Foundations of Physics Letters, Vol. 5, No. 3, 1992

W A V E - F U N C T I O N C O L L A P S E I N D U C E D BY T H E R M A L F L U C T U A T I O N S OF D E T E C T O R V A RIA BL E S. S T A T I S T I C A L C O L L A P S E OF WAVE F U N C T I O N S

Tsunehiro Kobayashi

Institute of Physics University of Tsukuba Ibaraki 305, Japan

Received October 31, 1991; revised March 17, 1992

We present a simple example where the wave-function collapse is re- alized by thermal fluctuations of detector variables. This mechanism for the wave-function collapse is essentially due to the indeterminacy of the initial internal-states of detectors by the thermal fluctuation. Two types of the collapse are shown: One is the mechanism of the col- lapse arising from the phase ambiguity of the initial detector state and another is that derived from the fluctuation of the number of particles in the detector actively working in single measurement process.

Key words: quantum measurement, quantum theory, wave-function collapse.

1. I N T R O D U C T I O N

The projection postulate for the quantum theory of measure- ment proposed by von Neumann [1], that is, the transition of a pure state to a mixed state (say the wave-function collapse), still remains one of the fundamental problems in quantum mechanics. Many ideas have been presented for solving the problem over the past decades, but only a few models can directly show the transition on the formula of the density matrix.

Recently we have shown that there are two different mechanisms for realizing the wave-function collapse [2-6]. One mechanism called "quantum collapse" is represented by the internal trace for detector variables, which is defined by the partial trace operations in terms of

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266 Kobayashi

detector variables ignored in the measurement process [2-6]. (Two simple examples of this mechanism are given in Ref. 2.) The other mechanism called "statistical collapse" stems from statistical averages over ensembles carried out in quantum measurement processes [3-5]. Many-Hilbert-space theory which was first proposed by Machida and Namiki [7] and has extensively been studied by Namiki and Pascazio [8] is one of the latter mechanism. In order to clarify the difference between the two mechanisms, I shall in this paper present a simple example where the quantum collapse is not realized but the statistical collapse occurs. Criteria for the statistical collapse will also be further clarified.

Before starting the evaluation, let us briefly summarize the fun- damental idea for the mechanism of the wave-function collapse in terms of the internal trace [2-6]

(1) Even though detectors have a huge number of quantum num- bers corresponding to the internal variables, only one of them is taken out as the physical quantity to be read in the detectors, and others are deliberately ignored.

(2) The role of these internal variables ignored in the measurement process is taken into account as the internal trace defined by the the partial trace operations with respect to them. Note that the internal trace plays the same role as the form factor introduced in vertices for composite systems [5, 6].

(3) The macroscopic limit of the physical quantity to be read in the detector gives rise to the transition of a pure state to a mixed state, that is, the disappearance of the off-diagonal elements in the density matrix (the wave-function collapse).

We can understand that the wave-function collapse in this mech- anism is induced by a huge accumulation of small quantum effects, each of which represents a quantum mechanical change of the internal variable in the detector. The internal trace introduces this accumula- tion in the density matrix. This mechanism may be called "quantum collapse," because all quantities introduced in this formalism are de- finable and calculable in the ordinary quantum mechanics. These are summarized as the criteria for the detectors in the quantum measure- ment processes [2-4]:

Criterion (1): The detector must have the mechanism to produce a measurement quantity which becomes a macroscopic number in a certain limit.

Criterion (2): The detector must have the mechanism to realize the wave-function collapse in the macroscopic limit which is defined via the process given in the Criterion (1).

Wave-Function Collapse 267

An apparatus can possibly be a detector even if the wave- function collapse is not realized in the mechanism discussed above. Since an apparatus containing a huge number of internal variables always has ambiguities arising from thermal fluctuations, we cannot uniquely determine the initial internal states of the apparatus. Con- sidering that in the quantum measurement the same experiment must repeatedly be performed under the same conditions, this ambiguity of the apparatus should be taken into account. That is to say, the same condition means the same except for the nonuniqueness of the internal state of the apparatus. The statistical average over the am- biguity has to be taken into account in the quantum measurement processes. Though the general framework for this problem has been given in Refs. 3 and 4, the criteria for realizing the wave-function col- lapse is still not clear. We shall, therefore, study this mechanism via a simple example and show the criteria explicitly.

