berry's phase in pilot-wave theory

Foundations of Physics Letters, Vol. 11, No. 2, 1998

BERRY'S PHASE IN PILOT-WAVE THEORY

Erik Sjoqvista and Henrik Carlsenb

aSub-Faculty of Philosophy, Oxford University,10 Merton Street, Oxford 0X1 4JJ, United [email protected]

bDepartment of Quantum Chemistry, Uppsala UniversityBox 518, S-751 20 Uppsala, [email protected]

Received 17 April 1997; revised 11 November 1997

The projective-geometric derivation of Berry's adiabatic geometricphase for a single trajectory in the de Broglie-Bohm pilot-wave the-ory is given. The relation between this phase and the first ordernature of pilot-wave theory is discussed. It is shown, in the casewhere the electromagnetic vector potential is a gradient, that thephase can be given in a locally phase invariant form. The appear-ance of the Berry connection is justified and its physical significancein pilot-wave theory is discussed.

Key words: pilot-wave theory, Berry's phase, adiabatic dynamics.

1. INTRODUCTION

The de Broglie-Bohm pilot-wave formulation of non-relativistic quan-tum theory [1-7] is based on the simultaneous existence at each in-stant of time of a configuration X = (X1,...,XN) and a complex-valued Schrodinger wave ij)(x) = ^(x1, ...,xN)> where N is the num-ber of particles in the system and we use 'bold-face' to designatethree-vectors. The theory is fundamentally a first order theory onconfiguration space [4,8,9]. This can be seen by considering the guid-ance relations

where S/h is the phase of the wave function, m* the mass of particle

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0894-9875/98/0400-0179$15.00/0 C 1998 Plenum Publishing Corporation

180 Sjoqvist and Carlsen

k, and k = 1,...,N; or, in the case of particles with charges qkmoving under the influence of an external classical vector potentialA = A(x,<),

2. BERRY'S PHASE FOR A SINGLE TRAJECTORY

Given a causal quantum theory with multi-particle trajectories D(t)(D(t) can, e.g., be in configuration space [1-7] momentum space [11]or phase space [12]), a geometric phase for an arbitrary ensemblehas recently been derived by Garcia de Polavieja [13] as follows.Suppose the ensemble is described by the density Q = g(d, t) andcr/H = ff(d, t)/H is the phase of the Schrodinger wave 0 expressedin the d—representation, the geometric phase for the complex-valuedfield g1/2eir/h can be derived from the Aitchison-Wanelik-Mukunda-Simon formula [14,15] yielding [13]

These equations entail that the velocity of the particle k is deter-mined by the 'force' fit = VXltS (or in the electromagnetic casefk = VXkS — gjfeA); a fact displaying a first order theory of dynamics.

A further fundamental property of pilot-wave theory is thephase covariance of the guidance relations. It can be seen thatboth (1) and (2) are unchanged under any global (x—independent)phase transformation and that (2) is locally phase covariant by de-manding A(xk,t) —> A(xk,t) + VXka(x,t)/qk if V* undergoes thex—dependent phase transformation i(>(x,t) —> eia(x,t)/ht^>(x,t). Thisguarantees that the trajectory X(t) is insensitive to the phase choiceof the guiding wave ip.

The intention of this report is to discuss Berry's adiabatic ge-ometric phase [10] in order to shed new light on the above propertiesof pilot-wave theory. The discussion is divided into two parts. Inthe first Berry's phase for a single trajectory of a quantum systemcoupled to classical degrees of freedom, the latter acting as a slowenvironment driving the former, is considered (Sec. 2). The sec-ond discusses the effect caused by Berry's phase on the pilot-wavedynamics of quantum-mechanical environment degrees of freedom(Sec. 3).

