The main lesson taught by Einstein’s theory of gravity is that the gravitational field has a geometrical nature, while that of quantum theory is that fields are quantized and come with fundamental excitations. This suggest a realization of quantum gravity in terms of quantum geometry.

In loop quantum gravity, this idea is realized explicitly, and observables encoding the intrinsic and extrinsic data of a spatial geometry are represented as quantum operators on a Hilbert space. This allows to discuss rigorously the quantum properties of geometrical operators measuring the area or the volume of a region of space, and their spectra turn out to be discrete.

This construction, which is due to Ashtekar, Isham and Lewandowski, was the first to achieve a mathematically rigorous representation of the field variables of general relativity. Moreover, it allows for a description of the action of the group of spatial diffeomorphisms, and thus for the isolation of diffeomorphism-invariant quantum states. These achievements are central to the theory of loop quantum gravity and key to many of its properties.

But is this representation of quantum geometry, constructed by Ashtekar, Isham and Lewandowski, the only possibly one? It has been argued that this is the case. We do however propose an alternative representation in our CQG paper, which circumvents a well established uniqueness theorem for the Ashtekar – Lewandowski representation.

In quantum field theory, it is known that there can typically be many unitarily inequivalent representations of a given algebra of observables. One way to understand this is through the fact that the vacua underlying the different representations have different properties. A Hilbert space, together with all of its (excited) states, can be constructed from the vacuum by (repeatedly) applying observables to this latter. However, one might be interested in (vacuum) states that are so highly excited with respect to the first vacuum that they are not anymore elements of the first Hilbert space. For example, vacua leading to unitarily inequivalent representations may have vanishing or non-vanishing expectation values of a scalar field operator (at all spatial points).

We therefore see that the choice of a Hilbert space representation is tightly related to a choice of preferred and most easily described state. For theories in which a notion of energy is available, the vacuum is chosen as the state of minimal energy, and states with finite energy can then be obtained from this vacuum by applying observables.

In general relativity, there is no preferred notion of energy (if one considers compact manifolds), and a longstanding question has been how to characterize the physical vacuum. Physical states should satisfy all the quantum equations of motions (which are constraints), and as such encode the full dynamics of the system.

Our goal is to eventually construct such physical states starting from a kinematical Hilbert space. However, in light of the above discussion on unitarily inequivalent representation, the following question arises: What is a good starting point for the construction of physical states. In other words, what is a good kinematical vacuum state?

The Ashtekar – Lewandowski vacuum seems to constitute a good starting point due to its independence from any classical background. It is maximally peaked on the configuration describing a totally degenerate spatial geometry, and also maximally spread in the canonically conjugated variable encoding the extrinsic geometry.

However, this property makes the construction of semi-classical states, which would describe non-degenerate geometries, extremely intricate. Another difficulty with the Ashtekar – Lewandowski vacuum is that (the semiclassical interpretation of) spin foam geometries, which are used to construct the path integral representation of loop quantum gravity, are rather based on states of opposite nature. These states are almost everywhere flat, except at defects carrying curvature.

Attempts at constructing alternative representations however have to take into account a celebrated uniqueness result by Lewandowski, Okolov, Sahlmann and Thiemann. This theorem states that the Ashtekar – Lewandowski representation is the unique representation of the kinematical (holonomy – flux) algebra of observable of loop quantum gravity. This result relies on a certain number of technical assumptions, two of which being as follows. The first is that spatial diffeomorphisms act as automorphisms and leave the vacuum invariant, the second is that the flux operators, encoding the spatial geometry, are well defined.

Koslowski and Sahlmann have introduced representations that violate the first assumption. In these representations the vacuum is `shifted’ to be peaked on a certain (spatial) background geometry. The peakedness property of the states does however not change. This leaves the questions whether there are other representations with a diffeomorphism invariant vacuum.

Given the fact that the Ashtekar – Lewandowski vacuum is a totally squeezed state, it is natural to search for squeezed states with the opposite peakedness property. Such a state, peaked on flat connections and maximally spread in spatial geometry, is what we have constructed in our CQG paper, together with a representation of the quantum geometry observable algebra, based on this (vacuum) state. This vacuum state has the nice property of being invariant under spatial diffeomorphisms. In fact, it corresponds to a physical state of topological BF theory. The new representation we construct in our paper can be understood as being dual to the Ashtekar – Lewandowski representation. Therefore, while the Ashtekar – Lewandowski representation has holonomies of the gauge connection as its configuration variables, the new representation is based on the flux variables. However, these fluxes also need to be “compactified”, and one has to consider exponentiated fluxes. This is the reason for which the uniqueness theorem is bypassed.

The key question is now whether this new vacuum and the associated representation constitute a better starting point for the construction of physical states for general relativity. There are a number of indications showing that this might be the case. In particular, (2+1)-dimensional gravity can be described as a BF theory, and therefore the new vacuum itself is a physical state. For (3+1)-dimensional gravity, the spin foam formalism is underlying the implementation of the dynamics, and spin foams are again constructed out of BF theory. Moreover, the geometric interpretation of spin foams supports the idea of having flat building blocks whose gluing leads to curvature defects. This precisely coincides with the interpretation of the states in our new representation.

Future work will provide more detailed studies of the new representation and of variants which can be derived from it. This will enable us to see which properties are independent of the representation being used, and which ones are representation-dependent.

A key crucial question remains open: Are there more representations with a diffeomorphism – invariant vacuum? The answer to this might be connected to a classification of topological (lattice gauge) field theories. Indeed, as explained in a paper by Dittrich and Steinhaus (NJP, ArXiv), the physical states of a topological field theory can serve as vacuum states. Defects violating the constraints of the topological field theory are excitations generated by the underlying kinematical observable algebra. This picture has been realized in our paper with the new vacuum based on BF theory. Other topological field theories, which for instance arise as phases in spin foams (PRD, ArXiv), might lead to other vacua.

This opens the possibility of exploring new types of quantum geometries, and could lead to many important developments in quantum geometry and quantum gravity. One or several of these realizations of quantum geometry might turn out to be preferred for the construction of a physical Hilbert space. It could also happen that all representations are useful in describing different sectors of the theory.

*Read the full article in Classical and Quantum Gravity:
A new vacuum for loop quantum gravity
Bianca Dittrich and Marc Geiller 2015 Class. Quantum Grav. 32 112001*

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