Lectures a brief history of mine
Does God Play Dice?
This lecture is about whether we can predict the future, or whether it
is arbitrary and random. In ancient times, the world must have seemed
pretty arbitrary. Disasters such as floods or diseases must have
seemed to happen without warning, or apparent reason. Primitive people
attributed such natural phenomena, to a pantheon of gods and
goddesses, who behaved in a capricious and whimsical way. There was no
way to predict what they would do, and the only hope was to win favour
by gifts or actions. Many people still partially subscribe to this
belief, and try to make a pact with fortune. They offer to do certain
things, if only they can get an A-grade for a course, or pass their
driving test.
Gradually however, people must have noticed certain regularities in
the behaviour of nature. These regularities were most obvious, in the
motion of the heavenly bodies across the sky. So astronomy was the
first science to be developed. It was put on a firm mathematical basis
by Newton, more than 300 years ago, and we still use his theory of
gravity to predict the motion of almost all celestial bodies.
Following the example of astronomy, it was found that other natural
phenomena also obeyed definite scientific laws. This led to the idea
of scientific determinism, which seems first to have been publicly
expressed by the French scientist, Laplace. I thought I would like to
quote you Laplace's actual words, so I asked a friend to track them
down. They are in French of course, not that I expect that would be
any problem with this audience. But the trouble is, Laplace was rather
like Prewst, in that he wrote sentences of inordinate length and
complexity. So I have decided to para-phrase the quotation. In effect
what he said was, that if at one time, we knew the positions and
speeds of all the particles in the universe, then we could calculate
their behaviour at any other time, in the past or future. There is a
probably apocryphal story, that when Laplace was asked by Napoleon,
how God fitted into this system, he replied, 'Sire, I have not needed
that hypothesis.' I don't think that Laplace was claiming that God
didn't exist. It is just that He doesn't intervene, to break the laws
of Science. That must be the position of every scientist. A scientific
law, is not a scientific law, if it only holds when some supernatural
being, decides to let things run, and not intervene.
The idea that the state of the universe at one time determines the
state at all other times, has been a central tenet of science, ever
since Laplace's time. It implies that we can predict the future, in
principle at least. In practice, however, our ability to predict the
future is severely limited by the complexity of the equations, and the
fact that they often have a property called chaos. As those who have
seen Jurassic Park will know, this means a tiny disturbance in one
place, can cause a major change in another. A butterfly flapping its
wings can cause rain in Central Park, New York. The trouble is, it is
not repeatable. The next time the butterfly flaps its wings, a host of
other things will be different, which will also influence the weather.
That is why weather forecasts are so unreliable.
Despite these practical difficulties, scientific determinism, remained
the official dogma throughout the 19th century. However, in the 20th
century, there have been two developments that show that Laplace's
vision, of a complete prediction of the future, can not be realised.
The first of these developments was what is called, quantum mechanics.
This was first put forward in 1900, by the German physicist, Max
Planck, as an ad hoc hypothesis, to solve an outstanding paradox.
According to the classical 19th century ideas, dating back to Laplace,
a hot body, like a piece of red hot metal, should give off radiation.
It would lose energy in radio waves, infra red, visible light, ultra
violet, x-rays, and gamma rays, all at the same rate. Not only would
this mean that we would all die of skin cancer, but also everything in
the universe would be at the same temperature, which clearly it isn't.
However, Planck showed one could avoid this disaster, if one gave up
the idea that the amount of radiation could have just any value, and
said instead that radiation came only in packets or quanta of a
certain size. It is a bit like saying that you can't buy sugar loose
in the supermarket, but only in kilogram bags. The energy in the
packets or quanta, is higher for ultra violet and x-rays, than for
infra red or visible light. This means that unless a body is very hot,
like the Sun, it will not have enough energy, to give off even a
single quantum of ultra violet or x-rays. That is why we don't get
sunburn from a cup of coffee.
Planck regarded the idea of quanta, as just a mathematical trick, and
not as having any physical reality, whatever that might mean. However,
physicists began to find other behaviour, that could be explained only
in terms of quantities having discrete, or quantised values, rather
than continuously variable ones. For example, it was found that
elementary particles behaved rather like little tops, spinning about
an axis. But the amount of spin couldn't have just any value. It had
to be some multiple of a basic unit. Because this unit is very small,
one does not notice that a normal top really slows down in a rapid
sequence of discrete steps, rather than as a continuous process. But
for tops as small as atoms, the discrete nature of spin is very
important.
