QHT – outline

In the following I will present the main ideas and results involved in the theory while avoiding too much technicality. For detailed information please check out these papers or contact either me or my colleague Johannes Aastrup.

Contents:

The basic principle

Quantum holonomy theory1 arises from an intersection of two different research fields in theoretical physics and modern mathematics, namely quantum field theory and non-commutative geometry.

This reseach project was initiated around 2004, when Johannes Aastrup and myself set out to apply the mathematics of non-commutative geometry to a general framework related to quantum gravity formulated in terms of Ashtekar variables. Initially we were interested in the question of finding a theory of quantum gravity, and after ten years of analysis we discovered a surprisingly simply algebra, which now provides the foundation of our work.

The algebra, which we call the quantum holonomy-diffeomorphism (QHD) algebra, contains two types of basic elements: the first type simply encodes how objects are moved around in a 3-dimensional space (we call them holonomy-diffeomorphisms) and the second type generate translations on the spectrum of the algebra generated by the first type of elements.

Together these two types of elements give rise to a generalised form of quantum mechanics of diffeomorphisms.

Why is this interesting?

There are several reasons, mathematically conceptually, and philosophically.

Mathematically:

  • The QHD algebra naturally involves what is called the canonical commutation relations of a gauge theory. Specifically, the canonical commutation relations emerge as the infinitesimal limit of the central algebraic relation in the QHD algebra. This implies that the QHD algebra automatically involves the key ingredient of a theory based on gauge fields, namely the kinematical sector. This can either be a Yang-Mills theory – the type of theory, that describes the strong and electro-weak interactions in the standard model of particle physics – or gravity formulated in terms of Ashtekar variables. It is surprising that an algebra with these basic characteristics has not previously been studied.
  • The QHD algebra is what is called non-commutative. This implies that it falls within the framework of non-commutative geometry, a branch of modern mathematics, which has proven exceptionally interesting as a framework of unification of the four fundamental forces.

Concerning the last point, Connes et. al. have demonstrated that the entire standard model of particle physics coupled to general relativity can – primarily at a classical level – be formulated as a purely gravitational theory, by employing the machinery of non-commutative geometry to a certain type of non-commutative algebras known as almost-commutative algebras.

Incidentally, the QHD algebra produces an algebra in a semi-classical limit, which resembles such an almost-commutative algebra. This means that the QHD algebra involves a mechanism, which – albeit not yet directly related to the standard model – could potentially entail a unification of the fundamental forces of Nature.

Conceptually:

  • The QHD algebra is almost completely canonical with the dimension of space being essentially the only free parameter2. This means that once the spatial dimension (dim = 3) is chosen everything else is essentially fixed. This is crucial because a mathematical principle with the prefix ‘fundamental’ should not come with a large number of free parameters.

Philosophically:

  • How can one possibly imagine a principle, that cannot be reduced to other deeper principles but at the same time rich enough to explain everything? That seems like a very tall order. The QHD algebra has, however, the characteristics of precisely such a principle. This adds a philosophical layer to the theory – we are aware of very few other candidates to such an ultimate principle.

What does all this mean? It means that quantum holonomy theory from the get-go has the key characteristics of a fundamental theory. Whether it really is such a theory is the subject of our present research.

The basics of quantum holonomy theory

Once we have decided to focus our attention on the QHD algebra the question arises how a candidate for a fundamental theory can be build around it. To answer this question we need to introduce the concept of a configuration space.

The point is that the QHD algebra can be understood as an algebra of functions over what is called a configuration space of gauge fields. A configuration space is a space in which each point is a particular configuration of a certain field, and a gauge field is a field that encodes information about how objects are moved around in space.

This means that once we have chosen to work with the QHD algebra we are immediately lead to consider the underlying configuration space.

Note that this configuration space in itself is not a novel object in theoretical high-energy physics. Configuration spaces of this type play a key role in quantum field theory, for instance, which can be understood as quantum mechanics on the configuration space. What is novel is what comes next.

