Quantum Gravity from Non-commutative geometry

An important problem to tackle in the 21st century is how to combine quantum field theory and general relativity into a theory of quantum gravity, that describes the fabric of space and time at the smallest scales. Some predictions in current physics, like the mass of the proton and mergers of black holes, are too complicated to calculate by hand, and the fundamental structure of space-time might well fall into that same category. Computer simulations allow us to study non-perturbative aspects of physics and thus make it possible to explore quantum gravity. These computer simulations are based on path integral approaches to quantum gravity, the path integral in this case means a sum over all possible geometries. To sum over geometries we need to discretize geometry, we introduce something like space-time atoms. One method to do this is to use non-commutative geometries, in these the discreteness has the odd side effect that on very small scales going left and then forward is not necessarily the same as going forward and then left. We can write these geometries as matrices, and use the computer to calculate the path integral over these. In this project we plan on pursuing five related points: 1) Code development: To undertake simulations and to further push the development of the field I will develop a modular, state of the art, open source code and a database of configurations for the community to work with. 2) Coordinates on the geometry: We can generate geometries, however to understand these geometries better we need to be able to identify points on them, for which we need coordinates. 3) Other observables: To understand the geometries better we plan on searching for more properties we can measure, to do this we will comb the mathematical literature, and use machine learning algorithms on the computer. 4) A better action: The action is a measure of the energy of the geometries, which the algorithm uses to sample likely geometric states. We currently use a simple approximate action, however in this part of the project we will examine other energy measures. 5) Adding matter: A theory of gravity should describe the interaction of matter and geometry. Until now we have only studied geometry, in this part of the project we will add matter. This work will further the search for a theory of quantum gravity based on non-commutative geometry. The code development will be an important task to enable the following research projects. The coordinates, other observables and the action will help to better understand the geometric structure of space and time at the lowest scale. The last part, including matter in the theory, will be the most important step, in taking our theory from a model of pure geometry to a true theory of gravity in which matter and geometry interact.

An important problem to tackle in the 21st century is how to combine quantum field theory and general relativity into a theory of quantum gravity, that describes the fabric of space and time at the smallest scales. Some predictions in current physics, like the mass of the proton and mergers of black holes, are too complicated to calculate by hand, and the fundamental structure of space-time might well fall into that same category. Computer simulations allow us to study non-perturbative aspects of physics and thus make it possible to explore quantum gravity. These computer simulations are based on path integral approaches to quantum gravity, the path integral in this case means a sum over all possible geometries. To sum over geometries we need to discretize geometry, we introduce something like space-time atoms. One method to do this is to use non-commutative geometries, in these the discreteness has the odd side effect that on very small scales going left and then forward is not necessarily the same as going forward and then left. We can write these geometries as matrices, and use the computer to calculate the path integral over these. One of the most impressive outcomes of the project was an algorithm that can recreate a coordinate embedding from a non-commutative geometry. This allows us to make an image of a non-commutative geometry, and as the saying goes, a picture speaks louder than a thousand words. This is particularly true in this very abstract field, since the image allows us a more direct understanding of the geometry. To describe a realistic universe, the universe should contain some matter. As a first step towards describing non-commutative geometry together with matter I have studied a simpler model, the so-called 2d orders, coupled to matter. In these simulations I found that matter and geometry are correlated, and change their behaviors at the same points. This is important, since it shows that matter and geometry in these simulations are truly coupled.