Professor
Shahn Majid

Professor
Shahn Majid

Lay Persons Guide to My Research on
Quantum Spacetime

You could say that my
research is about absolutely nothing and completely
pointless. Both statements are correct, but let me
explain!!

First of all, my research
is mainly about empty space with absolutely nothing in
it. But Riemann in 1854 and Einstein in 1917 taught us
that empty space still has structure; it’s curved and
this curvature makes objects moving in it bend around,
and this bending was Einstein’s way to understand
gravity. Einstein also taught us that space and time
are really parts of a single 4-dimensional spacetime
continuum. So my research is about the structure of
space and time.

Now the thing is that
Einstein never really agreed with quantum mechanics,
whereas we now know since 1924 that on a subatomic
scale physics is `fuzzy’ due to quantum effects. This
fuzziness is expressed in the famous Heisenberg
uncertainty relations that you cannot perfectly
measure the position and momentum (roughly speaking,
velocity) of a particle at the same. My research takes
these ideas much further and says that due to quantum
effects you cannot measure the location in space and
time of an event to perfect accuracy; spacetime is
itself intrinsically fuzzy due to quantum
effects. That means that the mathematical
concept of a point in space and time does not apply;
the geometry of the real world does not truly have
points but rather has a quantum structure. That’s what
I work on more precisely, a new conception of
geometry.

I suspect you
want proof, so here it is. The trick is to step back
and look at the following `big picture’.
We plot the mass and size of everything in the
known Universe on a log scale where each notch on the
axis is a factor of 10 billion. The thing is that
everything lies in the trian
gular region in the middle with everything to the left
forbidden by Heisenberg’s uncertainty relation and
everything to the right forbidden by Einstein’s theory
of gravity. On the left slope, if a quantum particle
has a smaller and smaller wavelength then it needs to
have more and more mass-energy-momentum, you just move
down the slope on the left. On the right, if you try
to put too much matter into a small space it forms a
black-hole and if you put in more mass you just move
up the slope on the right as the black-hole gets
bigger. Now think about trying to measure a location
in space, say, with greater and greater accuracy. You
would need probes of smaller and smaller wavelength -
a light microscope, electron microscope, proton
microscope (aka large hadron collider) etc - and these
would be made of wave-particles of greater and greater
mass-energy. Eventually you would reach the point
where their mass-energy curved spacetime so much that
they formed black-holes. This is the bottom tip of the
triangle where quantum theory collides with gravity.
Distances smaller than that scale (it is called the
`Planck scale’ and is about 10
billion-billion-billion-billionth of a metre) are
therefore intrinsically unknowable and hence have no
place at the foundations of science. I’m taking
the view here that Science is not about angels dancing
on pinheads.

There are two clues in the
figure about where to go from here. The first is that
we, humans, are somewhat in the middle. I take that to
suggest that we built science around ourselves and in
the process we boxed ourselves in. In truth, Nature
does not know what mathematics, in particular, is in
our current maths books, so to truly break out of the
box we need to think about Pure Mathematics itself.
That’s how come I am a pure mathematician. The second
clue is a kind of symmetry or duality between the left
slope, elementary particles, and the right slope,
black holes. The search for the correct structure of
space and time led me since the 1980s to focus on a
kind of self-duality in the nature of mathematics.

I’d now like to
give you a flavour of what that self-duality is about,
but you will need to fasten your seatbelt. The duality
I am referring to is a kind of mind-body duality or
duality between a property and that which measures the
property, between reality and the representation of
reality. For example, the state of the world might be
a point x in some space X of possibilities. A question
to ask about the world would be a function on X, i.e.,
something that assigns a numerical value f(x) for each
point x in X. Now the thing is that the set of
all possible functions f is itself some space, lets
call it X^, and one could equally well say that x is a
function or measurer of f as a point in X^. You would
reinterpret the same number f(x) as x(f). This means
that we can always reverse the interpretation of what
is the real thing and what is its measurer or
representation.

Let’s do that
in the context of space and time. Instead of working
with a spacetime continuum X let’s just regard the set
of functions on it as the `real thing’. It’s an
algebra of a certain special `commutative’ type. Now
it turns out that most of geometry, including gravity,
can be written in dual form in terms of this algebra,
never once mentioning X itself. At that point you can
let the algebra be a different `noncommutative’ type
of algebra, such as the matrix algebras that
occur in quantum mechanics. These can never be
functions on an actual spacetime but you can still
talk about gravity in the algebraic form. So what I’ve
been doing is `noncommutative Riemannian geometry’ as
my approach to solving that big unsolved problem of
quantum theory unified with gravity. Only such
radically new pure mathematics could hope to describe
physics at the Planck scale where quantum theory and
gravity collide. Examples of such quantum geometries
are `quantum groups’ and the discovery of one of the
two main classes of these was the first concrete
outcome of my research. My recent work has included
astronomical predictions coming from the above quantum
spacetime hypothesis.

You can read more online by exploring the menu at the top or my old Space and Time blog. You can read more in print in my book On Space and Time. You can read about my more mathematical work in algebra, geometry and category theory under the Research tab at the top.

My Space and Time Blog