The Emergent Copenhagen Interpretation of Quantum Mechanics - Timothy J. Hollowood

8/13/2019 The Emergent Copenhagen Interpretation of Quantum Mechanics - Timothy J. Hollowood

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arXiv:1312.3427v1

[quant-ph]12Dec2013

The Emergent Copenhagen Interpretation of Quantum Mechanics

Timothy J. Hollowood

Department of Physics, Swansea University,

Swansea, SA2 8PP, UK(Dated: December 13, 2013)

We introduce a new and conceptually simple interpretation of quantum mechanics based on re-duced density matrices of sub-systems from which the standard Copenhagen interpretation emerges

as an effective description of macroscopically large systems. Wave function collapse is seen to b e auseful but fundamentally unnecessary piece of prudent book keeping which is only valid for macro-systems. The new interpretation lies in a class of modal interpretations in that it applies to quantumsystems that interact with a much larger environment. However, we show that it does not sufferfrom the problems that have plagued similar modal interpretations like macroscopic superpositionsand rapid flipping b etween macroscopically distinct states. We describe how the interpretation fitsneatly together with fully quantum formulations of statistical mechanics and that a measurementprocess can be viewed as a process of ergodicity breaking analogous to a phase transition. Thekey feature of the new interpretation is that joint probabilities for the ergodic subsets of statesof disjoint macro-systems only arise as emergent quantities. Finally we give an account of theEPR-Bohm thought experiment and show that the interpretation implies the violation of the Bellinequality characteristic of quantum mechanics but in a way that is rather novel. The final con-clusion is that the Copenhagen interpretation gives a completely satisfactory phenomenology ofmacro-systems interacting with micro-systems.

I. INTRODUCTION

The central mystery of quantum mechanics is presenteven in the simplest measurement on a qubit. Solving theSchrodinger equation for a suitable Hamiltonian gives anevolution of the form

c+|z+ +c|z |A0

c+|z+ |A+ +c|z |A ,(1)

where the two states|A are macroscopically distinctstates of the measuring device. But how can this be

consistent with the fact that when any experiment ofthis type is performed a definite outcome occurs either|A+ or|A? The Copenhagen interpretation1 solvesthe problem by collapsing the wave function, i.e. choos-ing only one of the distinct terms on the right-hand sidewith probabilities |c|2, respectively, on the grounds thatthe measuring device is macroscopic. The universal suc-cess of applying the rule disguises the fact that it is re-ally only a rule of thumb: when is a system sufficientlymacroscopic that it qualifies for collapse? This arbitraryseparation of systems into microscopic and macroscopicis the famous Heisenberg cut.

We can measure how macroscopically distinct the two

states of the measuring device|A are by estimatingtheir inner product. Let us suppose the measuring de-vice has a macroscopically large number of microscopic

[email protected] The Copenhagen interpretation is not really a completely settled

set of ideas. We are using the term to stand for the way thatmost working physicists successfully use quantum mechanics inpractice without having to even think about foundational issues.

degrees-of-freedom N. States are macroscopically dis-tinct if all the microscopic degrees-of-freedom are sepa-rated by a macroscopic scale L. If is a characteristicmicroscopic length scale in the system and assuming, say,Gaussian wave functions for the microscopical degrees-of-freedom spread over the scale , the matrix elementsbetween macroscopically distinct states is roughly

A+|A exp N L2/2 . (2)

Just to get a feel for the numbers, suppose N 1020, 1010m (atomic size) and L 104m, giving

e1032

In the following we will use to denote a generic scalecharacterising the inner products of macroscopically dis-tinct states. The estimate above is intended as a guideand the fact that this generic scale is so small will playan important role in this work.

There is an interesting analogue of the measurementproblem in classical statistical mechanics. Consider anIsing ferromagnet above its Curie temperature. In a typ-ical configuration, the spins point randomly up or downand there is no net magnetization. In the standard in-terpretation of classical statistical mechanics, the ensem-ble average captures a time average of the dynamics ofthe underlying microscopic state of the system. This isa statement of ergodicity: over time, interactions ensurethat the microscopic state explores all the available stateswith a probability given by the usual Boltzmann factor.Now, if the temperature is reduced below the Curie tem-perature, there is a phase transition and the magnet findsitself in an ordered state where the spins tend to line upin the same direction, either up or down, and the systemdevelops a net magnetization. At a microscopic level, er-godicity is broken and for a given initial micro-state the
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satisfactory phenomenology of macroscopic systems andis closely allied to the modern understanding of statisti-cal mechanics as it arises from the quantum mechanics ofsystems interacting with large environments or baths [2330]. Classical behaviour, the Born rule and wave functioncollapse are not put in by hand but are seen to emerge asan approximate phenomenology of systems with a macro-scopic number of degrees-of-freedom. Heisenbergs cut is

replaced by a continuous spectrum of classicality. Fi-nally, the new interpretation leads to a solution of themeasurement problem via the mechanism of ergodicitybreaking.

The paper is organised as follows. In sectionII we in-troduce the new interpretation by defining ontic statesin section II A. In section II B, we discuss disjoint sub-systems and how their ontic states can relate to the onticstates of their union. In section II C, we discuss sub-systems in general and the extent to which quantum de-scriptions of a system are to be thought of as effectivetheories with an in-build ultra-violet cut off. SectionII D then considers the dynamics of ontic states. Thistakes the form of a stochastic process which must sat-isfy a number of conditions. Most importantly, as wediscuss in sectionII E, it must be coarse-grained at thescale of the ultra-violet cut off in the temporal domain toavoid problems of other modal interpretations. SectionIII discusses how a recognisable classical ontology canemerge for macro-systems. This involves a discussion ofhow a classical ontology involves a patching together ofontic states of a number of macro-systems embedded ina larger environment. We show that it is meaningful todefine joint probabilities for disjoint systems but only inan emergent sense. SectionIIIB explains how the newinterpretation is related to modern formulations of sta-tistical mechanics built on quantum mechanics. Section

IV is devoted to a discussion of measurement. We firstshow in section IV A how a simple model without anenvironment can account for some features of measure-ment but also has a number of problems. In section IV B,these problems are all resolved in a more realistic modelthat includes the environment as well as a measuring de-vice that is not 100% efficient. In sectionVwe discussthe classical experiment of Bohm based on the originalthought experiment of Einstein, Podolsky and Rosen andshow the new interpretation gives a description that re-produces that of Copenhagen quantum mechanics butwithout the need to collapse the wave function. Finally,in sectionVI, we draw some conclusions.

II. THE EMERGENT COPENHAGEN

INTERPRETATION

A. Ontic States

A key feature of the new interpretation, as in someother modal interpretations, is the focus on sub-systemsof larger systems A S: see figure 1. For simplicity,

we assume that S is large enough so that its quantumdynamics is to an excellent approximation unitary. Inmost cases, we can assume that the state of S is pure|(t); indeed, much of the what we say will be indepen-dent of the exact state ofSwhether mixed or pure.5 Foreach sub-system,A Sfor which the Hilbert space ofSfactorizes as HS = HA HE, we can define a reduceddensity matrix by tracing over the Hilbert space of the

complement E:

A(t) = TrE|(t)(t)| . (3)

It is a theorem that the reduced density A(t) has a dis-crete spectrum:

A(t)|i(t) =pi(t)|i(t) , (4)

where the{pi(t)} are a set of real numbers with 0 pi(t) 1 and

ipi(t) = 1. The reduced density matrix

A is the epistemic stateofA.

The ontic state: at a particular instant of time, thenew interpretation asserts that A is actually in oneof the eigenstates|i(t). The state that is actuallyrealised is called the ontic state.

This property assignment is done at an instant of time tand therefore a degeneracy in the{pi(t)}is not realistic.However, dealing with problems that arise from degen-eracies, or more realistically near-degeneracies, as timeevolves is key to building a successful modal interpreta-tion. Note, also that the time-dependence of the onticstate|i(t) refers to the time at which the decomposi-tion (4) is made and it is important that these states donot generally solve the Schrodinger equation.6

Unlike other modal interpretations, we do not directlyinterpretpi(t) as the probability that A is in the onticstate|i(t), although this will emerge in certain situa-tions. In fact, the more fundamental probabilities in thenew interpretation are conditional probabilitiespi|j(t, t0)that, given the system was in the ontic state |j(t0) at anearlier timet0 < t, the system is in the ontic state |i(t)at timet. It is a hypothesis that these conditional prob-ability are related to the single-time probabilities pi(t)via

pi(t) = jpi|j(t, t0)pj(t0) (5)

Given this, there are two ways that the pi(t) can be in-terpreted as single-time probabilities:

5 Note that the mixed states that we have are all improper mix-tures and we do not need the concept of a proper mixture.

6 In fact, sinceAis generally interacting with Ethere is no conceptof a Schrodinger equation applying within the sub-system A.


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Initial condition: the key equation (5) shows that is anunambiguous way to define the single-time proba-bilities pi(t) via an initial condition. If at t0 thestate of S is a tensor product state |0(t0) |0(t0), then the ontic state at t0 is uniquely|0(t0). In that case,pi(t) = pi|0(t, t0) is the prob-ability that the ontic state is |i(t) at timet > t0.