Pilot-Wave Theory 181

where {•) denotes integration over d and r is the final time. Promnow on we assume that Q fulfills the continuity equation

associated with the velocity field D(d,t). The explicit derivation of(3) can be found in the appendix. The geometric phase 79[^,<r] isreparametrization invariant, i.e., invariant under t —> t, where iis a monotonic function of t. It is also projective-geometric, i.e.,79[»7£,er + £] = yj[0,o], where £ and T/ are real d—independentfunctions of t. This is a generalization to an arbitrary ensembleof the global phase invariance of the conventional geometric phase7JIV-M [10,16,17]. _

Consider now, in the case of pilot-wave theory (where d = x),the wave function il>(x,t) initially being the energy eigenfunction(f>o(x) and forced to adiabatically follow some external classical pa-rameters Y(t). Thus 0 can be written as

where <p(x,Y(t)) = \tp(x,Y(t))\elt(*'YW>/h denotes a continuous fam-ily of instantaneous non-degenerate energy eigenfunctions with <p(x, Y(0)) = <po(x) and ft is determined by the Schrodinger equation [10].Furthermore assume if) is taken round a circuit Cy in parameter spaceduring the time interval |0,T]. If the trajectory X(t) arising fromthe wave function 0(a:,r ), with a fixed Y along Cy, constitute aclosed loop Cx of period T = r(Y) (i.e., X(t + T) = X(t)), then us-ing (3) we derive the geometric phase associated with Cx by taking1

Q = s ( x - X(t)) and a = S(x,t):

1 That 6(x — X(t)) is in fact a solution of the continuity equationalong Cx follows directly from the identity df/dt = —X . Vxf, validfor any function of the form f = f(x — X(t)).


where we have used ff(t+r) — g(t). Note that 7o[£, S] is only definedfor closed loops in configuration space since 6—functions centered atdifferent points have strictly zero overlap.

Now, given T ^C T (^> varies slowly compared to the changein the configuration), the factual evolution X(t) will approximatelycoincide with Cx- Under this condition (6) defines Berry's adiabaticgeometric phase associated with Cx, where the pilot-wave dynamicsfulfilling T <C T is said to be adiabatic [18]. In the stationary case(T infinite) the single-valuedness of ^> implies yg[£, S] = n27r, wheren is an integer being non-zero only when there is at least one wavefunction node inside Cx-

Note that the T .< T condition differs from the usual adia-baticity condition [19], which is based on the variation of the wavefunction being slow compared to a time-scale determined by the in-verse of the separation of energy levels [20,21]. That this differenceactually leads to different wording regarding a given physical situa-tion can be illustrated by considering a real-valued adiabatic wavefunction. In this situation it follows that the particle is dragged bythe wave function so that typically r ~ T and the above adiabaticityrequirement (T -C T) is violated.

We would also like to point out that a derivation of the ge-ometric phase for an arbitrary closed single trajectory has recentlybeen given by Parmenter and Valentine £22]. The present analysisclarifies the geometric significance of their result in the importantcase of adiabatic pilot-wave dynamics, viz. the projective-geometricproperty of the phase and its uniqueness with respect to the conven-tional geometric phase.2

Let us now analyze the physical meaning of fg[8,5] in thelight of the first order character of pilot-wave theory. Combining (1)and (6), we can write

or, similarly, in the electromagnetic case, using (2)

where we in analogy with classical electrodynamics have defined thefirst order 'Lorentz force' /L(X,<) = — (giA(xi,i),...,gjvA(xw,t)).We can therefore physically justify the definition of first order 'work'

2 We are grateful to Gonzalo Garcia de Polavieja for pointing thisout to us.

3 Note that the magnetic contribution A to the dynamics actuallydo work on the particle in pilot-wave theory. This is not the case inclassical electrodynamics in which the magnetic part of the Lorentzforce, i.e., qv x B, B being the classical magnetic field, is always per-pendicular to the particle motion leading to vanishing contributionto the mechanical work.

