Godel and the End
of Physics
This lecture is the intellectual property of
Professor S.W.Hawking. You may not reproduce, edit, translate,
distribute, publish or host this document in any way with out
the permission of Professor Hawking.
Note that there may be incorrect spellings, punctuation and/or
grammar in this document. This is to allow correct pronunciation
and timing by a speech synthesiser.
In this talk, I want to ask how far can we go in our search for
understanding and knowledge. Will we ever find a complete form
of the laws of nature? By a complete form, I mean a set of rules
that in principle at least enable us to predict the future to an
arbitrary accuracy, knowing the state of the universe at one
time. A qualitative understanding of the laws has been the aim
of philosophers and scientists, from Aristotle onwards. But it
was Newton's Principia Mathematica in 1687, containing his
theory of universal gravitation that made the laws quantitative
and precise. This led to the idea of scientific determinism,
which seems first to have been expressed by Laplace. If at one
time, one knew the positions and velocities of all the particles
in the universe, the laws of science should enable us to
calculate their positions and velocities at any other time, past
or future. The laws may or may not have been ordained by God,
but scientific determinism asserts that he does not intervene to
break them.
At first, it seemed that these hopes for a complete determinism
would be dashed by the discovery early in the 20th century; that
events like the decay of radio active atoms seemed to take place
at random. It was as if God was playing dice, in Einstein's
phrase. But science snatched victory from the jaws of defeat by
moving the goal posts and redefining what is meant by a complete
knowledge of the universe. It was a stroke of brilliance whose
philosophical implications have still not been fully
appreciated. Much of the credit belongs to Paul Dirac, my
predecessor but one in the Lucasian chair, though it wasn't
motorized in his time. Dirac showed how the work of Erwin
Schrodinger and Werner Heisenberg could be combined in new
picture of reality, called quantum theory. In quantum theory, a
particle is not characterized by two quantities, its position
and its velocity, as in classical Newtonian theory. Instead it
is described by a single quantity, the wave function. The size
of the wave function at a point, gives the probability that the
particle will be found at that point, and the rate at which the
wave function changes from point to point, gives the probability
of different velocities. One can have a wave function that is
sharply peaked at a point. This corresponds to a state in which
there is little uncertainty in the position of the particle.
However, the wave function varies rapidly, so there is a lot of
uncertainty in the velocity. Similarly, a long chain of waves
has a large uncertainty in position, but a small uncertainty in
velocity. One can have a well defined position, or a well
defined velocity, but not both.
This would seem to make complete determinism impossible. If one
can't accurately define both the positions and the velocities of
particles at one time, how can one predict what they will be in
the future? It is like weather forecasting. The forecasters
don't have an accurate knowledge of the atmosphere at one time.
Just a few measurements at ground level and what can be learnt
from satellite photographs. That’s why weather forecasts are so
unreliable. However, in quantum theory, it turns out one doesn't
need to know both the positions and the velocities. If one knew
the laws of physics and the wave function at one time, then
something called the Schrodinger equation would tell one how
fast the wave function was changing with time. This would allow
one to calculate the wave function at any other time. One can
therefore claim that there is still determinism but it is
determinism on a reduced level. Instead of being able accurately
to predict two quantities, position and velocity, one can
predict only a single quantity, the wave function. We have
re-defined determinism to be just half of what Laplace thought
it was. Some people have tried to connect the unpredictability
of the other half with consciousness, or the intervention of
supernatural beings. But it is difficult to make either case for
something that is completely random.
In order to calculate how the wave function develops in time,
one needs the quantum laws that govern the universe. So how well
do we know these laws? As Dirac remarked, Maxwell's equations of
light and the relativistic wave equation, which he was too
modest to call the Dirac equation, govern most of physics and
all of chemistry and biology. So in principle, we ought to be
able to predict human behavior, though I can't say I have had
much success myself. The trouble is that the human brain
contains far too many particles for us to be able to solve the
equations. But it is comforting to think we might be able to
predict the nematode worm, even if we can't quite figure out
humans. Quantum theory and the Maxwell and Dirac equations
indeed govern much of our life, but there are two important
areas beyond their scope. One is the nuclear forces. The other
is gravity. The nuclear forces are responsible for the Sun
shining and the formation of the elements including the carbon
and oxygen of which we are made. And gravity caused the
formation of stars and planets, and indeed, of the universe
itself. So it is important to bring them into the scheme.
