Does the standard model originate from the mathematics of empty space?

We have a new paper on the arXiv, in which we link the spectral formulation of the standard model to a candidate for a fundamental theory based on the geometry of a configuration space.

Together with the mathematician Johannes Aastrup I have just posted a new paper on the arXiv, in which we show that the structure of a so-called almost-commutative geometry, which is the mathematical backbone of Chamseddine and Connes’ formulation of the standard model of particle physics coupled to Einsteins theory of general relativity, emerges from a candidate for a fundamental theory that we have been developing during the past two decades. In the following I will explain what we have found and what we believe it means.

NOTE: I will try to keep the discussion in this post at a level where most readers can follow, but in a few places I will add some technical details, which I ask readers, who are not familiar with the subject, to simply skip over.

To derive the theories that we already know

A key task for any candidate for a new fundamental theory is to be able to explain the physics that is already known at lower energy levels. Thus, for a new theory in theoretical high-energy physics it must be possible to derive both Einsteins theory of general relativity and the standard model of particle physics — which includes quantum field theory — from it.

This task is especially important in a situation like the one we are in today, where we have very little empirical data to work with. In the assessment of the viability of a new theory it is a highly encouraging sign if it is possible to derive already established theories from it. In this way we can almost regard the theories that we already know as empirical data that needs to be explained (or post-dicted) by a more fundamental theory. At the end of the day there is of course no substitute for falsifiability, a new theory must ultimately be able to predict new physics to be accepted as a physical theory, but reproducibility of known physics is nevertheless a strong temporary measure.

It is in this light that I would like to discuss our recent paper. During the past two decades we have been developing a candidate for a fundamental theory — which I will say more about shortly — that naturally gives rise to the basic building blocks of both bosonic and fermionic quantum field theory, i.e. the forces and the matter degrees of freedom of nature, together with some elements of general relativity. In our new paper we show that the key elements of the spectral formulation of the standard model also emerges from our candidate theory in a semiclassical limit, and thus we have moved one step closer to an actual post-diction of the standard model.

The spectral standard model

To explain how all this works we need to begin with the standard model of particle physics. Arguably the most interesting development in theoretical high-energy physics since the discovery of the standard model in the 1970s is the realisation made Ali Chamseddine and Alain Connes that the standard model coupled to general relativity can be formulated as a single gravitational theory based on noncommutative geometry.

This result can be understood as a multistage rocket:

• First stage is the reconstruction theorem, which states that there exists an equivalent formulation of Riemannian geometry¹ based on a so-called spectral triple, which consists of a C*-algebra, a Dirac operator, and a Hilbert space. The theorem states that a spectral triple that satisfies a certain set of axioms constitute an equivalent formulation of a Riemannian spin-geometry. Roughly speaking, this means that there exists an equivalent formulation of Einsteins theory of general relativity² that is based on a the mathematics of quantum mechanics instead of a metric field.

• Second stage is the realisation that the axioms behind this equivalent formulation can be straightforwardly generalised to include also noncommutative algebras³. The point is that although the reconstruction theorem is based on a commutative algebra the axioms it employs do not require this restriction; it can simply be lifted. This opens the door to what is known as noncommutative geometry.

• Third stage is then the discovery that the standard model coupled to general relativity provides one of the simples examples of such a noncommutative geometry: the standard model emerges from a relatively minor modification of the spectral formulation of general relativity, where the algebra of functions is multiplied with a certain matrix factor⁴. This is called an almost-commutative algebra and the formulation of the standard model that it entails is known as the spectral standard model.

This new formulation of the standard model is interesting for several reasons. First of all, it casts the standard model coupled to general relativity as a single gravitational theory, in which key elements of the standard model are given a completely new conceptual interpretation. Thus, from a conceptual point of view it is far richer and more interesting than the ordinary formulation, which is important because it may provide us with important hints as to what lies beyond the standard model. Second, this new formulation comes with new empirical predictions as well as fewer free parameters than the original formulation.

Two fundamental questions

The new formulation of the standard model raises two key questions:

1. where does the almost-commutative algebra and geometry originate from? Why did Nature choose this particular algebra built out of functions and matrices?

2. how does quantum field theory fit into this spectral formulation of the standard model?

Concerning the second question, then the point is that Chamseddine and Connes’ formulation of the standard model does not involve quantum field theory as a primary ingredient. Rather, quantum field theory is more like an add-on applied in a secondary step to the standard model but not to gravity. But if the new formulation is supposed to be fundamental — and I certainly believe that that is the case — then surely quantum field theory must play a central role in it⁵?

