1. Introduction

At present our physical worldview is deeply schizophrenic. We have, not one, but two fundamental theories of the physical universe: general relativity, and the Standard Model of particle physics based on quantum field theory. The former takes gravity into account but ignores quantum mechanics, while the latter takes quantum mechanics into account but ignores gravity. In other words, the former recognizes that spacetime is curved but neglects the uncertainty principle, while the latter takes the uncertainty principle into account but pretends that spacetime is flat. Both theories have been spectacularly successful in their own domain, but neither can be anything more than an approximation to the truth. Clearly some synthesis is needed: at the very least, a theory of quantum gravity, which might or might not be part of a overarching `theory of everything'. Unfortunately, attempts to achieve this synthesis have not yet succeeded.

Modern theoretical physics is difficult to understand for anyone outside the subject. Can philosophers really contribute to the project of reconciling general relativity and quantum field theory? Or is this a technical business best left to the experts? I would argue for the former. General relativity and quantum field theory are based on some profound insights about the nature of reality. These insights are crystallized in the form of mathematics, but there is a limit to how much progress we can make by just playing around with this mathematics. We need to go back to the insights behind general relativity and quantum field theory, learn to hold them together in our minds, and dare to imagine a world more strange, more beautiful, but ultimately more reasonable than our current theories of it. For this daunting task, philosophical reflection is bound to be of help.

However, a word of warning is in order. 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. Philosophers must take what physicists say about quantum gravity with a grain of salt.

To lay my own cards on the table, I should say that as a mathematical physicist with an interest in philosophy, I am drawn to a strand of work that emphasizes `higher-dimensional algebra'. This branch of mathematics goes back and reconsiders some of the presuppositions that mathematicians usually take for granted, such as the notion of equality [8] and the emphasis on doing mathematics using 1-dimensional strings of symbols [12,19]. Starting in the late 1980s, it became apparent that higher-dimensional algebra is the correct language to formulate so-called `topological quantum field theories' [7,20,30]. More recently, various people have begun to formulate theories of quantum gravity using ideas from higher-dimensional algebra [6,11,16,22,23]. While they have tantalizing connections to string theory, these theories are best seen as an outgrowth of loop quantum gravity [24].

The plan of the paper is as follows. In Section 2, I begin by recalling why some physicists expect general relativity and quantum field theory to collide at the Planck length. This is a unit of distance concocted from three fundamental constants: the speed of light $c$, Newton's gravitational constant $G$, and Planck's constant $\hbar$. General relativity idealizes reality by treating Planck's constant as negligible, while quantum field theory idealizes it by treating Newton's gravitational constant as negligible. By analyzing the physics of $c,G,$ and $\hbar$, we get a glimpse of the sort of theory that would be needed to deal with situations where these idealizations break down. In particular, I shall argue that we need a background-free quantum theory with local degrees of freedom propagating causally.

In Section 3, I discuss `topological quantum field theories'. These are the first examples of background-free quantum theories. However, they lack local degrees of freedom. In other words, they describe imaginary worlds in which everywhere looks like everywhere else! This might at first seem to condemn them to the status of mathematical curiosities. However, they suggest an important analogy between the mathematics of spacetime and the mathematics of quantum theory. I argue that this is the beginning of a new bridge between general relativity and quantum field theory.

In Section 4, I describe one of the most important examples of a topological quantum field theory: the Turaev-Viro model of quantum gravity in 3-dimensional spacetime. This theory is just a warmup for the 4-dimensional case that is of real interest in physics. Nonetheless, it has some startling features which perhaps hint at the radical changes in our worldview that a successful synthesis of general relativity and quantum field theory would require.

In Section 5, I discuss the role of higher-dimensional algebra in topological quantum field theory. I begin with a brief introduction to categories. Category theory can be thought of as an attempt to treat processes (or `morphisms') on an equal footing with things (or `objects'), and it is ultimately for this reason that it serves as a good framework for topological quantum field theory. In particular, category theory allows one to make the analogy between the mathematics of spacetime and the mathematics of quantum theory quite precise. But to fully explore this analogy one must introduce `$n$-categories', a generalization of categories that allows one to speak of processes between processes between processes... and so on to the $n$th degree. Since $n$-categories are purely algebraic structures but have a natural relationship to the study of $n$-dimensional spacetime, their study is sometimes called `higher-dimensional algebra'.

Finally, in Section 6 I briefly touch upon recent attempts to construct theories of 4-dimensional quantum gravity using higher-dimensional algebra. This subjects is still in its infancy. Throughout the paper, but especially in this last section, the reader must turn to the references for details. To make the bibliography as useful as possible, I have chosen references of an expository nature whenever they exist, rather than always citing the first paper in which something was done.

© 1999 John Baez
baez@math.removethis.ucr.andthis.edu

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