REFERENCEIC/89/44

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

MARKOV PROCESSES IN HILBERT SPACEAND CONTINUOUS SPONTANEOUS LOCALIZATION

OF SYSTEMS OF IDENTICAL PARTICLES

Gian Carlo Ghirardi

Philip Pearle

and

Alberto Rimini

7

IC/89/44

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

MARKOV PROCESSES IN HILBERT SPACEAND CONTINUOUS SPONTANEOUS LOCALIZATION

OF SYSTEMS OF IDENTICAL PARTICLES •

Gian Carlo GhirardiInternational Centre for Theoretical Physics, Trieste, Italy

andDipartimento di Fisica Teorica, Universita di Trieste, Trieste, Italy

Philip PearleHamilton College, Clinton, New York 13323, USA

and

Alberto RiminiDipartimento di Fisica Nucleare e Teorica, Universita di Pavia, Pavia, Italy.

MIRAMARE - TRIESTEMarch 1989

* Work supported in part by the Istituto Nazionale di Fisica Nucleare,Sezioni di Pavia e Trieste.

ABSTRACT

Stochastic differential equations describing the Markovianevolution of state vectors in the quantum Hilbert space are studiedas possible expressions of a universal dynamical principle. Thegeneral features of the considered class of equations as well as theirdynamical consequences are investigated in detail. The stochasticevolution is proved to induce continuous dynamical reduction of thestate vector onto mutually orthogonal subspaces. A specific choice,expressed in terms of creation and annihilation operators, of theoperators defining the Markov process is then proved to beappropriate to describe continuous spontaneous localization ofsystems of identical particles. The dynamics obtained in such away leaves practically unaffected the standard quantum evolutionof microscopic systems and induces a very rapid suppression ofcoherence among macroscopically distinguishable states. Theclassical behaviour of macroscopic objects as well as the reductionof the wave packet in a quantum measurement process can beconsistently derived from the postulated universal dynamicalprinciple.

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T

1. Introduction

1.1. General considerations

The quantum description of physical phenomena meets some conceptual diffi-culties which motivate the uneasiness that many people fee! with this theoreticalscheme. The root of these difficulties can be traced back to the fact that the theoryincorporates, at its axiomatic level, two different principles of evolution. The firstprinciple, expressed by the Schrodinger equation, has to be applied in ordinary sit-uations in which a truly (juantuni system evolves undisturbed. Such an evolutionis deterministic and linear. The second principle has to be applied when a mea-surement takes place. The theory introduces a specific assumption to describe thissituation, i.e. the postulate of wave packet reduction (WPR), according to whichIhc wave function undergoes a sudden stochastic change. As all of us have learnedfrom the classic treatise of Dirac, "a measurement always causes the system tojump iiilo arr eigenstnte of the dynamical variable that is being measured". Notethat, at the level of the normalized wave function, WPR is a nonlinear process.

The fundamental difficulty of the theory can then be simply summarized. Ifone takes into account, that, after all, a measuring apparatus is a physical systemmade up of quantum constituents, one is led to expect that the measuring processshould be describable as the evolution of the closed physical system 5 |- A, S andA denoting the measured system and the apparatus, respectively. The problem isthat, in such a way, one gets, as the result of the measuring process, a superpo-sition of macroscopically distinguishable states which appears to contradict bothWPR, which provides a statistical mixture, and common sense. Removing suchcontradictions is the task of the theory of quantum measurement, but, in our aswell as in other people's opinion,1"4 a solution of the problem not implying somekind of breakdown of quantum mechanics in certain conditions does not seem tobe possible.

The above features render the theory not fully internally consistent and partiallyambiguous. On one side it requires acceptance of the idea that there are systemswhich are not truly quantum mechanical, leading therefore to a dualistic attitudein the description of physical phenomena; on the other, due to the impossibilityof a neat definition of the borderline between the two classes of physical systems,it compels the physicist to disregard from time to time the exact equations of thetheory and to supplement them with vague verbal assertions.6

At a more philosophical level one can remark that the adoption of two differentprinciples of evolution reflects the embarassing position of the orthodox quantuminterpretation with respect to the problem of the subject.-object distinction. Thetheory accepts in its postulates that some distinction must be made, but. it cannotprescribe where or when to put it.3

An important step towards greater physical exactness and logical consistencywould he made, if one could devise a manner to account for the stochastic jumps

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meutioued by Dirac in terms of precise dynamical processes governed by definitemathematical equations. Needless to say, since these jumps violate both the deter-ministic and the linear nature of the Schrodinger equation, such a program requiresone to accept a modification of it. Various attempts at introducing nonlinear andstochastic elements in the dynamical equations have been made. The crucial dif-ficulty one has to face when trying to follow this line derives from the miraculousaccuracy of the predictions of the theory for all microscopic systems. How can onedevise modifications of the standard theory such that they have a negligible impact,for small systems and at the same time they are able to yield a fast dynamical sup-pression of the unwanted superpositions of macroscopically distinguishable states?Nevertheless, dynamical reduction models whereby a superposition of states con-tinuously evolves into one of its terms have been considered by various authors.6 ~'J

These works have concentrated on achieving such an evolution, but. they have leftunsolved two basic problems:

i) the preferred-basis problem — which are the states to which the dynamicalreduction process leads?ii) the system-dependence problem — how can the coherence suppressing processbecome more and more effective when going from microscopic to macroscopicsystems?

A new approach to the problem of modifying the standard quantum dynamicshas recently been proposed.13'14'4 We shall refer to it as quantum mechanics withspontaneous localization (QMSL), The model is based on the assumption that eachconstituent of any system suffers, at randomly distributed times with an appro-priate mean frequency, a sudden collapse consisting in a localization of the wavefunction within an appropriate range. An important difference between dynamicalreduction models and QMSL is that in QMSL finite changes of the state vectoroccur instantaneously — quantum jumps really take place, even though they arespontaneous and their occurrence does not require the intervention of any observeror the interaction with a macroscopic apparatus. However, the model gives a pre-cise answer to the questions raised under i and ii above •••- the preferred basis isthat of localized states for the constituents and the correct system dependencefollows rigorously.

