Foundations of Physics, Vol. 22, No. 3, 1992

Many-Hilbert-Spaces Approach to the Wave-Function Collapse Mikio Namiki 1 and Saverio Pascazio 2 Received September 6, 199I The many-Hilbert-spaces approach to the measurement problem in quantum mechanics is reviewed, and the notion of wave function collapse by measurement is formulated as a dephasing process between the two branch waves of an interfering particle. Following the approach originally proposed in R e f 1, we introduce a "decoherence parameter," which yields a quantitative description of the degree of coherence between the two branch waves of an interfering particle. By discussing the difference between the wave function collapse and the orthogonality o f the apparatus" wave functions, we analyze critically two proposals, recently appeared in the literature, (z' 3~ and argue that neither one describes a dephasing process. We conclude that the concept of "wave function collapse," according to the conventional Copenhagen interpretation, is to be replaced by that of a statistically defined dephasing process.

1. I N T R O D U C T I O N The Copenhagen approach to the measurement problem in quantum mechanics is based on the conventional notion of wave function collapse by measurement. Many objections have been raised against the Copenhagen interpretation and von Neumann's postulate, which require the presence of an external observer in order to explain the evolution from a pure to a mixed state: Indeed, this is unsatisfactory from several points of view, because, for instance, the necessity of introducing classical concepts in order to explain the quantum postulates makes quantum mechanics a non self-contained theory. On the other hand, S. Machida and one of the present authors pointed out some years ago (a) that it is possible to explain the evolution from a t Department of Physics, Waseda University, Tokyo 169, Japan. 2 Dipartimento di Fisica, Universit~i di Bari and INFN, Sezione di Bari, 1-70126 Bari, Italy. 451

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pure state to a mixed state without resorting to classical concepts. In their many-Hilbert-spaces (MHS) approach, the wave function collapse is indeed described within quantum mechanics itself via a continuous superselection rule whose precise mathematical meaning was later discussed by Araki. (5) The MHS theory has also formulated a definite numerical criterion (1) for the wave function collapse in terms of an order parameter e, which gives an estimate of the degree of coherence of a quantum system. This criterion has been applied to the study of neutron interference experiments with an absorber. (6' 7) In this paper we shall review the main characteristics of the MHS formulation along the line of thoughts of Ref. 1, and shall comment on two papers recently written by ScuUy and Walther (2) and by Stern, Aharonov and Imry. (a) We shall argue that these papers do not describe the wave function collapse, but simply the vanishing of the interference term in a double-slit experiment. Let us start by summarizing the main ideas of the MHS approach.

2. THE " C O L L A P S E OF THE WAVE F U N C T I O N " IN THE MHS APPROACH Let us begin our discussion by analyzing a typical "yes-no" experiment in which an "emitter" E sends particles into a measuring apparatus. The wave function of each incoming particle is split into ~ = 01 + 02, where ~'1 and 02 are branch waves running through the two spatially separated routes I and II. This is the so-called spectral decomposition, and fully preserves the coherence between the two branch waves. ~t'4~ We place a "detector" D on route 2 and assume that it modifies the corresponding wave function according to @i -~ Zl//2

(1)

In the following discussion, for the sake of simplicity, we shall consider only the case of an ideal non-destructive detector for which T is a complex number with modulus very close to (eventually a little less than) 1. If the emitter E and the detector D click in coincidence, then, according to the conventional Copenhagen interpretation, the following transition of the wave function takes place ("wave function collapse"): 0 --+ @2

(~.t1 disappears)

(2)

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453

On the other hand, if E clicks but D does not, the wave function evolves according to ~/ "-)' I/11

(~t 2 and therefore ~ disappear)

(3)

Equations (2) and (3) are sometimes accepted as the "measurement postulate," under the basic assumption that the probability of detection by D is proportional to P2= I~,~12 (very close to 10212), while the probability of no detection by D is proportional to Pt = 1~112, We shall refer to this kind of behavior, characterized by the disappearance of one branch wave, as to the "naive Copenhagen interpretation." The process in Eqs. (2) or (3) is an acausat and probabilistic event and cannot be regarded as any kind of wave motion in which ~ is continuously shrinking into 01 or ~%. Moreover, it should be remarked that quantum mechanics can never predict the definite result of a single measurement on one dynamical system in a superposed state, but it gives only a probabilistic prediction for the accumulated distribution of measured values obtained by many independent single measurements on many independent systems, each of which is described by the same wave function. Hence, there is no way of talking about the measurement problem except in terms of an accumulation of many experimental results. In order to examine the physical effect of the change ~2 ~ ~; given by D, let us introduce the Young-type "two-step" experiment, in which the two branch waves are forwarded to small slits in order to produce two spherical waves travelling towards a screen. Each particle makes a single spot on the screen, and can be observed there with a probability distribution proportional to

