1. Description of the Collapse Scheme. 

Introduction.

In order to avoid the difficulties associated with standard many-branch quantum mechanics, one possibility is to suppose that the wave function collapses down to just one branch, with the collapse to the ith branch occurring a fraction of the time.  Ghirardi, Rimini and Weber, (GRW) and Pearle have devised an ingenious mathematical scheme that elegantly implements this collapse interpretation.  A brief description of the theory is that a random force on each branch of the wave function, biased in a certain direction (in Hilbert space) that depends on the , causes the branches to grow unevenly.  The GRW idea is that, to make the orders of magnitude come out correctly (so that one has quick collapse when a macroscopic detector is involved, but not when a single photon is involved), the effect must be per particle.  The time for collapse then becomes smaller as the number of particles involved (which is presumably the number of particles in the detector) becomes large.  The parameters are chosen so that, after at most a millionth of a second, only one branch is left, with an average fractional occurrence for the surviving branch i of . 

Details of the collapse process.

The growing or shrinking is carried out roughly as follows:  Space is divided into cells of size cm.  Each particle within the cell is subject to the same force, with the force being determined in a probabilistic manner.  If we think of the pointer on the dial of a detector, its position will be different (perhaps vertical for detection of an up spin and horizontal for the detection of a down spin) on different branches .  And so a particular cell will have different numbers of particles on the different branches (sometimes zero and sometimes not zero).  The force affects only the norm of each branch, not the shape of the wave function. 

The probability law for the "force" w_j applied to each atom in cell j is

(IIIB4-1)  

where is a white noise factor.  Note that each of the  as well as the overall norm is time-dependent, so the time evolution is not unitary.  Note also that the probability law for each cell depends on the overall norm squared (not just on the norm squared of a particular branch).

2. Problems. 

In spite of the fact that the theory “works,” there are a number of reasons to object to it.  We list them here. 

• There is no experimental evidence for collapse. 

• The theory is ad hoc; its sole purpose is to deal with the fact that we perceive only one branch with a probability in agreement with the quantum mechanical probability law.  Further, a few years after the theory was developed, it was found that it was in conflict with experimental results unless the random force depended on the mass of the particles involved, so the mass dependence was included in the theory.  This gives the theory an even more ad hoc flavor; it is adjusted by one means or another until the conflicts with experiment go away.

• The theory requires the introduction of two new physical constants, a frequency (in the white noise) and a  length (cell size).  These constants do not have a use elsewhere in physics.  From the lack of observable effects, they are forced by experiment to be in a certain small range.  (The lack of an observable effect in this regard adds to the ad  hoc feeling of the model.)

• The “agent” that applies the random force is unknown.  In Brownian motion, to which this scheme has been compared, the random force comes from collisions with atoms.  But there is no such picture for the GRW-P scheme.  The force cannot be attributed to anything in current quantum mechanics, such as the vacuum state, because nothing within linear quantum mechanics can change the norm of  any branch.

• 1. If the probability function of Eq. (IIIB4-1) for the random force is proportional to any function of  other than  itself—if it is proportional to , for example—then the probability law derived from the Pearle procedure will be something other than the observed  law.  Thus making the Pearle probability function proportional to  is essentially equivalent to  assuming  the quantum mechanical law to start with.  The Pearle procedure is therefore not really a derivation of the probability law unless one can explain why—in some specific model—the Pearle probability function is proportional to .

This will be a problem for collapse models generally.  The  law is the only probability law consistent with the quantum mechanical equations of motion.  What specifically is it within a given collapse model that forces it to give the only result that is consistent with quantum mechanics.  Coincidences are not acceptable in physics, so there must be an explicit explanation for why it meshes with quantum mechanics.

• A basic principle of quantum mechanics is relativity, and the GRW-P theory is not relativistic.

• A basic principle of quantum mechanics is linearity, and the GRW-P theory is not linear (because the time evolution depends on the norm squared). 

• 2. The forces applied to each cell are different for for the different cells, but their values are coordinated across branches (because of the norm squared factor in Eq. (IIIB4-1)).  This violates the quantum mechanical rule that the time evolution of each branch is independent of the time evolution of the other branches.  A full collapse theory must explain the mechanism that violates this (linearity) rule.

