Quantum theory and relativity

This is a difficult section, from which we shall learn little that has
obvious relevance to our theme. Nevertheless, the section must be
included since its subject is very important and is an extremely
successful part of theoretical physics. There is also the possibility,
or the hope, that it could one day provide the answers to our
problems.
The mysteries that we met in Chapter One arose from certain
experimental facts. We have learned that quantum theory predicts
the facts but does not explain the mysteries. Now we must learn
that quantum theory also meets another separate problem, namely
that it is not compatible with special relativity.
The reason for this'is that special relativity requires that the laws
of physics be the same for all observers regardless of their velocity
(provided this is uniform). This requirement implies that only relative velocities are significant, or, in other words, that there is no
meaning to absolute velocity. In practice this fact makes little
difference to physics at low velocity; it is only when velocities
become of the order of the velocity of light (3 x lo8 m s-’) that the
new effects of special relativity are noticed.
Quantum theory, as originally developed, did not have this
property of being independent of the velocity of the observer, and
is thus inconsistent with special relativity. Although the practical
effects of this inconsistency are very tiny for the experiments we
have discussed, there are situations where they are important, and
it is natural to ask whether quantum theory can be modified to take
account of special relativity, and even to ask whether such
modifications might provide some insight into our interpretation
problems. The answer to the first of these questions is a qualified
‘yes’; to the second it is a tentative ‘no’.
The relativistic form of quantum mechanics is known as relativistic
quantum field theory. It makes use of a procedure known
as second quantisation. To appreciate what this means we recall
that, in the transition from classical to quantum mechanics, variables
like position changed from being definite to being uncertain,
with a probability distribution given by a wavefunction, i.e. a
(complex) number depending upon position. In relativistic
quantum field theory we have a similar process taken one stage
further: the wavefunctions are no longer definite but are uncertain,
with a probability given by a ‘wavefunctional’. This wavefunctional
is again a (complex) number, but it depends upon the
wavefunction, or, in the case where we wish to talk about several
different types of particle, upon several wavefunctions, one for
each type of particle. Thus we have the correspondence:
First quantisation:
Second quantisation:
x, y, . . .
W(x), V(x ), . . .
replaced by W(x, y, . . . )
replaced by Z( W(x), U(x). . .),
The analogue of the Schrodinger equation now tells us how the
wavefunctional changes with time.
An important practical aspect of relativistic quantum field theory
is that the total number of particles of a given type is not a fixed
number. Thus the theory permits creation and annihilation of
particles to occur, in agreement with observation.
For further details of relativistic quantum field theory we must refer to other books. (Most of these are difficult and mathematical.
An attempt to present some of the features in a simple way
is made in my book To Acknowledge the Wonder: The story of
fundamental physics, referred to in the bibliography.) There is no
doubt that the theory has been enormously successful in explaining
observed phenomena, and has indeed been a continuation of the
success story of ‘non-relativistic’ quantum theory which we outlined
in 82.5. In particular, it incorporates the extremely accurate predictions
of quantum electrodynamics, has provided a partially unified
theory of these interactions with the so-called weak interactions,
and has provided us with a good theory of nuclear forces. In spite
of these successes there are formal difficulties in the theory. Certain
‘infinities’ have to be removed and the only way of obtaining results
is to use approximation methods, which, while they appear to
work, are hard to justify with any degree of rigour.
Do we learn anything in all this which might help us with the
nature of reality? Apparently not. If, in our previous, nonrelativistic,
discussion, we regarded the wavefunction as a part of
reality, we now have to replace this by the wavefunctional, which
is even further removed from the things we actually observe. The
wavefunctions have become part of the observer-created world, i.e.
things that become real only when measured.
We must now consider the problem of making quantum theory
consistent with general relativity. Since general relativity is the
theory of gravity, this problem is equivalent to that of constructing
a quantum theory of gravity. Much effort has been devoted to this
end, but a satisfactory solution does not yet exist. Maybe the lack
of success achieved so far suggests that something is wrong with
quantum theory at this level and that, if we knew how to put it
right, we would have some clues to help with our interpretation
problem. This is perhaps a wildly optimistic hope but there are a
few positive indications. Gravity is negligible for small objects, i.e.
those for which quantum interference has been tested, but it might
become important for macroscopic objects, where, it appears,
wavefunction reduction occurs. Could gravity somehow be the
small effect responsible for wavefunction reduction, as discussed in
$3.7?
Probably the correct answer is that it cannot, but if we want
encouragement to pursue the idea we could note that the magnitudes
involved are about right. The ratio of the electric force (which is responsible for the effects seen in macroscopic laboratory
physics) to the gravitational force, between two protons, is about
For larger objects the gravitational force increases (in fact it
is proportional to the product of the two masses), whereas this
tends not to happen with the electric force because most objects are
approximately electrically neutral, with the positive charge on
protons being cancelled by the negative charge on electrons.
Consider, then, the forces between two massive objects, each of
which has charge equal to the charge on a proton. The electric force
will be equal to the gravitational charge if the objects weigh about
10-6g. Thus we can see that gravitational forces become of the
same order as electrical forces only when the objects are enormously
bigger than the particles used in interference effects, but
that they are certainly of the same order by the time we reach
genuine macroscopic objects. (See also the remarks at the end of
Appendix 7.)
We end this section by noting a few other points. General
relativity is all about time and space, about the fact that our
apparently ‘flat’ space is only an approximation, about the
possibility that there are singular times of creation, and/or extinction,
about the existence of black holes with their strange effects.
Some of these facts could be relevant, but at the present time all
must be speculation. As an example of such speculation we mention
the suggestion of Penrose that there might be some sort of
trade-off between the creation of black holes and the reduction of
wave packets (see the acticle by Penrose, ‘Gravity and State Vector
Reduction’ in Quantum Concepts in Space and Time, ed C J Isham
and R Penrose [Oxford: Oxford University Press 19851).