It is these two questions that our new paper may be able to answer. But before I can discuss that I need to give you some more details about our candidate for a fundamental theory.

A fundamental theory from empty space

The basic idea behind our approach to a fundamental theory is that a theory that is truly final must be irreducible in terms of further scientific reductions. And for a theory to be irreducible it must be based on principles that are extraordinarily simple. Like, say, three-dimensional space.

And thus that is our first main idea: to base our theory on the mathematics of three-dimensional space. Just that.

One basic feature of space is that you can move things around in it. Mathematically, such movements are encoded in what we call the Holonomy-Diffeomorphism algebra, or the HD-algebra for short. Technically, the HD-algebra is generated by parallel-transports along flows of vector-fields.

The HD-algebra is interesting for several reasons:

1. it comes with a very high level of canonicity.

2. there exist a canonical extension of it⁶ called the Quantum-Holonomy-Diffeomorphism algebra, short: the QHD algebra, which naturally encodes the canonical commutation relation of a Yang-Mills gauge theory⁷.

3. the HD-algebra can be understood as a noncommutative algebra of functions on a configuration space of gauge fields.

This last point requires some explaining. A gauge field is a field that encodes information about how certain degrees of freedom are moved around in space. Gauge fields play a key role in the standard model, where they represent the three fundamental forces, i.e. the strong, weak, and electro-magnetic forces. And a configuration space is an enormous space that includes all possible configurations of a field. Configuration spaces are well known in theoretical high-energy physics where they play a central role in quantum field theory. One can think of a configuration space of gauge fields as a space that encodes all possible ways an object can be moved around in space.

So what we propose is to base a fundamental theory entirely on the HD-algebra. Or put differently, to derive a final theory from the mathematics of empty space.

Here is our second main idea: to consider the geometry of the configuration space that is associated to the HD-algebra.

What this means is that we introduce a metric on the configuration space. That is, a distance between different field configurations. Concretely, the way we do this is by constructing a Dirac operator on the configuration space, which means that we employ the key lesson learned from noncommutative geometry, which is that the geometry of a space is encoded in the Dirac operator on that space.

All this is rather technical and involves some new concepts that takes some time to get used to. We have spent the past years developing this framework and thus interested readers can check out our papers or read about it for instance on my blog. The key takeaway is that such a geometrical construction on a configuration space gives rise to the key building blocks of bosonic and fermionic quantum field theory, i.e. forces and matter fields. Specifically, we obtain the canonical commutation relations of bosonic and fermionic quantum field theory as well as Hamilton operators of Yang-Mills theory and Dirac theory.

A semiclassical limit

We are now ready to discuss the result in our new paper, which is that in a certain limit the geometrical construction of a configuration space gives rise to structures that are very similar to the almost-commutative spectral triple that Connes’ and Chamseddine have identified as the mathematical backbone of the standard model. Specifically, we find:

1. that the HD-algebra gives rise to an almost-commutative algebra in a semi-classical limit.

2. that the fermionic sector of the emerging quantum field theory interacts with this emerging almost-commutative algebra in a manner that is almost identical to the setup found in the almost-commutative geometry in the spectral formulation of the standard model. The fermionic Hamilton operator involves a Dirac operator on three-dimensional space⁸, and this Dirac operator interacts with both factors in the almost-commutative algebra, both the functions and the matrices.

3. that the Hilbert space, in which the emerging almost commutative algebra acts, involves a double-fermionic structure that is similar to the fermionic doubling structure found in Chamseddine and Connes formulation of the standard model.

This means that our construction provides us with possible answers to the two questions I raised above:

• First possible answer: that the almost-commutative algebra that Chamseddine and Connes have identified as the mathematical backbone of the standard model originates from the HD-algebra

• Second possible answer: that quantum field theory is an integrated part of the emerging almost-commutative algebra since it emerges from the geometrical construction on the configuration space.

It also means that we have a framework that binds together key ingredients of the three pillars of modern theoretical high-energy physics — quantum field theory, the standard model, and general relativity — in a simple framework based entirely on the HD-algebra, which simply encodes elementary features of three-dimensional space itself.

What we don’t know

Now, all this sounds very promising, and we certainly think it is, but there are many open questions that still must be addressed before we can declare victory and go fishing. Let me end this post by highlighting three of the most important ones:

1. Is the almost-commutative algebra and geometry that emerges from our construction identical to the one Chamseddine and Connes found?