I~1 ÷ ~12 = P1 + P2 + 2Re(O* ~'2)

(4)

in which for simplicity we have used the same symbols, I//1 and ~ , for the spherical waves as for the original branch waves. We can obtain a particle distribution proportional to Eq. (4) on the screen, if we accumulate many spots, one after another, corresponding to many particles brought into the experimental setup by a stationary beam. The last addendum of the r.h.s. is the interference term and reflects the coherence between the two branch waves. The naive Copenhagen interpretation enforces us to accept that one of the branch waves actually disappears in each measurement. The disappearance results in the erasing of the interference term in each measurement, by which the probability distribution in Eq. (4) becomes P1 + P2

(5)

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As was mentioned earlier, however, quantum mechanics never gives us a definite answer for the result of a single measurement on one system, so that no one can know whether a branch wave has disappeared or not in a single measurement. This implies that we can observe Eqs. (4) and (5) only on the accumulated distribution over many particles. Consequently, the "wave function collapse" is not to be considered as the afore-mentioned disappearance in a single measurement, but is rather to be formulated for the accumulated distribution as follows:

Re(O*O'2) = 0

(6)

accumulated

For this reason, in our approach, we describe the notion of "wave function collapse" by Eq. (6) (or equivalently, Eq. (5)) instead of Eqs. (2) and (3). This can be realized if the detector gives, for the ensemble of accumulated particles, a random sequence of phase shifts between the two branch waves in an experimental run. Under this notion of wave function collapse, therefore, we do not require a branch wave to disappear in a single measurement, but assume instead that both branch waves are still alive after the interaction with D. The essential ingredient to obtain Eq. (6) is not the disappearance of one branch wave but the decoherence between the two branch waves. In other words, a perfect detector is an apparatus that yields Eq. (6) by provoking a perfect decoherence between the two branch waves. By observing that Eq. (5), obtained by erasing the interference term, is a sum of probabilities of finding one of two mutually exclusive events, we can easily understand the natural result, usually referred to as "wave function collapse," that once we have found one event, another event should never occur. This is the same result obtained by the naive Copenhagen interpretation. In this context, one may say that, in the MHS approach, the notion of wave function collapse still remains, but within a wider framework than the Copenhagen one. It should also be emphasized that in our proposal the wave function collapse yielding Eqs. (5) or (6) takes place in the detection step but not in the spectral decomposition. The latter is only a preparatory step of the whole measurement, in which the phase correlation between the two branch waves is fully kept. As an additional reason against the disappearance of the branch waves, let us introduce a different kind of gedanken experiment, in the following way. Suppose that instead of D we place, in route II, equipment controlled by a parameter e; the equipment is empty if e = 0 but becomes a perfect detector in the case e = 1, through a sequence of intermediate steps, in which e changes continuously from 0 to 1. As an example, one can consider e to be proportional to the density of material in the detector. In

