EFTA02688804.pdf

A real ensemble interpretation of quantum mechanics Lee Smolin • to for Theoretical Physics, , Waterloo, Ontario NM 2Y5, Canada (Dated: April 15. 201O A new ensemble interpretation of quantum mechanics is proposed according to which the ensemble associated to a quantum state really exists: it is the ensemble of all the systems in the same quantum state in the universe. Individual systems within the ensemble have microscopic states, described by beables. The probabilities of quantum theory turn out to be just ordinary relative frequencies probabilities in these ensembles. Laws for the evolution of the beables of individual systems are given such that their ensemble relative frequencies evolve in a way that reproduces the predictions of quantum mechanics. These laws arc highly non-local and involve a new kind of interaction between the members of an ensemble that define a quantum slate. These include a stochastic process by which individual systems copy the beables of other systems in the ensembles of which they are a member. The probabilities for these copy processes do not depend on where the systems arc in space, but do depend on the distribution of beables in the ensemble. Macroscopic systems then are distinguished by being large and complex enough that they have no copies in the universe. They then cannot evolve by the copy law, and hence do not evolve stochastically according to quantum dynamics. This implies novel departures from quantum mechanics for systems in quantum states that can be expected to have few copies in the universe. At the same time, we are able to argue that the centre of masses of large macroscopic systems do satisfy Newton's laws. Contents I. Introduction 2 A. Basic hypotheses 3 B. More about beables and interactions amongst members of an ensemble 3 H. The real ensemble formulation of quantum mechanics 5 A. Kinematics and dynamics of individual systems 5 B. Restrictions on the evolution rules 6 I . Good large N limit 6 2. Time reversal invariance 7 HI. Recovery of the Schroedinger equation 7 A. Final form of the evolution rules 8 I.V. A possible approach to phase alignment 9 V. The classical limit 10 VI. Issues that require more investigation 12 VII. Conclusions 13 ACKNOWLEDGEMENTS 14 References 14 EFTA_R1_02022330 EFTA02688804  2 I. INTRODUCTION In this paper we propose a novel interpretation of quantum mechanics that offers new answers to some basic questions about quantum phenomena. I. Why do microscopic systems have indefinite values of observable, while macroscopic systems have definite values? 2. What is the meaning of the probabilities in quantum physics? 3. If the quantum state is associated to an ensemble, where are the members of the ensemble to be found? This new interpretation is a theory of beables, and hence solves the measurement problem by asserting that there is a real state of affairs in any quantum system given by the values of the beables. At the same time, we assert that the quantum state describes an ensemble of individual systems. Resolving the measurement problem by means of a theory of beables recalls existing hidden variables theories such as those of dcBroglic Bohm'I, 21, Vinkpt and Nelson(4]. However, we aspire to remove an awkward feature of those theories which is that the dynamics of the beables of individual systems depend on the wavefunction. In the formulations of de Broglie. Bohm and Vink this is expressed by an equation which asserts that the particle moves in a quantum potential, which is built from derivatives of the wavefunction. In Nelson's stochastic formulation of quantum theory the osmotic velocity depends on the wavefunctiont4, SI. This dependence of the dynamics of individual beables on the wavefunction is a characteristic, but most mysterious feature of quantum theory. This dependence is awkward because of a principle, which we can call the principle of explanaory closure: anything that is asserted to influence the behavior of a real system in the world must itself be a real system in the universe. It should not be necessary to postulate anything outside the universe to explain the physics within the one universe where we live'. This means that the wavefunction must correspond to something real in the world. In the de Broglie-Bohm interpretation this is satisfied by asserting that the wavefunction is itself a beable. This results in a dual ontology-both the particle and the wavefunction are real. But this violates another principle, which is that nowhere in Nature should there be an unreciprocated action. This means that there should not be two entities, the first of which acts on the second, while being in no way influenced