In Sec. 2 the statistical average is studied in the general frame- work. A simple and calculable example is presented and the criteria for the wave-function collapse are shown in Sec. 3. In Sec. 4 the prob- lem of distinguishing the two mechanisms are discussed. Remarks will be given in Sec. 5.

2. G E N E R A L F R A M E W O R K OF S T A T I S T I C A L C O L L A P S E

Since the apparatus consists of a huge number of constituents, it is impossible to determine its internal states completely. Let us consider this situation in quantum measurement processes. In the initial state the apparatus (say detector) must have a fixed value with respect to the physical quantity (QI) to be read in the detector, but its internal state cannot uniquely be determined. In general such a state is described in terms of the superposition of different internal states as

[ QI ) = ~ C ([q]i) l QI: [q]! ), (2.1)

where C ([q]i) is the coefficient normalized by

]C ([q]i)]2 = 1, [q],

[q]I = (ql, q2,.- .) represents a combination of all quantum numbers required to describe the internal state completely, and the sum must be performed over all different combinations compatible with the fixed value QI. As was mentioned in Sec. 1, the experiments must be re- peated over an ensemble to obtain the correct probability distribution

268 Kobayashi

of the measured quantity (Q). We cannot expect that the coeffi- dents C ([q]l) will be the same via all the repeated experiments. We therefore have to attach a label representing the order of the repeti- tion to the coefficients as C (t) ([q]I), where I denotes t he / th order in the repeated experiments.

The final state of the detector with a fixed value Qm also have to be labeled by 1 as

I Qm )(0 = y ~ C(O ([q]m)I Qm: [q]m ), [q] .~

(2.2)

because the final coefficients C g) ([q]m) are related to the initial ones via the S matrix representing the interaction of the detector with the object as

C g) ([q]m) = E ( Qm: [q]m ! SI QI: [q]z ) C(O ([q]i). [q] ..

(2.3)

Note that, as was shown in Ref. 2, the detector states with different Q are generally not orthogonal to each other, i.e.,

(Q': [q'] I Q: [q]) # 5Q,,Q, (2.4)

for fixed N < c~, but for Qm = QI = Q the orthogonality

(Q: [(] I Q: [q]) = 5[q,],[q] (2.5)

is kept (for details, see Ref. 2). Following this line of reasoning, the density matrix has to be

distinguished by the label l as

(2.6)

where the final state for the object and the detector in the lth exper- iment is given by

) } = (2.7) rt

In (2.7),)--~nCnlUn) stands for the spectral decomposition of the object state in terms of the quantum number to be measured, and I Qn )(l) is the detector state corresponding to the nth eigenvalue of

Wave-Function Collapse 269

the object. The density matrix with the internal trace is written down as

P (0 = I n t ' e f f TrQ(fl (l))

(2.s) = E E cne~n In) (m[ ® [[ Qn ) ( Qm 1[ 1(l) (Qn,Qm),

n m

where

II Qn ) ( Qm II 1(l) (Qnm) (2.9)

_= Tr[q] C g) ([q]n) C g) ([q]m)* I Qn: [q]n ) ( Qm: [q]m 1"

In (2.9) Tr[q] means the trace over the internal variables [q]. The density matrix must be averaged over a large number (say L) of the repetition; then we have

L pg)\ _--__ 1 x ~ (l) (2.10)

l= l

n m

where

IIQ ) (Q.~ II (I(t) (Qn,Qm)}L _ Tr[q] ( C (l) ([q]n)C g) ([q]m)*} L ]Qn: [q]n ) ( Qm: [q]m I"

and the average operation is defined by

(2.11)

L 1 (Ag)Bq)

l=1 (2.12)

We easily see that the averaged density matr ix/pg) \ coincides with \ e f f / L

the usual density matrix with the internal trace Peff if all the elements do not have the l dependence, i.e., there is no ambiguity in the ini- tial detector state. In the very special case, where the coefficients

270 Kobayaslfi

C (t) ([q]n) are completely random with respect to their phases and also their magnitudes, the average over l derives the relations in the limit of L ~ c~:

lim (C q) ([q]n) C q) ([q]m)* }L = ~Q,.,,Q,,,~[q],~,[q]m" (2.13) L-~cx~ \

These relations are nothing but the realization of the wave-function collapse [3-5].