1.2. Features of QMSL

QMSL assumes that each particle of a system of n distinguishable particleslabelled by index i experiences, with mean frequency A', a sudden spontaneouslocalisation described by15

(I-1) \>i') ~* \<i''r) -'- ^vW 1 ) -

In Eq. (l . l) , where state vectors refer to the n-particle system, L'x is a norm reduc-ing, positive, selfadjoint, linear operator representing the localization of particle

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/ around point a;. It contains a parameter, having dimensions of length, whichmeasures the size of the localization volume. In Refs. 12, 13, L'x was chosen tol>e a Gaussian function, centered in as, of the position operator of particle j ; thelength parameter was denoted by 1/y/a. Since the process (1.1) does not conservetlir norm, we rewrite it in the nonlinear norm-conserving form

and contextually assume that the probability density for the occurrence of z: is

Assumption (1.3) requires that

( 1 , 1 1 =1-

The process we have described is a Markov process in Hilbert space. Due to theoccurrence of sudden finite changes of the state vector, we call it a hitting process.It is to be noted that assumption (1.3), which makes the occurrence of the collapsesmore likely where the wave function is larger, is strictly analogous to the postulateabout, the probabilities of the outcomes of a measurement in standard quantuminechfuiics. As already stated, however, the QMSL collapses occur universally andspontaneously and do not. require any actual measurement to be performed.

We draw attention to the following features of QMSL.The stochastic modification of the standard dynamics introduced by QMSL

induces, according to the collapses which have occurred, the decomposition of astatistical ensemble described by a state vector into subensembles each describedby a state vector. So, any member of the ensemble is associated at all times witha. delinitr state vector. The possibility of associating with each individual closedsystem (in particular, with the system S + A considered by measurement theory)a definite wave function at any time opens the way to the interpretation of thewave luuction itself as a real property of the system, laying the foundations of anobjective description of physical phenomena.5'11114'4

Tin- collapses correspond to approximate localizations of the constituents. Thiscorresponds to a definite choice of the preferred basis. It is remarkable that this< tioiri- by itself solves the problem of system dependence. In fact, each localizationof ;t single constituent, when applied to a system for which the rentre-of-massposition is a collective variable (i.e. the wave function of the system is the product.<>f a centre of mass wave function tunes an internal wave function) is sufficient tolocalize thr whole system. It follows that the frequency of the process for the wholesystfin is the sum of the frequencies for the single constituents. This fact allows onetu choose the parameters in such a way that the process is completely ineffective for

microscopic systems and nevertheless it is sufficient to quickly suppress coherencebetween distant states of macroscopic systems. It has also been shown1' thatdifferent choices for collapse mechanisms (in particular the assumption that theyinvolve besides position also the momentum variable) do not share the cumulativeproperties of frequencies.

The QMSL model, as presented above, is consistent only in the case of systemsof distinguishable particles. In fact, the QMSL collapses (1.1) or (1.2) do notpreserve the symmetry properties of the state vector. A natural generalizationof QMSL to the case of identical particles can be obtained17 by using the wholeset of positions of all particles to distinguish among different configurations of thesystem. Such a version of QMSL, however, does not appear to have nice featureswhen translated into the language of second quantization. Another kind of QMSLmodel uses the set. of densities around all points of space to discriminate amongdifferent configurations. These models will be described in Sect. 4.3 below, but theapproach of Sect. 3, also based on densities, will turn out to be much preferable.

1.3. C'SL and other recent developments

It has been proved recently18 that it is possible to devise a dynamical reductionmodel which exhibits all the appealing features of QMSL. It will be referred toas the continuous spontaneous localization (CSL) model. In C'SL one assumes astochastic evolution equation of the Ito form

(1.5)

where dh is a random, selfadjoint, linear operator. It is built up with the operatorsrepresenting the density of particles around all points of space and contains in itsdefinition (which will be given in Sect. 3 below) a length parameter 1 /'*/a anda strength parameter 7. The process defined by Eq. (1.5) does not conserve thenorm. This fact needs an interpretation, which leads to the introduction of anotherprocess

(1.6)

which is norm conserving and nonlinear, due to the dependence of dh$ on \<p).Et|. (l.fi) embodies an assumption concerning the probabilities to be assigned tothe state vectors \d>) which is the counterpart of the assumption (1.3) fur a. hittingprocess. The procedure leading from the process (i...r») to (l.fi) will be explainedin detail in Sect. 2.

Other recent investigations19'20 deal with dynamical reduction models similarto the one considered in Ref. 18 and here. In Ref. 19 an equation very closeto Eq. (1.6) is introduced (without deriving it from a linear process), but it is

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~ T

not specialized to the use of densities around space points to discriminate amongdifferent configurations. The idea of using densities is considered in Ref. 20, where,however, the dynamical equation has a more complicated structure than here.

1.1. Aims and contents of the present paper

This paper deals first with a new presentation and discussion of the class ofMarkov processes in Hilbert space to which the CSL theory belongs. Then, thegeneral formalism is specialized to OSL in the framework of second quantizationand the physical consequences of the theory are thoroughly investigated. Finally,the relationship between the classes of stochastic processes used by QMSL andCSL is discussed.

In Section 2.1, linear Markov processes in Hilbert space are introduced and thebasic physical assumption concerning probabilities is stated and embodied into theevolution equation. Sect. 2.2 is devoted tu the genera! proof that the consideredMarkov processes lead to dynamical reduction of the state vector on the commoneigenspaces of the operators which define the process. In Sect. 2.3, the equationfor the statistical operator is derived.

Section 3.1 defines the CSL theory in second-quantized form. Sect. 3.2 investi-gates the physical consequences of CSL. First, it is shown that, under appropriateassumptions, the centre -of-mass and the internal motions of a system decouple,the stochastic terms in the dynamical equation do not affect the internal structureand the centre -of-mass wave function obeys a stochastic differential equation ofthe CSL type. Secondly, the rates of reduction on the approximate position eigen-states of a macroscopic object are evaluated and it is shown that it is possibleto choose the two parameters defining the stochastic process in such a way thatreduction is both effective for macroscopic objects and completely negligible formicroscopic particles. Finally, considering together the localization process andthe Schrridinger evolution, it is shown that the stochasticity introduced in thebehaviour of macroscopic bodies is also negligible.