by it2. But this is exactly what the double ontology of deBroglie-Bohm implies, because the wavefunction acts on the particles, but the positions of the particles play no role in the Schroedinger equation which determines the evolution of the wavefunction. A class of interpretations called "statistical interpretations" aim to overcome the double ontology by asserting that the wave- function corresponds to an ensemble of systems. But this falls short of satisfying the principle of explanatory closure unless that ensemble really exists in the world. It is not sufficient to posit that the wavefunction corresponds to an cpistcmic ensemble that is defined in terms of our ignorance of the world. Neither is it acceptable to imagine that there is a spooky way in which "potentialities affect realities:' If the behavior of individual systems is to depend on a wavefuction which corresponds to an ensemble, then the principle of explanatory closure demands that each and every member of that ensemble be a physical system in the universe. But if the elements of the ensemble the quantum state represents exist. then perhaps the apparent influence of the wavefunction on the individual entities could be replaced and explained by interactions between the elements of the ensemble. By so explaining the influence of the quantum state on the individual system in terms of a new kind of interaction posited to act between members of the ensemble that the quantum state represents, we satisfy both the principle of explanatory closure and the principle of no unreciprocated action. In interpretations in which the ensemble is epistemic it would not make sense to posit interactions amongst members of the ensemble because it would mean that physical particles-the distinguished member of the ensemble that are real- are interacting with shadows that reside only in our ignorance of their true motions. It would be to have reality depend explicitly on possibility. But if all the elements of the ensemble are real then there is no barrier to positing new kinds of interactions amongst them. These interactions arc certainly non-local. But we already have strong reason to assert that any theory of beables that reproduces quantum mechanics must be highly non-local. This leaves us with one more question to answer: where do the members of the ensemble corresponding to the ground state of the hydrogen atom reside? There is a simple, but novel answer that can be given to this question: in the universe. That is. the ensemble corresponding to a hydrogen atom in as ground state is the real ensemble ofall the hydrogen atoms in the ground state in the universe. The test of this general idea is whether a simple form can be proposed for the interactions amongst the members of the ensemble, that reproduces quantum kinematics and dynamics. In fact, we will see that a simple form of the interactions, in This argument and its implications are developed in 181. 1 ' Einstein invoked this princniple in a 1921 talk where he objected to "the postulation: in Newtonian mechanics. "of a thing (the spacetime continuum) which acts without being acted upon" 191. EFTA_R1_02022331 EFTA02688805  3 which the members of the ensemble interact in pairs, suffices. This simple interaction is that the beables of systems in the ensemble copy each other's states, with a probability that depends on the beables of the systems in the ensemble. Let us now proceed to make these ideas more concrete. This interpretation is based on several simple hypotheses: A. Basic hypotheses • Quantum mechanics applies to small subsystems of the universe which come in many copies. Thus, it applies to hydrogen atoms and ammonia molecules, but not to cats or people or the universe as a whole. Quantum mechanics is hence an approximation to an unknown cosmological theory. • For each microscopic system, there is an ensemble of systems in the universe with the same constituents, preparation and environment. A pure quantum state is a statistical description of one of these ensembles. The elements of the ensemble will be labeled .5/ where / = I ....,N. • Each individual microscopic system. Sr in the ensemble has two beables. The first is the value of some observables, which will be denoted b. The possible values of B are indexed by a = I,...P and are denoted tea. The second beable is a phase e4i. We then assert that the microscopic state of an individual system is the value of the pair of beables, (ahet ). • The beables evolve by a discrete and probabilistic Me. There is a probability in each unit time that each system sr copies the beables of system St. When this happens. al —1Oj, e4t —o est (I) The probability