It was some time before people realised the implications of this
quantum behaviour for determinism. It was not until 1926, that Werner
Heisenberg, another German physicist, pointed out that you couldn't
measure both the position, and the speed, of a particle exactly. To
see where a particle is, one has to shine light on it. But by Planck's
work, one can't use an arbitrarily small amount of light. One has to
use at least one quantum. This will disturb the particle, and change
its speed in a way that can't be predicted. To measure the position of
the particle accurately, you will have to use light of short wave
length, like ultra violet, x-rays, or gamma rays. But again, by
Planck's work, quanta of these forms of light have higher energies
than those of visible light. So they will disturb the speed of the
particle more. It is a no win situation: the more accurately you try
to measure the position of the particle, the less accurately you can
know the speed, and vice versa. This is summed up in the Uncertainty
Principle that Heisenberg formulated; the uncertainty in the position
of a particle, times the uncertainty in its speed, is always greater
than a quantity called Planck's constant, divided by the mass of the
particle.
Laplace's vision, of scientific determinism, involved knowing the
positions and speeds of the particles in the universe, at one instant
of time. So it was seriously undermined by Heisenberg's Uncertainty
principle. How could one predict the future, when one could not
measure accurately both the positions, and the speeds, of particles at
the present time? No matter how powerful a computer you have, if you
put lousy data in, you will get lousy predictions out.
Einstein was very unhappy about this apparent randomness in nature.
His views were summed up in his famous phrase, 'God does not play
dice'. He seemed to have felt that the uncertainty was only
provisional: but that there was an underlying reality, in which
particles would have well defined positions and speeds, and would
evolve according to deterministic laws, in the spirit of Laplace. This
reality might be known to God, but the quantum nature of light would
prevent us seeing it, except through a glass darkly.
Einstein's view was what would now be called, a hidden variable
theory. Hidden variable theories might seem to be the most obvious way
to incorporate the Uncertainty Principle into physics. They form the
basis of the mental picture of the universe, held by many scientists,
and almost all philosophers of science. But these hidden variable
theories are wrong. The British physicist, John Bell, who died
recently, devised an experimental test that would distinguish hidden
variable theories. When the experiment was carried out carefully, the
results were inconsistent with hidden variables. Thus it seems that
even God is bound by the Uncertainty Principle, and can not know both
the position, and the speed, of a particle. So God does play dice with
the universe. All the evidence points to him being an inveterate
gambler, who throws the dice on every possible occasion.
Other scientists were much more ready than Einstein to modify the
classical 19th century view of determinism. A new theory, called
quantum mechanics, was put forward by Heisenberg, the Austrian, Erwin
Schroedinger, and the British physicist, Paul Dirac. Dirac was my
predecessor but one, as the Lucasian Professor in Cambridge. Although
quantum mechanics has been around for nearly 70 years, it is still not
generally understood or appreciated, even by those that use it to do
calculations. Yet it should concern us all, because it is a completely
different picture of the physical universe, and of reality itself. In
quantum mechanics, particles don't have well defined positions and
speeds. Instead, they are represented by what is called a wave
function. This is a number at each point of space. The size of the
wave function gives the probability that the particle will be found in
that position. The rate, at which the wave function varies from point
to point, gives the speed of the particle. One can have a wave
function that is very strongly peaked in a small region. This will
mean that the uncertainty in the position is small. But the wave
function will vary very rapidly near the peak, up on one side, and
down on the other. Thus the uncertainty in the speed will be large.
Similarly, one can have wave functions where the uncertainty in the
speed is small, but the uncertainty in the position is large.
The wave function contains all that one can know of the particle, both
its position, and its speed. If you know the wave function at one
time, then its values at other times are determined by what is called
the Schroedinger equation. Thus one still has a kind of determinism,
but it is not the sort that Laplace envisaged. Instead of being able
to predict the positions and speeds of particles, all we can predict
is the wave function. This means that we can predict just half what we
could, according to the classical 19th century view.
Although quantum mechanics leads to uncertainty, when we try to
predict both the position and the speed, it still allows us to
predict, with certainty, one combination of position and speed.
However, even this degree of certainty, seems to be threatened by more
recent developments. The problem arises because gravity can warp
space-time so much, that there can be regions that we don't observe.
Interestingly enough, Laplace himself wrote a paper in 1799 on how
some stars could have a gravitational field so strong that light could
not escape, but would be dragged back onto the star. He even
calculated that a star of the same density as the Sun, but two hundred
and fifty times the size, would have this property. But although
Laplace may not have realised it, the same idea had been put forward
16 years earlier by a Cambridge man, John Mitchell, in a paper in the
Philosophical Transactions of the Royal Society. Both Mitchell and
Laplace thought of light as consisting of particles, rather like
cannon balls, that could be slowed down by gravity, and made to fall
back on the star. But a famous experiment, carried out by two
Americans, Michelson and Morley in 1887, showed that light always
travelled at a speed of one hundred and eighty six thousand miles a
second, no matter where it came from. How then could gravity slow down
light, and make it fall back.