The class of hitting processes (the discontinuous stochastic processes used intJMSL) is reconsidered in Section 4.1. An appropriate infinite frequency limit, un-der which n hitting process reduces to a continuous process of the type introducedin Sect. 2.1, is discussed in Sect. 4.2. Finally, two examples of hitting processesguinf!, into the CSL process defined in Sect. 3.1 in the infinite frequency limit are)>roscnted in Sect. 4.3.

Some concluding remarks are contained in Section 5.

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2. Markov processes in Hilbert space

2.1. Raw and physical processes

In the Hilbert space, we consider the Markov process |V !B(')) satisfying the Itustochastic differential equation

(2.1)

where C and A =such that

(2.2)

- (Cd1 + A-dB)\4<),

is a set of operators, fl s {BJ is a real Wiener process

d~B\ = 0,

dBidBj - 6tJ7rfr,

7 being a real constant., and the dot product has the obvious meaning

(2.3)<-—'i

The index i can be continuous, in which case the sum becomes an integral andthe Kronecker delta becomes a Dirac delta. Given an initial state |V'(0)), Eq. (2.1}generates at time ( an ensemble of state vectors ] VB( ' ) ) I where B denotes a partic-ular realization B{<) of the Wiener process. To simplify notation, the dependenceof |i/') on t and B will be often dropped, as in Eq. (2.1). The process (2.1) andthe ensemble generated by it will be called the raw process and ensemble. In theraw ensemble, each state vector I V ' B O ) ' l a s 'he same probability as the particularrealization B(f) which originates it through Eq. (2.1).

The raw process (2.1) does not conserve the norm, in general. In fact, using Itocalculus, one finds(2.4)

where we have used the notation rf|</-) — \dt]'). If the state vectors |i/'fl(')) wereof norm 1, their probabilities, given by the raw ensemble, could naturally beinterpreted as the physical probabilities. The vectors |V's(')) being not of norm1, let us consider the ensemble of the normalized vectors

having the same probabilities as the corresponding vectors |0/ j(f» (i.e. as therealizations B(t) of the Wiener process) and the ensemble of the normalized vectors

(2.6)

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whose probabilities are those of the vectors IV'B(')) tiroes their squared normsIII/ 'BOII3 ' We use different symbols for the vector functions IXB(*)) a l ld | ^ B ( ' ) ) Iin spite of the fact that the r. h. sides of Eqs. (2.5) and (2.6) coincide, because theirprobabilities are different, so that as random vector functions they are different.In fact, as we shall see and as is obvious, they obey different stochastic differential(•(Illations. We choose as the physical probabilities those of the vectors (2.6) ratherthan those of the vectors (2.5). The ensemble of the vectors |<£JJ(*)) and thestochastic process in the Hilbert space which generates it will be called the physicalensemble and process. As mentioned in the introduction, the prescription leadingto the physical ensemble is the counterpart of the assumption (1.3) of QMSL and oftht1 postulate of standard quantum mechanics on the probabilities of the outcomesof a measurement.

Let us now investigate the relation between the raw and the physical processes.Indicating by p(B(i,to)) the probability of the realization B(1,ta) of the Wienerprocess or equivalently of the state vector | I / 'B(0) a"d by q(B{t,t(,)) the proba-bility of the state vector |<Afl(f)), one has by definition

(2.7) = \\4>B{i,t0)[\-p(B{t,U)).

It is easily shown (hat, because of linearity of Eq. (2,1) together with the Markovnature of the Wiener process £?, the procedure Seading from the raw to the physicalensemble can be performed just at the considered final time or, in addition, anynumber of times between the initial and the final times. It follows that Eq. (2.7)can be replaced by its specialization to the infinitesimal time interval (fo,(n + di),

(2.H) q(dB) = (1 + \p(dB).

Tiie possibility of considering the physical ensemble depends on fulfilment of thecondition that the total probability associated with the distribution q is 1. Thisamounts to requiring that, for any IV1}, the average relative to the distribution p..r I he weighting factor \\4>\\2 is 1,'i.e. d\[<f\f = d\\\l<\\- = 0. From Eq. (2.4), onelinds

VVIicn this condition is taken into account, denoting by —i-H the anti-Hermlteanpurl. nfC, Kq. (2.1) becomes

(2.ID) d\r!') ••- {-Ufdi ±A-dB - \-,A^Adi) \i<).

This equa t ion is of I he form (1.5) for selfadjoint A. Eq . (2.4) simplifies to

C2 . l l ) d\U<f --- {4'\{A i-Al)\4')-dB.

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Then Eq. (2.8) becomes

(2.12) q(dB) = (1 + 2R-dB)p(dB)

where

(2.13) R = §{V| {A + ^ f ) W,

and the probability distribution q is now normalized. Indicating by dB' the ran-dom variable whose distribution is q, one has

(2.14)

so that

(2.15)

'i = 2R,f dt,

~dB\dB'i = hn

dB' = dB +2R-ydt,

and £?' is a diffusion process having the same diffusion as B and drift. 2R-y. Themeaning of the process B' and of its differential dB' follows from that of theprobability distribution q which defines them. The set of all realizations B'(1)coincides with that of all realizations B(t) (in fact both sets coincide with theset of all functions satisfying a given initial condition), but their probabilities,according to the definition (2.7) of q, are those of the. physical ensemble insteadof those of the raw ensemble. The stochastic differential equation for the physicalprocess can now easily be written. We first, write the equation for the processgenerating the normalized vectors \\). From Eqs. (2.10) and (2.11), by directevaluation, one gets(2.16)

d\\) -•-- \(-iH - - R)-dB]\x),

R -=

It is easily checked that Eq. (2.16) conserves the norm and that this feature doesnot depend on B having drift zero. The physical process is obtained by replacingeach realization B[t) of the r andom function B{ I) by an equal realization having

the appropr ia te different probability, i.e. by an equal realization D'(t) of the ran-

dom fnnclion B'(t). This amoun t s to replacing dB by dB' in Eq. (2.1fi), so tha t

we get

(2.17)

d\<p) ---- [( iH - \-,AH-A -7A R, +• hR-R) dt +{A - R)-dB'} \<j>),

-1-

T"

It is convenient to rewrite Eq. (2.17) in terms of the original Wiener process B.One gets(2.18)

d\<t>) = {[-M - | 7 ( ^ ' - R)-A+ LfiA - R)-R)dt + (A - R)-dB}\<f>),

J?.- \{<p\(A + A^)\^).

