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.