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the case e = 0, of course, we have full coherence between two branch waves. But what happens for any finite value of e, smaller than unity ? It is natural to assume that the two branch waves do not disappear and keep the phase correlation, at least up to a certain extent, even though the modulus and the phase of 02 are modified by the interaction with the constituents of the equipment. On the other hand, we can design the equipment so as to generate a signal for certain arbitrary values of e smaller than unity, through an appropriate physical process. In this case we are performing an imperfect measurement, which yields a partial collapse of the wave function, because the interference term, though reduced, is still nonvanishing. In an imperfect measurement, the two branch waves do not disappear, but only lose partially their relative phase correlation (their coherence). Even when e is close to 1, we cannot accept that one branch wave disappears in an imperfect measurement. Why must we accept the disappearance only in the limit e = 1 ? We must not and need not accept it. What actually happens in this limit is only the total dephasing or the complete decoherence between the two branch waves, and this is enough to yield the wave function collapse, given by Eqs. (5) or (6). This implies that the same equipment can work well as a measuring apparatus in some cases but simply acts as a phase shifter or an absorber in other cases, depending on the condition specified by parameters like the afore-mentioned e. The reason why we have a random sequence of phase differences between the branch waves in an experimental run is that different incoming particles will interact with the detector system or with some of its local systems (with a huge number of degrees of freedom) in different microscopic states and will then undergo different phase shifts, because the internal motion of the detector system will change its microscopic state during each intermediate time-interval between subsequent measurements on different particles. The whole ensemble of the microscopic states of the detector or of its local systems relevant to an experimental run cannot be represented within the framework of a single Hilbert space, so that we are inevitably led to a direct sum of many Hilbert spaces for their representation. (4) In the limit of infinite degrees of freedom, our wave function collapse is described by a continuous superselection rule. (4' 5) In the next section, we will introduce our decoherence parameter ~, in terms of which a definite criterion for the wave function collapse can be formulated. An outline of the density matrix description of the MHS theory will be given in Section 4, and we shall comment on Scully and Walther's and Stern, Aharonov and Imry's proposals in Section 5.

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3. A C C U M U L A T I O N OF SINGLE EVENTS Following the line of thought outlined in Section 2, we shall reformulate a "yes-no" experiment by taking into account the MHS structure of the detector. In doing so, we shall also find that it is possible to define an order parameter, that wilt be named &coherence parameter, in terms of which a precise and quantitative definition of wave function collapse can be given. In a "yes-no" experiment an incoming wave function is split into two branch waves ~1 and ~k2, corresponding to the two routes in the interferometer. (Henceforth, we will call "incoming particle" an incoming Schrrdinger wave function). We will place our detector along the second path, so that the relative wave function will be modified according to ~2 ~ T~/2, where T is the detector's "transmission coefficient." The wave function will be = ~1 + TO2

(7)

and the intensity after recombination

1~&I2=I~&1+TOzt2= IO112+ ITI21O212+ 2Re(t~ TtP2)

(8)

Let us take now the MHS structure of the detector into account: Equations (7) and (8) hold for every single incoming particle. Let us label the incoming particle with j (j = 1,..., Np, where Np is the total number of particles in an experimental run), and rewrite the transmission coefficient as

r-,Tj,

j = l ..... up

(9)

This reflects a fundamental property of our approach: every incoming particle is described by the sum of the same branch waves ~ = ~kl + ~2, immediately before interacting with the detector. But after the interaction, the detector transmission coefficient T will depend on the particular detector state at the very instant of the passage of the particle. Since the detector undergoes random fluctuations (which reflect the internal motion of its elementary quantum constituents and its MHS structure), the same macroscopic state of the detector will correspond to many different microscopic ones. Consequently, different incoming particles will be affected differently by the interaction with the detector, and will be described by slightly different values of T. Accordingly, Eq. (8) becomes

I~(J)12=lOa+ZjO212=l~,l=+lT~l=[O212+2Re(~?Zj~2)

(10)

If we define P(J)= [~O(J)[2 as the probability of detecting the j t h particle

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after recombination of the two branch waves, then, after many particles have been detected, the average probability will be given by 1 up - - ~ P(J)= I~I2+i[~212+2Re(O*T~2)

P=Np+:,

(11)

where we have defined the average transmission probability 1

N~

t-: g , ~ l

[T,[2

(12)

and the average transmission coefficient 1

Up

e = 7l,p E ~,

(13)

]TI2#{

(14)

j=l

Note that, in general,

Moreover, from Eq. (11), a sufficient and necessary condition for observing no interference (wave packet collapse) is T = 0 or equivalently [TI 2 =

0

(15)

Observe that Eq. (15) represents the total dephasing between the two branch waves: This is the condition for the total loss of coherence between the two interfering branch waves. In order to make our argument more precise, let us write IT]2= t-(1 - e )

(16)

where ~ can be shown to be 0 ~
, / ? x/1 - e Re(~b*eie02)

(17)

where we have written T = I TI e ;~. In Eq. (17) the interference term contains the factor x/1 - e, which is absent if the MHS structure of the detector is not taken into account: The standard quantum-mechanical formula for the intensity at the screen (Eq. (8), with t = ITI2), P = [@ll2 + t [~b2[2 + 2 x/7 Re(~b*e~e~b2)