Equilibrium:we will see, in section III B, that there isanother definition that is valid when A is a macro-

system in equilibrium with its environment E sothat A(t) is only slowly varying. In this case,for a characteristic time scale , pi|j(t, t0), witht = t0 + , becomes independent of j, the initialstate. In that case, pi(t) = pi|j(t, t0) (approxi-mately time-independent) is the probability of find-ing the system in the ontic state|i(t), indepen-dent of the initial state|j(t0).

One might imagine that the assignment of a particularontic state is tantamount to specifying a kind of hiddenvariable. As we will see this is potentially misleading

because the behaviour of ontic states is not at all likestandard hidden variables. In particular, the ontic statesofA are not global property assignments and the recog-nition that they are only intrinsic properties from theperspective ofA in relation to the rest of the total sys-tem is known in the literature on modal interpretationsas relationalism or perspectivalism [1215]. We want toemphasize that this is not really a philosophical stancebut is simply acknowledging what it means to performa trace that involves summing over states in a disjointsub-system to A.

There is an important additional detail to mentionhere. Since the total system A+ E is in a pure state

|, then assuming dA dE, where dA = dimH

A, etc.,because the environmentEis typically much bigger thanthe sub-system of interest A, each ontic states ofA, say|i, is precisely correlated with a particular ontic stateof E, which we call the mirror ontic state and denote|i. This follows because A and E have the samenon-vanishing spectrum and one way to exhibit the cor-relation is via the Schmidt decomposition of|:

| =i

pi|i |i . (6)

The states|i |i are a set ofdA orthogonal vectorsin the dAdEdimensional Hilbert space ofA +E. So the

property assignment of|i is always precisely correlatedwith the mirror assignment of|i to its complement. Wewill see in section II Dthat the mirror ontic states playan important role in the dynamics of ontic states.

Finally, we should point out that ontic states are ir-reducible in the sense that there is no further notion ofprobability on top of their inherent probability. In thisregard, we do not assume a priori the Born rule whichwe will have to ultimately derive from the behaviour ofontic states in realistic situations.

B. Disjoint Sub-systems

It is important that once we trace down to the Hilbertspace HA factor, we potentially forgo any knowledge ofthe ontic states of other disjoint sub-systems. Generallyworking with the sub-space A is only good for askinginclusive questions regarding the dynamics of ontic statesin a disjoint sub-system. In particular, as mentioned in

the last section, this means that ontic assignments for Acannot generally be taken as global property assignments.

The only information we can have on disjoint sub-systems A and B are the epistemic and ontic states ofthe combined system A + B . The ontic states of thelatter will not generally be related to products of onticstates ofA and B : see figure2. This means that it is not

A

|a

E

S

|i

B

FIG. 2. Two disjoint systems A and B interacting with alarge environment E. Ontic states of A and B are |i and|a, respectively, However, ontic states of A + B are notgenerally equal to tensor products |i|a and the ontologyis quantum.

generally consistent to make joint ontic assignments anddefine joint probabilities to sub-systems A and B . How-ever, in sectionIII A, we will see that under suitable cir-

cumstances such joint assignment and probabilities canemerge whenA and B are macro-systems embedded in amuch larger environment.

In addition, as we saw in the last section, the onticstate|i of a sub-system A is precisely correlated withthe mirror ontic state |i of theE, the complement ofAin the sense that A + Eis the total system. However, inthis case the pair of ontic states have no precise relationto that ofA +Ewhich is the pure state| in (6).

C. Sub-systems and Effective Theories

Since each sub-system A Senjoys its own set of on-tic states and, if the latter are to represent a propertyassignment, how are we to understand deformations inthe definition of the sub-system? This involves discretechanges when we decide to move degrees-of-freedom fromthe environmentEto A, and vice-versa. The new inter-pretation must explain how a recognisably stable classicalontology can emerge out of the myriad of different pos-sible sub-systems and their associated ontic states. Es-sentially it does this by recognising that classical states


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are not directly identified with a particular ontic stateof a particular sub-system but rather to coarse grainedhistories of ontic states and these are not sensitive to theprecise microscopic definition ofA. We may change thedefinition ofAby moving microscopic degrees-of-freedominto and out of A without affecting the collective be-haviour ofA that arises from coarse-grained time aver-ages of a time sequence of ontic states. In fact, we will

see that for the sub-systemA that is in equilibrium withthe environmentE, its classical state is identified withthe ensemble or epistemic state A(t). The collective dy-namics defined by the ensemble is expected to be stablewith respect to microscopic re-definitions ofA.

Before we describe time dependence within the newinterpretation, it is important that we establish the lim-itations of a particular quantum mechanical descriptionof a physical system. Many of the problems with exist-ing modal interpretations result from making unrealisticassumptions about the range of validity of the quantumdescription. Analysing a quantum system involves identi-fying an appropriate Hilbert space and Hamiltonian suchthat the resulting dynamics is unitary. However, suchdescriptions can only be approximately valid above someparticular length or time scale.7 Equivalently, using theuncertainty principle, below a particular momentum orenergy scale. For instance, consider a scalar particle. Atlow enough momenta, non-relativistic quantum mechan-ics is a good approximation and particle number is effec-tively a conserved quantity so it make sense to write downeffective theories by taking a sector of the full Hilbertspace with one particle H1. Such a description, however,will break down when the momentum increases and rel-ativistic effects become important. This is governed bya momentum scale mc, a length scale /mc(the Comp-ton wavelength), an energy scale mc2 and a temporal

scale /mc2. In fact, non-relativistic quantum mechan-ics based on the one-particle sector leads to violationsof causality for measurements based on finite spatial re-gions on the scale /mc. In that case, to recover a con-sistent and causal description one must allow particles tobe created and destroyed and the effective theory in theone-particle sector is no longer valid; one must insteadwork with a much larger Hilbert space containing all-particle sectors H0H1 in order to have a unitarydescription. Of course, this is where quantum field the-ory becomes the more appropriate formulation. At evenhigher momenta, this description could break down, forinstance, if the particle were a composite. At some high

momentum scale the constituents would become impor-tant and a different effective theory would be needed.The message here is this that, when we analyse a typ-

ical quantum system, we are inevitably doing a low mo-mentum (or low energy, large distance/time scale) ap-proximation. In this effective description there is no sense

7 This point was made in [1] and elaborated in discussions withJacob Barandes [46].

in which the effective Hilbert space HA is a factor of thetotal Hilbert space of the universeassuming that thelatter even makes sense. In other words, it is not evenclear thatAhas a well-defined parent system in the sensethat HS= HA HE. However, in the spirit of effectivetheory one can imagine that we can identify Ewith allthe degrees-of-freedom at the scale of the effective the-ory that directly interact, or are entangled, with A. The

only requirement is that S is chosen to be large enoughin order to achieve an approximately unitary descriptionof the overall dynamics. For overall consistency, it isimportant that the detailed nature of E, the so-calledenvironment or bath, is largely irrelevant for the be-haviour of the sub-system A. This turns out to be thecase as long as dE dA. One could say that it is crucialthat the environment is present but that its details arelargely irrelevant, including the overall state ofS, exceptin very special situations where the initial state is verynon-generic. The latter occurs when a carefully designedmeasuring device is interacting with a microscopic quan-tum system.

Given that the analyses of quantum systems areonly effective descriptions valid above some particularlength/time scale, the so-called ultra-violet cut off, it isimportant that that the new interpretation yields a for-malism that that is not sensitive to phenomena on thecut off scale. This does not means that the new inter-pretation is not applicable to shorter distances or times,but in order to be valid at more refined scales one wouldneed to apply it to the more fundamental effective theorythat takes over at this scale. The importance of this ob-servation is that many of the problems suffered by othermodal interpretations result from issues that involve ar-bitrarily short distance and time scales well beyond thevalidity of the effective theory. The new interpretation,

on the contrary, is immune from these difficulties becauseit acknowledges the inherent limitations of an effectivedescription. The fact that analyses of quantum systemsare only effective, means that there are intrinsic errors toany calculation which involve powers of the characteristiclength/time scale to the length/time ultra-violet cut off.

D. Ontic Dynamics

If the issue of how to define ontic states at a given in-stant of time is simple to state, the issue of how onticstates change in time is a subject fraught with problemsfor all modal interpretations. Our conclusion is that pre-vious approaches are fundamentally flawed because theymake unrealistic expectations as to the limits of the va-lidity of the analysis. They do not recognise the key fact,discussed in the last section, that an analysis of a quan-tum system is generally only an effective one and so isvalid only on distance and time scales that are greaterthan some specific ultra-violet cut off. It is then perhapsno surprise that problems arise when one tries to con-struct a theory of ontic dynamics that is continuous in


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time.A key principle, therefore, is that it only makes sense to

define ontic dynamics on a temporal coarse graining scalethat is the ultra-violet cut off scale. So for a particle, forinstance, this would be /mc2. As long as this scaleis smaller than any characteristic decoherence time scaleof the system under study then the effective descriptionis valid. We denote the

characteristic decoherence time scale =

Then the validity of the effective theory requires that

. (7)

The scalewill be defined more precisely later. The con-dition (7)ensures that (i) the effective theory is a validdescription and (ii) the coarse graining appears smooth atthe scale of time-dependent phenomena of the system. Tosummarise, our coarse graining scale lies at ultra-violetcut off scale of the effective theory and for consistency ofthe effective description, as long as (7) is satisfied and a

Markov condition respected, the whole formalism is theninsensitive to the exact value of since the discretizationerrors are of order /. However, we cannot attempt totake the cut off 0 and remain within the validityof the effective theory. Note that even the dynamics ofthe epistemic state governed by the Schrodinger equationcannot be considered more fundamental because it is alsoonly an equation valid within the effective theory.