3. A SIMPLE E X A M P L E OF T H E STATISTICAL COLLAPSE

We shall investigate a typical double-slit measurement in which the incident wave of the object is split into two branch waves CA and CS, corresponding to the two different paths. An apparatus is put only on the path A. After the interaction of the object and the apparatus the final state of the total system ] ~I/F } is written as

B

n = A

(3.1)

where I¢,t ) stands for the apparatus state corresponding to the nth eigenstate of the object. The density matrix with the internal trace is given by

Peff = Int. Tr[ # F ) ( q/F I

= CAC*A l e A ) (CA]® Int. Tr (I¢A) (CA i)

+ c8c~ I ¢B ) (¢81 ® int. T~ (I CB ) ( ¢8 i)

+ cac~ I Ca ) (¢8 I ® ~nt. T~ (I CA > ( ¢8 I)

(3.2)

+ CBC~4 I ¢B ) < CA [ ® Int. Tr (I CB ) ( CA I),

where [¢B ) coincides with the initial state, because the apparatus does not act when the object travels along the path B.

Wave-Function Collapse 271

3.1 M o d e l

Let us discuss the measurement process described by the inter- action between an extreme relativistic particle (object) and an emul- sion (apparatus) schematized as a one-dimensional array of scatterers. Following the model studied by several authors [9, 10], we take a sim- ple Hamiltonian as

H g = t7o + HI N, (3.3)

where N

Ho = c p , HI N = ela E 6(x - na)a} n) (3.4) n=l

and p and x are, respectively, the momentum and position operators of the particle, na(n = 1 . . . N) are the positions of the scatterers for the site-length a, el stands for a parameter with the energy di-

mension, and a~ n) is the Pauli spin-matrix for the nth scatterer. We neglect the energy difference between two states of the scatterers in the Hamiltonian.

In the interaction picture the time evolution is given by the unitary operator

uN( t, tt) = eiH°t/he-i f t, HU(t,,)dt,/he_iHot,/h (3 .5)

which satisfies

ih dUN (t ' t ') = HiNt(t)U N (t,t '), (3.6) dt

with U N (t, t) = 1. HiNt(t) is evaluated as

HiNt(t) ----- eiHot/hHIN(t)e-iHot/h

N (3.7) = eIa E 6(x + ct - an)a~ n) .

n = l

The formal solution is given by

uN(t , t l) = exp (--i f N tt Hint ( t ) , t ~

(3.8)

272 Kobayash[

and the S matrix is directly computed as

s N = t--,~,t,~-c~lim u N ( t , ff) = exp [ - z "~'c n=lE cr~n) "

We can rewrite S N a s N

sN -~ I I S(n)' n = l

where the S matrix for the n th scatterer S (n) is written as

S (n) = exp [ - i e ia o.(n)] hc l J

(eIa ~ _ ia~n) sin [' cIa = c o s \ hc ] \ hc ] "

The probability for the transition of one scatterer is given by

(~ia3. w -- sin 2 k ~ c ]

(3.9)

(3.10)

(3.11)

(3.12)

3.2 M o d e l for t h e Q u a n t u m Col lapse

Let us consider the apparatus, of which initial state is described by

N qQN _-- H ,~n ) (3.13)

n = l

that is, all scatterers are in ¢~n). After the interaction with the object, the final apparatus state is evaluated in terms of S N as

N (3.14) : II [ 1~-¢-~ ¢~o~- ~,(0o~ l

(n=l)

Let us go to the study of the wave-function collapse in terms of the internal trace for the apparatus variables. In this process the off- diagonal terms of the density matrix with the internal trace given in (3.2) are evaluated as

(p~)o~: ( ~'/r-~- ~ )~' (~.1~)

Wave-Function Collapse 273

where the coefficients like CAC* B and the elements for the object in (3.2) are neglected. Then the off-diagonal terms vanish in the macroscopic limit, tha t is,

lim ( p N ) = 0 , (3.16) N---~c~ off

for w ~ O. Quantum collapse does happen.