(18)

is recovered in the limit ~ = 0. On the other hand, from Eq. (17), we find that interference is lost, and hence the wave function collapse takes place,

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in the limit e = 1. For this reason, e has been named decoherence parameter. The intermediate cases, in which 0 < e < 1, correspond to partial losses of coherence, namely to the partial collapse of the wave function. The parameter e is in fact an order parameter for the wave function collapse. We shall conclude this section with an ergodic hypothesis: the average over many particles going through the detector (denoted hitherto with a bar) is assumed equal to the statistical ensemble average over all the possible detector's microstates. If we denote the latter with < . - - > , our assumption reads ....

<... >

(19)

This ergodic hypothesis leads to a natural interpretation of the formulae of the present section. It is worth stressing that our ergodic assumption makes sense only if Np, the total number of particles in an experimental run, is very large. Stated differently, it makes no sense to speak of wave function collapse for a single particle. Moreover, it can be shown that a random sequence T:, for which T = 0 or e = 1, can be obtained for a detector with a huge number of degrees of freedom.

4. D E N S I T Y MATRIX F O R M U L A T I O N OF THE WAVE F U N C T I O N C O L L A P S E I N THE M H S T I I E O R Y

We shall now give a formulation of wave function collapse in terms of density matrices. The aim of such an approach is twofold: First, the loss of coherence suffered by a quantum system is most simply formulated in terms of density matrices, where it is known to correspond to the absence of off-diagonal terms; Second, the formulae of this section will allow us to compare our results to other proposal in the literature. (2' 3) Let us denote the density matrices of the total system, the object particle and the detector system by ptOt, pQ and pD, respectively. The whole detection process can be written as

p rt°'= e

iHt/h p Q ® p D eiHt/h

t ~ ~ ) e - iH°t/h s { p 1Q ® p D } S "fe iHOt/h

(20)

where we have introduced the S-matrix defined by

e im/h , ~ ,

e-i~ot/~S

(21)

H and H o being the total and free Hamiltonian of the total system, respectively, and H o = H0Q + H g is given by the sum of the free Hamiltonians of Q and D respectively. The density naatrices p/e and p~ represent, respec-

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459

tively, the initial states of Q and D before the detection and after the spectral decomposition. In particular, we have

PQ= Z

I~k)(0,1

(22)

k , l = 1,2

corresponding to the decomposition O + 01 + ~2, given in Section 2. Following the discussion in Sections 2 and 3, we add the subscript j to all the relevant quantities in Eqs. (20) and (21), because thejth particle in an experimental run will meet the detector in a (micro)state described by p D.j and will undergo a transition described by Sj. Hence, the accumulated distribution on the screen, in an experimental run, is described by the following average of p ~ over j: ~.tot_ ~t

1

--Npj

N,

lim p,,j tot =

t~

~

~.k,l

(23)

k,l=l,2 ~ t

where Zkl

1

, = _ _ ) , e--,Ho*/hSj{[@k)(l@Zl ",~~''.J"o ~CtJt0*/h~,j,. Npj=l

(k, l = 1, 2)

(24)

As mentioned in Section 3, we can replace the average over j with the statistical ensemble average over all the possible detector's microstates on the basis of the ergodic hypothesis of Eq. (19). Taking into account the fact that the interaction takes place only between ~92 and D, we can write (by dropping the index j for simplicity)

s*=lO1)(Oll®p ~, s t = I~12F)( ~12F]@pD S t = [~l)(ff~F] ®(P~ St)

(25)

5{1¢'2)(011 ®P~} S* = 1~2F)(Ca I® (Sp~) where t@2F), (~/2FI and pD stand for the corresponding final states. On the other hand, the general structure of the S-matrix is known to be S = ei°S'e i°

(26)

in the channel representation (4' 8) in which 0 is a diagonal matrix representing the main part of the phase shift and is proportional to some parameter characterizing the size of the target (like, for instance, N, the number of elementary constituents of the target), and S' is responsible for possible channel-couplings including the reduction of the transmission probability.

