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.
Gravity matters 
Dear Professor Barrett :
Greetings. Really impressive paper , I would like to ask some questions : Maybe as you know in causal dynamical triangulations or CDT phase diagram consists of 4 distinct phases namely A, B, C and D. The basic properties of these phases are fundamentally different from each other and only in phase C (called de sitter phase) an extended 4D spacetime is found (this is a classical or semi-classical limit of CDT). Very recently It was found a further phase called D or bifurcation phase (because it shows a double-peak structure as function of the difference of the neighbouring 3-volumes and This is related to the appearance of a vertex of very high order on alternating spatial slices, collapsing into a single such vertex as we cross into phase B or crumpled phase ) also it was strongly suggested phase transition between phase C and phase D is 2th or higher order ,furthermore results imply transition from phase C to D ,a phenomenon so-called signature change ,in other words signature of spacetime semi-classical at low curvature regime is Lorentzian but at very high curvatures (say planck scale) it is not fixed and turns into Euclidean signature. Also recent work by Lisa Glaser and Sumati Surya in causal set theory or CST ,show phase diagram of CST resemble two phases of CDT ,namely phase C and B of CDT is in one to one correspondence to manifoldlike and non-manifoldlike (layered crystalline structure) phases of CST. My basic question is : do your new results about non-commutative geometry and its phase diagram and phase transitions have something in common with mentioned phase diagram of CDT and CST , and what do phase diagram of non-commutative geometry say about the signature of spacetime at very high and low curvatures( high and low parameters in non-commutative geometry which resemble manifoldlike and non-manifoldlike phase). I will be extremely grateful if let me know your interesting ideas about mentioned issues ,in particular issue of signature (mainly do you expect a tight connection between Euclidean and Lorentzian version of non-commutative geometry in which at very high densities Euclidean phase is dominated and at low curvatures or densities Lorentzian structure emerge ?) Have you explored also the spectral dimension of non-commutative geometry (what is the spectral dimension of non-manifoldlike and manifoldlike structures
Kind Regards ,
Kaveh
Kaveh,
So far we have found two phases, one very low-dimensional and one very high-dimensional, separated by a second-order phase transition. Maybe the low dimensional phase is analogous to branched polymers and the high-dimensional one to a crumpled phase in DT, but this is just speculation. We find the continuum behaviour is exactly at the phase transition, just as one gets QCD at the phase transition of lattice QCD.
We don’t yet know how to look for any Lorentzian behaviour, or whether one has to change the model to get this. It is an interesting question.
We have looked at the spectral dimension and similar things, and have some interesting results which I hope will appear in a paper soon.
John
Hello. Professor Barrett , I have been struggling with a non-trivial issue for a long time about path integral in quantum gravity , many QG models (Covariant QG ) utilize path integral technique. This formulation has proven crucial to the subsequent development of theoretical physics, because it is manifestly symmetric between time and space. Unlike previous methods, the path-integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system.The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. But my question is rather simple but unfortunately I can’t properly understand it , maybe you can help me to resolve this matter (which I guess is crucial for any covariant model of QG) .
As you know in the Euclidean Dynamical Triangulations ,the prescription for the ground state wave function over closed 3- geometries is the Euclidean functional integral over 4-geometries (M,g), thus this prescription is a functional over all closed 3-geometries and is the initial state of the universe from which further evolution of the wave function can be (uniquely) determined. As far as I know ,different discrete approaches to quantum gravity like simplicial quantum gravity, causal dynamical triangulations and causal set theory(CST) choose different regularisation schemes” . Exactly at this point my question arise , my questions is difference between path integral in CDT and CST (causal set theory) : inCDT , sum is taking over all restricted Lorentzian geometries (hence causal) but someone can rotate geometries to Euclidean ones ( in order to do Monte carlo simulations) and then rotate back to Lorentzian signature again ( It is said that the sample space preserves its Lorentzian signature during the simulations). But situations is different in CST , CST postulates a fundamental discreteness in contrast of CDT ,but my question is not this . It is said by proponents of CST that there is no way to Euclideanise” a causet ( it is fundamentally Lorentzian and it is said the sum will remain Lorentzian and unchanged ) ,though the dynamics can be Euclideanised ( by means of a Beta parameter as inverse temperature or wick rotation parameter : this Beta parameter controls the phase diagram of CST ,so that when Beta is in its maximum value CST shows a crystalline phase ,reminder of crumpled phase of EDT and CDT and when Beta decreases phase turns into a continuum semi-classical spacetime). In full 2d quantum gravity Beta is taken to be a Wick rotation parameter , being the physically relevant quantum regime. Calculations with the Euclidean measure are assumed to analytically continue to this quantum regime in a manner similar to quantum field theory. The points I don’t understand are : 1- Both CDT and CST are inherently Lorentzian but how can CDT can be Euclideanised but this is impossible for CST ? 2- If Euclideanised CST is impossible , what does it mean by saying that of course dynamics can be Euclideanised (by Beta parameter playing the role of wick rotation =inverse temperature)? It will be my pleasure if you let me know your points on this matter .
Best Regards ,
Kaveh