We do not know the answer to this question. First of all, the exact form of the matrix factor in the almost-commutative algebra that we obtain depends on some technical details, which are not yet resolved. Also, the whole geometrical construction on the configuration space is still being explored and developed and thus we do not yet have clarity about a number of technical issues. And finally, it is not completely straight forward to compare our framework to that of Connes since we work in a Hamiltonian setting whereas Connes works in a Lagrangian setting⁹.

2. Is our construction Lorentz invariant?

The Lorentz symmetry is a central symmetry in nature and evidence suggests that it is not broken even at very high energy scales. It is therefore an important question whether this symmetry is preserved in our construction. We do not know the answer to this question. Since we obtain the Hamilton operators of both Yang-Mills and Dirac theory it is clear that the Lorentz symmetry is present in our construction but we do not know if the whole construction is invariant¹⁰.

3. What about general relativity?

We have reasons to believe that at least parts of general relativity will also emerge from our construction, but we do not yet have decisive evidence that this is indeed the case.

Let me end this part by saying that I see no reason why all this couldn’t work out. We have not yet been able to analyse everything in depth, but so far I haven’t seen anything that makes my alarm-bells go off. The development of a new theory is an enormous task that can take decades, and we are only two people working on this project. So, if anyone wishes to join our efforts there are many intersting questions to work on.

A shout-out to my sponsors

I would like to give a very big thank you to my sponsors, who have made this work possible. I am financed entirely by private sponsors, without their support I would not be able to continue my work with Johannes Aastrup. I am deeply grateful for their generous support and for their faith in my work.

In particular, I would like to thank the Danish entrepreneur Kasper Bloch Gevaldig as well as the US entrepreneur Jeff Cordova for their generous support.

If anyone would like to support my work you can do so via PayPal or Ko-fi.

Have a nice spring (and try to stay sane)

With this I wish you all a very happy spring (if you live in the northern hemisphere), I hope that you all manage to stay sane in these strange times of ours. Take good care everyone.

Best wishes, Jesper

1. Riemannian (and pseudo-Riemannian) geometry is the mathematical machinery that Einstein used to formulate the general theory of relativity.

2. There are some important caveats here, such as the difficulty of treating a pseudo-Riemannian geometry. For details check out for instance Connes’ original paper.

3. A commutative algebra is an algebra where the elements commute. That is, it does not matter in which order they are multiplied. A noncommutative algebra, on the other hand, is an algebra where the elements do not commute. A simple example of a noncommutative algebra is a matrix algebra, for instance one generated by rotations in three-dimensional space.

4. In Connes’ original work the matrix factor in the almost-commutative algebra is basically determined by the gauge structure of the standard model. In more recent works it has been shown that under certain additional assumptions the matrix factor is singled out by the axioms of noncommutative geometry as being unique.

5. Note how complicated this question really is: because the new formulation is fundamentally gravitational — the spectral triple is a geometrical entity — it suggests that if quantum theory is to play a key role in it then it should somehow involve a quantisation of gravity, i.e. a theory of quantum gravity. But quantising gravity is of course something that we do not know how to do, hence the complication.

6. The QHD-algebra is generated by 1) holonomy-diffeomorphisms, which are parallel transports along flows of vector-fields on a manifold (i.e. moving stuff around), and 2) by translations on the associated configuration space of gauge connections. The point is that the HD-algebra can be understood as an algebra of operator-valued functions on a configuration space, and as such it is natural to consider also translation on that configuration space.

7. Yang-Mills theories play a key role in the standard model, where they describe the forces between elementary particles. The canonical commutation relation is the foundational equation in a quantum theory.

8. To be precise: it is a four-dimensional Dirac operator that is projected onto a three-dimensional space according to some space-time foliation.

9. The Hamilton and Lagrange formalisms are two equivalent yet very different ways to approach both classical and quantum physics. In our case we are developing a Hamiltonian formulation. We have not yet explored the transition to a Lagrangian formulation.

10. The key question here is whether the metric on the configuration space can be constructed in a way that preserves the Lorentz symmetry and if so, what would necessitate that. The point is that we cannot allow ourselves to impose structures like the Lorentz symmetry on our construction and still insist that it is a candidate for a final theory. If a theory is truly fundamental or final, then it must be possible to derive known physics from it — and the Lorentz symmetry is part of known physics — instead of imposing it on it.