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Let us restrict our attention to the simple case in which the particle is transmitted with probability 1 (elastic one-channel case). Then we can write !~2F) = T 1~92)

T = e2i°~'(~t2t S' tt/12)

(27)

Note that T is just the same coefficient given in Eq. (7) (apart eventually from a trivial factor). We have now to add the subscriptj to all the relevant quantities in Eqs. (25-27), and then take the averages of ~,='k't in Eq. (24) over j. Since, for large values of N (the previously mentioned parameter characterizing the size of the target), the phase shift 0e~ can become completely random, we have ~-t~k't-n-v for k ¢ l. The condition T = 0, and therefore e = 1, is representative of the vanishing of 2~ 't for k ¢ l. That is to say, all the off-diagonal components of Eqs. (23) and (24) (with respect to the routes), vanish under the condition ~ = 1 because they are proportional to T, while all the diagonal and auto-correlated components are kept nonvanishing because they depend only on I TI 2. Thus, we obtain from Eq. (23), under the condition e = 1, ~'tot

'~'1,1

~2,2

(28)

which is an explicit expression for the wave function collapse, characterized by the lack of off-diagonal and cross-correlated components. This formula will be the starting point for a comparison with some alternative proposals in the next sections.

5. D I F F E R E N C E BETWEEN WAVE F U N C T I O N COLLAPSE AND O R T H O G O N A L I T Y O F THE APPARATUS WAVE F U N C T I O N S Let us start by considering the spectral decomposition step of a typical Stern-Gerlach experiment given by a magnetic field: ~I"1-.~-(CaUa "~ ebUb)~ -+ Cabla~ a "-ItCbUb~ b

(29)

where u a and Ub are spin eigenfunctions, Co and Cb constants, ~b is the position wave packet before the magnetic field, and ~ba and ~b are the wave packets running in the upper and lower channels after the spectral decomposition. It is quite clear that (~ba [~bb) = 0, because the supports of the two wave packets never overlap. However, we cannot say that the process in Eq. (29) describes the wave function collapse, because the two branch waves still keep their phase correlation which will be observed after a possible recombination.

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461

Sometimes the process in Eq. (29) is identified with a typical measurement process (yon Neumann) by regarding uk ( k = a, b) as object eigenfunctions, and by replacing ~b and ~bk, ( k = a , b) with apparatus wave functions. The previous arguments therefore imply that the orthogonality between the ~bk's does not necessarily give the wave function collapse (refer also to the discussion in Section 6). From this point of view, we do not agree with the conclusions of Scully and Walther c2) and Stern, Aharonov and Imry, (3) namely that the orthogonality of the apparatus wave functions gives the wave function collapse. 5.1. Scully and Walther's Case Let us briefly review the main points of Scully and Walther's (SW) analysis, (2) who consider a polarized neutron interacting with two micromasers. The ith micromaser (i= 1, 2), whose initial state vector is [q~0), is placed along the ith arm of an interferometer. The initial neutron + micromaser state vector is ~ ( t = o) = [01(r, O) +

~'2(r, 0)] IT> ® I~°~°>

(30)

where ~'i(r, 0) is the neutron wave packet running through route i, and the initiaJ neutron spin is assumed to be "up." If the micromasers are prepared in a state that can provoke a spin flip on the neutron with probability close to one, the final state vector is ~(t) = ~t(r, t)J.l.> ® 14~{~b~> + ~2(r, t)[,I.> ® t,:/,°c/,(>

(31)

where lq~{> denotes the state of the micromaser after interaction. The interference term is

2Re~,* ~b2( ~(~°l q~0q~>

(32)

and one can see that the presence of the maser provokes a reduction of the interference term, which turns out to be multiplied by the factor f 0 0 f (q~,~21~1~2>

(33)

SW consider two interesting cases. If the micromaser cavity is prepared in a coherent state lq~°> = [~i>,

i = 1, 2

(34)

i = 1, 2

(35)

one can write, to a good approximation [q~f> ,,, [c~,>,

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because the "classical" coherent field is not changed much by the addition of a single photon associated with the neutron spin flip. In this way f 0 0 f ~1~21 qSt~b2) -~ (0q~2{0q~2) = 1

(36)

and interference is not reduced, even though spin flipping has occurred. On the other hand, if the micromaser cavity is initially in a number state 1~ °) = Inz),

i = 1, 2

(37)

things are very different, because after the neutron spin flip we have l ~ ' ) - - . I n f + t ),

i=1,2

(38)

and interference disappears, because f 0 0 f (qYlq521 q~l~b2) ~- @1+ 1, n21nl, n2 + 1 ) = 0