We note that the equations for the norm conserving processes (2.16) and (2.17) or(2.18), contrary to Eqs. (2.1) or (2.10), are nonlinear.

The rase in which A is a set of selfadjoint operators is of particular interest. Inthis case Eq. (2.18) becomes

(2.19)d\4>) =: {[--iff - l-y (A -R)2}dt + (A-R)-dB}\<t>),

R = (4>\A\<1>).

This equation is of the form (1.6).

'1.1. Reduction of the state vector

We shall now show that, in the case in which A is a set of commuting selfadjointoperators, the non-Schrodinger terms in Eq. (2.19) induce for large times thereduction of the state vector on the common eigenspaces of the observables A. Theinputs and the outcomes of the present section are essentially the same as thoseof the proofs of reduction given by one of the authors in a series of papers.8'7'18

The proof given here is somewhat more direct in that it works on the squaredamplitudes without making use of the Fokker-Planck equation for their probabilitydensity.

Since we are interested here in discussing the physical effects of the new terms,we disregard for the moment the Schrodinger part of the dynamical equation.Then Eq. (2.19) becomes simply

(2.20)

Let. us write

(2.21)

where the orthogonal projections PB sum up to the identity and it is understoodthat «„ / aT (i.e. nai ^ <iT; for at least one value of i) for a / T. We considerthe real non-negative variables

d\<f>) ^ [ - h {A - Rf dt + (A - R)-dB

R

(2.22) (<t>\P«\<t>) - za

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having the property

(2.23) Y.ns" = l-

In terms of such variables one finds

(2.24) R = YJaa°z°,

(2.25) (A-R)\4>) = Y^ V Zr(aa-aT)

(2.26) (A - R}2 |0) = ]T

It follows that the stochastic differential equation (2.20) CJUI be written

(2.27} dPa\<t,)

Use of this equation in the relation

Pa\<t>).

(2.28)

gives for the variables za the set of stochastic differentia! equations

(2.29) < f : r = 2 2 , ^ zT(av - ar)-dB.

Qualitatively, Eq. (2.29) shows that the diffusion of {z,,} vanishes when {z^}approaches the solutions of the set of equations

(2.30) ^ ^ ^ ( c - a r ) = 0,

so that the values of {za} eventually accumulate towards such solutions, A for-mal prnof of the fact that {z^} asymptotically reduces to one of the solutions ofEq. (2.30) is easily obtained. From Eq. (2.29) one finds

(2.31) rf;; = 2za

and in turn

(2.32) dzJ^T:

It follows that

(2.33)

zT[a.» - aT)\ -,<tt

^

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This result, together with the bouudedness property

cntiiils that, for i -» oo,

dt.

siiij; again Eq. (2.32), we gel

II is shown in Appendix A that the only solutions of the set of equations (2.30) areof 1 ho form ;i — 0, . . . , -ff = 1,. -., corresponding to \4>) lying in one of the commoncigeiispaces of I.lie observables A. Since Eqs. (2.27) do not change the Hilbert spacerivy to which each component Pn\(j>) belongs, we conclude that \4>) asymptoticallyreduces to one c>f its initial components P,,|$(0)) times a real normalizing factor.

The probabilities of the various possible issues are also easily calculated. In fact,since </"„ — fiif, 0, one has

On the other hand

CUH)

si) I lint one finds

(2.39)

- 1),

Pr ( ;„(*) - 1) --^(O),

[2, 10)

As i>tic r;m sec, (his result is a d.'-.„ 0.

direct consequence''' ' • l s of the martingale property

0.\\V luivr shown in tins section that t h<> mm Schmdiuger terms in Eq. (2-1 -̂))

produce in I tic long time limit the reduction uf the state vector on the commonrii;ctis]>iicfs nf tlic operators A. T I IP time rate of such a process and its competitionwith Ihr Sihrodinger evolution are most easily studied in the framework of thes t ; i t i s t i * i i l i .>|iPi+fit .(il ' .

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2.3. Statistical operator

The statistical operator corresponding to the physical ensemble and its evolutionequation are easily obtained from the definition

(2.41) P -

and Eq. (2.10), or from

(2.42) p =

and Eq. (2.18). Using once more the Ito calculus in evaluating dp, one gets

where the symbols |- ,-] and {•,•} denote the commutator and the anticommu-tator, respectively. This is the Lindblad form31 for the generator of a quantumdynamical semigroup. It is remarkable that the general Lindblad generator can beobtained from a stochastic process in Hilbert space. As we shall see in Sect. 4.1,hitting processes give rise to particular forms of the generator. The derivation ofEq. (2.43) we have presented provides a description of the ensemble to which p(i)corresponds such that, each member of it has a definite state vector at any time.

3. Continuous localization of a system of identical particles

3.1. Definition of the process

We now apply the general formalism introduced in Sect. 2 to the continuousspontaneous localization of a system of identical particles.

Let sis consider the creation and annihilation operators o'(a:,a), 0(1,.s) of apa.rticle at point x with spin component s satisfying canonical commutation oranticoninuilation relations. We define a. locally averaged density operator

13-1)

where g(:r.) is a spherically symmetric, positive real function peaked around IT 0,normalized ill such a way that

(3.2) fxg{v)=. 1,

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s o III a t

(3.3) fdixN(x) = AT,

/V being the total number operator. The operators JV(a;) are selfadjoitit and com-nns1e witli each other. In the following we choose

where o is a parameter such that n~3^2 represents essentially the volume overwhich the average is taken in the definition of N(x). The improper vectors

(3.5) ]q,s) ~:tfaHqu

arc the normalized common eigenvectors of the operators N(T.) belonging to theeigenvalues

Wv identify now, with reference to Sect. 2.1. the index i which labels the oper-ators A; with the space point, a; and the operators A, with the density operatorsN(x), Then Eq. (2.10) becomes

(3.7)

where

(3.8)

d\tj') Uidi \ d:ixN(x)dBix) • ±7 d*xN2(x)dt\ |t/'},

dB{x) •-- 0.

