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
- Brief tour of quantum holonomy theory
- Connection to loop quantum gravity
- Open questions
The basic principle
Quantum holonomy theory arises from an intersection of two different and hitherto largely unrelated research fields in theoretical physics and modern mathematics, namely canonical quantum gravity and non-commutative geometry.
By applying the mathematics of non-commutative geometry to the question of how to reconcile Einsteins theory of relativity with quantum mechanics 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 conceptionally, and philosophically.
- The QHD algebra naturally involves what is called the canonical commutation relations of quantum gravity. 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 of quantum gravity, namely the kinematical sector. It is surprising that an algebra with this 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 emploing the machinery of non-commutative geometry to a certain type of non-commutative algebras known as almost-commutative algebras.
Incidentially, 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 possible 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 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 theory of unified quantum gravity. Whether it really is such a theory is the subject of our present research.
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.
In addition to these two fundamentals we add two additional requirements:
- A semi-classical limit.
- A check of internal, mathematical consistency.
1. A Hilbert space representation
A Hilbert space is the mathematical stage on which the algebra acts in order to produce the numbers, which may eventually be measured in experiments.
The Hilbert space, that 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 at a very short scale.
One consequence of this non-locality is that very extreme field configurations — such as singularities — are assigned zero probability. This implies that the initial big bang singularity and the space-time singularities purported to exist at the centre of black holes cannot exist 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. We believe that such an operator will give rise to a time-evolution that is physically relevant — a conjecture that is backed by our recent study.
3. A semi-classical limit
Another key feature of the Hilbert space representation 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 non-commutative 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 Connes and Chamseddine 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 gravitational setup. It is the mathematical structure of the holonomy-diffeomorphisms – i.e. moving stuff in space – that gives rise the 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 must be determined by further analysis.
[To be written]
Connection to loop quantum gravity
Another approach to quantum gravity is the research program of loop quantum gravity (LQG). Since LQG is in some ways similar to quantum holonomy theory I shall here briefly compare the two.
Basically LQG and QHT share the same classical setup. Both theories are based on Ashtekar variables and holonomies of the Ashtekar connection. The main differences between them lie in the quantization. Let me list some of the main points:
Scope: the aim of LQG is primarily to quantize general relativity as a constrained system. Thus to derive canonical matter couplings or to unify the fundamental forces is not one of its primary goals.
In contrast, QHT is a top-down approach that precisely aims at all the above: to derive canonical matter couplings and a unification of forces from a single principle, namely the QHD algebra.
The algebra: The key technical tool in LQG is a choice of an algebra generated by parallel transports restricted to a projective system of piece-wise analytic graphs. This means that the holonomies in LQG all live on graphs and act pointwise.
In contrast, the QHD algebra involves holonomy-diffeomorphisms, where the parallel transports act along flows of vector fields and not pointwise.
This means that LQG and QHT treat diffeomorphisms in completely different ways. Where LQG treat diffeomorphisms in terms of constraints QHD has them built into the algebra. This issue is critical since the choice of algebra determines what theory you are dealing with. All differences between LQG and QHT can be traced back to the choice of algebra.
The Hilbert space: the kinematical Hilbert space in LQG is constructed as the inductive limit of intermediate Hilbert spaces associated to finite graphs.
The Hilbert space representation in QHT is very different from the one used in LQG: it is separable and strongly continuous. As already pointed out, this means that we have access to infinitesimal operators. Another significant difference between the two theories is that the Hilbert space in QHT is a priori background dependent, an issue that we discussed in a recent paper, where we suggest that our construction should be understood in terms of a semi-classical phase.
The constraint algebra: the computation of the constraint algebra is the technical term for the check of internal consistency of the theory, i.e. whether general covariance is preserved in the quantum theory.
In LQG these computations are in general not meaningful since the theory does not permit infinitesimal diffeomorphisms. Since LQG predicts a discretised space-time this makes sense and can be traced back to the construction of the kinematical Hilbert space.
In contrast, due to the different construction of the Hilbert space QHT does permit infinitesimal diffeomorphisms and therefore the computation of the constraint algebra is readily accessible and has been commenced.
The semi-classical limit: In LQG the construction of a semi-classical state is a serious challenge, which again can be traced back to the construction of the kinematical Hilbert space. In stark contrast, the semi-classical limit is accessible in QHT and thus does not appear to pose any problem whatsoever.
The nature of space-time: LQG predicts a granular structure of space, where fundamental geometrical quantities such as area and volume are quantized.
The picture that emerges from QHT appears to be very different — more analysis is needed to determine precisely how, but the granular structure seen in LQG is absent . The Hilbert space representation comes with a parameter of dimension ‘length’, which reflects a dampening of ultra-violet degrees of freedom.
The QHD algebra was discovered 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)