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. Whenever I write ‘we’ I am referring also to Johannes Aastrup.
The basic principle
Quantum holonomy theory arises from an intersection of two different research fields in theoretical physics and modern mathematics, namely quantum field theory and non-commutative geometry.
By applying the mathematics of non-commutative geometry to a general framework related to quantum gravity formulated in terms of Ashtekar variables and to the question of finding a theory of quantum gravity we discovered after ten years of analysis a surprisingly simply algebra, which now provides the foundation of our work.
This 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 an abstract 3-dimensional space (holonomy-diffeomorphisms) and the second type generates translations on the spectrum of the algebra generated by the first type of elements.
Together these two types of elements generate a generalized form of quantum mechanics of diffeomorphisms.
Why is this interesting?
There are several reasons, mathematically conceptually, and philosophically.
- 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.
Specifically, Connes et. al. have demonstrated that the entire standard model of particle physics coupled to general relativity can — 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 such a type of algebra in a semi-classical limit. 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.
- The QHD algebra is almost completely canonical with the dimension of space being essentially the only free parameter. 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.
- 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.
A brief tour of quantum holonomy theory
Once the QHD algebra has been chosen as the foundation of a theory there are two ingredients, which must be added to obtain a quantum theory. These are
- A Hilbert space representation.
- A dynamical principle, i.e. a Hamiltonian.
1. A Hilbert space representation
A Hilbert space is a mathematical stage on which the algebra acts in order to produce the numbers, which may eventually be measured in experiments.
The Hilbert space, which we find, has a number of interesting features. First of all, it is non-local, which means that the probability for a quantum transition between two field configurations depends on the scale, where these field configurations differ. If they differ mostly at a large scale, then the probability will be relatively large compared to the case, where they differ on a very short scale.
One consequence of this non-locality is that very extreme field configurations — such as singularities — are assigned zero probability. This appears to imply that the initial big bang singularity and the space-time singularities purported to exist at the center of black holes are ruled out within our framework.
2. A dynamical principle
The dynamical principle is what makes the theory move — it introduces time evolution.
There are several approaches to obtain a Hamilton operator. One is to use the toolbox of non-commutative geometry. Normally one thinks of geometry as an assignment of distances and curvature to a finite-dimensional space — i.e. the geometry of a donut or an elephant. But in this case, we use non-commutative geometry to construct a geometry over a space, where each point itself is a 3-dimensional geometry (this is called a configuration space).
Specifically, what we propose is to construct a Dirac type operator on this infinite-dimensional configuration space. This operator has a number of surprising characteristics:
- Its square gives rise to the Hamilton operators of a non-perturbative Yang-Mills quantum field theory coupled to a fermionic sector on a curved background. Thus, the theory involves the time-evolution of a Yang-Mills-Dirac theory.
- The construction of the Dirac type operator requires an infinite-dimensional Clifford algebra, which in turn gives us the fermionic content of the theory. What this means is that in our setup the fermions play an intrinsically geometrical role.
Note here that all this involves a background, that remains classical. That is, we propose that Einstein’s theory of general relativity should not be quantized.
A semi-classical limit
A key feature of the Hilbert space representation, which we work with, is that it is strongly continuous. What this means is that we can construct infinitesimal operators — not only operators, that for instance give us finite transitions between different field configurations, but also the limiting case where these transitions are infinitesimal.
This implies that the construction carries information about the underlying space — our 3-dimensional space — and in particular it means that a semi-classical limit is relatively accessible within this framework, a statement that is supported by recent findings.
Furthermore, we find that the algebra generated by the holonomy-diffeomorphisms produces in the semi-classical limit what is known as an almost commutative algebra.
This is interesting. The Fields medalist Alain Connes, who has pioneered the field of noncommutative geometry, together with the physicist Ali H. Chamseddine have shown that the entire standard model of particle physics coupled to gravity can — at a classical level — be formulated as a single gravitational theory over a certain almost commutative algebra.
Now, the algebra, which we obtain in the semi-classical limit, is not the same as the one Chamseddine and Connes identify, but nevertheless, there is a first connection between quantum holonomy theory and the mathematics of the standard model of particle physics.
What is interesting here is the way matter degrees of freedom as well as Higgs and gauge fields (i.e. forces and mass) appear to arise from what is a priori a purely geometrical setup. It is the mathematical structure of the holonomy-diffeomorphisms — i.e. moving stuff in space — that gives rise the basic algebraic setup, which Connes and Chamseddine have identified as the source of the structural richness of the physical reality. Whether this link between quantum holonomy theory and the standard model of particle physics can be substantiated is not known at the moment, it must be determined by further analysis.
Normally quantum theories involving fields are constructed perturbatively. The name of that game is called perturbative quantum field theory and within that game, it is crucial to check that the theory, which one constructs, is internally consistent. The point here is that perturbative quantum field theories are constructed as perturbations around free theories and for such constructions, one must make sure that those perturbations do not break critical symmetries.
The theory, which Johannes and myself have constructed is of a different kind. It is not a perturbative theory, which means that we do not face the same dangers as one does within the ordinary framework. This does, however, not imply that key symmetries cannot be broken within our framework — in fact, the Lorentz symmetry will be broken, or modified, at the Planck scale — but this does not jeopardize the consistency of our theory. In a non-perturbative framework as ours, the key task is to prove that the theory exists in a strict sense and once this is achieved (and we have proven this in a special case) we know that it is mathematically consistent.
We found the QHD algebra in 2013 and thus quantum holonomy theory is still in its very infancy. This means that although we have a fairly good handle on its basic ingredients and 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 an almost-commutative algebra in a semi-classical limit. It remains to show that this algebra is related to Connes’ work on the standard model. This, however, is a considerable task since Connes’ work is set in a Lagrangian framework whereas quantum holonomy theory operates with a Hamilton framework.
- Tomita-Takesaki theory: the theory of Tomita and Takesaki states that under certain circumstances a state on a von Neumann algebra gives rise to a canonical time flow in the form of a one-parameter group of automorphisms. We suspect that the application of Tomita-Takesaki theory to the setup of quantum holonomy theory will provide a natural dynamical principle.
- (to be continued)