**Introduction**

The story starts at the end of the nineteenth century, when the first serious blows were dealt to the assumption that the world around us is inherently continuous. Physicists had found that electromagnetic radiation, such as light, behaved in a way that could not be explained by classical physics, which imagined it to take the shape of waves. In an attempt to find a new mathematical explanation, the physicist Max Planck took a radical step: he assumed that the energy of electromagnetic radiation did not vary continuously, as the wave picture suggested, but came in little packages called quanta. Albert Einstein later picked up on this idea, and by 1905 the notion that light travelled in continuous waves had been replaced by Einstein’s photons: little units of light that can behave both like isolated particles and like waves.[**Marianne Freiberger**-plus.maths.org/content/quantum-geometry]

The idea of describing the physical world entirely in terms of geometry has a history dating back to Einstein and Klein in the early decades of the century. This approach to physics had early success in general relativity but the appearance of quantum mechanics guided the development of theoretical physics in a different direction for a long time. During the past quarter of a century the programme of Einstein and Klein has seen a renaissance embodied in gauge theories and, more recently, superstring theory. During this time we have also witnessed the happy marriage of statistical mechanics and quantum field theory in the subject of Euclidean quantum field theory, a development which could hardly have taken place without Feynman’s path integral formulation of quantization. In this book we shall work almost exclusively in the Euclidean framework.

The unifying theme of the present work is the study of quantum field theories which have a natural representation as functional integrals or, if one prefers, statistical sums, over geometric objects: paths, surfaces and higher-dimensional manifolds. Our philosophy is to work directly with the geometry as far as possible and avoid **parametrization and discretizations** that break the natural invariances. In this introductory chapter we give an overview of the subject, put it in perspective and discuss its main ideas briefly.

Lagrangian field theories whose action can be expressed entirely in terms of geometrical quantities such as volume and curvature have a special beauty and simplicity.[Jan Ambjørn, University of Copenhagen, Bergfinnur Durhuus, University of Copenhagen, Thordur Jonsson, University of Iceland, Reykjavik Publisher: Cambridge University Press]

### Basic problems in quantum gravity

Formulating a theory of quantum gravity in dimensions higher than two leads to a number of basic questions, some of which go beyond those encountered in dimension two. Among these are the following:

(i) What are the implications of the unboundedness from below of the **Einstein–Hilbert action**?

(ii) Is the non-renormalizability of the gravitational coupling a genuine obstacle to making sense of quantum gravity?

(iii) What is the relation between Euclidean and Lorentzian signatures and do there exist analogues of the **Osterwalder–Schrader axioms** allowing analytic continuation from Euclidean space to Lorentzian space-time?

(iv) What is the role of topology in view, for instance, of the fact that higher-dimensional topologies cannot be classified?

We do not have answers to these questions and our inability to deal with them may be an indication that there exists no theory of** Euclidean quantum gravity** in four dimensions or, possibly, that quantum gravity only makes sense when embedded in a larger theory such as **string theory**. [Jan Ambjørn, University of Copenhagen, Bergfinnur Durhuus, University of Copenhagen, Thordur Jonsson, University of Iceland, Reykjavik Publisher: Cambridge University Press]

To be continued