(39)

SW argue that this provides a counterexample to Bohr and Heisenberg "random-phase" argument, according to which interference is lost in a double-slit experiment when the interaction between the particle and the experimental apparatus yields a random phase shift between the two branch waves of the former. According to SW, the loss of interference due to the vanishing of the scalar product in Eq. (39) reflects a toss of coherence between the two branch waves of the particle. We think that the last conclusion is incorrect: The vanishing of the interference term does not imply at all a loss of coherence. A loss of coherence corresponds to what is commonly called "collapse of the wave function," and this is not what happens in the example proposed by SW. This can be easily shown by writing the final density matrix for the neutron + micromaser syste/n p , = i~,,l=®l~,Yq~o)(~{45ol +lg,=12® 105,~2)(~,05210 f o Jr + t~* qJ2® 1~°052f) (~lf~°l + h.c.

(40)

where the common term 1{ )(~L has been suppressed for simplicity. One sees immediately that the off-diagonal part of the density matrix does not vanish, as a result of the interaction. This conclusion is true independently of the final state of the maser cavity, which can be given indifferently by Eq. (35) or (38). Equation (40) should be compared to Eq. (28), obtained by us in the MHS framework. In this sense, we can state that the process analyzed by SW correspond to a loss of interference, and not to the wave function collapse. Indeed, in Eq. (40), the quantum-mechanical coherence is fully kept between the two branch waves of the neutron +micromaser

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system. We shall come back to this important point, that has already been a source of deep misunderstanding in the past, °) in our final discussion. 5.2. Stern, Aharonov and Imry's Case

The model studied by Stern, Aharonov, and Imry (SAI) (3) is an electron interacting with an environment. As a guiding example, one can think of an Aharonov-Bohm interference experiment on a ring. The electron, whose coordinate is x, is split into two branch waves l(x) and r(x), crossing the ring along the right and left side, respectively. The initial wave function of the electron + environment system is written as gJ(t = 0 ) = El(x) + r(x)] ®)~o(t/)

(41)

where t1 denotes the set of coordinates of the environment and Zo its initial wave function. At time %, when interference is observed, the wave function is

7t(Zo) = l(x, Zo) ® Zl(r/) + r(x, %) ® Z,(t/)

(42)

where Zl.r(tl) are the states of the ring after interaction. The interference term is given by

2Re(l*(x, zo)r(x, zo)fdtlz~(q)Zr(q) )

(43)

and it is reduced by the factor f

dt/x*(t/)Zr(t/)

(44)

Notice that, since the environment is not observed, its coordinates q are integrated upon: Mathematically, this corresponds to taking the scalar product between the two final environment states, exactly like in the example proposed by Scully and Walther. Observe that also in this case the final density matrix of the electron + environment system is p,0 =

It(x, %)12® IZzO1)) ()~t(q)l + Ir(x, ~o)I2® lZr(~))(Z,(tt)l + l*(x, to) r(x, %) I;G(t/)) (Zl(~/)l + h.c.

(45)

so that the off-diagonal part does not vanish, as a result of the interaction. As already pointed out in connection with the previous example, this

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process describes a (partial) loss of interference, and not the wave function collapse. Unlike Scully and Walther, SAI argue that this loss of interference can alternatively be ascribed to a dephasing of the two electron states. But even in this case, we do not understand the reason why we can take the partial inner product or the partial trace with respect to the apparatus states. We stress again that we have to show the vanishing of the off-diagonal components of the total density matrix, without resorting to the partial inner product or the partial trace. Indeed, the task of a satisfactory theory of measurement is to derive an evolution of the type in Eq. (28) (or, in general, a condition similar to Eq. (6)), instead of simply postulating it.