that this happen will be assumed to be a function of the beables of the two systems as well as the number of systems with the same values of A in the ensemble. It does not depend on where the members of the ensemble are in the universe. • The phases also evolve continuously according to a law that also depends on the distribution of beables in the ensemble. • We hypothesize that there is a process of phase alignment, by which the phases of two systems with the same values of evolve to become equal. The dynamics as first posited below conserves the alignment of phases. After that I present a model for the dynamic alignment of phases. • Finally, we hypothesize that these ensembles arc well mixed by the dynamics just described, so that the probability to a make a measurement of the beable on any member of the ensemble and get a particular value,ba, is given by the relative frequency with which that value appears in the ensemble. We will expand on the motivation for these hypotheses shortly, and then show how they may be expressed in a form that is equivalent to quantum mechanics. But what we have said is sufficient to answer the questions with which we opened. I. Microscopic systems have indefinite values of beables. while macroscopic systems have definite values, because micro- scopic systems come in many copies, and so are subject to the copy rule, in which they evolve stochastically by copying the beables ofmembers of the ensemble they share. Macroscopic systems are those that have no copies, anywhere in the universe, hence they are nor subject to the copy dynamics. 2. The probabilities in quantum physics refer to ordinary relativefrequencies in an ensemble ofreal. existing systems. 3. The members of the ensemble are to befound spread throughout the universe. B. More about beables and Interactions amongst members of an ensemble Before we go on to develop the hypotheses just stated it would be good to revisit some aspects of the motivation. We begin with the similarities and differences to other theories of beables. This proposal shares with hidden variables theories such as deBroglie-Bohm,Vink and Nelson the idea that there am real beables. It shares with Nelson also the idea that pure quantum states correspond to ensembles of individual systems. However, it differs from all of these interpretations in asserting the ensemble be physically real, as well as in several other respects. First, it eliminates the need to pick the configuration space as a beable. In what follows there is assumed to be a beable A observable, but its choice is inessential. That this is possible was shown by VinkI3), by giving a deBroglie-Bohm like formulation for a general choice of beables. Indeed, some of the formal development that follows was inspired by Vink's paper13]. Whether there is a preferred choice for it is a subject for future work. EFTA_R1_02022332 EFTA02688806  4 Second, we eliminate the double ontology which requires that both the positions of the particles and the wavefunction be beables. This can be criticized as an extravogent hypothesis, which makes the world as ontologically bizzare as interpretations such as many worlds that posit the reality of the quantum state. However, the lesson of Nelson's formulation 141, is that, as explained in l5l, one cannot succeed in making the whole wave- function just a derived property of an ensemble, derived from the values of configurations of individual systems. Given the form of the wavefunction, %Kass) = VI: eis(" ) (~) it is certainly appropriate to regard the probability density p(x,t) as a property of the ensemble and we will do so. But it is much more challenging to regard the phase .5(x,r) as derived from an ensemble. For one thing, the deterministic evolution equation for the position beable of deBrogle-Bohm theory has the velocity depend on S(.r,t). But, if the rates of change of beables depend on S(x,i) it seems that by our principle of explanatory closure, S(.r,i ) must also be a beable, or must be determined by beables. But then this contradicts our second principle of no unreciprocated influence and we find ourselves in trouble. To get out of trouble we take a new approach to this conundrum. We posit that each individual microscopic system has a second beable, which is a phase. e*. We also posit that the dynamics forces these to a class of configurations in which they E. come to depend on the other beables Hence " A , where at is the value of the beable in the system /. Once that is the case the information to determine the function SC;0 is to be found distributed in the phase beables of all the individual systems in the ensemble. An interaction between the beables