It can be verified that (10) is projective-geometric and reparametriza-tion invariant in the above sense. As an illustration consider a parti-cle with charge q moving on a unit circle threaded by an Aharonov-Bohm solenoid of strength $/(2ir). The first order Lorentz force inthis case can be written as fL, = — g$/(27r)Vx0. Taking <f>(0) =

where S should fulfill

That is yg[£, 5] is not invariant under local phase transformationsand therefore (6) is not satisfactory in the electromagnetic case.This is perhaps not surprising since the conventional geometric phasey b [ | V | 2 , s ] , > which like Yg[£, S] is derived from (3), fails to be invariantunder local phase transformations [16,23]. However we would like toemphasize that the reason for the non-invariance is different in thetwo cases: 7f l[|V|2> S] is non-invariant because the path in projectiveHilbert space depends on the local value of the phase, while 7S[£, 5]is non-invariant due to the subtraction of the first order Lorentz forcefl from the total force in (8).

In the case where //, is a gradient of a scalar function we candefine a locally phase invariant Berry phase for Cx as

(having the dimension of action instead of energy) on the particleassociated with the non-electromagnetic force3 by its proportionalityto Berry's phase for a single trajectory.

Consider next the local phase transformation V* (x, t) —*e t a(x,t)/hil>(x,t), where a can be any differentiable function of x andt. Clearly, as was pointed out in the introduction, Cx is invari-ant by requiring the vector potential to undergo the transformationA(xfc,<) —> A(Xfc,t) + VXfca(a:,t)/<7jf On the other hand, we have



eike/i/2ir, where 6 is the polar angle of the particle and k is aninteger, we obtain jg[S,S] = 2nk and ig[6,S] = 2irk — g$. Thetransformation <p(0) —> eiie(f>(6) leads to 7, [£,5] —> 7,[tf,S] + 2irlwhile %[£, 5] is invariant since q$/(2n) —>. g$/(2?r) + /.

3. THE BERRY CONNECTION

In the absence of external interactions the Hamiltonian for a quan-tum system, which we take to be dividable into a light and a heavypart, can be written as

where (p is an y—differentiate eigenfunction of h, being normalizedfor all y with respect to integration over x. The Schrodinger equationassociated with H entails that (13) is a good approximation whenever\ ( f ' ( y } \ ^ y h \ f ( y } } | < k(y) -£ '(j/)l> where 9?' is any eigenfunction ofh different from y, e and e' are the corresponding non-degenerateenergy eigenvalues and (.} denotes integration over x. It is the pur-pose here to analyze the pilot-wave dynamics of the y—system arisingfrom such a wave function and show that this dynamics is related tothe conventional Berry phase.

We start by writing x(y>0 = |xG/i*)lelS^y>t^ft an<^ (f(x^y) =

|^(ar, y)|e"(x,y)/h yielding the guidance relation of the heavy system

It is now possible to give a plausibility argument for replacing (14),which in fact requires solving the guidance relation for the light sys-tem, by a simpler equation involving an averaged first order forcewhich only depends on y.

where h = h(Vx,x,y) is the Hamiltonian for the light system (a;)and M is the mass of the heavy system (y) measured relative to themass of the light system. A paradigmatic example of such a systemis a molecule where the electrons and the nuclei constitute the lightand heavy degrees of freedom respectively. On the condition thatM is large (which is what we mean by 'light' and 'heavy') it maytranspire that the total wave function can approximately be writtenin the Born-Oppenheimer form (see, e.g., Ref. [24], pp. 641-642)

It is perhaps tempting to say that (18) establishes a link betweenBerry's phase JB and first order work associated with the loop Cy.However this is misleading for the following reason. The time de-pendence of the force / on the RHS of (18) entails that the workdepends on the actual time it takes for the loop to be traversed. Onthe other hand Berry's geometric phase depends only on the loop Cyitself. In fact, as mentioned in the previous section, JB = 7j[M2>3],i.e., 7s, in addition of being projective-geometric, is independent ofthe speed at which Cy is traversed (reparametrization invariance).

Regarding the physical properties of the Berry connection ABwe first note that there is no explicit charge involved in (17). How-ever the Schrodinger equation describes the dynamics of interacting

as the averaged guidance relation, which can be solved without know-ing X(t).