• The model does not supply enough detail to explain why the random processes from macroscopically separated cells, and different branches of the wave function, are coordinated.

• 3. Finally, since the force applied at each instant depends on the norm squared at each instant, whatever supplies the force must somehow “sense” the value of the norm at each instant.  This sensing involves an integral over the wave functions of all the macroscopic number of particles involved and over a macroscopic volume in space, so it makes the theory highly nonlocal.  It also involves what in conventional quantum mechanics would be a sensing of the norm in entirely different universes (each branch is in its own universe).  It is difficult to imagine the physical mechanism whereby such an instantaneous, nonlocal sensing is carried out.

Further, the wave function of the whole universe branches when there is a branching anywhere.  So apparently the norm factor must be taken over all particles and all space.  This makes it even more difficult to imagine the physical mechanism responsible for the random force.

The GRW-P Model Cannot be a Primary Theory

Because of these difficulties, it seems impossible to have Eq. (IIIB4-1) be a primary equation in a theory of collapse.  Instead it must be a consequence of the physics.  If it were a full theory, it would explain why the probability of Eq. (IIIB4-1) is proportional to the norm squared; it would explain why the probability of a force on a localized cell depends on the highly nonlocal norm; and it would explain what physical mechanism applies this most peculiar force.

3. Conclusion. 

My conclusion is that, in spite of its seeming elegance, there is no persuasive reason to suppose the GRW-P collapse theory is correct.  Further, the difficulties pointed out for the specific model, especially points 1, 2, and 3, also indicate that there are formidable barriers to showing that any collapse theory can provide an acceptable interpretation of quantum mechanics. (Note that there are currently no reasonable collapse theories which are substantially different from the GRW-P model.)

Thus, as for "particle" interpretations, I would not like to be in the position of having to base my view of reality on the slight chance that the collapse interpretation is correct.

3A.  Summary of arguments against collapse.

• In spite of a number of attempts, no experimental evidence for collapse has been found.

• To have collapse, there must be coordination between the wave functions for the different versions of reality, and this is impossible in the current formulation of quantum mechanics.  So this would represent a radical departure from current theory.

• Another way to say the same thing is that a collapse theory must be nonlinear.

This implies the "force" changing the a's must have instantaneous knowledge of the a's. It is difficult to imagine a way this could be carried out physically.

• There is no known physical mechanism for carrying out the collapse.

• A theory of collapse would be ad hoc; its only application would be to the collapse problem.

A mathematically elegant theory of collapse has been proposed by Ghirardi, Rimini, and Weber, and Pearle.  But it has problems.

• It uses a random force applied by we-know-not-what.

• It is nonlinear, in violation of linearity, the most basic principle of quantum mechanics.

• The form of the nonlinear force is simply assumed to have a particular form, so that the conjectured theory will lead to the  probability law.

• The force law has two parameters in it, a length and a frequency.  The absence of experimentally detectable effects limits these parameters to a narrow range.

• The force law must be assumed, without justification, to be proportional to the mass of the particles involved.

In summary, there is no encouragement, either experimentally or theoretically, for the idea that the wave function collapses down to just one version.

3B. Proof of Nonlinearity of any Collapse Scheme.

We gave one version of the necessity for nonlinearity above.  But there is a better way of showing this.  We write

(X)  

where U is for a particular time evolution.  (There will be many possible U’s, corresponding to different random forces.)  As t goes to infinity,  can have only one non-zero component for any starting .  But if U is independent of the a’s, there is no such matrix.  Thus U must depend on the a’s.

4. Collapse Interpretations and Particles. 

I don't believe there is collapse.  But if there were a valid collapse theory, this would eliminate the need for the concept of particles.  The collapse theory would explain the probability law and why only one particular branch of the wave function is perceived.  And the reasoning of section IIIA, using only collapse and the principles of quantum mechanics, would explain all the particlelike properties of physical existence.

The fact that the wave function is all that physically exists in collapse models has an interesting consequence which is in agreement with section IIIB3.  Those physicists who subscribe to the collapse interpretation must agree that (1) what we are aware of is the wave function itself, and (2) the wave function must give a complete description of the physical world.