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Lectures a brief history of mine
Does God Play Dice? Cont...
This was impossible, according to the then accepted ideas of space and
time. But in 1915, Einstein put forward his revolutionary General
Theory of Relativity. In this, space and time were no longer separate
and independent entities. Instead, they were just different directions
in a single object called space-time. This space-time was not flat,
but was warped and curved by the matter and energy in it. In order to
understand this, considered a sheet of rubber, with a weight placed on
it, to represent a star. The weight will form a depression in the
rubber, and will cause the sheet near the star to be curved, rather
than flat. If one now rolls marbles on the rubber sheet, their paths
will be curved, rather than being straight lines. In 1919, a British
expedition to West Africa, looked at light from distant stars, that
passed near the Sun during an eclipse. They found that the images of
the stars were shifted slightly from their normal positions. This
indicated that the paths of the light from the stars had been bent by
the curved space-time near the Sun. General Relativity was confirmed.
Consider now placing heavier and heavier, and more and more
concentrated weights on the rubber sheet. They will depress the sheet
more and more. Eventually, at a critical weight and size, they will
make a bottomless hole in the sheet, which particles can fall into,
but nothing can get out of.
What happens in space-time according to General Relativity is rather
similar. A star will curve and distort the space-time near it, more
and more, the more massive and more compact the star is. If a massive
star, which has burnt up its nuclear fuel, cools and shrinks below a
critical size, it will quite literally make a bottomless hole in
space-time, that light can't get out of. Such objects were given the
name Black Holes, by the American physicist John Wheeler, who was one
of the first to recognise their importance, and the problems they
pose. The name caught on quickly. To Americans, it suggested something
dark and mysterious, while to the British, there was the added
resonance of the Black Hole of Calcutta. But the French, being French,
saw a more risqu� meaning. For years, they resisted the name, trou
noir, claiming it was obscene. But that was a bit like trying to stand
against le weekend, and other franglais. In the end, they had to give
in. Who can resist a name that is such a winner?
We now have observations that point to black holes in a number of
objects, from binary star systems, to the centre of galaxies. So it is
now generally accepted that black holes exist. But, apart from their
potential for science fiction, what is their significance for
determinism. The answer lies in a bumper sticker that I used to have
on the door of my office: Black Holes are Out of Sight. Not only do
the particles and unlucky astronauts that fall into a black hole,
never come out again, but also the information that they carry, is
lost forever, at least from our region of the universe. You can throw
television sets, diamond rings, or even your worst enemies into a
black hole, and all the black hole will remember, is the total mass,
and the state of rotation. John Wheeler called this, 'A Black Hole Has
No Hair.' To the French, this just confirmed their suspicions.
As long as it was thought that black holes would continue to exist
forever, this loss of information didn't seem to matter too much. One
could say that the information still existed inside the black hole. It
is just that one can't tell what it is, from the outside. However, the
situation changed, when I discovered that black holes aren't
completely black. Quantum mechanics causes them to send out particles
and radiation at a steady rate. This result came as a total surprise
to me, and everyone else. But with hindsight, it should have been
obvious. What we think of as empty space is not really empty, but it
is filled with pairs of particles and anti particles. These appear
together at some point of space and time, move apart, and then come
together and annihilate each other. These particles and anti particles
occur because a field, such as the fields that carry light and
gravity, can't be exactly zero. That would mean that the value of the
field, would have both an exact position (at zero), and an exact speed
or rate of change (also zero). This would be against the Uncertainty
Principle, just as a particle can't have both an exact position, and
an exact speed. So all fields must have what are called, vacuum
fluctuations. Because of the quantum behaviour of nature, one can
interpret these vacuum fluctuations, in terms of particles and anti
particles, as I have described.
These pairs of particles occur for all varieties of elementary
particles. They are called virtual particles, because they occur even
in the vacuum, and they can't be directly measured by particle
detectors. However, the indirect effects of virtual particles, or
vacuum fluctuations, have been observed in a number of experiments,
and their existence confirmed.
If there is a black hole around, one member of a particle anti
particle pair may fall into the hole, leaving the other member without
a partner, with which to annihilate. The forsaken particle may fall
into the hole as well, but it may also escape to a large distance from
the hole, where it will become a real particle, that can be measured
by a particle detector. To someone a long way from the black hole, it
will appear to have been emitted by the hole.
This explanation of how black holes ain't so black, makes it clear
that the emission will depend on the size of the black hole, and the
rate at which it is rotating. But because black holes have no hair, in
Wheeler's phrase, the radiation will be otherwise independent of what
went into the hole. It doesn't matter whether you throw television
sets, diamond rings, or your worst enemies, into a black hole. What
comes back out will be the same.