6. D I S C U S S I O N In order to make our point clearer, let us consider a yon Neumann measurement process

~u(t = O) = ~b® q)o ~ ~(t) = ~ CkUk® ~bk k

(46)

where ~, = Y'.k CkUk is the particle state vector, and q~o and ~k are, respectively, the initial and final states of the detector. The notion of wave function collapse has been often identified with the asymptotic orthogonality (~)k, ~l)=(~kl"~O(e); • N~ov), 0

(47)

where N represents the degrees of freedom of the detection system. This is simply not correct: Indeed, if we decompose the final state density matrix into the sum of its diagonal and off-diagonal parts P = I ~u) ( ~uI = Pdi,g + Po~

=2[Ckl2lUk)(Ukl®lC[)k)((fPk[ q- 2 CkCl~]Uk)(Ul](~]~k){(J~ll k

(48)

k~l

we know that by calculating the trace with respect to the D-states before taking the limit for N ~ o% we obtain

Tr Dpoff = O(e) (49)

TrDp2off=~ Ickl 2 (1 --Ickl 2) Iu~)(ukl + O ( e ) k

This means that even though its trace vanishes, Port itself does not vanish, even in the infinite N limit. Therefore this kind of approach can never give

Many-Hiibert-Spaces Approach to the Wave

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the exact wave function collapse as formalized by us in Section 3, and in Eq. (28) in particular. This conclusion applies to the two cases considered in the previous Section, (2' 3) as welt as to other attempts at formulating the wave function collapse, in which the disappearance of the off-diagonal components is not explicitely shown. (1°) Most recently, in his comment on the quantum Zeno effect, (11) Ballentine argued that Eq. (29) or Eq. (46) (together with Eq. (47)), are quite enough to describe the wave function collapse. We do not agree with his argument for the same reasons as mentioned earlier. The quantum Zeno effect itself will be discussed in a forthcoming paper. (12) 7. CONCLUSIONS We have summarized the MHS approach to the quantum measurement problem and have shown that it is capable of explaining in a simple way the loss of quantum mechanical coherence as a result of the interaction with macroscopic apparata. Our approach has been compared to some alternative proposals in the literature, and it has been argued that these do neither describe the wave function collapse, nor imply a loss of quantum mechanical coherence. We have proposed that coherence is lost at a statistical level, when many interfcring particles interact with a macroscopic object: This is ascribable to the effect described by Eq. (14), which is in fact responsible for the loss of coherence and which allows us to define the parameter e of Eq. (16). In terms of this decoherence parameter it has been possible to formulate a definite quantitative criterion for the wave function collapse, which is obtained in the limit t : 1. We also showed that this condition corresponds to the vanishing of the off-diagonal terms of the local density matrix, as shown for instance in Eq. (28). We emphasize that the loss of quantum mechanical coherence (i.e. the vanishing of the off-diagonal terms of the density matrix) is shown explicitely, in the MHS approach, and is not to be simply ascribed to the orthogonality of the apparatus wave functions and/or to be postulated, like in the original theory of yon Neumann, in which the wave function is collapsed due to the intervention of an external observer who is supposed to provoke an acausal change of the wave function by simply observing the quantum system. No external observer is needed in our case, in order to explain the collapse of the wave function, for the evolution from a pure state to a mixture is simply a statistical effect due to the macroscopicity of the detector. This work was partially supported by Italian CNR under The bilateral project Italy-Japan. n. 91.00184.CT02

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REFERENCES 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 1t. 12.

M. Namiki and S. Pascazio, Phys. Rev. A 44, 39 (1991). M. O. Scully and H. Walther, Phys. Rev. A 39, 5229 (1989). A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 4!, 3436 (1990). S. Machida and M. Namiki, Prog. Theor. Phys. 63, 1457 (1980); 63, 1833 (1980); M. Namiki, Found. Phys. 18, 29 (1988). H. Araki, Prog. Theor. Phys. 64, 719 (1980). J. Summhammer, H. Rauch, and D. Tuppinger, Phys. Rev. A 36, 4447 (1987); H. Rauch, Proceedings of the Third International Symposium on the Foundations of Quantum Mechanics, S. Kobayashi et aL, eds. (Physical Society of Japan, Tokyo, 1990), p. 3. M. Namiki and S. Pascazio, Phys. Lett. 147A, 430 (1990). K. M. McVoy, Ann. Phys. (lv2Y.) 54, 552 (1969). E. P. Wigner, Am. J. Phys. 31, 6 (1963). See for example, K. Hepp, Helv. Phys. Acta 45, 237 (1972); R. Fukuda, Phys. Rev. A 35, 8 (1987); A 36, 3023 (1987); Prog. Theor. Phys. 81, 34 (t989). L. E. Ballentine, Phys. Rev. A 43, 5156 (1991). S. Inagaki, T, Tajiri, I. Nabekawa, and M. Namiki (in preparation).

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