^ 7 c.3 (;c - y) A-

This is, in a different notation, the process already considered in Ref. 18 for iden-tical particles. The generalization to several kinds of particles is immediate. Theprocess (3.7) can also be presented in terms of nonsingular, correlated randomvariables, as shown in Appendix B.

The equation for the statistical operator (2.43) reads in the present case

( ' ! " ' 'dii\H,p\ M fd^Nlx^pNix) - h i /rf3

In t h e r e p r e s e n t a t i o n of v e c t o r s ( 3 . r i ) . E<|- ( 3 . 9 ) b e c o m e s

(3.1(1)

where

(3.11)G(y -y") = Jd3xg{y' - x)g(y" ~ x)

For a single particle, Eq. (3.10) reduces to

d , „ , ,

(3 .12) d<

We note that, taking

(3.13) \ --- T ( r i / 4 7 r ) J ' = ,

this etjiiation coincides with the equation for a single particle considered in Ref. 13.

3.2. Physical consequences

.S'epanilion n{ the cei\tre-of-mass (notion

We discuss the physical implications of the modified dynamical equation (3.7|under the assumption that the order of magnitude of the length parameter a•~iJ2

is such that it can reasonably be admitted that the internal wave function of amacroscopic body is sharply localized with respect to ci~1!2.

Let Q be the centre-of-mass (cm.) coordinate of the system of identical particleswhich constitutes the considered macroscopic body,

(3.H) Q [ V «;.

a.nd write

q, = Q

The coordinates q, with resjiect to the cm. sum up to zero, so 1ha,l tliey atefunctions of 3A' - 3 independent internal variables, which we indicate by r. Theinternal variables r, together with the cm. coordinate Q, are functions of thecoordinates q,. The internal variables, as defined here, also describe rotations ofthe ii-particle system. We arbitrarily assume that the orientation of the systemis sharply defined in the wave function similarly to the truly internal struriure.

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*

Instead, one should consider 3 orientation variables, to be treated along the samelines as the cm. coordinate, and 37V — 6 truly internal variables, to be assumedsharply localized in the wave function. The problem would then be considerablymore complicated without gaining very much as regards physical insight. So, weconsider the wave function

where the symbol I , I means symmetrization or antisymmetrization with re-

h|n-ct. to interchanges of the arguments (q;,st). The wave functions <J and \ areunderstood to he separately normalized. The function A(r,s) is assumed to besharply (with respect to a"1 '2) peaked around the value rj of r.

The fiction of the operator N(x.) on the wave function (3.16) is easily workedmil. One finds

( 3 . I T ) - xf)

According to our assumptions, the factor in front of the function A varies muchinori" slowly than A itself, so that we can take r — r0 in the factor. In other words,we hea t the factor as if A(r,.°i) were of the form f<3N 3{r ~ r0 )£(*). Then

(3.IS)

where

(.•!.!'•))

--•• F(Q - T

F{Q *) = ^[n/2*)3

to FJ(). (3.18} the operatorre, on thr assumption that

( - H Q + g , - ( r 0 ) - * ) ' ) •

acts only on the factor "t of V>-

x.)dB{x)

rch|)<Mtivflv, tli<- wave function (.'i.l(j) satisfies Kq. (3.T). It is seen that , underimi' assumptions, flic c m . nnd the internal motions decouple as in the absence of

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the stochastic terms in Eq. (3.7). Furthermore, the stochastic terms do not affectthe internal structure, while the cm. wave function obeys a stochastic differentialequation, again of the type (2.10), whose consequences will be discussed below.

Reduction rates

The operators F{Q - x) appearing in Eq. (3.21), which correspond to the oper-ators A, of Eq. (2.10), are real functions of the cm. position operator Q. They area set of commuting selfadjoint operators, so that, as we know from the results ofSect. 2.2, the noii-Schrodinger terms in Eq. (3.21) induce the reduction of the statevector to the eigenvectors of the position Q. Of course, such a process requiresan infinitely long time while, in finite times, only the reduction to approximateeigenstates of Q takes place. We discuss here the time rate of the localizationprocess by studying the time dependence of the off-diagonal elements of the sta-tistical matrix {Q'\p\Q"). Again, we disregard the effect of the Schrodinger term,this approximation being justified by the fact that, for the values of \Q' - Q"\ inwhich we are interested, the reduction process will turn out to be very fast.

Equation (2.43) becomes in the present case

(3.23) ~{Q'\P\Q") = -T{Q\Q"){Q'\p\Q"),at

x \\F2(Q' - T.) + \F2{Q" -x)- F[Q' -x)F{Q" - x)] .

Eq. (3.23) gives

(3.25) (Q'W)IQ") =exp(-r()(Q1|p(0)|Q").

It is oasily found that T is an even function of Q' ~ Q". Since it is assumed thatvery many constituents of the considered body are contained in a. volume ft"3 '3 ,we can use the macroscopic density approximation, consisting in replacing the sumby an integral in Eq. (3.19). Then one writes

(.-J.2H) F(Q - x) «(Q f it

where D(y) is the number of particles per unit volume in the neighborhood of thepoint y --•- Q f y,

A further approximation, which we call the sharp scanning approximation, canbo used, since we ate not. interested here in the details of t he function P for Q' - Q"

-17-

T

•••* 0. The sharp scanning approximation consists in replacing the normalizedC;uissian function appearing in Eq. (3.26) by the corresponding delta function.Then (me has

F(Q - *)"- D{x-Q)>

V(Q' - Q") -- 7 jd3x \D2[x)- D(x)D(x + Q' - Q")] ,

(3.27)

resulting in

(3.28)

where suitable changes of the integration variable have also been made. The.physical meaning of F is easily understood by making reference to a homogeneousmacroscopic body of density Do. Then

iinul being the number of particles of the body in the cm. position Q' which do not.lir'isi the volume occupied by the body in the cm. position Q". The ratio betweenihe macroscopic frequency (3.29) and the microscopic frequency appearing in frontof the second term at the r.h.s. of Eq. (3.12) is novlDa (An/a)3/2.