So what has all this to do with determinism, which is what this
lecture is supposed to be about. What it shows is that there are many
initial states, containing television sets, diamond rings, and even
people, that evolve to the same final state, at least outside the
black hole. But in Laplace's picture of determinism, there was a one
to one correspondence between initial states, and final states. If you
knew the state of the universe at some time in the past, you could
predict it in the future. Similarly, if you knew it in the future, you
could calculate what it must have been in the past. The advent of
quantum theory in the 1920s reduced the amount one could predict by
half, but it still left a one to one correspondence between the states
of the universe at different times. If one knew the wave function at
one time, one could calculate it at any other time.
With black holes, however, the situation is rather different. One will
end up with the same state outside the hole, whatever one threw in,
provided it has the same mass. Thus there is not a one to one
correspondence between the initial state, and the final state outside
the black hole. There will be a one to one correspondence between the
initial state, and the final state both outside, and inside, the black
hole. But the important point is that the emission of particles, and
radiation by the black hole, will cause the hole to lose mass, and get
smaller. Eventually, it seems the black hole will get down to zero
mass, and will disappear altogether. What then will happen to all the
objects that fell into the hole, and all the people that either jumped
in, or were pushed? They can't come out again, because there isn't
enough mass or energy left in the black hole, to send them out again.
They may pass into another universe, but that is not something that
will make any difference, to those of us prudent enough not to jump
into a black hole. Even the information, about what fell into the
hole, could not come out again when the hole finally disappears.
Information can not be carried free, as those of you with phone bills
will know. Information requires energy to carry it, and there won't be
enough energy left when the black hole disappears.
What all this means is, that information will be lost from our region
of the universe, when black holes are formed, and then evaporate. This
loss of information will mean that we can predict even less than we
thought, on the basis of quantum theory. In quantum theory, one may
not be able to predict with certainty, both the position, and the
speed of a particle. But there is still one combination of position
and speed that can be predicted. In the case of a black hole, this
definite prediction involves both members of a particle pair. But we
can measure only the particle that comes out. There's no way even in
principle that we can measure the particle that falls into the hole.
So, for all we can tell, it could be in any state. This means we can
not make any definite prediction, about the particle that escapes from
the hole. We can calculate the probability that the particle has this
or that position, or speed. But there's no combination of the position
and speed of just one particle that we can definitely predict, because
the speed and position will depend on the other particle, which we
don't observe. Thus it seems Einstein was doubly wrong when he said,
God does not play dice. Not only does God definitely play dice, but He
sometimes confuses us by throwing them where they can't be seen.
Many scientists are like Einstein, in that they have a deep emotional
attachment to determinism. Unlike Einstein, they have accepted the
reduction in our ability to predict, that quantum theory brought
about. But that was far enough. They didn't like the further
reduction, which black holes seemed to imply. They have therefore
claimed that information is not really lost down black holes. But they
have not managed to find any mechanism that would return the
information. It is just a pious hope that the universe is
deterministic, in the way that Laplace thought. I feel these
scientists have not learnt the lesson of history. The universe does
not behave according to our pre-conceived ideas. It continues to
surprise us.
One might not think it mattered very much, if determinism broke down
near black holes. We are almost certainly at least a few light years,
from a black hole of any size. But, the Uncertainty Principle implies
that every region of space should be full of tiny virtual black holes,
which appear and disappear again. One would think that particles and
information could fall into these black holes, and be lost. Because
these virtual black holes are so small, a hundred billion billion
times smaller than the nucleus of an atom, the rate at which
information would be lost would be very low. That is why the laws of
science appear deterministic, to a very good approximation. But in
extreme conditions, like in the early universe, or in high energy
particle collisions, there could be significant loss of information.
This would lead to unpredictability, in the evolution of the universe.
To sum up, what I have been talking about, is whether the universe
evolves in an arbitrary way, or whether it is deterministic. The
classical view, put forward by Laplace, was that the future motion of
particles was completely determined, if one knew their positions and
speeds at one time. This view had to be modified, when Heisenberg put
forward his Uncertainty Principle, which said that one could not know
both the position, and the speed, accurately. However, it was still
possible to predict one combination of position and speed. But even
this limited predictability disappeared, when the effects of black
holes were taken into account. The loss of particles and information
down black holes meant that the particles that came out were random.
One could calculate probabilities, but one could not make any definite
predictions. Thus, the future of the universe is not completely
determined by the laws of science, and its present state, as Laplace
thought. God still has a few tricks up his sleeve.
That is all I have to say for the moment. Thank you for listening.
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