Now suppose the trajectory Y(t) traces out a loop Cy in con-figuration space and integrate AB along Cy yielding the conventionalBerry phase [10]

where we have averaged with the weight |<^>|2 and As(y) = »{v(l/)l^» \V>(j/)) is the Berry connection (see, e.g., Ref. [24], p. 581). Then from(16) we can take

In order to eliminate the a;—dependence, which solely appears in thesecond term on the LHS of (15), a fact emerging from the approx-imate nature of the Born-Oppenheimer wave function, we replace(15) with the averaged equation

Taking the imaginary part of the Schrodinger equation (withthe Hamiltonian (12)), of which ty given by (13) is assumed to be asolution, we obtain


charged systems so charge enters implicitly in the Berry connection.Secondly by analogy with the electromagnetic case, (17) is phase co-variant provided As(y,t) —> AB(y,t) + V y a(y , t ) / n if x undergoesthe local phase transformation x(y^) —> e'^'^xG/). It then fol-lows that the trajectory can be made phase independent and if, inaddition, a is differentiable everywhere in the y—space then JB alsobecomes phase invariant. Thirdly coming to the spatio-temporal de-pendence of the Berry connection it is clear that a local phase canalways be chosen in which AB becomes time independent4 and thatAB is a non-trivial function on configuration space of the heavy sys-tem. In the context of the latter remark we simply note that it is ingeneral impossible to write AB(y) = (Aa(yi),..., AB(YN}) (assum-ing the y—system consists of N particles). This entails an importantdifference between the Berry connection and a classical vector po-tential, viz. in contrast to the former a local phase can always bechosen in which the latter separates into JV independent parts.

4. CONCLUSION

We have analyzed physical properties of pilot-wave theory in thelight of Berry's adiabatic geometric phase. An expression for theBerry phase for a single trajectory obeying the projective-geometricproperty and reparametrization invariance has been derived. Wehave justified the physical significance of first order work in pilot-wave theory by its relation to this phase. The Berry phase for asingle trajectory has been shown not to be locally phase invariantwhich led us to introduce a locally phase invariant modification ofthis phase, in the case where the electromagnetic vector potential is agradient. We have shown that the averaged pilot-wave dynamics forthe heavy part of a light-heavy system naturally includes the Berryconnection.

ACKNO WLED CEMENT

We wish to thank Gonzalo Garcia de Polavieja for telling us about hiswork on geometric phases in causal quantum theories. We also thankHarvey R. Brown, Gonzalo Garcia de Polavieja and Osvaldo Goscin-ski for critical reading of the manuscript and helpful discussions.ES acknowledges a post-doctoral scholarship from The Wenner-GrenFoundation. Partial financial support from the Swedish Natural Sci-

4 This is not the case for classical vector potentials, which can benon-trivially time dependent as a consequence of time dependentdistributions of moving charges.


which gives the desired result (3).

REFERENCES

1. L. de Broglie, J. Phys. (Paris) 8, 225 (1927).2. D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952).3. A. Valentini, "On the Pilot-Wave Theory of Classical Quantum

and Subquantum Physics," Doctoral Thesis (1992), InternationalSchool for Advanced Studies, Trieste.

4. D. Diirr, S. Goldstein, and N. Zhangi, J. Stat. Phys. 67, 843(1992).

so we end up with

But

Now integrating over d yields

Let $(d, t) = Q1/2(d, t)eio(d,t)/h and consider the quantity J = I(d, t)inside the brackets of the second term on the RHS of (A.1):

A kinematic formulation which includes all existing definitions of thegeometric phase [10,16,17] was given independently by Aitchison andWanelik [14] and Mukunda and Simon [15]. For a complex-valuedfield $ they defined the geometric phase as

APPENDIX

ence Research Council (NFR) is also acknowledged.


5. P.R. Holland, The Quantum Theory of Motion (Cambridge Uni-versity Press, Cambridge, 1993).

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1962), Vol. 2, Chap. 17.20. I.J.R. Aitchison, Physica Scripta T23, 12 (1988).21. A. Bohm, B. Kendrick, and E. Loewe, Intern. J. Quantum Chem.

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berry's phase in pilot-wave theory

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