Recent Developments of Quantum

Models with explicit collapse
In 53.7, and Appendix 7, we considered how the measurement problem
of quantum mechanics could be solved by changing the theory so that a
wavefunction would evolve with time to become a state corresponding
to a unique value of the observable that was being measured. Two
difficulties with this approach were noted, namely, it seemed to require
prior knowledge of what was to be measured (since a state cannot in
general correspond to a unique value of several observables), and also
it had to happen very quickly in circumstances involving observation,
but at most very slowly in the many situations where the Schrijdinger
equation is known to work very well.
An explicit model, in which both these difficulties were overcome,
was proposed by three Italians, GianCarlo Ghirardi, Albert0 Rimini
and Tullio Weber (now universally known as GRW), in a remarkable
article published in 1986 (Physical Review D 34 470). They noted, first,
that all measurements ultimately involve the position of a macroscopic
object. (The special role of position is already used implicitly in the de
Broglie-Bohm model, as was noted in 55.2). Thus the measurement
problem can be solved if wavefunctions evolve so as to ensure
that macroscopic objects quickly have well-defined positions. By a
macroscopic object we here mean something that can seen by the naked
eye, say, something with a mass greater than about gm. Similarly,
a well-defined position requires the spread of the wavefunction to be
less than an observable separation, say, less than about cm.
In order to achieve this end, GRW postulated that all particles suffer
(infrequent) random ‘hits’ by something that destroys (makes zero)
all their wavefunction, except that within a distance less than about lO-%m from some fixed position. This position is chosen randomly
with a probability weight proportional to the square magnitude of the
particle’s wavefunction, i.e., to the probability of its being found at that
position if its position were measured (see 82.2).
GRW assumed that the typical time between hits was of the order
of lo%, which ensures that the effects of the hits in the microscopic
world are negligible, and do not disturb the well established agreement
between quantum theory and experiment. However, even the small
macroscopic object referred to above, with mass gm, contains
about 10l8 electrons and nucleons, so typically about one hundred of
these will be hit every second. Although it might at first sight seem
that hitting a few particles out of so many would have a negligible
effect, it turns out that, in a measurement situation, just one hit is
enough to collapse the whole state: when one goes, they all go! This
is the real magic of the GRW proposal. To see how it comes about we
imagine that the macroscopic object represents some sort of detector
(a ‘pointer’) which tells us whether a particle has, or has not, passed
through a barrier (see Chapter 1). Explicitly, suppose the pointer is
in position 1, with wavefunction D’, if the particle has been reflected,
and in position 2, with wavefunction D2, if it has not. Note that, for
example, D’ corresponds to all the particles of the object being close
to position 1. We assume that, in a proper measurement, the separation
between the two positions is greater than both the size of the object
and the GRW size parameter lo4 cm. The wavefunction describing
this situation has the form (cf 54.5):
Now we suppose that one of the particles is hit. The centre of the
hit will most likely occur where the wavefunction is big, i.e., in the
neighbourhood of either position 1 or position 2 (with probabilities
IPR12,l P~lre~sp ectively). Suppose the random selection chooses the
former. Then the whole wavefunction given above will be multiplied
by a function which is zero everywhere except in the neighbourhood
of position 1. Since the second term in the above state is zero except
when all the particles are near position 2, it will effectively be removed
by this hit (there are no values for the position of the hit particle
for which both factors, the hitting function and the wavefunction D2,
simultaneously differ from zero). In other words the wavefunction
will have collapsed to the state in which the particle was reflected. Notice that it is something that happens in the detector that establishes
whether or not the particle is transmitted; without a detector no such
determination is made (except within a time of around 10l6 s, the
average collapse time for a single particle).
Since, as we have seen, even for a small detector the typical time
between the collapses is of the order of s, which is less than the
time it takes for a person to respond to an observation, it is clear that the
GRW model has the desired effect of giving outcomes to measurements.
As a working, realistic, model of quantum theory it is important. It
provides insight into the theory; it raises fascinating questions relating
to when a conscious observation has actually occurred, particularly
because the disappearance of the unwanted terms is only approximate
and so-called ‘tails’ always remain; it also gives a structure in which
questions like the relation with relativity can be discussed. Whether
it is true is another question. It seems very unnatural, although more
satisfying versions in which the hitting is replaced by a continuous
process (similar to that discussed in Appendix 7) have been developed
by GRW, Philip Pearle and others. A review of this work, and further
references, is given in the articles by Ghirardi and Pearle published in
Proceedings of the Philosophy of Science Foundation 2 pp 19 and 35
(row).
The predictions of collapse models do not agree exactly with those
of orthodox quantum theory; for example, they give a violation of
energy conservation. It is this that puts limits on the parameters-the
process must not happen too quickly. Any bound system, initially in
its stable, lowest energy state, will have a certain probability of being
excited to a higher energy state if one of the constituents is ‘hit’. Thus,
for example, hydrogen atoms will spontaneously emit photons. Philip
Pearle and I have recently shown that the best upper limit on the rate
(i.e., lower limit on T), probably comes from the fact that protons are
known to be stable up to something like years. These protons are
in fact bound states of three quarks, and every time a quark is ‘hit’
there is a very small probability that the proton will go to an excited
state which will spontaneously decay. The fact that such decays have
not been observed puts severe restrictions on GRW-type models (and
may even rule out some simple versions).
In one sense it is an advantage for a model that it gives clear,
distinctive predictions, because this allows the possibility that it might
be verified. On the other hand, in the absence of any positive evidence
for the unconventional effects, the fact that the free parameters of the model have to be chosen rather carefully-to make the process
happen fast enough in a measurement situation, but not too fast to give
unobserved effects elsewhere-is a negative feature; why should nature
have apparently conspired so carefully to hide something from us?
A partial answer to this last question might lie in the possibility that
the parameters of the collapse are not in fact independent of the other
constants of the physical world, but arise in particular from gravity, as
suggested in Appendix 7. In his wide-ranging book, The Emperor’s
New Mind (Oxford University Press, 1989), Roger Penrose gives other
reasons for believing that gravity might be associated with collapse.
He also develops the idea that the human mind’s ability to go beyond
the limits of ‘algorithmic computation’, i.e., the use of a closed set of
rules, shows that it can only be explained by really new physics, and
that such new physics, which would be ‘non-computable’, might well
be associated with the collapse of the wavefunction.