Since it is the interaction between A and E that isresponsible for the time dependence of the probabilities

pi(t), on the time scale , and the reason that the onticstates |i(t) do not satisfy the Schrodinger equation, thefact that the coarse graining scale

means that over

a time step the set of probabilities and ontic states donot change much. The implication is that there exists aunique one-to-one mapping between ontic states at timetwith those att + . We can use the freedom to permutethe labels of the ontic states at each time step to ensurethat the mapping associates |i(t) with |i(t+) in thesense that

j(t+)|i(t) ij+ O(/) , (8)

in which case, the associated individual probabilities onlychange by a small amount

pi(t+) pi(t)1 + O(/) . (9)It is important to emphasise that this continuity condi-tion (8) is defined at the temporal scale and we arenot at liberty to take the limit 0 since this wouldgo beyond the domain of applicability of the effectivetheory. This is fortunate because it saves the new inter-pretation from the scourge of macro-flips, a disease thatinfects other modal interpretations. A macro-flip occurswhen the eigenvaluespi(t) andpj(t), for two macroscop-ically distinct ontic states, try to cross. But generically

the eigenvalues repel each other and this leads to anextremely rapid flip, over a time scale of order , ofthe ontic state from|i to the macroscopically distinctstate|j, or vice-versa. Given the importance of thisissue, we describe it more fully in section II E. Havingsaid that, apart from solving the problem of macro-flips,having a small but finite cut off will not affect the dy-namics over the physically relevant time scale since the

discretisation errors involve powers of/ 1.Given that the total systemS= A+Eis assumed to bea pure state |(t), the time dependence of the epistemicstate A(t) is determined by solving the Schrodingerequation forSgiving

A(t) = TrE

U(t, t0)|(t0)(t0)|U(t0, t)

. (10)

where U(t, t0) is the unitary time evolution operator in S.We want to emphasize that it is really only meaningful todescribe this epistemic dynamics on the coarse grainingscale .

The problem before us is to write down a similar dy-

namical equation for the probability that the system Ais in the ontic state|i(t+) given that it was in theontic state|j(t):

pi|j(t) pi|j(t+, t). (11)

These conditional probabilities, along with the Markovproperty discussed below, define a stochastic process. Inorder to be consistent with the probability constraint (5),we must have

pi(t+) =j

pi|j(t)pj(t) , (12)

but this does not determine the stochastic processuniquely so its definition is a hypothesis. However, wewill argue there are some additional natural constraintswhich lead us to a unique process:

Markov: ontic states carry no memory of their previoushistory and so the conditional probability to be inthe ontic state|i(t+) should only depends onthe ontic state|j(t) and not on ontic states atearlier times. This condition is fundamental to ourwhole approach since it ensures that we can buildup the more coarse grained dynamics in terms ofthe microscopic transitions (11) at the ultra-violetscale .

Locality: the stochastic process should be driven by thelocal interaction between A and Eand so in orderthat it leads to a local form of dynamics we requirethat it depends only on the initial and final onticstates|j(t) and|i(t+) ofA and their onticmirrors|j(t) and|i(t+) ofEas well as Hintthe part of the total Hamiltonian

H= HA IE+ IA HE+ Hint , (13)


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that describes the local coupling between A andE.8

Ergodicity: generically we require that any state canreach any other state in a finite number of steps.However, this must break down for two macroscop-ically distinct states|i and|j, for which werequire that there should only be a minute prob-ability of order (a typical inner product of twomacroscopically distinct states described in sectionI) of a transition between them over a time scale .

As long as the stochastic process has these properties,then its actual microscopic details are largely irrelevantto the behaviour of macro-systems. The Markov prop-erty is very natural given that in the new interpretationthe ontic state of a system is just a property assignmentat a particular time and has no memory of previous onticstates. Moreover imposing this condition is fundamentalbecause it means that the ultra-violet dynamics pi|j(t)determines the whole stochastic process since over a se-ries of time steps tn = t+ (n 1), according to theMarkov property,

pjN|j1(tN, t1) =

j2,...,jN1

N1n=1

pjn+1|jn(tn)

. (14)

The resulting stochastic process is therefore a conceptu-ally simple discrete-time Markov chain.

We now turn to the definition of the ultra-violet dy-namics. A useful observation involves the matrix ele-ments

Vij(t) =pi(t+)pj(t)

1/2

Re

i(t+)

| i(t+)

|U(t+, t)

|j(t)

|j(t)

,

(15)

where |i are mirror ontic states ofEdefined in (6) andU(t+, t) is the unitary time evolution operator inA+E.Note that these matrix elements are completely symmet-rical between A and Eand the resulting dynamics willconsequently respect the exact correlation between theirontic states. One finds from this definition and (6), that

pi(t+) =j

Vij(t) , pj(t) =i

Vij(t), (16)

so it it tempting to relate

Vij(t) pi|j(t)pj(t). (17)

But we cannot have equality here, because the matrix el-ementsVij(t) are not necessarily valued between 0 and 1.

8 Issues involving locality should properly be formulated in termsof relativistic quantum field theory, so our notion here is moreprimitive.

However, we can proceed as follows. If we have labelledthe states to be consist with the continuity condition (8),it follows that since

pi(t+) pi(t) =j

Vij(t) Vji(t)

, (18)

the right-hand side must be small of order /. We candefine the ultra-violet process by taking, fori

=j ,9

pi|j(t) = 1

pj(t)max

Vij(t) Vji(t), 0

(19)

and

pi|i(t) = 1 j=i

pj|i(t). (20)

This process satisfies the probability constant (12).It is important to realize that there is no guarantee

that the process defined above is consistent in the sensethat the conditional probabilities pi|j(t) are valued in theinterval [0, 1]. In fact, the consistency conditions are

j=i

pj|i(t) 1 , i . (21)

As 0, the elements pi|j(t), i= j, can be made ar-bitrarily small and so the process can always be madeconsistent in this limit. Hence, there is an upper boundon how big the cut off can be taken. In order to inves-tigate the this, we can interpretpi for a single time stepas being due to a mismatch between the flows into andout of the ith state, that is

pi=

j=ipi|jpj

j=ipj|ipi . (22)

The net flow out of the ith state involves the sum on theleft-hand side of (21). If we define the decoherence timescale by

=

supi

j=i

pj|i

1. (23)

So consistency of the process requires . Note thatif then this does not imply a breakdown of theformalism but rather a breakdown of the validity of theeffective theory: one should go to a more fundamentaleffective theory valid at smaller distance/time scales.

Generically, both terms on the right-hand side of (22)will be of the same order, so that

pi Opi/

. (24)

9 Using these expressions it is easy to see that the process definedin (19) and (20) above agrees with the one defined in [1]: to

compare formulae the quantities p(n)ij in [1] are the conditional

probabilities pi|j(tn) here.


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However, later, we will describe the situation when A isin equilibrium with the environment, in that case the piwill be approximately constant on account of a balancebetween the flows into and out of the states in (22). Itis important to notice that ontic states with very smallprobabilitiespj(t) do not give anomalously large values of

pi|j(t) as might be inferred from (19) because the factorof pj(t) in the denominator is generally balanced by a

factor of a similar order in the numerator.In the limit , we can evaluate the matrix elements

(15) in perturbation theory:

Vij =

pipj

Im i| i|Hint|j |j + . (25)

The fact that the matrix elements in (25) only depends

on the coupling Hint and the ontic states|i and|jand their mirrors|i and|j encapsulates the localityrequirement. Note that at this leading orderVij = Vji.

In addition, the fact that the transition probabilitiesdepend on a matrix elements (25) involving the states

|i |i means that the stochastic process will be seento satisfy the ergodicity requirement. Essentially, if thetwo states are macroscopically distinct states, then wecan expect pi|j(t) will be suppressed by a factor of order relative to the generic situation. The probability thatthe system will make a transition between the ontic stateswill therefore be vanishing small. In fact, over a time Tthe chance that the system will make a transition fromone macroscopically distinct state to another would beorder T/. It is clear, given the crude estimate of

in sectionI,we would have to wait of the order ofe1032

times the age of the universe to see such a transition.It is worth emphasising that the ultra-violet dynam-

ics we have defined is not unique. The ambiguity corre-sponds to changing

pi|j =ijpj

,i

ij =j

ij = 0 , (26)

subject to the constraint (21). However, there are noobvious quantities ij that could be defined that are atthe same time consistent with the ergodicity requirement.We take the process that we have defined as being a hy-pothesis on the same level as the Schrodinger equationthat determines the dynamics of epistemic state. How-ever, it is possible to take an agnostic point-of-view and

avoid a concrete microscopic definition of the stochasticprocess because:

As long as the ergodicity condition is satisfied, alongwith the key probability relation (5), the microscopicdetails of the stochastic process are actually irrele-vant for reproducing standard Copenhagen interpre-tation phenomenology of macro-systems which are inequilibrium with their environment (as described insectionIII B).