3.3 M o d e l for t h e S t a t i s t i c a l C o l l a p s e

Let us s tudy the case where the initial apparatus state is wri t ten in terms of the superposition of the two states (¢0 and ¢1) for each scatterer, that is,

N ¢~N= I-I ~ + , (3.17)

(~,=1)

where a (n) and a~ n) are taken to be real for simplicity. After the interaction with the particle, the state is evaluated as

+~ = s~+? (3.18)

N , ~ , ~ ) + ,~,o~:~)~o~], (n=i)

where c - x/1 - w and s = v/-W. The off-diagonal term in the density matr ix with the internal trace is given by

( p N ) o ff = Int. Tr ( ~ f l ~ / N )

N (3.19)

where the coefficients and the elements for the object are neglected. We est imate its magni tude as

N L(~)o~ -- n [~ + ~co~ (o~o~_ o~oO~)] ~ (~.~o)

n = l

274 Kobayashi

It is trivial tha t in the case with

for all n (3.20) coincides with (3.15), and then the collapse occurs as was discussed in (3.16) in the limit N --~ ~ for w # 0. In general, however, (3.20) does not disappear in the limit.

(i) Random Phase Limit. In the free states of the apparatus

the energy of the state given by q~N does not depend on ~(n) and

~n) , even if there is some energy difference between the two states. That is to say, those phases generally have the l dependence such as Olin)(1) and oi~n)(l)," as was discussed in Sec. 2. Here we study the

extreme case where all the phase parameters are completely random in the repetit ion of the measurements. The average defined in (2.10) for (pN)o ff of (3.19) results in

g < (peff)off>L (C) N (3.21) in the large L limit. Then we have

lira < (PN)otT>L = 0 (3.22) N._.4 o O

for c = v / 1 - w < 1 (i.e., w # 0). The wave-function collapse is realized. This is nothing but a simple example of the statistical collapse. Of course, in the case with

cos 2 (a~ n)- a~ n)) = 1

for almost all of n we do not have such a collapse mechanism. Let us go to the next study.

(ii) Fluctuation of the Number of Active Scatterers in the apparatus. The maximum of (3.20) is given in the case with

c° s 2 = 1

for all n = 1 , 2 , . . . , N. Actually we have

= (3 .23)

Wave-Function Collapse 275

where 0 = eia/hc and a~ n) - a~ n) = 21rm (m = 0 or an integer) are taken. It is trivial that the internal trace operation and the average over the phases, keeping

does not generate the collapse. We, however, have also to take ac- count of the fluctuation of the active scatterer number in the appa- ratus, as was pointed out by Narniki and Pascazio [8, 10]. In every repetition of the measurement process the number of the scatterers (N) actively working in every measurement, which essentially depends on the density of the scatterers in the apparatus, has some thermal fluctuation at non-zero temperatures. In general such a fluctuation may be described by the Gaussian distribution for the large-N limit as

PN = ~ e -(N-N°)2/2a2, (3.24) 4F o

where No = (N), the mean value of N, is a macroscopic number proportional to N, and the dispersion a 2 may be proportional to N0.

The statistical average for pN eft can be represented by the average in terms of the distribution (3.24) in the large-L limit as

L

L~c~ off L L~oo L P off /=1

[ . e t a ] - - ~ - - N l • - = - Z P N ' e x p [ hc

J N,

The sum over Nt is well approximated by the integration as

oo

--Z - - M ~ dNlPNte -i(ela/hc)Nl PN, e x p [ hc

Nl -oo

(3.25)

(3.26)

= e-(o~/2)(~,,~/h&e-i(~la/hc)Yo.