5. Decoherence Theory. 

Decoherence theory is sometimes cited as a possible interpretation of quantum mechanics.  If my understanding of this interpretation is correct, its thesis is that the dynamics of quantum mechanics itself cause collapse. To illustrate, suppose we do a Stern-Gerlach experiment on a single silver atom.  There are two branches to the wave function, one in which the silver atom wave function travels along path A and one in which the wave function travels along path B.  Detectors DA and DB are put in the two paths.  Just before the atom hits the detectors, the wave function is

                                 [silver atom wave function on path A][DA,no][DB,no]

                                                              +                                                        

                                 [silver atom wave function on path B][DA,no][DB,no]

And just after the silver atom hits the detector(s), the wave function is

                                 [silver atom wave function on path A][DA,yes][DB,no]

(IIIB4-2)                                                +                                           

                                 [silver atom wave function on path B][DA,no][DB,yes]

Decoherence theory appears to say that when the atomic wave function hits or does not hit detector DA, the internal quantum mechanical dynamics of the detector will cause it to jump either into the yes state or into the no state; it will not stay in two different states.  That is,

               [silver atom wave function on path A][DA,yes][DB,no]

or

               [silver atom wave function on path B][DA,no][DB,yes]

But this cannot be, for  once the silver atom wave function has split into two branches, the two branches must evolve entirely independently, with no change in norm for either branch.  The argument is exactly the same as the "separate universes" argument of section IIIA3, subsection 5, with atom i there corresponding to the silver atom here.  Thus the dynamics of quantum mechanics imply, beyond a doubt, that the state will always stay in the superposition of Eq. (IIIB4-2) rather than evolving to a single branch.  So the decoherence approach cannot solve the mysteries of quantum mechanics.

6. Proposed Experimental Test of Collapse.

(Note: This section appears to be incorrect.

It is apparently not possible to deduce "intermediate" or "partial"

probabilities from over-all probabilities.)

How does one test for collapse?  The traditional method is to look for interference effects between two branches of the wave function.  If they are predicted by conventional quantum mechanics but are not experimentally observed, then collapse must have occurred.  The problem with this method is that GRW theories suggest that collapse occurs only when a large number of particles are involved.  And interference experiments with “large” systems are notoriously difficult to do.

So we will propose a different type of test under the (Pearle) assumption that the coefficients are functions of time (that fairly rapidly evolve to 0 or 1).  I do not know whether such an experiment can be carried out.  One interesting possibility, however, is to do a variation of the Aspect experiment (IIIA7) as explained in subsection 7 of this section.  The time lag would be introduced by having one set of detectors farther away from the source than the other.

The Experiment

To illustrate the basic principles, we will use a half-silvered mirror experiment, but there are probably many ways to make use of the idea.  A half-silvered mirror splits a light wave function into two parts.  One part, with amplitude , is reflected upwards on a vertical path, while the other part, with amplitude , is transmitted through the mirror and travels on a horizontal path.  There is one detector, V, on the vertical path and a second detector, H, on the horizontal path.  The detectors indicate detection by 1 and no detection by 0. 

At time t=0, just after the detectors are tripped and the photon annihilated, the wave function of the detectors is

(IIIB4-3)  

Now in continuous collapse schemes such as Pearle’s, the coefficients are functions of time (determined by a random force).  So the wave function at time t is

(IIIB4-4)  

where we obtain the normalization condition in Pearlelike theories by “renormalizing” at each instant.

At time t, we send a second photon wave function through the apparatus and assume the detectors record both results.  Then just after the second photon hits and is annihilated, the wave function is

(IIIB4-5)  

The probabilities of the four results, using the readings on the V detector,  are then (presumably)

(IIIB4-6)  

Analysis

Because of the random force, different for each run, the norm squared values of  and  will be different on each run.  They will presumably have Gaussianlike distributions.  Their average values will  presumably be  and  resp., so we probably cannot learn anything by simply observing the averages.  But there must be some way to tease out the fact that there is a distribution of values of

 and .  I suggest the following method.

First we run the two-photon experiment m times, with m, say, on the order of 10 or 20, and record the number of occurrences,  of the four possibilities.  (The sum of the n’s is m.)  Then we run it  more times, with N large, and record the results,  and so on, for the k^th  run.  Next, for each of the four possibilities, we calculate the total number of times that the result comes up i,i’.