Finally, it is worth making clear the point that, al-though we have introduced an auxiliary stochastic pro-cess to define the dynamics of ontic states, this is com-pletely different from dynamical collapse models dis-cussed in the literature; for example in the review[43]. The latter involve stochastic modifications ofSchrodingers equation itself, in other words they involveintroducing stochastic dynamics for the epistemic state

which is a completely different philosophy from the onewe are setting out here.

E. The Continuum Process and Macro-Flips

It is tempting, even though it runs counter to themethodology of effective theory, to take the stochasticprocess that we defined in the last section and take thecut off 0, in order to define a continuum pro-cess. However, typically one should expects pathologiesto arise when effective theories are pushed beyond theirrange of validity. Indeed, in the present case, a pathology

manifests as the existence of micro-flips that represent asevere problem for existing modal interpretations thatinsist on following ontic states continuously in time. Inthis section, we will describe how they arise and how thenew interpretation avoids them.

The ontic states of a sub-system|i(t) are definedcontinuously in time and so it is tempting to define onticdynamics that is also continuous in time. In fact, thecontinuum limit of (18)takes the form

dpidt

=j

Jij , Jij = Jji , (27)

with

Jij =2

pipj

Im i| i|Hint|j |j , (28)

which is equal to the perturbative form of (Vij Vji)/using (25). Written in this form, manifests the fact thatifA does not interact withEthenJij vanishes.

10 In thatcase, a continuum Markov process can be defined whosemaster equation takes the form

dpidt

=j=i

Tijpj Tjipi

, (29)

corresponding to transitions into and out of|i. Forthe stochastic process satisfying the ergodicity condition

10 Note that the expression for Jij seems to be missing a terminvolving a time derivative compared with [1,10]. However, it iseasy to see that, since t(|j |j) = t|j |j + |j

t|j and the sets of states {|j} and {|j} are orthonormal,this term actually vanishes. In addition, the total Hamiltonianmay be replaced by Hintfor the same reason.


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10

More fundamentally, the new interpretation makesstatements about ontology that are precise but they arenot necessarily of a classical kind. For instance, as dis-cussed in sectionII B, an interpretation based on reduceddensity matrices cannot generally make joint property as-signments to two disjoint systems A and B let alone as-sociate probabilities to them. The only statements thatcan be made involving both A and B are via the com-

bined systemA +B and in this case its ontic states willgenerally not be a tensor product of ontic states of thesub-systems, i.e.|i |a. The relation between on-tic states of the three systems A, B and A+B will bemore fuzzy and potentially contradictory. One could saythat the fragments of reality cannot be drawn togetherto form a consistent whole. We call this a quantumontology.

However, a familiar classical ontology can be anemergent phenomenon in the following sense. SupposeA and B are two disjoint weakly-interacting or causallyseparated macro-systems in, or close to, equilibrium andhence strongly entangled with the environment E withdA

, dB

dE

. We expect in these circumstances that theontic states ofA + B will indeed approximately factorizeinto a tensor product of the ontic states ofA and B. Themismatch will involve typically minute order effects:

|m(i,a) = |i |a + O(), (35)

where m = m(i, a) is a 1-to-1 map. What this meansis the descriptions provided by A, B and A+B can beintegrated into a consistent whole, at least to high degreeof accuracy.

In this context, it is meaningful to make joint prop-erty assignments forA and B and we can interpret, in anemergent sense, the probabilitypm(i,a) as the joint prob-

ability for a pair of ontic states|i and|a ofA andB:

p(i, a) emergent

= pm(i,a) . (36)

It is the view from A+B that is needed to follow anypotential correlations between the ontic states ofA andB. Note also that the ontic states ofA or B still cannotgenerally be taken as global property assignments outsideof the triplet of sub-systems A, B andA +B.

The emergent joint probabilities satisfy the usual prob-ability relations, but only to order ,

a

p(i, a) = pi+O

(),

i

p(i, a) = pa+ O(),(37)

although

iap(i, a) = 1.In general, the ontic states ofA and B can be corre-

lated in a classical (non-entangled) sense when A+B=A B meaning that

p(i, a) =pipa . (38)

The case when A and B are not correlated correspondsto when A+B = A B andp(i, a) = pipa.

Note that we will meet an example in section V anexample where the ontic states ofA + Bare tensor prod-uct states ofA and B but the factors of one of them arenot the ontic states of the corresponding sub-system. Sosimply being a tensor product state is not sufficient tohave a classical ontology.

The picture above of emergent joint ontic assignmentsand joint probabilities can be generalized to many weaklyinteracting or causally disconnected macro-systems A1+ +An. An ontic state of the parent system will be,to order , a tensor product state|m(i(1),...,i(n)) =|(1)

i(1) |(n)

i(n), and so emergent joint probabilities

can be defined of the form

p

i(1), . . . , i(n) emergent

= pm(i(1),...,i(n)) . (39)

The picture we have here is that the emergent classi-cal world involves patching together very slightly differ-ent descriptionsdiffering at

O()of the same systems

from the point-of-view ofAi, Ai+ Aj, Ai+ Aj+ Ak, etc.For more microscopic systems this integration of onticstates becomes more ambiguous and a classical descrip-tion evaporates to be replaced by an ontology that istruly quantum. We can quantify the degree of classical-ity in terms of the generic scale . So we can expectsystems to exhibit quantum fuzziness when is not sosmall so that the relation between the ontic states be-comes ambiguous and joint ontic assignments and jointprobabilities cannot be consistently defined.

B. Link with Statistical Mechanics

A key requirement of an interpretation of quantummechanics is to explain how the classical behaviour ofmacroscopic systems emerges. Macro-systems with manydegrees-of-freedom are complicated systems whose collec-tive behaviour is captured by the techniques of statisticalmechanics. It seems natural that any quantum origin ofclassical behaviour must, at the very least, be able to givea consistent foundation to classical statistical mechanics.In fact, we might hope that such an understanding wouldput classical statistical mechanics on a firmer conceptualfooting given that it is still, somewhat surprisingly, a con-troversial subject. In particular, there is no consensus on

the role of probability, the meaning of entropy and therelation of ensemble averages to time averages.

In the last few year a rather different and intrinsicallyquantum approach to the subject has been developed[2330]. In this approach, ensembles arise at the quantumlevel when a system A is entangled with a large thermalbath, or environment, E. So the total system S= A + Ecan be in a pure state but, nevertheless, the sub-systemAhas a reduced density matrix that defines an ensembleA. In this point-of-view, the thermodynamic entropy


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11

of A is precisely the entanglement entropy of the sub-systemS= Tr(Alog A) which is non-vanishing whenA is non-trivially entangled with the bath.

When the bath is much bigger than the system dEdAthere are some very powerful principles that emerge.

11

For almost any pure state of the total system in somesubspace HR HSdescribed by some global constraintR (preserved under time evolution) the reduced density

matrix of the system A is approximately equal to

A TrEPR

dR, (40)

where PR is the projection operator on the subspaceHR. If the global constraint is on the energy and theinteraction between the system A and the environmentEis sufficiently weak then it is straightforward to showthat A is approximately the canonical ensemble,

A e HA

Z , Z= TrAe

HA . (41)

However, it is important that the principle applies toother possible constraints R on the total system, includ-ing the absence of a constraint, and also to cases wherethe interaction between the system and bath is not small.

The implications for statistical mechanics are evident.If the total system starts out in almost any state in thesubspace HR, then the state of the sub-system A is ap-proximately the state (40) independent of time. Thisdescribes a situation where the system is in equilibriumwith its environment. In this analysis, the entropy be-comes an objective property of the state ofA caused byentanglement with E.

More general questions involve situations when the

system starts off in a state which is not in equilibrium[28,29]. The results are summarised below:12

Equilibration: subject to some reasonable conditions,every pure state of S is such that a small sub-system A Swill equilibrate meaning that A(t)approaches a limit which fluctuates about a con-stant. Note that the initial state does not need tobe typical, in other words even though the over-whelming number of states ofSare such that A isin equilibrium already, it is also true of initial stateswhereA is far from equilibrium. This includes ten-sor product states| = | |.

Bath independence: in the case that the initial stateis a tensor product | = | |, the equilibriumstate ofA is independent of the state of the bath|.

11 The following discussion here is taken mainly from Popescu,Short and Winter [23] and Linden, Popescu, Short and Winter[28,29].

12 It is also possible to incorporate the constraint R on the system.

Sub-system independence: there are general condi-tions under which the equilibrium state ofA is in-dependent of| the initial state ofA. However,there are also non-generic situations for which theequilibrium state depends sensitively on|.

It is clear that the focus on sub-systems means that thisre-formulation of statistical mechanics is closely related

with modal quantum mechanics. Moreover, the new in-terpretation adds a new and important detail to the storythrough the existence of the ontic state of the sub-systemA. This is analogous to the micro-state of classical sta-tistical mechanics. However, it is important to point outthat its dynamics, described by the stochastic process de-scribed in sectionII D, is conceptually simpler than thedynamics of micro-states in classical statistical mechanicsbecause in the quantum case the number of ontic statesis always finite and the stochastic process is a simplediscrete-time Markov chain.