The so called weak nuclear forces have been unified with the
Maxwell equations by Abdus Salam and Stephen Weinberg, in what
is known as the Electro weak theory. The predictions of this
theory have been confirmed by experiment and the authors
rewarded with Nobel Prizes. The remaining nuclear forces, the so
called strong forces, have not yet been successfully unified
with the electro weak forces in an observationally tested
scheme. Instead, they seem to be described by a similar but
separate theory called QCD. It is not clear who, if anyone,
should get a Nobel Prize for QCD, but David Gross and Gerard ‘t
Hooft share credit for showing the theory gets simpler at high
energies. I had quite a job to get my speech synthesizer to
pronounce Gerard's surname. It wasn't familiar with apostrophe
t. The electro weak theory and QCD together constitute the so
called Standard Model of particle physics, which aims to
describe everything except gravity.
The standard model seems to be adequate for all practical
purposes, at least for the next hundred years. But practical or
economic reasons have never been the driving force in our search
for a complete theory of the universe. No one working on the
basic theory, from Galileo onward, has carried out their
research to make money, though Dirac would have made a fortune
if he had patented the Dirac equation. He would have had a
royalty on every television, walkman, video game and computer.
The real reason we are seeking a complete theory, is that we
want to understand the universe and feel we are not just the
victims of dark and mysterious forces. If we understand the
universe, then we control it, in a sense. The standard model is
clearly unsatisfactory in this respect. First of all, it is ugly
and ad hoc. The particles are grouped in an apparently arbitrary
way, and the standard model depends on 24 numbers whose values
can not be deduced from first principles, but which have to be
chosen to fit the observations. What understanding is there in
that? Can it be Nature's last word? The second failing of the
standard model is that it does not include gravity. Instead,
gravity has to be described by Einstein's General Theory of
Relativity. General relativity is not a quantum theory unlike
the laws that govern everything else in the universe. Although
it is not consistent to use the non quantum general relativity
with the quantum standard model, this has no practical
significance at the present stage of the universe because
gravitational fields are so weak. However, in the very early
universe, gravitational fields would have been much stronger and
quantum gravity would have been significant. Indeed, we have
evidence that quantum uncertainty in the early universe made
some regions slightly more or less dense than the otherwise
uniform background. We can see this in small differences in the
background of microwave radiation from different directions. The
hotter, denser regions will condense out of the expansion as
galaxies, stars and planets. All the structures in the universe,
including ourselves, can be traced back to quantum effects in
the very early stages. It is therefore essential to have a fully
consistent quantum theory of gravity, if we are to understand
the universe.
Constructing a quantum theory of gravity has been the
outstanding problem in theoretical physics for the last 30
years. It is much, much more difficult than the quantum theories
of the strong and electro weak forces. These propagate in a
fixed background of space and time. One can define the wave
function and use the Schrodinger equation to evolve it in time.
But according to general relativity, gravity is space and time.
So how can the wave function for gravity evolve in time? And
anyway, what does one mean by the wave function for gravity? It
turns out that, in a formal sense, one can define a wave
function and a Schrodinger like equation for gravity, but that
they are of little use in actual calculations.
Instead, the usual approach is to regard the quantum spacetime
as a small perturbation of some background spacetime; generally
flat space. The perturbations can then be treated as quantum
fields, like the electro weak and QCD fields, propagating
through the background spacetime. In calculations of
perturbations, there is generally some quantity called the
effective coupling which measures how much of an extra
perturbation a given perturbation generates. If the coupling is
small, a small perturbation creates a smaller correction which
gives an even smaller second correction, and so on. Perturbation
theory works and can be used to calculate to any degree of
accuracy. An example is your bank account. The interest on the
account is a small perturbation. A very small perturbation if
you are with one of the big banks. The interest is compound.