(IIIB4-7)  

where the  is the Kronecker delta.

We now set

(IIIB4-8)  

If  a(t) were not a function of the time lag, we would expect, from standard probability theory,

(IIIB4-9)    

(IIIB4-10)    

(IIIB4-11)    

So the procedure is this:  We do the measurements for some m, N and determine the .  Then, remembering that   is a function of , we find the value of   that minimizes the sum of the squares in Eq. (IIC4-11).  If, for that minimizing value, the sum of the squares is significantly greater than N (because some of the a’s depend on the time lag and will therefore have a spread of values) , then we know there is a spread in the a’s caused by a Pearle-like collapse.  Conversely if no deviation from the expected spread is observed for all t in the range appropriate for collapse, then we have experimental evidence that collapse is unlikely.

Note that there is a similar test for collapse in section IIIE, subsection 7.  The experimentally simpler test of that section simultaneously tests for collapse and the existence of particles.

Order of Magnitude Calculation

It would be useful to know what values of N and m give significant deviations from classical results.  To do this, we need an equation for the probabity when the amplitudes a(t) have a range of values.  If we let

(IIIB4-12)  

and let the probability of value  at time t be , then the probability of observing j occurences of the state ii’ in m runs is

(IIIB4-13) 

One way to approach this equation is to do a computer simulation, using the Gaussianlike probabilities of the a(t) of reference 3. 

We can also do a rough approximate calculation assuming a spread of in the a’s.  If we use

(IIIB4-14)  

then

(IIIB4-15)    

with the second derivative evaluated at .

If we want

(IIIB4-16)  

so that Eq. (IIIB4-10) is substantially violated (which would imply there is a Pearle collapse process going on), then the condition on N  is

(IIIB4-17)  

For j=m,

(IIIB4-18)  

and so

(IIIB4-19)  

Similarly, for j=0,

(IIIB4-20)  

The smallest N is

(IIIB4-21)  

So if we choose , we must have N>8,000 to get an appreciable deviation from the classical result.  This value for  seems reasonable, but one really needs to look at the details of the spread in the coefficients a(t) as a function of time in the Pearle model to check.

7. Test of Collapse Using an Aspect-like Experiment

The pairing of photons and control of the time lag are perhaps most easily implemented in a variation of the Aspect experiment.  Two photons with opposite polarization states are emitted (within 5ns of each other) from an atomic source.  One travels along the positive z axis, to the right, and the other travels along the negative z axis, to the left.  If x and y as the two polarization directions, the state of the two photons is

(IIIB4-22)  

The photon traveling to the right enters a polarimeter.  This splits it into its x and y polarization states and each travels on a different path.  There are detectors, [Rx] and [Ry] on the two paths.  At time t=0, the two light wave functions trigger their respective detectors, with 1 denoting triggered and 0 not triggered.  The state of the system is then

(IIIB4-23)  

At time  later (L for time lag), the photon traveling to the left hits another polarimeter and splits into two parts, with the plane of polarization rotated by an angle .  We can re-express the left photon state in terms of polarization states along the rotated  axes as

(IIIB4-24)    

The two branches of the photon wave function traveling to the left hit their respective detectors (and are annihilated) with the resulting state

(IIIB4-25)  

Note that the coefficient in front of each of these four states is a function of the time lag.  Because of the random force, these coefficients will be different for each run.

This gives us a four-state wave function analogous to that of Eq. (IIC4-5).  The analysis of the probabilities goes through in the same way as for that equation if we make the association

(IIIB4-26)  

If the collapse times that need to be probed are as great as , then the distance the left polarimeter must be moved is .  This will make the experiment quite difficult.  Perhaps a similar experiment can be done with massive spin ½ particles, so the distances are not so large.

Special Case.

 One interesting special case is that of  (the axes are not rotated).  Then there are only two final states instead of four, and the value for N of Eq. (IIIB4-20) becomes smaller because there is no factor of  in the denominator.

References.

1. G. C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34, 470 (1986).

2. P. Pearle, Phys. Rev. A 39, 2277 (1989).

3. See Akihiro Nakano and Philip Pearle, Foundations of Physics 24, 363 (1994)  for a clear, simplified version of the collapse theory.