zeno paper

8/10/2019 Zeno Paper

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Zeno Paradox for Bohmian Trajectories: The

Unfolding of the Metatron

Maurice A. de Gosson

Universitat Wien, NuHAG

Fakultat fur Mathematik

A-1090 Wien

Basil Hiley

TPRU, Birkbeck

University of London

London, WC1E 7HX

June 9, 2010

Abstract

We study an analogue of the quantum Zeno paradox for the Bohmtrajectory of a sharply located particle. We show that a continuouslyobserved Bohm trajectory is the classical trajectory.

1 Introduction

I am glad that you are deeply immersed seeking an objective de-scription of the phenomena and that you feel the task is muchmore difficult as you felt hitherto. You should not be depressedby the enormity of the problem. If God had created the world hisprimary worry was certainly not to make its understanding easyfor us. I feel it strongly since fifty years.[10]

When David Bohm completed his book, Quantum Theory [3], whichwas an attempt to present a clear account of Bohrs actual position, hebecame dissatisfied with the overall approach [5]. The reason for this dis-satisfaction was the fact that the theory had no place in it for an adequatenotion of an independent actuality, that is of an actual movement or activityby which one physical state could pass over into another.

In a meeting with Einstein, ostensibly to discuss the content of his book,the conversation eventually turned to the possibility of whether a determin-istic extension of quantum mechanics could be found. Later while exploringthe WKB approximation, Bohm realised that this approximation was giv-ing an essentially deterministic approach. Surely truncating a series cannot

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turn a probabilistic theory into a deterministic theory. Thus by retaining

all the terms in the series, Bohm found that one could, indeed, obtain adeterministic description of quantum phenomena. To carry this through,he had to assume that a quantum particle actually hada well defined butunknown position and momentum and followed a well-defined trajectory.

In Bohms approach, the Schrodinger equation can be cast into a formthat brings out its close relationship to the classical Hamilton-Jacobi theory,the only difference being an addition term which can be regarded as a newquality of energy, called the quantum potential energy. It is the propertiesof this energy that enables us to account for all quantum phenomena suchas, for example, the two-slit interference effect where the trajectories areshown to undergo a non-classical behaviour [32].

In this paper we will show that if we continuously observe a Bohm tra-jectory, it becomes a classical trajectory. Thus, in a sense, continuous ob-servation dequantizes quantum trajectories. This property is, of course,essentially a consequence of the quantum Zeno effect, which has been shownto inhibit the decay of unstable quantum systems when under continuousobservation (see [7, 11, 17, 18]).

The idea lying behind the Bohm approach (Bohm and Hiley [7], Hiley[21], Hiley and collaborators [24, 25], Holland [26]) is the following: let = (r, t) be a wavefunction solution of Schrodingers equation

i

t =

2

2m2

r+ V(r)

.

Writing in polar formeiS/Schrodingers equation is equivalent to thecoupled systems of partial differential equations:

S

t +

(rS)22m

+ V(r) + Q(r, t) = 0 (1)

where

Q = 2

2m

2r

|||| . (2)is Bohms quantum potential (equation (1) is thus mathematically a Hamilton-Jacobi equation), and

t + rrS

m

= 0 (3)

which is an equation of continuity. The trajectory of the particle is deter-mined by the equation

mr = rS(r, t) , r(t0) =r0 (4)

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wherer0 is the initial position.

Since the quantum potential depends only on the wave function, and thelatter is ultimately a property of the metaplecticrepresentation [13, 14, 16],we proposed in [15] to call the entity whose motion is governed by theequation (4) a metatron. We chose this name because the particle, ratherthan being a classical object, is essentially an excitation induced by themetaplectic representation of the underlying Hamiltonian evolution.

The question we will answer in this paper is the following:

What do we see if we perform a continuous observation ofthe metatrons trajectory ?

If the observed trajectory is smooth, we will see the classical trajectory

determined by the Hamiltonian function

H(r,p) = p2

2m+ V(r).

Does this mean that Bohmian trajectories are therefore not real, thatthey are surrealistic? No, they are not surreal simply because we aremaking a distinction between what is, and what is observed by a physicalmeasurement. For example, in the two-slit experiment referred to above,we find that if we observe the motion of the particle as it passes throughone of the slits we will see no wave-like behaviour, but a classical trajectoryshowing on interference effects.