That is, there is interest on the interest, and interest on the
interest on the interest. However, the amounts are tiny. To a
good approximation, the money in your account is what you put
there. On the other hand, if the coupling is high, a
perturbation generates a larger perturbation which then
generates an even larger perturbation. An example would be
borrowing money from loan sharks. The interest can be more than
you borrowed, and then you pay interest on that. It is
disastrous.
With gravity, the effective coupling is the energy or mass of
the perturbation because this determines how much it warps
spacetime, and so creates a further perturbation. However, in
quantum theory, quantities like the electric field or the
geometry of spacetime don't have definite values, but have what
are called quantum fluctuations. These fluctuations have energy.
In fact, they have an infinite amount of energy because there
are fluctuations on all length scales, no matter how small. Thus
treating quantum gravity as a perturbation of flat space doesn't
work well because the perturbations are strongly coupled.
Supergravity was invented in 1976 to solve, or at least improve,
the energy problem. It is a combination of general relativity
with other fields, such that each species of particle has a
super partner species. The energy of the quantum fluctuations of
one partner is positive, and the other negative, so they tend to
cancel. It was hoped the infinite positive and negative energies
would cancel completely, leaving only a finite remainder. In
this case, a perturbation treatment would work because the
effective coupling would be weak. However, in 1985, people
suddenly lost confidence that the infinities would cancel. This
was not because anyone had shown that they definitely didn't
cancel. It was reckoned it would take a good graduate student
300 years to do the calculation, and how would one know they
hadn't made a mistake on page two? Rather it was because Ed
Witten declared that string theory was the true quantum theory
of gravity, and supergravity was just an approximation, valid
when particle energies are low, which in practice, they always
are. In string theory, gravity is not thought of as the warping
of spacetime. Instead, it is given by string diagrams; networks
of pipes that represent little loops of string, propagating
through flat spacetime. The effective coupling that gives the
strength of the junctions where three pipes meet is not the
energy, as it is in supergravity. Instead it is given by what is
called the dilaton; a field that has not been observed. If the
dilaton had a low value, the effective coupling would be weak,
and string theory would be a good quantum theory. But it is no
earthly use for practical purposes.
In the years since 1985, we have realized that both supergravity
and string theory belong to a larger structure, known as M
theory. Why it should be called M Theory is completely obscure.
M theory is not a theory in the usual sense. Rather it is a
collection of theories that look very different but which
describe the same physical situation. These theories are related
by mappings or correspondences called dualities, which imply
that they are all reflections of the same underlying theory.
Each theory in the collection works well in the limit, like low
energy, or low dilaton, in which its effective coupling is
small, but breaks down when the coupling is large. This means
that none of the theories can predict the future of the universe
to arbitrary accuracy. For that, one would need a single
formulation of M-theory that would work in all situations.
Up to now, most people have implicitly assumed that there is an
ultimate theory that we will eventually discover. Indeed, I
myself have suggested we might find it quite soon. However,
M-theory has made me wonder if this is true. Maybe it is not
possible to formulate the theory of the universe in a finite
number of statements. This is very reminiscent of Godel's
theorem. This says that any finite system of axioms is not
sufficient to prove every result in mathematics.
Godel's theorem is proved using statements that refer to
themselves. Such statements can lead to paradoxes. An example
is, this statement is false. If the statement is true, it is
false. And if the statement is false, it is true. Another
example is, the barber of Corfu shaves every man who does not
shave himself. Who shaves the barber? If he shaves himself, then
he doesn't, and if he doesn't, then he does. Godel went to great
lengths to avoid such paradoxes by carefully distinguishing
between mathematics, like 2+2 =4, and meta mathematics, or
statements about mathematics, such as mathematics is cool, or
mathematics is consistent. That is why his paper is so difficult
to read. But the idea is quite simple. First Godel showed that
each mathematical formula, like 2+2=4, can be given a unique
number, the Godel number. The Godel number of 2+2=4, is *.