When the system A equilibrates, A(t) and its eigen-values will fluctuate around a slowly varying quasi-equilibrium and the underlying Markov chain becomes

approximatelyhomogeneous: that is the transition ma-trixpi|j(t) becomes time independent over time scales oforder . Of course, there may be much slower time de-pendence for on scales . The rate of flow into and outof each ontic state in (22) approximately balances. Undergeneric conditions, although the microscopic transitionsbetween a pair of states only go one way, over a finitenumber n of time steps pi|j(t+n, t) is a matrix whoseentries are all > 0 and hence the equilibrium process isregular. The meaning of this is that any ontic state isonly a finite number of time steps away from any otherstate. The fact that the process is a regularhomogeneousMarkov chain implies that it is also ergodicand then it

is a standard result that

limn

pi|j(t+n, t) =pi(t+n), (42)

independent ofj , and so, whatever the initial state, aftera large number of steps, the probability distribution isequal to pi(t). More precisely, one can show that thenumber of time steps must be at least order /; in otherwords, one must wait for a time of the order of , thedecoherence time defined by (23), for the memory of theinitial state to be lost:

equilibrium: pi|j(t+, t) pi(t+) (43)

This means that, in equilibrium, the actual ultra-violetdetails of the stochastic process are hidden over timescales of order , the decoherence time.13

13 As a simple example of an equilibrium process, suppose that atequilibrium A is maximally entangled with E, so pi 1/dA.If we define the process by taking one of each pair {pi|j, pj|i}randomly and giving it the value p, while the other vanishes.


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In addition, ergodicity means that, in equilibrium, thetime average of a temporal sequence of ontic states overa time scale of order is well approximated by the en-semble average described by A(t). This gives us anotherway to interpret the single-time probabilities pi(t): whenAhas reached equilibrium with the bath, the single-timeprobabilities pi(t) are approximately constant and areequal to the probability that, in a suitably coarse-grained

time average, the system is in the ontic state|i(t), in-dependent of the initial state in the past. From the point-of-view of the emergent classical view, the ontic state ofa macro-system in equilibrium is hidden and the proba-bilities pi(t) can be given the ignorance interpretation:

When a system is in equilibrium with its en-vironment, the ensemble associated to theapproximately constantreduced density matrixA(t) captures the time average over the dynamicsof the ontic state. So we can associate the classi-cal description of a macro-system with the collectivebehaviour of the ensemble A(t).

It is noteworthy that the equilibrium state A(t) doesnot depend on the details of the initial state of the bath,however, the same cannot be said in all circumstances forthe initial state ofA. In certain situations, discussed in[28], the final equilibrium epistemic state ofAcan dependvery sensitively on the initial state. This is connected toa breakdown of ergodicity of the stochastic process. Wecan expect this to happen when the equilibrium stateof A has sets of ontic states which are macroscopicallydistinct. In these conditions, the matrix elements Vij(t)between states in different sets will be minute of order/. In this case, ergodicity of the stochastic process isbroken and once the system has equilibrated the resulting

time average of a temporal sequence of ontic states isthen only captured by a sub-ensemble of A(t). We couldsummarise the situation by saying that:

In the new interpretation, both the correlatathe ergodic subsets of states described by the sub-ensemblesand the correlations, in the form of the

joint probabilities (36), are emergent quantities.

In this situation, the dynamics of the underlying onticstate as described by the stochastic process is very sensi-tive to the initial state ofA. This is precisely what hap-pens in a phase transition in a statistical system like theIsing magnetic described in the introduction as the tem-

perature is lowered. But it also describes what happens

In this case, if one takes pi|j(t+ T , t) pi|k(t+ T , t) for somerandomly chosen distinct i, j and k, then this approaches zeroas exp(T/), with the time to approach equilibrium =2/(pdA), valid when . On the other hand, the decoher-

ence time is defined in (23) as

j=ipj|i1

2/(pdA) =so this confirms that the equilibrium time is equal to the deco-herence time.

in a quantum measurement where the measuring device isdesigned to be very sensitive to the quantum state of thesystem being measured. We will argue in IV, that whenthe measuring device equilibrates with the environmentthere is a breaking of ergodicity of the underling Markovchain such that each ergodic component corresponds toa distinct measurement outcome. Collapse of the wavefunction corresponds to a tidying up exercise in which one

removes the ergodic components that are not reachablefrom the component that corresponds to the particularmeasurement outcome that is realized, as illustrated infigure4.

ontic state

epistemic state A

t

sub-ensembles

(1)A

(2)A

FIG. 4. An illustration of the process of ergodicity breakingduring a measurement. Here, the set of of ontic states splitsinto two distinct sub-ensembles between which the probabil-ity of transition is vanishingly small. Two ontic histories areshown that end up in different sub-ensembles after the mea-surement.

IV. MEASUREMENTS AND THE COLLAPSEOF THE WAVE FUNCTION

In this section we show how the new interpretation cangive a convincing account of the measurement process.There are some benefits in starting with a simple model,showing how it successfully describes certain aspects butalso how it has certain limitations. Then we introduce amore sophisticated model which preserves the good fea-tures and solves the problems. This shows that modelsneed to be realistic otherwise one can be led to misleadingconclusions. Along the way, we will need to use intuitionfrom the theory of decoherence, e.g. [1622].

A. A Nave Model

The simplest set up involves an idealized von Neumannmeasurement on a microscopic system. Let us supposethe systemPis some finite dimensional quantum systemwith some basis states |i that are eigenstates of the op-erator associated to the observable we want to measure.The system P is then coupled to a measuring device A


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13

through a Hamiltonian which acts as

H|i | = |i H(i)| , (44)

If the initial state of the combined system is taken to bethe non-entangled state

|(0) = i

ci|i |(0) , (45)then using the linearity of the Schrodinger Equation wehave

|(t) =i

ci|i |i(t) , (46)

where |i(t) is the state that evolves from |(0) via theeffective Hamiltonian H(i). In order for A to be effica-cious, it must be that the states|i(t) become macro-scopically distinct after a macroscopic time T. This isthe process of decoherence [1622]. We expect the in-ner product of the states|i(t) to exhibit a behaviourroughly of the formi(t)|j(t) exp N X(t)2/2 , (47)fori =j , whereX(t) describes how the distance betweenthe microscopic constituents of the measuring device be-haves between the two measurement outcomes. We ex-pect that this goes from 0 to the macroscopic scale L atthe end of the measurement att = T, and soi(T)|j(T) , i =j . (48)We can estimate how fast decoherence occurs comparedwith the macroscopic measuring timeTby assuming that

X(t) is linear in t. As an example, by taking the samevalues for N, and L as in sectionI, we find

T

L

N e16 . (49)

Now we apply the new interpretation to the model.The reduced density matrix of the measuring device is

A(t) =i

|ci|2|i(t)i(t)| . (50)

We can now follow the ontic states in time as decoher-ence occurs [1, 34]: see figure 5. In this case, one canverify that only the transition probabilitiesp2|1(t),p3|1(t)andp3|2(t) are non-vanishing and so the histories of onticstates are rather simple as illustrated in figure 5. Afterthe measurement is complete t = T, the ontic states ofA are approximately equal to the|i(T) up to termswhich are order . But, as emphasised earlier, is somuch smaller than any other dimensionless scale in theproblem and in particular much smaller than the errorsintrinsic in the effective theory and so can safely be ig-nored. Given that initially the measuring device has a

t

1

0

p1(t)

p2(t)

p3(t)

|(0)

probability

decoherence

|j(t)macroscopically

distinct

T

FIG. 5. An example of how the probabilities pi(t) might be-have for the case dP = 3. During the decoherence period0 t , the states |i(t) become macroscopically distinctwhile the probabilities change in time. At the end of the mea-surement at T , the eigenstates of the reduced densitymatrix are approximately the states |i(T) that occur withprobabilitypi(T). Also show in bold is a hypothetic solutionof the stochastic process in this case with two transitions.

unique ontic state|(0), the corresponding probabilityto be in the ontic state|i(T) after the measurement is

pi(T) = |ci|2 + O(). (51)

So the new interpretation yields a satisfactory phe-nomenology in this simplest measurement model and,furthermore, yields the Born rule for the probabilities.There are, however, problems with the model:

(i) When the initial microscopic system has a degen-eracy|ci| =|cj | for some i= j . In the case of exact de-generacy one can easily show that the associated eigen-

vectors of A(T) after the measurement are no longer|i(T) and |j(T), but rather involves the macroscopicsuperpositions [34]

12

|i(T) |j(T). (52)There is a superficial answer to this problem which saysthat a degeneracy of the state ofP requires infinite finetuning that is unrealistic in practice [46]. Whilst thisis true, one can show that even when there is no exactdegeneracy the onset of decoherence can be delayed inan artificial way by having nearly degenerate states. If

|c1

|2 = 1

2

+ es and|c2

|2 = 1

2 es then the decoherence

time is proportional tos and so can be increased bytuning close to the degenerate point.

(ii) The model does not satisfy our requirement thatthe ontic states of macroscopic systems are robust againstmicroscopic re-definitions of the sub-system. In this case,if we re-define A to include the microscopic system Pitself then the ontic state of A + P now becomes thepure state of the total system (46) with probability 1. Sothe slightly re-defined system, with the addition of onemicroscopic degree-of-freedom, has led to a completely


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different ontology involving unacceptable superpositionsof macroscopically distinct states.

