Introduction to Non-Commutative Geometry

I gave a course of lectures at the ESI in Vienna on Non-Commutative Geometry in October 2022. The draft lecture notes are uploaded here. A similar series of lectures will be given at the Galileo Galilei Institute June 12-16th 2023.

Chapters: 1. Introduction 2. Fuzzy spaces with symmetry 3. Commutative analogues 4. Dirac operator 5. Real structures 6. Real spectral triples 7. Standard model charges 8. Standard Model masses 9. The fuzzy sphere 10. The fuzzy torus (+4 more chapters to come)

The old October version is here for those that did the coursework.

Please let me know any corrections or improvements via email.

Penrose on palatial twistors

The main purpose of this post is to publish a draft manuscript version of the research paper Chameleon Twistor Theory: a Geometric Programme for Describing the Physical World by Roger Penrose, with his permission. This will eventually appear in a book later this year and I will add the bibliographic information when I have it.

Roger Penrose talking about the paper in 2018

Abstract Original motivations are recalled, for the introduction twistor theory, as a distinctive complex-geometric approach to the basic physics of our world, these being aimed at applying specifically to (3+1)-dimensional space-time, but where space-time itself is regarded as a notion secondary to the twistor geometry and its algebra. Twistors themselves may be initially pictured as light rays—with a twisting aspect to them related to angular momentum. Twistor theory provides an economical conformally invariant description of quantum wave functions for massless particles and fields, best understood in terms of holomorphic sheaf cohomology, subsequently leading to a non-linear description of anti-self-dual (“left-handed”) gravitational (and Yang-Mills) fields. Attempts to remove this anti-self-dual restriction (the googly problem) led to a 40-year blockage to the development of twistor theory as a possible overall approach to fundamental physics. However, in recent years, a more sophisticated approach to this problem has been developed—referred to as palatial twistor theory—whose basic procedures are described here, where a novel generating-function approach to Λ-vacuum Einstein equations is introduced.

 

I will add my own thoughts on this paper here later.

Dichromatic state sum models

Manuel Bärenz and I have just finished a paper on 4-dimensional topological state sum models. It is posted here (since 12 Jan 2016), and on arXiv. The idea of the paper is to squeeze more out of the Crane-Yetter state sum model and perhaps indicate how to get a viable quantum gravity model (or condensed matter model) from it.

The original CY was defined just using the quantum group version of SU(2), and turns out to be rather too simple be an interesting physics model. The reason is that the group SU(2) is used to “colour” both one- and two-dimensional dual edges. This has the effect that the quantum theory can’t “see” the difference between the one- and two-dimensional stuff.

All this is best seen using a handle decomposition rather than a triangulation. There’s an operation that changes 1-handles into 2-handles (thus changing the topology of the manifold) and the problem with the original CY is that it is invariant under this operation – which is a property that isn’t wanted. As a consequence, CY is the same on lots of different manfolds, which is why it is “too simple”.

One of the things that Manuel and I have is an efficient translation between the triangulation picture and the handle picture. In the handle picture, Jerome Petit had the idea that the 1- and 2-handles can be coloured differently, to give a new set of models that he called “dichromatic”. We have understood that in the triangulation picture, this corresponds to the CY invariant being “nonmodular” (the original CY is “modular”). Interestingly, there are also models in the handle picture that don’t have a CY description at all. We calculated a few simple examples and found that one of them has configurations that are a plausible analogue of “teleparallel gravity” in the formulation given by Baez and Wise. This doesn’t yet mean we have a new quantum gravity model because, firstly, we only used finite groups instead of Lie groups (to keep things simple) and secondly, it isn’t clear that the action will be the gravity action. Still, it is an interesting direction.

What it needs next is to do a lot more examples. Probably any really interesting examples will involve representations of a group or quantum group that are non-unitary. New territory indeed!

Matrix Quantum Gravity

The idea behind the project outlined in my Como lectures is to model a space-time geometry with a (finite-dimensional) matrix. Then quantum geometry (or random geometry) can be modelled by a suitable integral over the space of matrices, which is again a finite-dimensional integral, and thus well-defined.

In general terms, integrating over a space of matrices has been studied a lot, and goes by the name of a “matrix model”. However, in this case the matrix variable is a very specific type of matrix, namely a Dirac operator. The Dirac operator satisfies a number of algebraic conditions, so my new quantum gravity can be thought of as a matrix model for a very specific type of matrices.

The point of using Dirac operators is that a Dirac operator specifies a geometry. In the classical case, the Dirac operator is a differential operator on a manifold and one can extract the metric tensor from it. In the case of a matrix, the number of spinor “modes” that the Dirac operator operates on is finite, so that there is a maximum energy. Physically, if there are matrix quantum geometry models that behave like quantum gravity, this maximum energy should be identified as the Planck scale. So the mysterious Planck-scale cutoff that is needed to make sense of quantum field theory is built in to the model in a fundamental way. Whether there really are viable matrix quantum gravity models is very much an open question, and the point of The Project is to try to answer this.