We are often asked if Bohm believed that there was an actual classicalpoint-like particle following these quantum trajectories. For Bohm there wasno solid particle but instead, at the fundamental level, there was a basicprocess or activity so that the track left in, say, a bubble chamber couldbe explained by anenfoldingunfoldingof this process [6]. Thus rather thanseeing the track as the continuous movement of a material particle, it can beregarded as the continuity of a quasi-local, semi-stable autonomous formevolving within the unfolding process [21]. As we will see, this is exactlywhat happens when we observe continuously the unfolding process, and wecan regard the visible track as arising from the evolution of the metatron.

2 Bohmian Trajectories Are Hamiltonian

We will show that in the particular case of a metatron initially localized at apoint, the Bohm trajectory is Hamiltonian (the general case is slightly moresubtle; we refer to the papers by Holland [27, 28] for a thorough discussion

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of the interpretation of Bohmian trajectories from the Hamiltonian point of

view).We will consider systems ofNmaterial particles with the same mass

m, and work in generalized coordinates x= (q1,...,qn) and p= (p1,...,pn),n = 3N. Suppose that this system is sharply localized at a point x0 =(q1,0,...,qn,0) at time t0. The classical Hamiltonian function is

H(x, p) = p2

2m+ V(x) (5)

hence the organising field of this system is the solution of the Schr odingerequation

i

t =

2

2m2

x+ V(x)

, (x, t0) =(x x0) (6)where x is then-dimensional gradient in the variablesq1,...,qn. The func-tion is thus just the propagatorG(x, x0; t, t0) of the Schrodinger equation.We write G in polar form

G(x, x0; t, t0) =

(x, x0; t, t0)ei

S(x,x0;t,t0).

The equation of motion (4) is in this case

mx = xS(x, x0; t, t0) , x(t0) =x0. (7)

2.1 Short-time estimates

We are going to give a short-time estimate for the function S. The interest ofthis estimate is two-fold: it will not only allow us to give a precise statementof the Zeno effect for Bohmian trajectories, but it will also allow us to provein detail the Hamiltonian character of these trajectories.

We will assume that the potential V is at least twice continuously dif-ferentiable in the variables q1,...,qn.

In [15], Chapter 7, we established the following short-time formulas fort t0 0 (a similar formula shave been obtained in [29, 30, 34, 35]):

S(x, x0; t, t0) =n

j=1

m(qj q0,j)22(t

t0

) V(x, x0)(t t0) + O((t t0)2) (8)

where V(x, x) is the average value of the potential on the segment [x, x]:

V(x, x0) =

10

V(x + (1 )x0)d. (9)

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We observe that the quantum potential is absent from formula (8); we

would actually have obtained the same approximation if we had replaced Swith the solution to the classical HamiltonJacobi equation

Sclt

+(xS)2

2m + V(x) = 0

while Sis a solution of the quantum HamiltonJacobi equation

S

t +

(xS)22m

+ V(x) + Q(x, t) = 0. (10)

How can this be? The reason is that if we replace the propagator G(x, x0; t, t0)by its classical approximation

Gcl(x, x0; t, t0) =

cl(x, x0; t, t0)e

i

Scl(x,x0;t,t0)

where cl is the Van Vleck density (i.e. the determinant of the matrix ofsecond derivatives ofScl) then we have

G(x, x0; t, t0) Gcl(x, x0; t, t0) =O((t t0)2)(cf. Lemma 241 in [15]) from which follows that

2

2m

2xGG

2

2m

2xGclGcl

=O((t t0)2);

the difference between the two terms O((t t0)2) in is thus absorbed by theterm (8). [We take the opportunity to remark that when the potentialV(x)is quadratic in the position variables q1,...,qn then Gcl = G; we will comeback to this relation later].

Moreover, formula (8) can be twice continuously differentiated with re-spect to the variablesqj andq0,j. It follows that the second derivatives ofSare given by

2S

qjq0,k=

m

t t0 jk + O(t t0)

and hence the Hessian matrix Sx,x0 (i.e. the matrix of mixed second deriva-tives) satisfies

detSx,x0 =

m

t

t0

n+ O(t t0). (11)

Formula (8) is the key to the following important asymptotic version ofBohms equation (7):

x =x x0t t0

1

2mxV(x0)(t t0) + O((t t0)2)). (12)

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Let us prove this formula. Using the expansion (8), formula (7) becomes

x =x x0t t0

1

mxV(x, x0)(t t0) + O((t t0)2)). (13)

Let us show that

xV(x, x0)(x, x0) =12xV(x0) + O(t t0); (14)

this will complete the proof of formula (12). We first note that (13) impliesin particular that

x =x x0t t0 + O(t t0)

and thus x

is given by

x(t) =x0+p0m

(t t0) + O((t t0)2) (15)

wherep0is an arbitrary constant vector. In particular we have O(xx0) =

O(t t0) and hencexV(x, x0) = xV(x0, x0) + O(x x0)

= xV(x0, x0) + O(t t0)from which it follows that

xV(x

, x0)(x

, x0) = 10 xV(x0+ (1 )x0)d + O(t t0)

=1

2xV(x0) + O(t t0)

which is precisely the estimate (14).