The result (3.25), (3.29) should be compared with the result

(3.30) ( Q ' I P O I Q " ) ~ e*P( ^nicto'l {Q'\pW\Q")<

(3.31) Amncro •=- "A,

valid for \Q' - Q"\ > a ' I-, obtained in Ref. 13 for the case of distinguishableparticles. We note that in the present case an additional factor Do ("Wo)3

appears in the macro-to-micro ratio, but such a factor is multiplied by the numberof uncovered particles nollt rather than by the total number n. Clearly, this is aconsequence of indistinguishability of particles and of the choice of density as thedynamical variable governing the process. In Ref. 13, the length parameter or"1'2

was chosen to be of the order of 10~5rni and the microscopic frequency \ wassuggested to be of the order of 10 '""sec"1 with the aim of obtaining Amncro ==l()7sec ' for a typical macroscopic number ti = 102S. We repeat here the same

(3.32) n~"' = 10 "cm

and look for a. valur of 7 such that the macroscopic frequency V is again of theorder ol l O V c x for Jiollt = 1013. Since Da = 1024cm 3 , we get

(3.33)

- 18-

corresponding, according to Eq. (3.13), to J s 10 ' 'sec ' . This value is suchthat nothing changes in the dynamics of a microscopic particle even in the case inwhich it has an extended wave function.

Position and momentum spreads

According to Eq. (3.28) or to the original expression (3.24), the diagonal ele-ments {Q\p\Q) of the statistical operator in the position representation are notaffected by the reduction process, as a consequence of the process being a localiza-tion. Of course, this does not. mean that the time evolution of (Q\p\Q) is the sameas given by the pure Schrodinger dynamics, because this dynamics is different, formixtures of localized states, from that for nonlocalized states. So, some changesare expected in the tune dependence of both position and momentum spreads, asa consequence of the presence of the localization process. An explicit evaluation ofthese effects is necessary in order to check that no unacceptable behaviour arises.

The equation for ihe statistical operator, in operator form, is written

(3.34) ~p - - i H,p] f •{F(Q - x)pF(Q-x)

where we retain now the Schrodinger term. We consider the case of a free macro-scopic body, so that, in our notation, H = P2 /(2Mh), M being the total mass.For a dynamical variable S, we define the mean value

(3.35) (S) -. U-[SP\.

The time derivative of (S), according to Eq. (3.34), is given by

7 \{F[Q -x)SF(Q - x) - 1 {S, F2(Q

From Ecj. (3.36), through tedious but elementary calculations, one gets

d 1(3.37a) - (Qi) ••-• —-</ ' ;} ,

(3.37b) - ' (P;) •= 0,

and

(3.3SH.) - {Q~) =(Q,Pr\ P<Q,),

(3.38b) — {QiP, + P,Q,} - 2 - (P,2),

(3.38c) !Lip?)^Uf,tf

-19-

(.•{.:*())

System (15.37) is the same as in the case of the pure Schrodinger (p.S.) evolution,si> that.

<<?.) = {Pi) = {Pi),

where the suffix 0 indicates the p.S. solution satisfying the same initial conditions.System (3.38) differs from the corresponding p.S. system for the non-zero r.h.s. inR<|. |USf | . One easily finds

(3,1.1 a

(3,11c) {P") - {P?) •+">P; ~ ' .

Tlie new feature of Eqs. (3.4i) with respect to the standard case is the momentumdiffusion coefh'cieut \ff>ih" appearing in the third equation. The extra terms intlie two first equations simply reflect the consequences of momentum diffusion on111 i- of her dynamical variables through the standard evolution. The same set ofequations was obtained in Ref. 13 in tlie case of distinguishable particles, the factor-) <*>, bring replaced there by nXa.

To evaluate tin- quantities &; tlie sharp scanning approximation is no longersulhcient, because here the derivative of the function F is required. We then usethe macroscopic density approximation (3.26). For definiteness and simplicity, weuinke reference to a homogeneous macroscopic parallelepiped of density Do havingedges of lengths L, parallel to the coordinate axes. Then, as shown in Appendix(\ one has with high accuracy

where S, /.] l]->Li/L, is tlie transverse section of the macroscopic parallelepiped.If ilir choice (3.:!2), (3,33) is used for n and •)• together with Do = 102<lnii 3 ,

(.in- gi-fs from Kq. (3.42) for the momentum diffusion coefficient.

(3.13) i i V ^ 10 3 2(gcmsec ' ) ' src"1 ,S',cm "2.

l'".ir ;\.n ordinary in;vcrc>sc«pic body, this value appears too small to give detectable• •fleets. K>r ii very small macroscopic particle, due to the 1/AJ1 factor in the extra

-20-

term of Eq. (3.41a), a non-negligible stochasticity could appear. For L, sa 10 cm(this is the smallest order of magnitude for which the approximations leading toEq. (3.42) remain valid), a time of the order of 102sec is required to make theextra term of the order of 10~I0cm!. We do not know whether this kind of effectcould be used to provide a significant experimental bound on the product i\facontained in the momentum diffusion coefficient. We note, however, that the value(3.42) could overestimate £;, because of the assumption of a rectangular profile fordensity.

4. Connection between continuous and hi t t ing processes

4.1. Hitting processes

We come now back to the type of stochastic process in Hilbert space which wasthe basis of QMSL. Let us consider the collapse process

(4.1) IV) {A1 - b) 2\

where A1 • {'4|;(' -- 1,...,A"} are commuting selfadjoint operators and b =\h,;i - 1, . . . ,A"} are real random variables. Index I identifies the process (4.1)within a family of similar processes. The process (4.1) does not conserve the norm,so we consider the process

(4.2) , j

and assume that the probability density for the occurrence of 6 is

(1-3) Vi{b)-\]t'b]\2.

