The QHD algebra
The cornerstone in quantum holonomy theory is the algebra HD(M), which is generated by holonomy-diffeomorphisms on a 3-dimensional manifold M. A holonomy-diffeomorphism maps a connection into an operator acting on spinors on M
where is the space of all smooth connections in a certain bundle and where denotes a holonomy-diffeomorphisms along the flow of a vector-field .
Thus, an element in HD(M) will parallel transport a vector along the flow of a diffeomorphism via the holonomies as prescribed by the connection . Because the holonomy-diffeomorphisms in HD(M) are local this action can be localised but it is important to note that it is not pointwise.
In this way the HD(M) algebra simply encodes how tensor degrees of freedom – i.e. stuff – is moved around on a manifold.
The HD(M) algebra depends on the manifold M as well as a choice of gauge group corresponding to the bundle over M. We choose a 3-dimensional manifold and the corresponding rotation group SU(2). For simplicity we choose the trivial bundle.
Once we have the HD(M) algebra it is natural to consider translations on the space of connections
where is a one-form with values in the Lie-algebra of SU(2). Note that these translation operators are canonical. We immediately find the relation
This relation is in fact an integrated version of the canonical commutation relations of a Yang-Mills theory and of quantum gravity formulated in terms of Ashtekar variables1. To see this one must expand first in terms of infinitesimal translations on M and secondly in terms of infinitesimal translations on .
This means that the algebra of holonomy-diffeomorphisms leads us directly into the realm of quantum gauge theory. A theory build over this algebra will inevitably involve the kinematical sector of a gauge theory.
Finally, the Quantum Holonomy-Diffeomorphism algebra — in short QHD(M) — is the algebra generated by HD(M) and all the translation operators .
Note that the QHD(M) algebra comes with a very large degree of canonicity. The HD(M) sector depends, as already noted, on the dimension of the manifold and a choice of rotation group, and the translation operators are canonical. Thus, if a fundamental theory does indeed arise from this algebra it will come with a very large degree of canonicity.
1To be precise, with SU(2) this corresponds to general relativity with an Euclidian signature. The Lorentzian signature corresponds to a connection, which takes values in the self-dual section of the Lie-algebra of .
The spectrum of HD(M)
The spectrum of an algebra is defined as the irreducible representations modulo unitary equivalence. The spectrum is the underlying configuration space on which elements in the algebra act as functions. By knowing the spectrum we gain important insight into what kind of theory we are building.
We have two key results on the spectrum of HD(M).
The first key result is that the separable part of the spectrum of HD(M) is given by so-called measurable connections, which are essentially connections that map a non-trivial volume (a Lebesgue measurable set) into another non-trivial volume.
This result first of all reaffirms our interpretation of the HD(M) algebra as an algebra related to quantum gauge theory — either a Yang-Mills theory or quantum gravity formulated in terms of Ashtekar variables. Secondly this also brings us in contact with loop quantum gravity, which too has Ashtekar variables as its classical point of reference.
The second key result gives us information on how the HD(M) algebra differs from the corresponding algebra found in loop quantum gravity. This result states that the so-called generalised connections, which are connections with support on piece-wise analytic graphs, are not part of the spectrum of HD(M).
To understand what this means we first need to know that the bulk of the spectrum found in loop quantum gravity is precisely given by generalised connections with support on a projective system of piece-wise analytic graphs. The second result therefore tells us that the loop quantum gravity spectrum is excluded from the spectrum of the HD(M) algebra. This means that quantum holonomy theory and loop quantum gravity are distinctly different theories since their underlying configuration spaces are very different.
The difference is that the HD(M) algebra captures information on the local measurable structure of the manifold M whereas the algebra in loop quantum gravity does not.
Interestingly, the generalized connections appear in our construction if we discretise HD(M). This means that the loop quantum gravity spectrum in a certain sense arises from a discretized version of quantum holonomy theory.
A metric structure on
By using the mathematics of non-commutative geometry we build a Dirac-type operator over the configuration space of connections. Where the ordinary Dirac operator is build from space-time derivatives, this Dirac-type operator can be thought of as a sum over infinitely many infinitesimal translation operators on the space of connections.
Such an operator is a natural object to consider since it constitutes a canonical metric structure over the configuration space . In plain english this means that we are constructing a geometry over an infinite-dimensional space in which each point itself is a 3-dimensional geometry. As crazy as this might sound it is in fact well-defined.
So why is this Dirac-type operator important? Well, there are several reasons, but first of all it provides us with a natural candidate for a dynamical principle. When you construct a quantum theory you have two basic tasks: 1) construct the kinematics and 2) construct the dynamics in the form of a Hamilton operator. In our case the first task amounts to finding Hilbert space representations of the QHD(M) algebra. As for the second task we need a natural candidate for a Hamilton operator and here the square of the Dirac-type operator is interesting. We have shown that there exist a natural Bott-Dirac operator (a variant of a Dirac operator related to the harmonic oscillator), which in our case is related to the Hamilton of a Yang-Mills theory.
What is particularly intersting here is that this Bott-Dirac operator naturally gives rise to a fermionic quantum field theory for which its square also gives the correct Hamilton operator. This results provides us with a completely new interpretation of fermionic quantum field theory as an ingredient in a geometrical construction over a space of field configurations.
The semi-classical limit of HD(M)
Let us now consider the semi-classical limit of the HD(M) algebra, which we find is equal to the algebra
where is the group of diffeomorphisms on M. This is so because the holonomies on a fixed classical geometry generate a two-by-two matrix algebra. Thus we find the almost-commutative algebra as a subalgebra.
As already mentioned this is interesting because the mathematician Alain Connes and the physicist Ali Chamseddine have shown that the entire standard model of particle physics coupled to general relativity can – at a classical level – be formulated as a single gravitational theory via the machinery of non-commutative geometry. The key mathematical ingredient in this intriguing formulation is an almost-commutative algebra
where is a finite-dimensional matrix algebra related to the gauge structure of the standard model. Now, the almost-commutative algebra, which we obtain, is not identical to the one Connes and Chamseddine identify. There are, however, a number of reasons why we believe there may nevertheless be a connection between the two:
- First of all, the work of Connes and Chamseddine is formulated in a Lagrangian framework, whereas our approach is in a Hamiltonian setting. This means that these two pieces of mathematical theory are not straight-forwardly comparable.
- Secondly, the above result deals with the HD(M) algebra alone. There is, however, additional structure to be taken into account, such as for instance the CAR-algebra that comes with the Dirac-type operator.
- Thirdly, the construction, which we have worked with so far, has SU(2) as a gauge group. This corresponds to a real Ashtekar connection and thereby general relativity with a Euclidian signature. The Lorentzian signature arises from the self-dual sector of . It is our belief that this issue is related to the introduction of a so-called real-structure, which is a key mathematical ingredient in the formalism of non-commutative geometry. This may very well influence the structure of the almost-commutative geometry, which arises in the semi-classical limit.
These technicalities aside the emergence of an almost-commutative algebra opens up for a fascinating interpretation of the standard model in terms of a semi-classical limit of a purely quantum gravitational theory. If this interpretation is correct then the gauge and Higgs sectors will be generated in part by the non-commutativity of the holonomy-diffeomorphisms.