## Final Report Summary - EQU (Exploring the Quantum Universe)

The research theme of this project was "quantum geometry", i.e. trying to understand how geometry can be quantised. The reason this is important is that presently we describe the space and time aspects of the universe in terms of geometry.

According the Einsteins classical theory of relativity, gravity IS a theory of geometry. Thus quantising geometry is also a quantisation of gravity. A classical particle follows a well defined (classical) path. A quantum particle does not have a well defined path, but the probability amplitude for moving from one point to another is obtained by summing over all paths between the points with certain weights which are functions of the so-called action. The same will be true in a generalised sense for the quantum universe: we obtain the quantum universe by summing over all spacetime histories of the classical universe. If one considers a model where the dimension of spacetime is only two one can perform this sum analytically. If the dimension of spacetime is four (our real world) one can perform the sum numerically, using large scale computer simulations.

The solution of the two-dimensional models suggest the following picture: at the shortest distances one has to give up the concept smooth geometries. To the extent one can talk about geometries these will be fractal at all small scales but in a universal way. Further, the fractal properties will depend on the matter content of the universe: the more matter, the more fractal the universe will be at the shortest distances. There can even be so much matter that the macroscopic universe is torn apart.

When we study the sum over four-dimensional geometries we have to rely on computer simulations. This means that we have to discretise our system, and to obtain "continuum" results we have to remove the discretisation (the cut-off) again.

Effectively this implies that we have to increase our discretised system to infinite size and study what in statistical physics is called second order phase transitions of the system. The outcome of these computer simulations suggests the following picture: our present universe exists just at the wedge between an isotropic, homogenous universe and a universe dominated by "condensed" spatial geometries, a kind of "black holes" with very limited extension. In the isotropic and homogenous phase there is a constant creation and annihilation at a microscopic scale of such (virtual) "black holes". However, the breaking of homogeneity and isotropy is precisely related to these virtual microscopic structures growing to manifest themselves as real, macroscopic geometric configurations.

These pictures of two- and four-dimensional universes inspired us to develop a mathematical model of the universewhere the "natural" state is pre-geometric with no existence of time and space. Time and space only emergence when a certain so-called W3-symmetry is spontaneously broken. In this symmetry breaking, time will appear together with a so-called Hamiltonian which allows the creating and evolution of space. It is thus a pre-Big Bang theory which allows in a natural way for the Big Bang, and for certain choices of the symmetry breaking parameters one does not need the concept of dark energy to explain the present expansion of the universe.

According the Einsteins classical theory of relativity, gravity IS a theory of geometry. Thus quantising geometry is also a quantisation of gravity. A classical particle follows a well defined (classical) path. A quantum particle does not have a well defined path, but the probability amplitude for moving from one point to another is obtained by summing over all paths between the points with certain weights which are functions of the so-called action. The same will be true in a generalised sense for the quantum universe: we obtain the quantum universe by summing over all spacetime histories of the classical universe. If one considers a model where the dimension of spacetime is only two one can perform this sum analytically. If the dimension of spacetime is four (our real world) one can perform the sum numerically, using large scale computer simulations.

The solution of the two-dimensional models suggest the following picture: at the shortest distances one has to give up the concept smooth geometries. To the extent one can talk about geometries these will be fractal at all small scales but in a universal way. Further, the fractal properties will depend on the matter content of the universe: the more matter, the more fractal the universe will be at the shortest distances. There can even be so much matter that the macroscopic universe is torn apart.

When we study the sum over four-dimensional geometries we have to rely on computer simulations. This means that we have to discretise our system, and to obtain "continuum" results we have to remove the discretisation (the cut-off) again.

Effectively this implies that we have to increase our discretised system to infinite size and study what in statistical physics is called second order phase transitions of the system. The outcome of these computer simulations suggests the following picture: our present universe exists just at the wedge between an isotropic, homogenous universe and a universe dominated by "condensed" spatial geometries, a kind of "black holes" with very limited extension. In the isotropic and homogenous phase there is a constant creation and annihilation at a microscopic scale of such (virtual) "black holes". However, the breaking of homogeneity and isotropy is precisely related to these virtual microscopic structures growing to manifest themselves as real, macroscopic geometric configurations.

These pictures of two- and four-dimensional universes inspired us to develop a mathematical model of the universewhere the "natural" state is pre-geometric with no existence of time and space. Time and space only emergence when a certain so-called W3-symmetry is spontaneously broken. In this symmetry breaking, time will appear together with a so-called Hamiltonian which allows the creating and evolution of space. It is thus a pre-Big Bang theory which allows in a natural way for the Big Bang, and for certain choices of the symmetry breaking parameters one does not need the concept of dark energy to explain the present expansion of the universe.