The total probability is 1 since

) ' V / 3 [d -fi'b2) =

The process (4.2), in conjunction with the assumption (4.3), can he railed tliephysical collapse process. We next assume that the /-th physical collapse processtakes place rd random tunes according to a Poisson process with mean frequency/(', being understood that in the interval between two collapses the system evolvesaccording to the Schrodinger equation. The stochastic process in Hilbert. spaceconstructed in such a way enjoys the Markov property but, contrary to the processconsidered in Sect. 2.1, is such that the state vector associated to a member of

-21-

the statistical ensemble does not evolve continuously in time. We call it a hittingprocess.

Due to the random distribution in time of collapses, the evolution of the sta-tistical ensemble is at any rate smooth, in the sense that the weight of any statevector is a differentiable function of time. In particular, a differential equation forthe statistical operator can easily be written12'13 and turns out to be

(4.5)

where

(4.6) r, i'/v) ldIlbexp ( -\jil (A1 - p exp ( - ' - bf) .The generator appearing in Eq. (4.5) is a particular Lindbiad generator. We:mte that the second of Eqs. (4.2) could be written more generally iu the formIV'jj,) = ^(,1^)1 ' n l ' i ' " order that the process have a physical interpretation as a

}) = 1.hitting process, the operators L^ must satisfy the sum rule Jdh b (•£},This fact entails the simple structure numberxp of the last term in Eq. (4.5).Therefore, contrary to the case of continuous processes considered in Sect. 2, themost general Fjiudblad generator cannot be obtained from a hitting process.

•1.2. Infinite frequency limit

It is natural to ask whether there is a relation between the hitting process de-scribed in the foregoing section and the continuous process considered in Sect. 2.1.The answer is affirmative. It is possible to show"2 that if one takes the infinitefrequency limit

(4.7) ,,! >oo, >?' — 0, § ^ ' = 7,

the hitting process (4.2), (4.3) goes into a continuous process of the type (2.19),with A having the structure

(4 .H) A ^ {A\\l^ 1 , 2 . . . ; ; = 1 , . . . , A ' } ,

and similarly for J3.Here we confine ourselves to notice that the statistical operator equation (4.!))

goes, in the infinite frequency limit, into Eq. (2.43) with A* — A, A having thestructure (4.8). In fact, as is shown in Appendix D,

(4 -3 ) T'W^fy^.^W-

so t h a t , in t h e l imi t ( 4 . 7 ) ,

This equation coincides with Eq. (2.43) with A^ - A, A having the structure(4.8).

- 2 2 -

4.3. Systems of identical particles

We consider here two different instances of hitting processes inducing the lo-calization of a system of identical particles, which go into the continuous processconsidered in Sect. 3 in the infinite frequency limit.

In the first instance,13 a collapse process is considered at each space point a:,

(4.11) f ( * ) - ; ) 2 ]W,

where N(x) is the density operator defined by Eq. (3.1) and z is a real randomvariable. We then assume that the probability density for the occurrence of zgiven .T is

(4.12) Vx(z) •- Holl-

and that the process at x occurs with frequency density ft[x). The infinite fre-quency limit is defined by

(4.13)

In the second instance, we consider the collapse process

H ) :- W1'1 exp ( - \(4.14)

where 11 {x) is a real random function. The collapse is assumed to occur with meanfrequency ft and the functional probability density for the occurrence of n (a;) isassumed to be

(4.!!))

witli respect to a suitably defined functional measure such that

W /rf|j)]expf-^ /r(4.1(5)

The infinite frequency Limit is defin

(.1.17) , i - - o c ,

- 2 3 -

In the limits (4.13) or (4.17), both hitting processes just considered become thecontinuous physical process

d]4>) = -iff - S d3x {N(x) - R(x)) dt[\ J I

( L 1 S ) + fd'xiNix) - R{x))dB{x)] \4>),•J J

R{x) = (4>\N(x)\4>)

which corres]>onds to the raw process (3.7). The statistical operator obeysRq. (3.9).

5, Conclusions

Wo have presented a mathematically precise and logically consistent modifica-tion of non-relativistic quantum mechanics aiming at solving the problems raisedby quantum measurement. The modification consists in superimposing upon theordinary Schrodinger evolution a Markov process in Hilbert space giving rise to acontinuous spontaneous localization of the wave function. The necessary require-ments which are to be satisfied by any modified version of quantum mechanics areiiii-t by the theory. In fact, as shown in Sect. 3, the proposed modification doesnot affect in any appreciable way the predictions of standard quantum mechanicsfor microscopic objects. On the other hand, when macroscopic systems are in-volved, the modification of the dynamical equation has, as its practically uniqueeffect, that of inducing a very fast suppression of coherence among macroscopicallydistinguishable states.

The theory discussed here allows a natural description of quantum measurementprocesses by dynamical equations valid for all physical systems. It is worth whilerepeating that, in this theoretical scheme, any member of the statistical ensemblehas at all times a definite wave function. As a consequence, the wave functionitself can be interpreted as a real property of a single closed physical system.

The continuous localization process considered in the present work constitutesprogress with respect to the localization by a hitting process, both because itavoids ilie consideration of sudden collapses of the wave function and because itallows one to express synthetically the law of evolution in the form of a stochasticdiJirirnlinl equation for the wave function. We note however that, as shown inScci. 1, there are hitting processes having a physical content as close as one wantslo >I nivrn continuous process of the type considered here.

Tin- challenging problem remains open of getting a satisfactory relativistic gesi-I'l'iilization of the present formalism.

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Acknowledgments

We thank John S. Bell for his interest in our work and for having put us incontact. We are indebted to Italo Guarneri for useful discussions and suggestions.

One of us (P. P.) is grateful to Hamilton College for a faculty fellowship and ac-knowledges financial support from the Istituto Nazionale di Fisiea Nucleare, Sezionidi Pavia e Trieste, where this work was initiated.