2.2 The Hamiltonian character of Bohmian trajectories

Letp0= (p1,0,...,pn,0) be an arbitrary momentum vector, and set

p0= x0S(x, x0; t, t0). (16)In view of formula (11), the Hessian ofS in the variables x and x0 is in-vertible for small values of t, hence the implicit function theorem impliesthat (16) determines a functionx = x(t) (depending onx0 andt0 viewed asparameters), defined by

p0= x0S(x(t), x0; t, t0). (17)

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Setting

p(t) = xS(x(t), x0; t, t0) (18)we claim that the functions x(t) and p(t) thus defined are solutions of theHamilton equations

x= pH(x,p,t) , p= xH(x,p,t) (19)

and that we have x(t0) = x0, p(t0) = p0. We are actually going to useclassical HamiltonJacobi theory (see [2, 12, 15, 16] or any introductorytext on analytical mechanics). For notational simplicity we assume thatn= 1. The function Ssatisfies the equation

S

t + 1

2mSx2

+ V(x)

2

2m

1

2

x2 = 0; (20)

introducing the quantum potential

Q = 2

2m

1

2

x2 (21)

we set H = H+Q so that (20) is just the quantum HamiltonJacobiequation

S

t + H

x,S

x, t

= 0. (22)

Differentiating the latter with respect to p= S/x yields, using the chainrule,

2S

x0t+

H

p

2S

x0x= 0 (23)

and differentiating the equation (17) with respect to time yields

2S

x0t+

2S

xx0x= 0. (24)

Subtracting (24) from (23) we get

2S

xx0H

p x= 0which produces the first Hamilton equation (19) since it is assumed that wehave 2S/xx0= 0. Let us next show that the second Hamilton equation

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The time-dependent flow (ft,t) consists of canonical transformations;that is, the Jacobian matrix offt,t calculated at any point (x, p) whereit is defined by a symplectic matrix. This is an immediate consequenceof the fact discussed above, namely, that the flow determined by anyHamiltonian function has this property.

We have seen that the Bohmian trajectory for a particle initially sharplylocalized at a point x0is Hamiltonian, and in fact governed by the Hamiltonequations (19):

x= pH(x,p,t) , p= xH(x,p,t). (27)The discussion of short-time solutions of Bohms equation of motion allowsus to give approximations to the solution. First, the solutions of the equationx= pH(x,p,t) are given by the simple relation

x(t) =x0+p0m

(t t0) + O((t t0)2)

as was already noticed in (15). Then we proved that the momentum p(t) =mx(t) is given by equation (12):

mx(t) =m(x(t) x0)

t t0 1

2xV(x0)(t t0) + O((t t0)2)). (28)

However we cannot solve this equation by inserting the value ofx(t) abovesince this would lead to an estimate moduloO(tt0) notO((tt0)2). Whatwe do is the following: differentiating both sides of the equation (28) withrespect to t we get

x(t) =x(t) x0

(t t0)2 +x(t)

t t0 1

2mxV(x0) + O(t t0)

that is, replacing x(t) by the value given by (28),

p(t) =mx(t) = xV(x0) + O(t t0).Solving this equation we get

p(t) =p0 xV(x0)(t t0) + O((t t0)2)).Summarizing, the solutions of the Hamilton equations (27) forH =H+Q

are given by

x(t) =x0+p0m

(t t0) + O((t t0)2)p(t) =p0 xV(x0)(t t0) + O((t t0)2)).

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The observant reader will have noticed that (up to the error term O((t t0)

2

))) there is no trace of the quantum potential Q

in these short-timeformulas. Had we replaced the function H with the classical HamiltonianH we would actually have obtained exactly the same solutions, up to theO((t t0)2)) term.

3 Bohmian Zeno Effect

3.1 The case of quadratic potentials

Here is an easy case; it is in fact so easy that it is slightly misleading:the Bohmian trajectories are here classical trajectories from the beginning,because the quantum potential vanishes.