Taking account of a 2 ~ No at finite temperatures, we have the dis- appearance of the off-diagonal terms in the macroscopic limit

N0-~oo off

276 Kobayashi

where Nl << - <

The wave-function collapse is realized via the statistical average for the thermal fluctuation. Note that this type of the statistical collapse does occur even without taking the internal trace in the density ma- trix. Actually we can show that this type of the collapse is applicable to all the initial detector states. Such a discussion in a special limit for N ~ co and N w 2 = finite was done by Namiki and Pascazio [10] as an example of the many-Hilbert-space theory. We see that the following criteria are required for the realization of this type of the statistical collapse:

(1) A phase which diverges in the macroscopic limit must be pro- duced in the off-diagonal elements of the density matrix with the internal trace via the process where a macroscopic measur- able quantity is produced by the interaction of the apparatus with the object. Usually the phase is proportional to the num- ber N of constituents of the apparatus. It should, however, be noted that N indicates the number of the constituents really acting in one measurement.

(2) The dispersion of the distribution for N arising from thermal fluctuation must increase and diverge in the macroscopic limit N --* co. Of course, the mean value of N has also to diverge in the limit. We know that the Criterion (2) ( a 2 --* co in N -~ co) is a quite general feature for most physical quantities at finite and not very low temperatures.

4. H O W T O F I N D D I F F E R E N C E S B E T W E E N Q U A N T U M A N D S T A T I S T I C A L C O L L A P S E S

The important remaining problem is to find the difference be- tween the two different mechanisms for the wave-function collapse, i.e., quantum and statistical collapses. How to distinguish them in real measurement processes is very important. We see that the sta- tistical collapse is meaningless if we can make detectors of which ini- tial states are uniquely determined, that is, no fluctuation at all. It is, however, very difficult and may be impossible for many particle system-like detectors at finite temperatures. In order to extract the difference we can consider the following two cases:

(1) Perform experiments at very low temperatures where the dis- persions hold finite even in the macroscopic limit. It may be very hard to realize it.

Wave-Function Collapse 277

(2) Detect temperature dependences of the off-diagonal elements in the density matrix for finite N. The complete wave-function collapse is not realized at finite N. The T dependence on the suppression of the off-diagonal elements appears only in the statistical collapse.

(3) We can find another possibility:

Make a detector, of which the macroscopic limit is defined by the macroscopic limit of the external field, as was presented in Model I of Ref. 2. In realistic processes it is, of course, impossible to realize the macroscopic limit of the external field. Then we have to measure the external-field dependence on the suppression of the off-diagonal elements, which is predictable in the quantum collapse.

5. R E M A R K S

We now know that there are two quite different mechanisms for the wave-function collapse in the quantum theory of measurements. The discrimination between them is not easy, as was discussed in the last section, but we see a clear difference between them. That is, the statistical collapse happens only via the statistical average. This fact means that we can not say anything about one event in a measurement process. The quantum collapse, however, harbors the possibility to explain it, because the internal trace defined by a large number of trace operations may be interpreted as the different expression of the ergodic process in each event. The transition from the wave nature to the particle may be understood as the result of this ergodic property of the internal trace.

R E F E R E N C E S

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2. T. Kobayashi and K. Ohmomo, Phys. Rev. A 41, 5798 (1990). 3. T. Kobayashi, in Proceedings, Symposium on the Foundations of

Modern Physics 1990, P. Lahti and P. Mittelstaedt, eds., p. 170. 4. T. Kobayashi, preprint, University of Tsukuba, UTHEP-209

(1990); Nuovo Cimento B (1992), in press. 5. T. Kobayashi, talk presented at the Second International Wigner

Symposium, Goslar, Germany, 1991; preprint, University of Tsukuba, UTHEP-220 (1991).

6. T. Kobayashi, preprint, University of Tsukuba, UTHEP-222 (1991).

278 Kobayashi

7. S. Machida and M. Namiki, Prog. Theor. Phys. 63, 1457, 1833 (1980); M. Namiki, Found. Phys. 18, 29 (1988).

8. M. Namiki and S. Pascazio, Phys. Lett. 147A, 430 (1990); Phys. Rev. A44, 39 (1991).

9. K. Hepp, Helv. Phys. Acta 45, 237 (1972); S. Kudaka, S. Mat- sumoto, and K. Kakazu, Prog. Theor. Phys. 82, 665 (1989).

10. M. Namiki and S. Pascazio, Found. Phys. Lett. 4, 203 (1991).