The geometry of the configuration space

The key new idea in our research project is to apply the machinery of noncommutative geometry to a setup based on the QHD algebra. What this means, essentially, is that we will consider the geometry of the configuration space of gauge fields.

What does that mean? It means that we are asking whether it makes mathematical sense to assign a distance to the space between two different points in the configuration space, i.e. two different gauge fields? A gauge field is basically a recipe for moving an object between two arbitrary points in the physical 3-dimensional space (turn it twice to the right, twist it forwards three times, or turn it 23166 times to the left?), and thus what we are asking is whether it makes mathematical sense to have a distance between different recipes? Today we know that the answer is yes.

Now, within the framework of noncommutative geometry the geometry of a space is encoded in a so-called Dirac operator, which is a special kind of gradient operator. We know Dirac operators from the Dirac equation, which Paul A.M. Dirac discovered in 1928. The Dirac equation is a relativistic wave equation, which plays a key role in fermionic quantum field theory. But from a mathematical perspective the Dirac operator is a much more general object, which encodes geometrical information about the space on which it is formulated.

This is the reason why we construct a Dirac operator on the configuration space we are working with. During the past few years we have been occupied with the question whether this can be done in a mathematically meaningful way, and recently we have published results that confirms that this can indeed be done.

One of the key features of the geometry, which we construct on the configuration space, is that it is dynamical. That is, it has a time-evolution. This is similar to Einsteins theory of general relativity, where the geometry of the three-dimensional space is also dynamical; the geometry of space changes over time.

What we propose is in fact a natural generalisation to Einsteins two theories of relativity. First there is special relativity, which is only concerned with flat space-time, and secondly, there is general relativity, which also involves gravity. We are now proposing a third theory of relativity, which is an Einsteinian theory of relativity of the configuration space that emerges from the QHD algebra. Due to this analogy we sometimes call our research project for ‘configurational relativity‘.

The emergence of quantum field theory

One of the most interesting features of the Dirac operator, which we construct, is that it naturally gives rise to the basic building blocks of bosonic and fermionic3 quantum field theory on a curved background:

  1. the canonical commutation relations, which is the foundation of bosonic quantum field theory, automatically emerge from the interaction between the Dirac operator and the HD algebra4.
  2. the canonical anti-commutation relations, which is the foundation of fermionic quantum field theory, automatically emerge from the Clifford algebra, which is used to construct the Dirac operator.
  3. the Yang-Mills Hamilton operator, which encodes the dynamics of Yang-Mills gauge theory, emerges from a Bott-Dirac operator, which is a natural extension of the Dirac operator.
  4. the Hamilton operator of a fermionic quantised field also emerges from the square of the Bott-Dirac operator.
  5. In addition to this the bosonic and fermionic Fock spaces naturally emerge when we construct Hilbert space representations of the Dirac operator and the QHD algebra.

This means that in this framework quantum field theory emerges from a purely geometrical construction related to the QHD-algebra. Note that these elements of quantum field theory all emerge in a certain local limit. The geometrical framework itself is, as I shall discuss shortly, inherently non-local.

Note here that all this implies that we have found a possible answer to the question concerning the origin of quantum theory. Within the framework discussed here quantum theory is an output, it emerges from a basic geometrical construction over a configuration space, which in turn can be arrived at from the basic mathematics of moving stuff around in empty space.

The emergence of key elements of general relativity

A priori quantum holonomy theory does not involve a geometry of the underlying three-dimensional space and thus it does not, a priori, have anything to do with general relativity. It turns out, however, that in a certain limit where the geometry of the configuration space becomes flat, key elements of general relativity do emerge.

To be specific, in that particular limit we see the emergence of a geometry of the underlying three-dimensional space, and most importantly, we find that this geometry is dynamical. In other words, we find that the cornerstone in Einsteins theory of relativity emerges from our construction.

