- sterdam, 1960).

7. S. W. Hawldng and G. F. R. Ellis, The large-~cale strncture of space-tirne (Cambridge University Press, Cambridge, 1973).

8. J. P. Vigier, Found. Phys. 21 (1991) 125.

222 Wood and Papin|

9. H. Weyl, Sitzungsber. Preuss. Akad. Wies. (1918) 465. 10. E. Santamato, Phys. Rer. D 29 (1984) 216; J. Math. Phy~. 25

(1984) 2«7. 11. E. Santamato, Phys. Rer. D 32 (1985) 2615; Phys. Lett. A 130

(1988) 199. For a detailed discussion of Santamato's approach see: C. Castro, Found. Phys. 22 (1992) 569.

12. F. London, Z. Phys. 42 (1927) 375. 13. J. T. Wheeler, Phys. Rer. D 41 (1990) 431.

a " ~' 14. W. R. Wood and G. P pim, Breakmg Weyl invariance in the interior of a bubble," Phys. Rer. D 45 (1992) 3617.

15. D. Gregorash and G. Papini, Nuovo Cimentõ B 63 (1981) 487; G. Papini, in High-Energy Physics, S. L. Mintz and A. Perlmutter, eds. (Plenum, New York, 1985), p. 179.

16. P. A. M. Dirac, Roe. R. Soc. London Ä 333 (1973) 403. 17. G. Papini, in Proceedings of the Fiflh Grossmann Meeting, D. G.

Blair and M. J. Buckingham, eds. (World Scientific, Singapore, 1989), p. 787.

18. G. Papini, Phyo. Lett. A 107 (1985) 26. 19. A. Kyprianidis, Phys. Lett. 111 (1985) 111; C. Dewdney, P. R.

Holland, A. Kyprianidis, and J. P. Vigier, included note in W. Mückenheim, Phys. Rep. 133 (1986) 337.

20. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).

21. C. Barrabès and W. Israel, Phya. Rer. D 43 (1991) 1129. 22. G. Papini and W. R. Wood, "Quantum mechanics and bubbles in

Weyl spacetime," in Proceedings on the Conference on Structure: from Physica to General Systems, G. Scarpetta, ed. (1993, in press).

23. P. R. Holland and A. Kyprianidis, Arm. Inst. Henri Poincaré 49 (1988) 325.

24. E. van Beveren, C. Dullemond, and T. A. Rijken, Phys. Rer. D 30 (1984) 1103; A. Aurilia, R. S. Kissack, R. Mann, and E. Spallucci, Phys. Rer. D 35 (1987) 2961.

25. M. J. Duft, B. E. W. Nilsson, and C. N. Pope, in Relativity, Cosmology, Topological Mass and Supergravity, C. Aragone, ed. (World Scientific, Singapore, 1984), p. 57.

26. P. Breitenlohner and D. Z. Freedman, Arm. Phys. (New York) 144 (1982) 249.

27. M. S. Morris, K. S. Thorne and U. Yurtsever, Phys. Rev. Lett. 61 (1988) 1446; J. Friedman, M. S. Morris, I. D. Novikov, F. Echeverria, G. Klinkhammer, K. S. Thorne, and U. Yurtsever, Phys. Rer. D 42 (1990) 1915.

28. J. R. Gott, III, Phys. Rer. Lett. 66 (1991) 1126. 29. D. Deutsch, Phys. Rer. D 44 (1991) 3197. 30. R. Bach and H. We yl, Math. Z. 13 (1992) 134; F. I. Cooperstock,

Gen. Rel. Grav. 6 (1975) 91.

Causai Interpretation of Quantum Mechanics 223

31. M. Aryal, L. H. Ford, and A. Vilenkin, Phy~. Rer. D 34 (1986) 2263.

32. D. Bohre, B. J. Hiley, and P. N. Kaloyerou, Phys. Rep. 144 (1987) 321.

33. L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1985).

NOTE

1. The issue of negative probabilities associated with second-order ware equations in the causal interpretation is discussed in Ref. 19.

A geometric formulation of the causal interpretation of quantum mechanics

Documents

Transcript of A geometric formulation of the causal interpretation of quantum mechanics