of individual systems that make up an ensemble that is described by the quantum state may seem a strange and novel idea. But once we regard the members of the ensemble as all physically real, this is just another interaction between systems in the universe. Certainly these interactions are highly non-local, but we already know from the experimental tests of the Bell inequalities that any theory of beables that reproduces quantum theory must be highly non-local. After all, at one time the idea of an interaction between the Sun and the planets seemed bizzare, because it was a non-local action at a distance. Once one accepts this general idea, the next step is to ask how the dynamics of an individual system can depend on the beables of other members of the ensemble in such a way that the predictions of quantum mechanics can be obtained. This is accomplished in the next section. We will see that to match the quantum evolution in this picture there must be both a stochastic and a continuous evolution rule. There is a stochastic process by which one member of the ensemble can copy the beables of another member of the ensemble. This stochastic process realizes an idea that the beables of a system we prepare here becomes unpredictably shuffled up with the beables of all the similarly prepared systems in the universe. There is also a continuous evolution of the phase beables. Both the stochastic and continuous evolution rules depend on relative frequencies in the ensemble. One motivation for the copy rule is the idea that space is an emergent property, as suggested by several proposals for quantum gravity. If space is emergent. then so is locality. From this perspective, two systems with the same constituents, preparations and environment,but only distinguished by their location in space. may be more closely related than is usually thought. Indeed. we already know that quantum statistics allows us to give a list of positions where hydrogen atoms in their ground states are to be found, but does not permit us to assert which hydrogen atom is in which position. If this extends to the level of the beables, then distinct beable configurations may not be stably located with respect to distinct positions in space. The whole ensemble of beable states of identical subsystems may then evolve in a way that is not captured by the usual local interactions. The copy rule is a simple suggestion for this new kind of interaction, which has a simple realization that reproduces quantum mechanics. Other rules might be contemplated, but as we will see the copy rule suffices for our purposes. What is nice about the copy rule is that it by itself gives all the dynamics the beables need. Imagine making a series of measurements of the beable of an atom you hold in your laboratory. The first measurement is 00. The second is different, it is an. The explanation is not that there was a process by which as evolved to at but that au was copied from another version of that atom somewhere in the universe. Evolution occurs because on subsequent observations you will be seeing beables copied from the ensemble. This appears to be like motion as a consequence of the rule that gives the probability for the copy process. Indeed, we will see in Section V that in an appropriate limit in which h can be ignored this can account for classical motion of large bodies. In the next section we put the hypotheses we stated above into precise mathematical form and impose several reasonable physical assumptions on the evolution rules. In section III we show that a very simple form of the rules then leads to the derivation of Schrodinger quantum mechanics. Section IV presents a model for phase alignment. This is a dynamics for the phases e4' which has a set of degenerate zero energy solutions that impose both phase alignment and Schrodinger dynamics. There are however issues of the stability of these solutions that remain a subject for further work. In section V we raise and resolve a question unique to this conception of quantum mechanics, which is whether we can derive the fact that large macroscopic bodies obey Newton's laws, while respecting the assertion that their precise microscopic states may be unique, and hence not part of a large ensemble. A list of open questions is the substance of section VI. and the conclusions are stated in section VII. EFTA_R1_02022333 EFTA02688807  5 II. THE REAL ENSEMBLE FORMULATION OF QUANTUM MECHANICS A. Kinematics and dynamics of individual systems The hypotheses we enunciated above become a formulation and interpretation of quantum mechanics, when we give them a precise instantiation. • Kinematics: description of individual