Let us assume that the potentialV(x) is a quadratic form in the positionvariables, that is

V(x) =1

2Mx x

whereMis a symmetric matrix. Using the theory of the metaplectic repre-sentation [13, 14, 15, 16] it is well-known that the propagator G is given bythe formula

G(x, x0; t, t0) = 12i

n/2im(t,t0)

|(t, t0)|e

i

W(x,x0;t,t0) (29)

whereW(x, x0; t, t0) is Hamiltons two-point characteristic function (see e.g.[2, 12]): it is a quadratic form

W =1

2Px x Lx x0+1

2Bx0 x0

whereP =P(t, t0) andB =B(t, t0) are symmetric matrices andL = L(t, t0)is invertible; viewed as function ofx it satisfies the HamiltonJacobi equation

W

t +

(xW)22m

+1

2Mx x.

Moreover,m(t, t0) is an integer (Maslov index) and(t, t0) is the determi-nant ofL = L(t, t0) (the Van Vleck density). Sincem(t, t0) and (t, t0) donot depend on x, it follows that the quantum potential Q determined by the

propagator (29) is zero. Since we have H =H+ Q, we see immediatelythat the quantum motion is perfectly classical in this case: the quantumequations of motion (19) reduce to the ordinary Hamilton equations

x= p

m , p= Mx (30)

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which can be easily integrated: in particular the flow (ft) they determine is a

true flow (because H=H

is time-independent) and consists of symplecticmatrices ([2, 15, 16, 12]). In fact,

ft= etX , X=

0nn

1mInn

M 0nn

.

Thus, in the case of quadratic potentials the Bohmian trajectories associatedwith the propagator are the usual Hamilton trajectories associated with theclassical Hamiltonian function of the problem.

Suppose now that we observe continuously the time evolution of themetatron which is so far quantum and try to find out what is recordedby our observation process. Practically this is done by performing repeated

position measurements at very short time intervals t. We assume thatthe recorded trajectory is, in the limit t 0, continuous and moreoversmooth; by this we mean that we can assign at every point a velocity vector(we are thus excluding Brownian motion-type behavior). Let us choose atime interval [0, t] (typically t = 1 s) and subdivide it in a sequence ofNintervals

[0, t] [t, 2t] [2t, 3t] [(N 1)t,Nt]with t = t/N; the integer N is assumed to be very large (for instanceN= 1018). Assume that a measurement at timet0= 0 localizes the particleat a point x0 it will be detected at a point x1 after time t; its momentumis p1 and we have (x1, p1) = ft(x0, p0). We now repeat the procedure,

replacing x0 byx1; since the observed trajectory is assumed to be smooththe initial momentum will be p1 and after time t a new measurement isperformed, and we find the particle at x2 with momentum p2 such that(x2, p2) = ft(x1, p1) = ftft(x0, p0). Repeating the same process untiltime t = Nt we find a series of points in space which the particle takesas positions one after another1 that (xN, pN) = (ft)

N(x0, p0). But in viewof the group property ftft =ft+t of the flow we have (ft)

N =fNt =ftand hence (xN, pN) =ft(x0, p0). The observed Bohmian trajectory is thusthe classical trajectory predicted by Hamiltons equations.

3.2 The general case

In generalizing the discussion above to arbitrary potentials, V(x), thereare two difficulties. The first is that we do not have exact equations for

1In conformity with W. Heisenbergs statement: By path we understand a series ofpoints in space which the electron takes as positions one after another [19]

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the Bohmian trajectory, but only short-time approximations. The second

is that the Hamilton equations for x

and p

no longer determine a flowhaving a group property because the Hamiltonian H is time-dependent.Nevertheless the material we have developed so far is actually sufficient toshow that the observed trajectory is the classical one.

The key will be the theory of LieTrotter algorithms which is a powerfulmethod for constructing exact solutions from short-time estimates. Themethod goes back to early work of Trotter [37] elaborating on Sophus Lies

proof of the exponential matrix formulaeA+B = limNeA/NeB/N

N; see

Chorin et al. [9] for a detailed and rigorous study; we have summarized themain ideas in the Appendix B of [15]); also see Nelson [31]. (We mentionthat there exists an operator variant of this procedure, called the Trotter

Kato formula; it can be used to derive the Schrodinger equation; see [15],Ch. 7)Let us begin by introducing some notation. We have seen that the datum

of the propagator G0 = G(x, x0; t, t0) determines a quantum potential Q

and thus Hamilton equations (19) associated with H =H+ Q. We nowchoose t0 = 0 and denote the corresponding quantum potential by Q

0 andsetH0 =H+ Q0. After time twe make a position measurement and findthat the particle is located atx1. The future evolution of the particle is nowgoverned by the new propagator G1 = G(x, x1; t, t0), leading to a new quan-tum potentialQ1 and to a new Hamiltonian H1; repeating this until time twe thus have a sequence of pointsx0, x1,...,xN=x and a corresponding se-quence of Hamiltonian functionsH0, H1,...,HN determined by the quantumpotentials Q0, Q1,...,QN. We denote by (f0t,t0), (f

1t,t1),...,(f

N1t,tN1

) the time

dependent flows determined by the Hamiltonian functions H0, H1,...,HN;we have set here t1= t0+ t, t2= t1+ tand so on.