Several layers of unification

The emergence of the central building blocks of bosonic and fermionic quantum field theory, which I discussed in the above, represents a new type of unification between bosons and fermions. Also, the emergence of a dynamical three-dimensional geometry represents yet another element of unification, this time between elements of general relativity and quantum theory. In other words, we find that the geometrical framework over the configuration space involves several layers of unification, the two mentioned here as well as one more, which I shall discuss shortly.

We already know the concept of unification from high-energy physics, where for instance supersymmetry was supposed to unify bosons and fermions through a certain extension of space-time symmetries. There are also the various types of GUT’s, grand unified theories, where the idea is to unify the three forces in the standard model in a single force at a higher energy. And then there is string theory, where the fundamental particles including a graviton are supposed to emerge from tiny vibrating strings living in 10 dimensions.

Mechanisms of unification are important. If you want to go from a simple starting point to a relatively complex endpoint like for instance the standard model of particle physics, where there are several types of fields that obey various symmetries, then you need some mathematical mechanism that can generate all that complexity. That is what unification does for you. The aim is to identify natural mechanisms that can generate rich mathematical structures or, equivalently, explain how the complex structures we observe originate from something simpler, without adding more assumptions to our starting point.

In our framework we have identified (or outlined) three different types of unification:

  1. First, there is the unification between bosons and fermions, which we already discussed.  
  2. Secondly, there is the emergence of a dynamical geometry of the underlying three-dimensional space.
  3. Thirdly, there is an intrinsic type of unification that comes from the HD-algebra being noncommutative5. What this means is that there exists an additional mathematical structure related to the so-called inner automorphisms of the algebra. In the case of the standard model of particle physics it is known, that it is this mechanism, that gives rise to the entire bosonic sector including all the gauge fields and the Higgs field. In our case we do not yet know what the effect of this additional structure will be, only that it must be there.

It is important to note here that there is a significant difference between the different layers of unification, which turn up in our framework, and the ones I mentioned earlier and which are already known in high-energy physics. The unification found in supersymmetry, in GUT’s, and in string theory all take place after quantum field theory has been introduced. Thus, they are post-QFT unifications. In contrast to this the unification that takes place in our framework are all pre-QFT. That is, they take place at a point that is deeper and therefore more fundamental than quantum field theory. In our framework quantum field theory is something that emerges, quantum theory is here an output, not an input. And the different elements of unification, which we encounter, all take place before quantum field theory emerges.

How to unify general relativity with quantum theory?

This is a question that theoretical physicists have been struggling with for almost a century. How to reconcile Einsteins theory of relativity with quantum theory.

So far the approach has almost always been to apply the machinery of quantum theory to the framework of general relativity. That is, to construct a theory of quantum gravity. There exist, however, the other logical possibility, which is to apply the machinery of general relativity, i.e. pseudo-Riemannian geometry, to the framework of quantum theory. This is precisely what we are suggesting.

Within quantum theory — quantum mechanics, quantum field theory — the fundamental ‘space’ is the configuration space. In quantum mechanics of a single particle this is simply three-dimensional space, but for more complicated configurations (say, multiple particles) it is more involved. And in quantum field theory the configuration space is an infinite-dimensional space. What we are suggesting is to apply the machinery of pseudo-Riemannian geometry, and in particular the machinery of non-commutative geometry, to the configuration space related to a quantum field theory of gauge fields.

In other words, we have discovered a new approach to the old question of reconciling gravity with the quantum. And what we know today, is that this approach can be formulated rigorously.

Dynamical ultra-violet regularisation

One of the central technical ingredients in the geometrical construction over the configuration space is what we call a dynamical ultra-violet regularisation.

We know ultra-violet regularisations from perturbative quantum field theory, where they are used to make sense of quantities, which would otherwise be infinite. An ultra-violet regularisation removes the infinite parts related to the far-ultra-violet limit from various physical entities, which permits various computations to be performed.

Now, a central point here is that the ultra-violet regularisations one encounters in perturbative quantum field theory are not physical. They are always computational artifacts.