states The state of an individual microscopic system, Sr consists of the pair of beables. (abets) • The ensemble of similarly prepared states. This system is one ofN similarly constituted systems in the universe, which have been prepared in the same state and are subject to the same external forces as they evolve. These are labeled by I. I, ...,N. The state of the ensemble is specified by the collection of pairs, Hal (t).e4 (0)}. • Ensemble state variables. The individual system evolves partly by a stochastic process. Because of this, an observer studying a particular member of the ensemble, cannot predict with certainty which beables she will measure if she makes a measurement at a later time. She can predict probabilities for different beables to be observed, which are derived from relative frequencies for the states in the ensemble. The relative frequencies n°(t) are defined to be the number of systems in the ensemble which have beable value a at time r. These are normalized to L A, = N. We will also write al for the state of the I'th system and n1= na, for the number of copies in the ensemble of the beables of system Sr . • Dynamics of individual systems There are two modes of evolution of the beables of a system. Stochastic evolution rule. There is a stochastic evolution by means of which the system 5 / can copy the beables of the system 51. The rate by which system / copies the beables of system J is assumed to be of the form P(/copy./) = F(abObni3Ohabaj) (3) When this happens the properties of the system Sr inherits the properties of system 51 so that a/ —) aJ, es, (4) We note that the rate a system I copies the state of system J is determined entirely by the beables of the two systems NiCOPM1b=ba=a) = F(ea nap Sap = F(nap Oa nor tai )ab (5) This defines the rates of copying F(nap+a,,naj otai )on as functions of the beables. We note that by definition the compo- nents of Fab must be all positive. Continuous evolution rule. When this mixing up or copying of the individual states does not happen. the phase evolves continuously in a way that depends on the ensemble. This must have the general form = ;G(irb(libn.r.4ij,aba.r) (6) • Evolution of the occupation numbers na We define the occupation numberssia, to be the number of members of the ensemble in state a. They evolve as follows = ;; 6aal( — kin) IPVcoPY-0 nicoPY I)) = ; lb.16,,..,bb,„[P(Icopyi)—P(Icopy I)) = nbna [Fd, — Pk) (7) 7a • Evolution of the probability densities From this we can write down a law for the evolution of the probability densities, defined by no Pa = — 8 N EFTA_R1_02022334 EFTA02688808  6 These must evolve as[3] Po = (Phrb-ea Para-.b) (9) where Tb.0 are transition rates. From (3) above we see that Tb_ki = F(naripai,no,.. bi)ddla (10) This is because the probability to copy a beable value a will be proportional to how many members of the ensemble presently have that value. Phase alignment. There is a specialization of the evolution rules which we will have to make to derive quantum mechanics from this general framework. This is that = (II) ie the phases are functions of the variables at. This will be called phase alignment. This is a stable condition, because once set as an initial condition it is preserved by the evolution rule (6). This is because we have then r[ri = = ;nbG(nbi .Ob,•noi •taj' abaj) = ;Gin ar cpai.n„.,..„.,,abaj) (I2) This implies that = ;G i(rta•tams•Ss)as• (13) where 64(rlaita,nsotb)„,, = nbG(naptar naj ,4101 (Thai). In section V we will describe and study a more general evolution law has solutions which achieve phase alignment, but in this and the next section we assume the phases have been aligned initially. B. Restrictions on the evolution rules We can introduce some physical considerations which will allow us to restrict the form of F and G. I. Good large N limit First, we do not have any evidence the probabilities for quantum states to evolve depend on the size of the ensemble of similarly prepared systems. So we require that Tb-40 and C depend on ratios Z. We can also posit that only relative phases are relevant, so that Ti -.a and C depend on #(41-40. These together give us FQII,Sba.,,Mabno= r( 111 ,ei*s-4'.0)a (14) so and similarly for C. G(no*islo•So)ab = G (t ,e'f4"-4a))ab (15) nb These equations assume all the nb >> I. There can be additional terms that go away in the limit no >> I EFTA_R1_02022335 EFTA02688809  7 2. Time reversal invariance It is easy to see that these forms are constrained by time reversal invariance. To see the implications of this let us consider an ansatz, which will be sufficient to recover quantum theory. 9 ri(°",e4-1-0- ) ).b= fael(41" -.11))ab (16) lib a(a,e(0.-406 = (a)r v 00.