The quest for a quantum gravity theory began more than fifty years ago; but so far, despite the hard efforts of some of the greatest minds of the twentieth century, a quantum theory of time and gravity remains a tantalizing and elu- sive problem. Numerous lines of research have been proposed, some of them involving the work of a considerable number of scientists within the physicists and mathematicians community; however, none of these dominant approaches have provided conclusive answers or any new fundamental insights about our classical theories. The situation results even more dramatic if we consider that there is not experimental evidence that can guide our search or that can be used to confront the new proposals. On the other hand, our mathematical machin- ery has become so rich that almost any theoretical speculation can find a sound mathematical framework to be formulated. Thus, today we find ourselves in a very particular situation, using the words of John Baez :

“The paucity of experimental evidence concerning quantum gravity has allowed research to proceed in a rather unconstrained manner, leading to divergent schools of opinion. If one asks a string theorist about quantum gravity, one will get utterly different answers than if one asks someone working on loop quantum gravity or some other approach. To make matters worse, experts often fail to emphasize the difference between experimental results, theories supported by experiment, speculative theories that have gained a certain plausibility after years of study, and the latest fads”.

Whatever the solution of the quantum gravity problem will be, this will probably revolutionize our conception of physical reality in a far more profound way than general relativity and quantum mechanics did. Where to find our answers, considering that we have two very successful theories that have been confirmed to a high degree of accuracy, is then an important question to answer.

When Einstein conceived special and general relativity his ideas were not based primarily on experimental considerations, they originated from his in- terest to understand reality, from his interest to find simple explanations; a characteristic that conditioned all his career and probably the main reason because he was always reticent to accept quantum theory in the instrumentalist form which was dominant along the first half of the twentieth century. On the other hand, quantum theory was conceived from a mix of experimental evidence, classical physical intuition and deep mathematical insights. The relation of the theory with reality and its explanatory character remain still open, almost a taboo.Unfortunately, the position assumed by the pioneers of quantum the- ory closed the debate about interpretational issues for a long period; and the concept of reality, which was fundamental in the conception of relativistic the- ories, subsided to a “combination of instrumentalism and studied ambiguity". Along these years the quantum gravity program has been dominated by this instrumentalist interpretation; but our inability to conciliate this form of the theory with general relativity is revealing that there is something deeper about the structure of quantum reality that the instrumentalist form of quantum mechanics has not been able to capture.

In 1952 in a lecture in Dublin Schrödinger suggested the idea that his equation seemed to be describing different histories that do not represent alternatives, but that are actually happening simultaneously. This “lunatic” idea, as Schrödinger himself defined it, was probably the first attempt to try to understand the relation of quantum formalism with reality outside the imposed ambiguity of the instrumentalist dominance. Schrödinger did not pursue this idea further, but five years later H. Everett arrived to the same conclusion independently.

During the eighties and the nineties D. Deutsch settled the foundations of quantum computation in closed relation with a deeper development of Everett ideas. The understanding of reality developed by Deutsch through Everett’s many-worlds and quantum computation can just be compared with the deeper insights that characterized Einstein’s original work. However, even if quantum computation is probably the most remarkable result in theoretical physics in the last thirty years, Deutsch’s ideas about the Everett interpretation have been almost completely ignored.

Unfortunately, despite Deutsch has obtained important results pursuing Everett ideas, so far the only mathematical insight we have of the structure of this quantum multiverse is still based just on the clas- sical quantum formalism. For this reason, some of the proposed experimental tests of Everett interpretation, as for example quantum computing parallelism, are still open to numerous alternative interpretations and even if experimentally verified will not help to settle this interpretation. Only the construction of a sound mathematical formulation able to settle the main controversial features of Deutsch-Everett ideas and able to provide incontrovertible experimental predictions, will close the debate in favour or against this interpretation.

This work originated from the search of alternative mathematical methods to tackle some of the main open problems in general relativity. The inadequacy of classical formalism to give a satisfactory formulation of some of these problems revealed not just the limitations of this formalism, but also the limi- tations of the conception of reality derived from these tools. This brought us to Deutsch-Everett ideas and to the conception of quantum geometry derived from this perspective. Based on this approach we have found a mathematical framework where these ideas seem to find a natural formulation and that provides the tools to find a natural extension of classical relativistic formalism. This framework opens the possibility to settle a definite interpretation of quantum theory revealing also interesting new pathways to affront the quantum gravity program. Our research program  is  then directed to obtain a sound mathematical formulation of the quantum multiverse contained in Deutsch-Everett ideas and to study the role of such model in the search of a quantum gravity theory.