One reason why this must be the case is that there are infinitely many ways you can regularise the ultra-violet limit, and thus one is immediately faced with the problem of choosing between them, should one decide to interpret the ultra-violet regularisation as physical. And since there is no meaningful way to choose between the different types of regularisation such an interpretation is not possible.

This changes, however, if the ultra-violet regularisation is dynamical. If the regularisation changes in time then one will not have to pick one since there will be a progression between different ultra-violet regularisations.

This is similar to what happens in general relativity. The idea that space has a curvature would at first appear to be meaningless precisely because there are infinitely many ways in which it can have a curvature. And since there is no meaningful way to choose one particular curvature of space over another it seems impossible that space should have a non-trivial geometry. The reason why this argument is wrong is precisely that the curvature is dynamical. It changes over time. This means that one does not have to choose one particular geometry since there is a progression between them.

Now, it turns out that an ultra-violet regularisation, which is compatible with the gauge symmetry of the configuration space, must be dynamical. Thus, any ultra-violet regularisation, that is gauge-covariant, can be interpreted as being physical since one is not faced with the problem of choice.

The concept of a physical ultra-violet regularisation is a novel concept in theoretical high-energy physics. To the best of our knowledge it has never been studied before.

There are several points worth noting here:

  1. if an ultra-violet regularisation is interpreted as being physical it implies that the theory at hand is non-local.
  2. a physical ultra-violet regularisation puts into question whether gravity itself needs to be quantised. After all, a key purpose of quantising gravity is precisely to regularise the ultra-violet limit at the Planck length, but if we already have one ultra-violet regularisation there is no need for a second one coming from a theory of quantum gravity.
  3. a dynamical ultra-violet regularisation would work as an anti-gravitational forcing.

Points number 1. and 3. are related: the non-locality that the ultra-violet regularisation gives rise to, implies that arbitrary localisation, which for instance occurs within a black hole, is prohibited. This means that the ultra-violet regularisation pushes against gravity, i.e. it works as an anti-gravitational forcing.

Open questions

We found the QHD algebra in 2013 and thus quantum holonomy theory / configurational relativity is still in its very infancy. This means that although we have a fairly good handle on its basic ingredients, it’s mathematical formulation, as well as several key results, there are many open questions. Let me list some of the most important ones:

  • Connection to Connes’ work on the standard model: We know that the algebra of holonomy-diffeomorphisms produces something that looks like an almost-commutative algebra in a semi-classical limit. It remains, however, to show that this algebra is related to Connes’ work on the standard model.
  • The emergence of general relativity: We know that a dynamical geometry of the underlying three-dimensional space emerges in a certain limit. What we do not know, however, is whether this geometry has precisely the time-evolution that Einsteins theory of relativity prescribes. There are certain signs that looks promising, but it still remains an unknown.
  • The Lorentz symmetry: a key ingredient in Einsteins theory of relativity is the Lorentz symmetry, which is the (local) rotation symmetry in a geometry with a Minkowski signature. At the moment it is unclear what role the Lorentz symmetry plays in our construction. We believe that certain adjustments must be made in order to see what role this critically important symmetry plays.
  • There are of course numerous technical issues, which I cannot elaborate on here.
  • (to be continued)
  1. We also use the term ‘configurational relativity’ for our research project. ↩︎
  2. There is also a choice of gauge group. However, if we consider holonomy-diffeomorphisms related to translation in space, then the choice of SU(2) or SO(3) becomes a canonical choice. A priori it is, however, possible to consider any compact gauge group. ↩︎
  3. bosons represent forces while fermions are matter fields. ↩︎
  4. The HD algebra is the same as the QHD algebra, just without the translation operators on the configuration space. Thus, the HD algebra is generated by the holonomy-diffeomorphisms alone. In other words, it is the part of the QHD algebra, that encodes how stuff is moved around in space. ↩︎
  5. That an algebra is non-commutative means that it matters in which order you multiply its elements. Numbers form a commutative algebra since it does not matter in which order you multiply them, but the HD-algebra is non-commutative. ↩︎