To define a matrix Dirac operator in a consistent way, the notion of geometry is generalised to “non-commutative geometry”. In this setting, a lot of geometric constructions still make sense. One hopes, for example, that there is a suitable notion of the limiting case of large matrices in which a sequence of matrix Dirac operators converges to one of the usual Dirac differential operators (“taking the Planck scale to infinity”). More loosely, one can understand constructions in non-commutative geometry as the algebraic analogues of familiar constructions in ordinary metric geometry on a manifold. So it is important to have all of the machinery of non-commutative geometry (technically, the Dirac operator is part of a “real spectral triple”).

The basic set-up for matrix geometries is explained in my paper “matrix geometries and fuzzy spaces as spectral triples”. The paper contains the general definitions and a lot of useful data, such as operations on gamma matrices and explicit forms for the Dirac operator. I’m hoping it will be a resource for future efforts at modelling quantum geometry.

Como lectures

I’m giving some lectures at the SIGRAV school in Como on “Non-commutative geometry and quantum gravity”. I’ll put the slides up here as they happen.

Lecture 1: The Planck scale. This lecture reviews the experimental evidence for a fundamental Planck scale. Various theoretical approaches to modelling the Planck scale are mentioned very briefly, mainly concentrating on the predictions of the spectral action suggested by non-commutative geometry.

Lecture 2: Non-commutative geometry: axioms, examples. The axioms of non-commutative geometry are given in this lecture alongside an explanation of how they are satisfied by an ordinary (commutative) manifold. The example given by the standard model internal space is also summarised.

Lecture 3: Matrix geometries and fuzzy spheres. The lecture explains my project to replace the usual space-time manifold with a non-commutative geometry based on finite-dimensional matrices. Examples with spherical symmetry are considered.

Lecture 4: State sum models, using 2d examples. A different topic is introduced in this lecture: 2d topological state sum models defined using a diagrammatic calculus. Some recent progress in defining models that are sensitive to the spin structure of the surface (and are thus fermionic) is explained.

Lecture 5: Non-commutative geometry in 2d state sum models. The final lecture connects the topic of state sum models and non-commutative geometry. It shows how defects in a 2d state sum model can carry a non-commutative geometry.

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Giving the lectures turned out to be enormously useful. I’m writing a paper called “matrix geometries and fuzzy spaces as spectral triples“, which I hope will appear soon is out now. The paper will be much better for having had to explain the ideas in the lectures.

Hope and glory

In a given field of research, there’s a big distinction to be made between the scientific achievements in that field and the future prospects for the field. The “glory” is a matter of record, the stuff of review articles, whilst the “hope” is a matter of opinion. So what’s the role for opinion in scientific research?

In very pure mathematics there’s a presumption against expressing your opinions (in print anyway) in case it misleads readers, who would rather construct their own (and perhaps very different) picture of how the subject might develop. This certainly means that opinion and fact are never confused, but the problem is that there is consequently no debate about the aims of a subject or about the value of what has been achieved.

One might think that opinion has no place in science but that isn’t really right. Treating every possible avenue of future research as equally worthy of attention is not what we do; it would be a good way to waste resources. Every scientist has to make a judgement about what is worth doing and what is not. So we have opinions on the way forward, and also often share them when we communicate science. It seems to succeed in a subject it is necessary to be optimistic about its prospects, and to convince others that what you are doing is worthwhile you have to communicate that optimism.

My view is that expressing that optimism is quite legitimate but one should be careful not to let it be confused with the story about what has been achieved.

So where are we in quantum gravity research? As far as the “glory” is concerned: nowhere. Approaches based on quantum field theory have lots of “approximation” schemes but no central definition (so one cannot say they are actually approximating anything). Superstring theory does not even have a complete perturbation series, and a general non-perturbative formulation does not exist. Loop quantum gravity does not exist either; while there is a description of  a state space that is “too big”, there is no definition of the constraints that should reduce this space to the physical states. Discrete approaches such as spin foam models avoid troubles with the continuum, but the models that have been investigated so far are not supposed to be a complete physical theory; there have both technical and conceptual shortcomings.

As for the “hope”, there are plenty of reasons to continue to work in these fields. For example, superstring perturbation theory is the subject of current research and there are grounds to hope that a more complete perturbation theory will emerge. And maybe this will help in looking for the non-perturbative theory. String theorists could do themselves a favour by dropping the pretence that a complete string theory exists. After all, how can you recruit people to work on this problem if the wider scientific community thinks it has all been done already?

However I’m not going to get excited about superstrings because experimental tests of supersymmetry have a habit of coming out negative. Whilst this does not prove the absence of supersymmetry, in my opinion it is not looking very likely.

I’m continuing to work on the overlap between higher category theory and quantum physics. Some of this is work on understanding the spin foam models of quantum gravity. But I think there is more physics to be extracted from higher category theory. Indeed the mathematics is already deeply involved with both topological quantum field theory and conformal field theory, both of which are mathematically well-defined.

Moreover, there is a lot of unexplored territory. For example, there are “defects” in topological models that are point-like, string-like, brane-like etc., and there is a role there for deformations of geometry that are non-commutative, non-associative or other such things that don’t even have names yet. There are non-commutative geometries in particle physics that look a whole lot more enticing than supersymmetry. How do these examples fit into categorical quantum field theories? I hope to find out.