(iii) As it stands the model describes the case of a per-fect measuring device that makes no errors. In realisticsituations one could imagine that A makes errors whichmanifests by less than perfect correlation between thestates ofP andA in (46). A more realistic situation has

|(t) = i

fij(t)|i |j(t) . (53)

with fij(t) f(0)ij , a constant, after the measurementtime t = T. Now the reduced density matrix of themeasuring device is

A(T) =ijk

f(0)ij f(0)ik |j(T)k(T)| (54)

and since the matrix f(0)ij need not be diagonal the ontic

states will involve superpositions of the macroscopicallydistinct states|i(T) [35].

B. A More Realistic Model

We now turn to a more sophisticated but realisticmodel which solves all three problems above. The newfeature is that the measuring device is also interactingwith an environment Eand so the complete system isP+A+E. The initial state of the system is

|(0) =

i

ci|i

|0(0) . (55)

Here, |0 is the initial state of the measuring device plusthe environment from which one deduces the reduceddensity matrix ofA by tracing out the environment:

A(0) = TrE |0(0)0(0)| , (56)

with eigenstates and eigenvalues

A(0)|a(0) =pa(0)|a(0) . (57)

We assume that the measuring device and environmentare in equilibrium and so in a highly entangled state.In that case, the emergent classical view is ignorant ofthe exact ontic state ofA. At equilibrium, the stochas-tic process described in section II D is ergodic and it isthe macroscopic time average of this process that corre-sponds to what we think of as the initial classical state ofthe measuring device and this is captured by performingthe ensemble average over A(0).

Evolving forward in time, and including measuring in-efficiencies, we expect to have a state of the form

|(t) =ij

fij(t)|i |j(t) . (58)

After the decoherence time t > and so certainly at theend of the measurement t = T, the states ofA+B aremacroscopically distinct:i(T)|j(T) , fori =j (59)andfij(t) f(0)ij , a constant matrix. Note that unitarityrequires that the matrixf

(0)ij satisfies

ij

f(0)ij 2 = 1 + O(). (60)The reduced density matrix of the measuring device plusenvironment A +Etakes the form

A+E(T) =ijk

f(0)ij f(0)ik |jk| , (61)

and so the ontic states ofA +Earej

Z(i)j |j(T) + O(), (62)

where the Z(i)j are eigenvectors of the matrix with ele-ments:

Mjk =i

f(0)ij f(0)ik ,

j

MjkZ(i)k =

(i)Z(i)j .

(63)

So, unfortunately, the ontic states involve macroscopicsuperpositions of states ofA + Eof the kind that we sawin the simple measurement model in the last section.

However, if we examine the ontic states of A, whichis the physically relevant sub-system, a more satisfactorypicture emerges. Since the states of the environment areto a high degree orthogonal for i =j , the reduced densitymatrix forA takes the form

A(T) =ij

f(0)ij 2(j)A (T) + O(), (64)where we have defined the component reduced densitymatrices, one for each of the outcomes,

(j)A (T) = TrE |j(T)j(T)| . (65)

Since the states|i(T) have such a small inner product(59), the component density matrices commute to highdegree of accuracy

[(i)A (T),

(j)A (T)] = O() . (66)

The ontic states |a(T), the eigenvectors of A(T), will,therefore, split up into dP mutually ergodically inacces-sible setsE(i), as well as a setE0 of approximately nulleigenvectors that play no important role. The componentdensity matrices take the form

(i)A (T) =

aE(i)

p(i)a|a(T)a(T)| + O(). (67)


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Therefore the ontic state|a(T), a E(j), has A(T)eigenvalue

pa(T) =i

f(0)ij 2p(j)a + O() . (68)It does not involve a macroscopic superpositionat leastto order and so compared to (54), one can see thatthe environment has the effect of yielding an acceptablephenomenology.

Under the stochastic process, even for macroscopictime scales, there is only a minute probability for |a(T)to make a transition into another state|b(T) in a dif-ferent ergodic set a E(i) and b E(j), for i= j. Forfixed j, the group of ontic states|a(T), a E(j), arenot macroscopically distinct, and so ontic states can ef-fectively only make transitions within this set. We can,in principle, use the stochastic process to work out theprobability pa|b(T, 0) that an initial ontic state|b(0)evolves to |a(T) using (14). However, according to ourdiscussion below equation (57), since the measuring de-vice is assumed to be in equilibrium with the environment

its ontic states are undergoing constant transitions. Thetime average of this process is captured by taking the en-semble average over A(0). With this initial averaging,we can use (14) to compute the probabilities that thesystem is finally in the ontic state|a(T):

b

pa|b(T, 0)pb(0) =pa(T). (69)

The result is simply the single-time probability in (68).Turning to the final state, as far as the emergent classi-cal ontology is concerned, the ontic states in an ergodicsetE(j) cannot be distinguished and are also undergo-ing constant transitions between themselves. Thereforeit is only meaningful to compute the inclusive probabilitythat the system ends up in the j th set:

p(j)(T) =

aE(j)

pa(T) =i

f(0)ij 2 + O(), (70)where we used the fact that Tr

(j)A = 1 so that

aE(j)p(j)a = 1 +O(). It is particularly notewor-

thy that the final result here is actually independent ofthe detailed form of the microscopic stochastic processthat we introduced in sectionII Dbecause we averagedover the initial ontic state and the calculated an inclu-sive probability for a sub-ensemble in the final state. The

non-trivial role of the stochastic process is that it was de-fined in such a way so as to lead to ergodicity breaking.Note that for a perfect measuring device fij(t) = ciijand so p(j)(T) = |cj |2 which is just the Born rule.

Notice that, although the individual probabilitiespa(t)can depend on time as a result of the interaction ofAwith the environment, the sum over the group is inde-pendent of time to order , assuming, as we have, thatthe measuring device settles down and ceases to makeerrors fort > T. It is clear that the more realistic model

solves problems (i) and (iii). Firstly, (iii) is solved be-cause the ontic states |a(T) do not involve macroscopicsuperpositions. In addition, (i) is solved because the neardegeneracy|ci| |cj | no longer leads to macroscopic su-perpositions or artificially extended de-coherence times.

Now we turn to problem (ii). The reduced densitymatrix ofA +P is

A+P(T) =ij

f

(0)

ij2|(j)(j)| (j)A (T) + O(),

(71)

where

|(j) =

i

f(0)ij 21/2i

f(0)ij |i (72)

and so the ontic states are

|(j) |a(T) + O(), a E(j) . (73)

The probabilities are identical to (68). Note that the

ontic states(73) are those ofA not entangled with statesofP. Hence, focussing on A + P rather thanA does notlead to the disastrous change in the ontology seen in thesimple model. However, unless the measuring device isperfect, the states|(j) are not the ontic states of P,they are not even orthogonal.

C. Collapse of the Wave Function

After the measurement has been completed, thestochastic process ensures that there are effectively notransitions between the states in different ergodic sets.

As far asA is concerned, if the state |a(T) hasa E(j)

then for all practical purposes for calculating future dy-namics, it would be prudent, but not necessary, to removethe other terms i= j from As reduced density matrix(64) leading to the replacement

A(T) (j)A (T) . (74)

So the collapse of the wave function in the new interpre-tation is just the innocuous process of removing termswhich could only have an effect of order in the future.As we have repeatedly emphasised such effects are manyorders of magnitude smaller than other kinds of system-atic errors that are inherent in the model. So if we con-

dition the future of a system on its present statethe er-godic sub-ensemble of ontic states at a given timethencollapse of the wave function is a harmless procedure ofremoving terms that are ergodically inaccessible.

D. Measurement of Continuous Quantum Systems

Measurements on microscopic systems with continuouseigenvectors, like the position of a particle, have caused


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problems for modal interpretations [3739]. In this caseone might expect a generalization of (46) to be

|(t) =

dx (x)|x |x(t) . (75)

Here,|x(t) is a macroscopic state ofA indicating thatthe particle is at x. The reduced density matrix of themeasuring device is then

A(t) =

dx |(x)|2|x(t)x(t)| . (76)

The potential problem is that even though the states|x(t) are expected to become approximately orthog-onal, for instance, one expectsx(T)|y(T) exp N(x y)2 , (77)the eigenvectors of A(T), contrary to nave expectations,are not localized in x. In fact, if the original wave func-tion of the particle (x) is spread out over a range xthen the ontic states of the measuring device involve aspread of the states |x(t) over the same range. In otherwords, the ontic states ofA would involve macroscopicsuperpositions. This is obviously a potential disaster forany modal interpretation.

The loop-hole in this thinking was identified in [1].In fact we should be on alert for any argument, likethat above, that relies on the smoothness of wave func-tions down to be arbitrarily small scales as this is notrealistic and in the spirit of effective theory. In factnon-relativistic quantum mechanics based on the one-particle truncation of the multi-particle Hilbert spacebreaks down on length scales /mc. So any attempt tomeasure the position of the particle down to these scalewill inevitably involve particle creation and annihilation.But in a more practical sense, any realistic description

would acknowledge the fact that A would have some in-herent finite resolution scale. A simple way to model thisis to imagine that the states ofA respond to the parti-cles position in finite bins [xj , xj+1], wherexj =x0+j,where is the resolution scale. As along as /mcthe description of the measurement process within non-relativistic quantum mechanics will be valid. In that case(75) should be replaced by

|(t) =j

xj+1xj

dx (x)|x |j(t) . (78)

After the measurement, the ontic states of A are ap-proximately one of the discrete macroscopically distinctstates |j(t) and so a satisfactory phenomenology with-out macroscopic superpositions ensues.

