Archive for the ‘non-commutative geometry’ Category
Lisa Glaser and I have started to investigate random non-commutative geometries in our latest paper
Monte Carlo simulations of random non-commutative geometries. The geometries are all specified by a Dirac operator that is a finite-dimensional matrix, so in this theory the partition function is a finite-dimensional integral. We investigated the phenomena that occur by calculating this integral on a computer, in much the same way as you would measure data from an experiment in a lab.
The most fascinating finding is that by tuning a parameter in the action we are able to find a phase transition (audible here!), and at this transition the spectrum of the Dirac operator looks a lot like the spectrum one would expect for a typical manifold in the usual sense of differential geometry. This is surprising, because non-commutative geometries generally look nothing like differential geometries.
The first plot shows the eigenvalue spectrum for the fuzzy sphere using an algebra of 10×10 matrices. The plot has the eigenvalues of the Dirac operator on the horizontal axis and the density of eigenvalues vertically.
The spectrum for the fuzzy sphere is the same as for an ordinary two-sphere in differential geometry, except that the spectrum is cut off after a finite number of eigenvalues (in this case, 200). These spheres are special in that the eigenvalues are in fact all integers, and have a multiplicity proportional to the eigenvalue, due to the spherical symmetry.
A Dirac operator on an arbitrary metric on a surface would not have this symmetry or the degeneracy, but the approximate V-shape of the plot should be the same, at least when the eigenvalues are large enough.
This is exactly what one sees in the second plot, an approximate V-shape with some slightly different detail for small eigenvalues. But the second plot is generated by an ensemble of random non-commutative Dirac operators!
Actually, we nearly didn’t find this. We looked at a range of different random NCG models with a free parameter and accumulated a large number of eigenvalue plots that looked nothing like a manifold. Only when we started to write a section of the paper explaining this did it become clear that a very small number of the plots did look something like a manifold. These plots occur only close to a phase transition, where the properties of the ensemble change sharply when the parameter in the model is changed.
Our findings fit into a general theme for discrete theories, like lattice gauge theory, where continuum-like behaviour occurs at, or near, a phase transition. Thus it seems like a great place to look for quantum gravity in non-commutative models.
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.
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.
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.