Repeating the observation procedure explained in the case of quadraticpotentials, we get in this case a sequence of successive equalities

(x1, p1) =f0t1,t0(x0, p0)

(x2, p2) =f1t2,t1(x1, p1)

(x, p) =fN1t,tN1(xN1, pN1)

which implies that the final pointx = xNobserved at timet is expressed interms of the initial point x0 by the formula

(x, p) =fN1t,tN1 f1t2,t1f0t1,t0(x0, p0).

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Denote now by (g0t,t0), (g1t,t1),...,(g

N1t,tN1

) the approximate flows determined

by the equations

(x1, p1) = (x0+p0m

t, p0 xV(x0)t)

(x2, p2) = (x1+p1m

t, p1 xV(x1)t)

(x, p) = (xN1+pN1

m t, pN1 xV(xN1)t).

Invoking the LieTrotter formula, the sequence of estimates

f0tk,tk1(xk1, pk1)

g0tk,tk1(xk1, pk1) =O(t

2)

imply that we have

limN

gN1t,tN1 g1t2,t1g0t1,0(x0, p0) = limNfN1t,tN1

f1t2,t1f0t1,0(x0, p0)

The argument goes as follows (for a detailed proof see [15]): since we havegktk,tk1 =f

ktk,tk1

+ O(t2) the product is approximated by

gN1t,tN1 g1t2,t1g0t1,t0 =fN1t,tN1 f1t2,t1f0t1,t0+ NO(t2)

and since t = t/N we have NO(t2) = O(t) which goes to zero whenN

.Now, recall our remark that the quantum potential is absent from the

approximate flowsgktk,tk1; using again the LieTrotter formula together withshort-time approximations to the Hamiltonian flow (ft) determined by theclassical Hamiltonian H, we get

limN

gN1t,tN1 g1t2,t1g0t1,0(x0, p0) =ft

and hencelimN

fN1t,tN1 f1t2,t1f0t1,0(x0, p0) =ftwhich shows that the observed trajectory is the classical one.

4 Conclusion.

We have shown that if a quantum particle is watched continuously, it willfollow a classical trajectory. In terms of the Bohm model, what this implies

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that the quantum potential is forced to remain zero so that no quantum

effects can occur. This result supports the conclusions reached for the tran-sition in an Auger-like particle discussed in Bohm and Hiley [7]. There itwas shown that the perturbed wave function, which is proportional to tfortimes less that 1/E, (E is the energy released in the transition) willnever become large and therefore cannot make a significant contribution tothe quantum potential. For this reason no transition will take place.

From these results we see that in the Bohm approach, it is the magnitudeof the quantum potential energy that distinguishes the quantum behaviourfrom the classical. Indeed this conclusion is quite obvious if we examineequation (1) since when Q is negligible compered with the kinetic energy,the equation is simply the classical Hamilton-Jacobi equation. Hiley and

Aziz Mufti [20] give an interesting demonstration of how the quantum po-tential can become negligible over time. They give a simplified cosmologicalexample of how, in an inflationary scenario, quantum behaviour can becomeclassical at later stages of the inflation.

This last example provides us with a very different way of arriving atthe classical limit than the prevailing view based on decoherence. In ourview the main difficulty in using decoherence is that it merely destroysthe off-diagonal elements of the density matrix but it does not give riseto the classical equations of motion. It continues to describe classical ob-jects by wave functions, a criticism that has already been made by Primas[33]. Furthermore it does not show how the Schrodinger equation becomesHamiltons equations of motion. Our method shows that it is the relation

between the symplectic and metaplectic representations that shows how theclassical is related to the quantum. It is when the global properties of thecovering group become unimportant that the classical world emerges. Ashas been pointed out by Hiley [22] [23], the Bohm approach is much closerto the Moyal approach using a deformation Poisson algebra. In the Moyalapproach the classical limit emerges in a very simple way, namely, in thosesituations where the deformation parameter can be considered to be small.

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