In[1] a similar issue was shown to arise in an experi-ment involving the monitoring of a decaying system. Inthis case if the measuring device is taken to have infinitetemporal resolution then superpositions of macroscopi-cally distinct states arise. However, once the finite tem-poral resolution scale of a realistic measuring device istaken into account the problem with macroscopic super-positions evaporates.

V. EPR-BOHM AND BELL

In this section, we discuss the new interpretation inthe context of Bohms classic thought experiment [16,40], based originally on the classic paper by Einstein,Podolsky, and Rosen[41], and the implications for Bellstheorem[42].

It is useful for simplicity to avoid introducing a sep-

arate environment. Our experience from section IV Asuggests that we may do this as long as (i) we avoid de-generacies in the state of microscopic system (ii) haveperfect measuring devices. The point is that the micro-system acts as a surrogate environment for the measuringdevices and the ontic states of the measuring devices actas proxies for the ergodic sets of ontic states of the morerealistic situation with a large environment.

A. The Thought Experiment

The EPR-Bohm set-up begins with a pair of qubits

initially prepared in the entangled state that we take tobe

| =c+|z+z +c|zz+ . (79)

Note that we assume that c are generic to avoid degen-eracies. The ontic states of 1 or 2 are each one of the pair|z with probabilities |c|2, or vice-versa, respectively.

BA21

mn

FIG. 6. The EPR-Bohm thought experiment. Two qubits inthe entangled state | are produced at the source and thenrecoil back-to-back towards 2 qubit detectors A and B de-signed to measure the component of the spin along directionsn and m, respectively. In our set up, we choose an inertialframe for which the interaction between A and 1 happensbefore B and 2.

The ontic state of 1 + 2 is uniquely |. Given ourdiscussion in sectionIIIA,we can say that the systems1, 2 and 1 + 2 are related in a quantum way and thereis no notion of a joint property assignment or associatedprobability for the ontic states of 1 and 2.

In order to perform measurements, we add twospatially-separated spin detectors A and B , whereA de-tects the spin of particle 1 and B detects the spin ofparticle 2 as in figure6. The complete system consists ofqubits 1 and 2, measuring devices A and B and the en-vironment, although as mentioned above in the presentcontext we will ignore the environment in this simplifiedanalysis. It is important that even though there is nogenuine environment, the ontic state ofA +B is alwaysrelated to A and B in a classical way, that is as a tensor


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product of the ontic states ofA and B as in (35) but inthis simplified model exactly. So the description from thepoint-of-view ofA,B and A + Bcan be integrated into aconsistent global description throughout the experiment.Another important point is that the mirror ontic statesto those ofA + B are the ontic states of 1 + 2 themselvesso that the ontic dynamics ofA +B depends implicitlyon 1 + 2 as is clear from the matrix elements in (25).

Let us analyse what happens if A is set to measurethe spin component in the n direction, at an angle tothe z-axis in the (x, z) plane, and B is set to measurethe spin in a direction m at an angle to the z-axis inthe (x, z) plane.14 This set-up is generic enough for ourpurposes.

The initial state of the overall system is

|(t1) = |A0B0 | . (80)Suppose, in a certain inertial frame, the interaction be-tween A and 1 happens first at time tA > t1. After ashort decoherence time, the state of 1 becomes entangledwith A and the state of the total system becomes for

t2 > tA

|(t2) = |A+B0 |n++ + |AB0 |n , (81)where

| =ccos(/2)|z csin(/2)|z . (82)Note that these states are neither normalised or orthog-onal.

Assuming in this simple model that the states |A areexactly orthogonal, the ontic state of 1 + 2 has changedfrom | to either of the non-entangled (and non-normalized states)

|n

with probabilities

|

.

The ontic state of 1 is now one of the pair |n and theseare now perfectly matched with the ontic states of 1 + 2.

The ontic states of 1 + 2, after the local interactionbetween A and 1, are, of course, the analogues of thecollapsed wave functions of the Copenhagen interpreta-tion. So the puzzle seems to be that a local interactionbetween A and 1 has changed the ontic state of 1 + 2and this seems to have led to a non-local change of theontic state of qubit 2. But it turns out this is not cor-rect. The key point point is that the interaction betweenA and 1 changes the ontic states of 1 and 1 + 2 but not2. To see this, the reduced density matrix of 2 before theinteraction betweenA and 1 is

2(t1) = |c+|2|zz| + |c|2|z+z+| , (83)while after the interaction,

2(t2) = |++| + || . (84)

14 The eigenstates are |n = cos(/2)|z sin(/2)|z and|m = cos(/2)|z sin(/2)|z.

But the states | are neither normalized or orthogonaland using (82) one can show that actually

2(t1) 2(t2) , (85)reflecting the fact that ontic assignments are local andan interaction between A and 1 cannot affect the onticstate of the causally separated qubit 2. The implicationis that the ontic state of qubit 2 remains one of the pair

|z throughout the interaction between A and qubit 1.To be clear, the interaction between A and 1 does notchange the possible ontic states|z of 2 but also theactual ontic state of 2. The latter fact follows from thestochastic process describing the ontic states of 2 giventhat the interaction Hamiltonian Hintis itself local, thatis of the form of a sum of two local interactions:

Hint= HA,1int +

HB,2int , (86)

where the first (second) term is non-vanishing for tin theneighbourhood oftA (tB). This implies

pz|z(t2, t1) = 1 . (87)

So although the interaction betweenA and 1 changes theontic state of 1 + 2 from| to one of the pair|nand this seems to have fundamentally changed the stateof 2, however, the ontic states of 1 + 2 cannot be brokendown in terms of the ontic states of its sub-systems 1and 2at least for t < tB. Consequently, there is anapparent duality for qubit 2 from the point-of-view of1 + 2 compared with 2:

2

1 + 2 | |nii|zk |zk

tA

It is tempting to say that the behaviour of the state of1 + 2 here is non-local. However, this is potentially mis-leading because in order to talk about locality within thesystem 1 + 2 requires us to break 1 + 2 down in termsof its sub-systems 1 and 2. But, to reiterate, we cannotbreak down the states|nii ontically in terms of 1 and2 because the descriptions via 1 + 2 and 2 cannot be in-tegrated into a consistent whole. We will have more tosay about this and its implications in sectionV B.

To emphasize, for t < tB, the ontic state of 1 + 2 isnot a tensor product of ontic states of 1 and 2 so there

is no sense in which the ontology is classical in thesense described in section IIIA. So in figure 7, whichsummarizes the ontic dynamics of 1, 2 and 1 + 2, welabel it as quantum.

On the other hand the behaviour ofA + B,A and B isalways classical as illustrated in figure8. The initial onticstate ofA + B is|A0B0 while after the measurement atA the ontic state is then one of the pair|AB0 withprobabilities

pAB0|A0B0(t2, t1) = | , (88)


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1 |z |n+

|n

|n+

|n

2 |z |z |m+

|m

1 + 2 | |n++

|n

|n+m+|n+m

|nm+

|n

m

tA tB

quantum quantum classical

FIG. 7. Snapshots of the ontic states of 1, 2 and 1+2 at timest1 < tA < t2 < tB < t3 in the given inertial frame. For t = t1andt2, the ontic states of 1 + 2 are not tensor products of theontic states of 1 and 2 and so the ontology is quantum. Inthe last time step t = t3 the relation between states becomesclassical in the sense that the ontic states of 1+2 are tensorproducts of those of 1 and 2.

for t1 < tA < t2. These follow from applying (5) alongwith the uniqueness of the initial ontic state.

Finally, at timetBin this frame,B interacts with qubit2 and after a further short decoherence time, the statebecomes

|(t3) =ij=

|AiBj |nimjmj |i . (89)

After this interaction the triplet of systems 1, 2 and 1+2now has classical ontology in the sense that the onticstates of 1 + 2 are one of the quartet|nimj and theseare tensor products of the ontic states of 1 and 2. For

the measuring devices, the final ontic states are one ofthe quartet|AiBj. It is important is that (i) the firstinteraction between A and 1 does not change the onticstate ofB and (ii) the second interaction between B and2 does not change the ontic state ofA. This is guaranteedby the locality of the interaction Hamiltonian (86):

pB0|B0(t2, t1) = 1 , pAi|Aj (t3, t2) = ij . (90)

The implication is that the ontic dynamics has the tree-like structure as shown in figure8. Therefore, once againusing (5), the only non-vanishing probabilities are

pAiBj |AiB0(t3, t2) =m

j

|i

2

i|i , (91)

for t2 < tB < t3. The final probabilities follow from (5),or by composing (88)and (91),

pAiBj |A0B0(t3, t1) =mj |i2 . (92)

As we have emphasised, the ontic states of A, B andA+ Bare just tensor products of those ofA andB for allt. In the more realistic model with an environment, this

relation will be emergent as in (35). Given the classicalontology, it is meaningful to define joint probabilities asin (36):

p(AiBj)emergent

= pAiBj |A0B0(t3, t1) . (93)

These emergent probabilities are exactly what wouldhave been predicted on the basis of Borns rule. So dur-

ing the experiment the entanglement between 1 and 2 isconverted into an emergent classical correlation betweenA and B. But what is crucial is that this correlationarises after the local interaction between 1 and A andseparately between 2 and B and not through any non-local interaction between A and B .