-•6)6 (171 nb nb Note that it(eaal -0°t ))„1, must be positive. We have then. because & = &, ) = T (Pb')epot(e(*.-6)),,,b- (aPa vPo9t_ (e1(16-‘ ))ba) (IS) Time reversal sends t -t but pa -> pa. Let us suppose it also send q„ (jk,. Then we have under time reversal Oa —) — Oa = (( 12 ) qPA- V ia-ea )ab Pb Pa ) (Pa vPtifCfritered))129) (19) We have time reversal invariance if this returns the same equation for fia. Recalling the positivity of Itjettrat - eat ))ab this can only be solved if q = 3 and 4k, = —4.a. We also have to impose (20) We have then Au =;:Wn et(e4"b )Lb — R.(e*-41 4/ ))ba) (21) Insisting on time reversal invariance of P in (13) then implies that (z)mb = •11(2)01p (22) However the power r is not fixed by time reversal invariance. HI. RECOVERY OF THE SCHROEDINGER EQUATION Let us summarize where we are as a result of our ansatz's plus the imposition of a good large N limit and time reversal invariance. We have two evolution equations pa = Z(P0Fal, PbFba) = %)/1. b(gt.(eA(41-416))tin 9(.4•44.14-4‘1))ba) (23) = Loa + Z ( ° n) ti(e",-.6)),„b (24) nb where ok, = Va. and Rafr and 'lid, satisfy the properties above. We can now expand & I, and thd, in Fourier series. R.(e1(‘ -ta)ab = ?MG RI,s111`.(n(4ki + 0:1,b) (25) EFTA_R1_02022336 EFTA02688810  8 CO u (eOrt)),,b = I Cb cos0(4). — (kb) + eat) n=a (26) To preserve the positivity of Fat, and hence t,b, we have sin(0) when that is positive sin+(0) = (27) 0 otherwise It is remarkable that just the first term with the further simplifications Rail, = fealb and &alb = bob suffices to reproduce quantum mechanics. 1(..(144°—+a )ab = Rub sin+ (Oa — +a+ bah) V (el* -3 4 / ))0b = RabCOS(ta — + bob) where, Rai, = Rb,, are positive constants. bat are constant phases which are odd under time reverse and, This gives us evolution rules = ‘riPbRab sin(4). — +fr + kb) +a = wo+71 Rabcos(fa — +6 ± kb) (nn It is easy to check that with the choice of r = —₹ this reproduces Schroedinger quantum mechanics. To see this we write the general quantum state. fi le .301 Vrs2e—df Pp >= (32) A fe- dmf" which clearly is a property of the ensemble and not of an individual physical system. Here we have defined Sa = (33) Equations (30) and (31) and hence the evolution rules we posited are then equivalent to evolution via the Schroedinger equation, dqi rh = Aqr (34) di driven by the hemtition Hamiltonian To 612 = (AT2 (35) here we have set aab = &beak (36) A. Final form of the evolution rules The final form of our evolution rules is P(Icopyi) = ivtWi Ram, sin+(@r— + 60,6,)+ ELI (37) 40/ = Of = Wa, Ram COS($r — ell bah) + 2frbr (38) f#r“Thrti It must be emphasized that we have derived a correspondence to quantum mechanics only with the proviso that no >> I and nr > > 1. When these are not satisfied other terms could come into the evolution rules. I have added terms it/ and Du to indicate these. EFTA_R1_02022337 EFTA02688811  9 IV. A POSSIBLE APPROACH TO PHASE ALIGNMENT The elimination of S(x,t) as a function of beable variables, and hence as an ontological entity in its own right, rests on the postulation of a dynamics which achieves phase alignment. This means that the phases. tly originally assigned independently to each member of the ensemble, become aligned so they depend only on the value of the beable. le 10/ Sal • (39) As we have shown, phase alignment is a fixed point of the dynamics we have postulated in (37,38). But is it an attractor? My investigations of this question have so far been inconclusive. But this is not the only option. It may be that the evolution described in (38) is an approximation to another dynamical law which achieves phase alignment. We now describe a possible model for such dynamics. We shall see that it is easy to show that this model has solutions which achieve phase alignment, but there remains an open question as to the stability of these solutions. Consider the following dynamical system. put in Hamiltonian first order form for simplicity. /2 11 S= f dr; [19 + — (2/(4),n)) — —2 (n- ) — — V sin2(4/ — 4(;)] 2 fr (40) ea, where the model depends on a new paranater, the frequency j. Q, is defined by eq (38), and the notation J E al means the subsystem J shares the beable value with I. We find the momenta are given by nr = Of($,n) (41) which satisfy the Poisson brackets {4,/,71j} = (42) with the Hamiltonian f2 H= ; [ iI (Ir/ )2 + n-QiWn)+ / sin2 (4r - 4r) (43) 2 cri The Hamilton equations of motion follow from the Poisson brackets and include (41) and = —.J2 sin(4r — $2) cos(ipi — _ p in°ilk (4'.n) (44) ativ Let us take f very large compared to w and the components of Rat, and consider this evolution in the approximation where the second term can be