Second, the meta mathematical statement, the sequence of
formulas A, is a proof of the formula B, can be expressed as an
arithmetical relation between the Godel numbers for A- and B.
Thus meta mathematics can be mapped into arithmetic, though I'm
not sure how you translate the meta mathematical statement,
'mathematics is cool'. Third and last, consider the self
referring Godel statement, G. This is, the statement G can not
be demonstrated from the axioms of mathematics. Suppose that G
could be demonstrated. Then the axioms must be inconsistent
because one could both demonstrate G and show that it can not be
demonstrated. On the other hand, if G can't be demonstrated,
then G is true. By the mapping into numbers, it corresponds to a
true relation between numbers, but one which can not be deduced
from the axioms. Thus mathematics is either inconsistent or
incomplete. The smart money is on incomplete.
What is the relation between Godel’s theorem and whether we can
formulate the theory of the universe in terms of a finite number
of principles? One connection is obvious. According to the
positivist philosophy of science, a physical theory is a
mathematical model. So if there are mathematical results that
can not be proved, there are physical problems that can not be
predicted. One example might be the Goldbach conjecture. Given
an even number of wood blocks, can you always divide them into
two piles, each of which can not be arranged in a rectangle?
That is, it contains a prime number of blocks.
Although this is incompleteness of sort, it is not the kind of
unpredictability I mean. Given a specific number of blocks, one
can determine with a finite number of trials whether they can be
divided into two primes. But I think that quantum theory and
gravity together, introduces a new element into the discussion
that wasn't present with classical Newtonian theory. In the
standard positivist approach to the philosophy of science,
physical theories live rent free in a Platonic heaven of ideal
mathematical models. That is, a model can be arbitrarily
detailed and can contain an arbitrary amount of information
without affecting the universes they describe. But we are not
angels, who view the universe from the outside. Instead, we and
our models are both part of the universe we are describing. Thus
a physical theory is self referencing, like in Godel’s theorem.
One might therefore expect it to be either inconsistent or
incomplete. The theories we have so far are both inconsistent
and incomplete.
Quantum gravity is essential to the argument. The information in
the model can be represented by an arrangement of particles.
According to quantum theory, a particle in a region of a given
size has a certain minimum amount of energy. Thus, as I said
earlier, models don't live rent free. They cost energy. By
Einstein’s famous equation, E = mc squared, energy is equivalent
to mass. And mass causes systems to collapse under gravity. It
is like getting too many books together in a library. The floor
would give way and create a black hole that would swallow the
information. Remarkably enough, Jacob Bekenstein and I found
that the amount of information in a black hole is proportional
to the area of the boundary of the hole, rather than the volume
of the hole, as one might have expected. The black hole limit on
the concentration of information is fundamental, but it has not
been properly incorporated into any of the formulations of M
theory that we have so far. They all assume that one can define
the wave function at each point of space. But that would be an
infinite density of information which is not allowed. On the
other hand, if one can't define the wave function point wise,
one can't predict the future to arbitrary accuracy, even in the
reduced determinism of quantum theory. What we need is a
formulation of M theory that takes account of the black hole
information limit. But then our experience with supergravity and
string theory, and the analogy of Godel’s theorem, suggest that
even this formulation will be incomplete.
Some people will be very disappointed if there is not an
ultimate theory that can be formulated as a finite number of
principles. I used to belong to that camp, but I have changed my
mind. I'm now glad that our search for understanding will never
come to an end, and that we will always have the challenge of
new discovery. Without it, we would stagnate. Godel’s theorem
ensured there would always be a job for mathematicians. I think
M theory will do the same for physicists. I'm sure Dirac would
have approved.
Thank you for listening.