A |A0 |A+

|A|A+|A

B |B0 |B0 |B+

|B

A + B |A0B0 |A+B0|AB0

|A+B+

|A+B|AB+|AB

tA tB

classical classical classical

FIG. 8. Snapshots of the ontic states of the measuring devicesA, Band A+Bat t1,t2and t3. The relation between the threesystemsA,B and A +B is always classical, as the ontic statesofA + B are just tensor products of the ontic states ofA withthose of B. However, for t = t3, the measuring devices are,nevertheless, correlated in a classical sense because A+B =A B.

B. Bells Theorem

Now we turn to a discussion of Bells theorem [31,42].The first point is that, since the predictions agree withthe conventional analysis of the Copenhagen interpreta-tion and the Born rule, it must be that the new interpre-tation implies a violation of the Bell inequality.

In fact it is easy to see that the new interpretationviolates outcome independence:

p(Bj |Ai, n , m) = p(Bj |n, m). (94)

To see this note that the conditional probability abovecan only be defined in the context of the ontic dynamicsof sub-systemA +B; indeed,

p(Bj |Ai, n , m) pAiBj |AiB0(t3, t2)

=

mj |i2i|i .

(95)


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We can now pinpoint where the dependence on the stateofA arises in the interaction between B and 2. Althoughthe sub-systemA+ B can be broken down in terms ofits component parts A and B because their ontology isaways classical, the ontic dynamics ofA +B dependson the mirror ontic state of 1+2, and 1+2 does not havea classical ontology for t < tB. The relation betweenthe ontic states ofA + B and the associated mirror ontic

states of 1 + 2 are as follows:

|A0B0 |AiB0 |AiBj

| |nii |nimj

tA tB

It is the entanglement of the mirror ontic state| thatleads, after the first interaction at t= tA, to the mirrorontic states|nii. Including only the important tensorproduct factors, we have

HA,1int

|A0 | |Ai |ni |itA

It is important to remember that|i are not the onticstates of qubit 2 even though they are tensor productfactors of the ontic states of 1 + 2 for t = t2. This inturn implies that when B interacts with qubit 2 at tB ,the ontic dynamics becomes implicitly dependent on theontic state of A through the mirror ontic state|nii.In this interaction, locality ensures that only the tensor

product factors of B and qubit 2 are relevant but thestate ofA influences the dynamics via this initial tensorproduct factor of qubit 2, that is|i, and this leads totheAi dependent probabilities (95):

HB,2int

|Ai |B0 |i |Ai |Bj |mjtB

Note that during the interaction between B and qubit 2the state of causally separated |Ai is inert but is includedin the above to see the correlation with the states

|i

.

On the other hand, the probabilities satisfy parameterindependence:

p(A|n, m) = p(A|n), (96)

since

p(A|m, n) j=

pABj |A0B0(t3, t1)

= |c|2 cos2(/2) + |c|2 sin2(/2),(97)

independent ofm, i.e. . This latter condition expressesthe fact that what happens atAdoes not depend on whatis measured atB. The violation of outcome independencebut observance of parameter independence is just as inthe Copenhagen interpretation and a violation of eitherimplies a violation of the Bell inequality. So what wehave shown is that the new interpretation violates Bellsinequality for the same reason the Copenhagen interpre-

tation does because the initial state of the qubits| isentangled. The novelty is that it does this without in-voking the collapse of the wave function. Instead whathappens is a curious and apparently non-local change inthe ontic state of 1 + 2 from| to|nii whenA and 1interact. On closer inspection, though, it is hard to saythis is a non-local process because it does not make senseto analyse the states|nii in terms of the ontic statesof their constituent qubits. So this is the spooky actionat a distance of quantum mechanics laid bare. In anyevent there is no violation of causality.

Another point to make is that the assignment of on-tic states clearly depends on the inertial frame chosen to

view the experiment. For instance in our chosen frame,where the measurement is made at A beforeB, the onticstates ofA +B after the first measurement are|AB0.But in another frame, where the measurement is madeat B before A, the ontic states ofA+B after the firstmeasurement are |A0B. This, of course, is no surprise,but modal interpretations have been claimed to be in-consistent with special relativity [44]. Fortunately theseclaims have shown to rest on the false assumption thatthe ontic states of a sub-system are global property as-signments and so are unwarranted [12].

VI. DISCUSSION

The emergent Copenhagen interpretation is a com-pletely self-contained interpretation of the quantum me-chanics that builds on and modifies earlier proposals thatare collectively known as modal interpretations. The keyingredient is the fact that one cannot isolate a quan-tum system from its environment. The interaction be-tween the two allows one to define the dual notion of theepistemic and ontic state of the sub-system. The for-mer evolves according to the Schrodinger equation (forthe total system), whilst the latter evolves according toa stochastic process. This dualism is analogous to theensemble and micro-state of classical statistical mechan-ics and allows for the solution of the measurement prob-lem by invoking an ergodicity argument familiar from thediscussion of a phase transition. Although we have pre-sented the quantum interpretation as an analogue of clas-sical statistical mechanics, in fact the former should betaken as a proper definition of the latter. In this regard,we are taking the viewpoint of [2330] but reintroducingthe notion of a micro-state in the form of the ontic state.However, unlike the situation of micro-states and ergod-icity in classical statistical mechanics, the dynamics of


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ontic states follows from a very simple Markov process.The most striking feature of the interpretation is that it

reproduces the phenomenology of the Copenhagen inter-pretation for macro-systems and the collapse of the wavefunction is just an innocuous process of removing ergod-ically inaccessible parts of the epistemic state. However,there is no sharp Heisenberg cut between microscopic sys-tems and macroscopic systems. One can quantify the

degree of classicality as the time required to see a transi-tion between two hypothetically macroscopically distinctstates; as order /, where is the magnitude of theinner product of the two states. A shortened descriptionof new interpretation will appear in [45].

Finally, we can put Schrodingers cat out of it misery.According to the new interpretation, it is either alive ordead but these states are not pure states rather they are

associated to ergodic subsets of ontic states that are, forall physically relevant time scales, mutually inaccessable.So the cat may be either alive or dead but in either casein equilibrium withhence strongly entangled withtheenvironment.

ACKNOWLEDGMENTS

I would like to thank Jacob Barandes for a very fruitfulexchange of ideas; many of my ideas arose from and aremuch sharper as a result of our communications. I amsupported in part by the STFC grant ST/G000506/1.

[1] T. J. Hollowood, The Copenhagen Interpretation as anEmergent Phenomenon, J. Phys. A 46, 325302 (2013)

[arXiv:1302.4228[quant-ph]].[2] H. P. Krips, Two Paradoxes in Quantum Mechanics,

Phil. Sci. 36, 145 (1969).[3] H. P. Krips. Statistical interpretation of quantum the-

ory, Am. J. Phys. 43, 420 (1975).[4] H. P. Krips. The Metaphysics of Quantum Theory,

Clarendon Press Oxford, 1987.[5] B. C. Van Fraassen, A Formal Approach to the Philos-

ophy of Science, In: R. Colodny, editor, Paradigms andParadoxes: The Philosophical Challenge of the QuantumDomain, pages 303366. University of Pittsburgh Press,1972.

[6] N. D. Cartwright, Van Fraassens Modal Model of Quan-tum Mechanics, Phil. Sci. 41, 199 (1974).

[7] B. C. Van Fraassen, Quantum Mechanics: An Empiris-ticist View, Oxford University Press, 1991.

[8] J. Bub, Quantum mechanics without the projection pos-tulate, Found. Phys. 22, 737 (1992).

[9] P. E. Vermaas and D. Dieks, The Modal Interpretationof Quantum Mechanics and Its Generalization to DensityOperators, Found. Phys. 25, 145 (1995).

[10] G. Bacciagaluppi and W. M. Dickson, Dynamics forModal Interpretations, Found. Phys. 29, 1165 (1999).

[11] P. E. Vermaas, A Philosophers Understanding ofQuantum Mechanics: Possibilities and Impossibilities ofa Modal Interpretation, Cambridge University Press,1999.

[12] J. Berkovitz and M. Hemmo, A New Modal Interpre-tation of Quantum Mechanics in Terms of Relational

Properties, In: W. Demopoulos and I. Pitowsky, ed-itors, Physical Theory and its Interpretation Essays inHonor of Jeffrey Bub, pages 128. The Western OntarioSeries in Philosophy of Science, Springer, 2006.

[13] C. Rovelli, Relational Quantum Mechanics, Interna-tional Journal of Theoretical Physics 35, 1637 (1996).

[14] D. Dieks, Objectivity in Perspective: Relationism in theInterpretation of Quantum Mechanics, Foundations ofPhysics 39, 760 (2009).

[15] G. Bene and D. Dieks, A Persp

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The Emergent Copenhagen Interpretation of Quantum Mechanics - Timothy J. Hollowood

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