References and notes

1. A. Shimony, American Journal of Physics 31 (1963), p. 755.2. E.P. Winner, in Fondanienii di meccanica quantistica (Varenna, 1970), edited

by B. d'Espagnat (Academic Press, New York and London, 1971), p. 122.3. J.S. Bell, in The Physicist's Conception of Nature, edited by Jagdish Mehra

(Reidel, Dordrecht, 1973), p. 687.4. G.C. Ghirardi, A. Rimini, T. Weber, Foundations of Physics 18 (1988), p. 1.5. J.S. Bell, in Schrodinger — Centenary celebration of a polymath, edited by

C'.W. Kilmister (Cambridge University Press, Cambridge, 1987), p. 41.6. P. Pearle, Physical Review D 13 {1976), p. 857.7. P. Pearle, Foundations of Physics 12 (1982), p. 249.fi. N. Gisin, Physical Review Letters 52 (1984), p. 1657; P. Pearle, Physical

Review Letters 53 (1984), p. 1775; N. Gisin, Physical Review Letters 53(1084), p. 1776.

9. P. Pearle, Physical Review D 20 (1984), p. 235.10. P. Pearle, Physical Review D 33 (1986), p. 2240.11. P. Pearle, in Quantum Concepts in Space and Time, edited by R. Penrose and

C.J. Isham (Clarendon Press, Oxford, 1986), p. 84; in New Techniques andIdeas in Quantum Measurement Tjjeory, edited by D.M. Greenberger (Ann.N.Y. Acad. Sci, 1986), p. 539.

12. G.C. Ghirardi, A. Rimini, T. Weber, in Quantum Probability and Applica-tions U (Heidelberg, 1984), edited by L. Accardi and W. von Waldenfels,Lecture Notes in Mathematics, vol. 113(i (Springer, Berlin, 1985), p, 223; inFundamental Aspects of Quantum Theory (Coma, 1985), edited by V. Goriniand A. Frigerio (Plenum Press, New York and London, 1986), p. 57.

13. G.C. Ghirardi, A. Rimini, T. Weber, Physical Review D 34 (1986), p. 470;Physical Review D 36 (1987), p. 32X7.

14. V. Bctiiitti. G.C. Ghirardi, A. Rimini, T. Weber, 11 Nuovo Cimento 100 B(1987), p. 27.

15. Though implicit in the formulation of Refs. 12, 13, the presentation of lo-calization as a process acting on the wave function has first been given by

-25 -

.I.S. Bell. See Ref. 5.16. K. Benatti, G.C. Ghirardi, A. Rimini, T. Weher, II Nuovo Cimento 101 B

(1988), p. 333.17. G.C. Ghirardi, O. Nicrosini, A. Rimini, T. Weber, II Nuovo Cimento 102 B

(1988), p. 383.18. P. Pearle, Trieste preprint IC/88/99.19. N. Gisin, preprint.•It). L. Diosi, preprint KFKI-1988-55/A.21. G. Limlblail, Coinniuiiications in Mathematical Physics 48 (1976), p. 119.22. V. BenaHi, G.C. Ghirardi, 0. Nirrosini, A. Rimini, in preparation.23. This type of collapse process has been suggested to us by j.S. Bell.

Appendix A

Wo are interested in solving the set of equations

(A.I) ^ ^ M a ^ - a i O ^ O

for refil, non-negative values of the unknown variables ;„, under the condition

(A.2) £^1.

Note that Eqs. (A.I) are not independent, each being a consequence of the re-maining ones.

It is immediately verified that, for any <r, the set of values z\ = 0, . . . , za = 1,...is an acceptable solution. It is also easily shown that such solutions are the onlyones, provided o , ^ aT for a f- T. In fact, suppose that, say, i\ / 0 and z2 ^ 0.The two first equations of the set (A.I) can then be written

- a T ) = 0, - a T ) = 0.

By subtraction on** get

Owing to the condition (A.2), Eq. (A.4) gives a^ = a2, contrary to the hypothesis.

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Appendix B

The process (3.7) can also be presented in terms of nonsingular, correlated,random variables dw(y). In fact, the second term at the r.h.s. of Eq. (3.7) can bewritten

(B.I) dh = Y,3 Jd3yaHy,s)a{y,3)dw(y)t

where

(B.2) dw(y) r, fexgly -

The variables dw(y) have the properties

1 •' S7(V7MV) - x)g(y' -x)dt = i G(y - y')dl,

where the function G is given by Eq. (3.11). The last term at the r.h.s. of Eq. (3.7)is - ^ times

(B.4) (dhf =

Appendix C

We evaluate here the factor (3.39) for the case of a homogeneous macroscopicparallelepiped. In the considered case, the function F, in the macroscopic densityapproximation (3.26), is given by

F(y) = Do(«/ exp ( - ±

(C.I)- i £>n TT [erf ((u/2)= {y, +

i

Using this expression in Ea,. (3.39) one finds

(C.2) o, = i/?J = £=) E {{af2)'

- 2 7 -

where

(C'i) £(*)

Since :r -.- (o/2) l / 2 /,, » 1, we can take E{x) = 4z, so that

Appendix D

The operation T'(p) defined by Eq. (4,6) can be written

' I ) - I ' Whf1- j<ih'bnp (-f3b2) expWA-b)pexp (-jib2) exp

hnvinp; suppressed the index I. Inserting the expansion

ink) Kq. (D.I) and takhig into account, that

(l).:ia) W/^f!- f<fKbnp (- /.?62) b, - 0,

(D.3I,) (^/TT)"72 /rfAbexp (-^63) 6,-fej = ^ * ; j ,

(l).:!r) (;J/ir)K/! f<iKbexp[-l3b2)bilbii---b,u = 0 (odd n),

• >ti<- l i ixls

' / '(,)) •:• exp { !;:iA2)

\p I i ^ V (.'l,7'-4r 4- ̂ {.4;,̂ }-) -

Finally, using I he expans ion

( ] * . . r > ) exp f kfiA2) — 1 - v,i3

;inil restuiing the index /, one gets Eq. (4.9).

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