neglected. Then we can approximate (44) for small phase differences as = —.1.2nr* + where is the average value of the phases in the subencernble that shares the value of the beable with system 1. The Hamiltonian in this approximation is H (9)2 + cm* - ()ad (47) = In this approximation Si is driven to the minimum of the potential where = oaf (4g) so the phases align to their average values for each value of the beable. Once there we have from (44) the full equations of motion. = ; *Ar arbc(4),n) (49) EFTA_R1_02022338 EFTA02688812  10 A solution to this is rci = itr =0 (50) This implies =i 2i(Cn) (51) which recovers (38), and hence the Schroedinger equation is satisfied. Hence our model has a degenerate set of zero energy solutions which achieves both phase alignment (48) and the Schroedinger dynamics. What we are not, however, able to show is that the solutions (485051) are stable. We can get a bit more insight by solving the action (40) for of and writing it in terms of complex variables zi = ?Pi which satisfy 42../ = 1 S = f dit [ i (41-1,20,n))2 — /2 7. sin2 Op/ — 2. &fa = f dr V [- I (t; g."1/(Z,n)z; - irlAzinkt)- .1.2 - V sin2 — 40.0] (52) to 2 2 fr0€, o This shows that the Wallstrom objection(10) is not relevent here, because the theory depends on the phase 2/ = en rather than on (Or directly; . Finally, we can note that when phase alignment is satisfied, the whole system becomes a lagrangian system, with an action principle given by S= ME (Pa(sia Wa) 1 d/T457,Rabcos(0.-44,+ 9i a bah)) (53) This suggests that the pa and qta are conjugate quantities in the phase of the more general theory in which phase alignment is satisfied. V. THE CLASSICAL LIMIT Once the conditions are met which are required to derive quantum mechanics, one can continue from there and consider the effect of taking h 0. This should allow us to recover classical mechanics as a limit of quantum mechanics, in the usual way. But notice that the same conditions we require to get quantum mechanics, which are large numbers of copies and large occupation numbers, are needed to recover classical mechanics through this route. This raises the question of whether the theory described here can account for the fact that large macroscopic bodies obey classical dynamics, when we assert that they do not obey quantum mechanics. Can we still derive the classical dynamics of large bodies, while still respecting the distinction that the exact quantum states of macroscopic bodies will often be unique? The following argument shows that it can. To show this we can start from the action principle (53). Let us consider a simple model of the translational degrees of freedom of the atoms in a body in one dimension, given by a one dimensional array of sites, with periodic boundary conditions, with a = I,...P labeling the sites. Let us multiply (53) by h to define an action S. We also can define the energy Ea = hwa, and the Hamilton-Jacobi function Sa = MN. We want to construct a coarse grained model of a macroscopic body so we choose the transition rates to give nearest neighbor interactions, defined with lattice spacing a, h Rea (bag, f. I + 606- I) (54) 27702 We define the potential energy to be h2 V(a). 4+ ma2 (55) 3 Thanks to Antony Valcntini for suggesting this was the case. EFTA_R1_02022339 EFTA02688813  I The action (53) then becomes S= di 2 pa(t+ 21 (a,3)2 _ v(a)_ ve+ o(a)) (56) where the quantum potential is h2 V2 Vrt — 72 , (57) Neglecting the quantum potential or, equivalently, taking li —* 0, we have the following equations of motion = — ax(Pa.$) (58) m S= — —n(axS)2 + V(a) (59) We recognize (58) as the conservation of probability, with current velocity v = la,s, and (59) as the Hamilton-Jacobi equation. Thus. we recover an ensemble of classical systems obeying the Hamilton-Jacobi equation. Note that if classical mechanics is construed to be an approximation to quantum dynamics. and the latter is a probabilistic theory of real ensembles, then so must be the former. That is why we derive classical mechanics in the form of an ensemble of systems whose probabilities evolve in a way that is driven by the Hamilton-Jacobi equation. There appears to be a puzzle here. It seems that an ensemble is required to derive classical mechanics as an approximation to the copy dynamics proscribed by (3738). But we have argued that macroscopic bodies have distinct quantum states. And yet. the derivation of classical dynamics depends on the beable occupation numbers being large. That is a consequence of the fact that we derived classical mechanics as an approximation to quantum mechanics, and therefor require the same conditions for its validity. Is there a contradiction here?