Next: Lessons from Topological Quantum Field Theory Up: Quantum Quandaries

1. Introduction

Faced with the great challenge of reconciling general relativity and quantum theory, it is difficult to know just how deeply we need to rethink basic concepts. By now it is almost a truism that the project of quantizing gravity may force us to modify our ideas about spacetime. Could it also force us to modify our ideas about the quantum? So far this thought has appealed mainly to those who feel uneasy about quantum theory and hope to replace it by something that makes more sense. The problem is that the success and elegance of quantum theory make it hard to imagine promising replacements. Here I would like to propose another possibility, namely that quantum theory will make more sense when regarded as part of a theory of spacetime. Furthermore, I claim that we can only see this from a category-theoretic perspective -- in particular, one that de-emphasizes the primary role of the category of sets and functions.

Part of the difficulty of combining general relativity and quantum theory is that they use different sorts of mathematics: one is based on objects such as manifolds, the other on objects such as Hilbert spaces. As `sets equipped with extra structure', these look like very different things, so combining them in a single theory has always seemed a bit like trying to mix oil and water. However, work on topological quantum field theory theory has uncovered a deep analogy between the two. Moreover, this analogy operates at the level of categories.

We shall focus on two categories in this paper. One is the category ${\rm Hilb}$ whose objects are Hilbert spaces and whose morphisms are linear operators between these. This plays an important role in quantum theory. The other is the category $n{\rm Cob}$ whose objects are $(n-1)$-dimensional manifolds and whose morphisms are $n$-dimensional manifolds going between these. This plays an important role in relativistic theories where spacetime is assumed to be $n$-dimensional: in these theories the objects of $n{\rm Cob}$ represent possible choices of `space', while the morphisms -- called `cobordisms' -- represent possible choices of `spacetime'.

While an individual manifold is not very much like a Hilbert space, the category $n{\rm Cob}$ turns out to have many structural similarities to the category ${\rm Hilb}$. The goal of this paper is to explain these similarities and show that the most puzzling features of quantum theory all arise from ways in which ${\rm Hilb}$ resembles $n{\rm Cob}$ more than the category ${\rm Set}$, whose objects are sets and whose morphisms are functions.

Since sets and functions capture many basic intuitions about macroscopic objects, and the rules governing them have been incorporated into the foundations of mathematics, we naturally tend to focus on the fact that any quantum system has a set of states. From a Hilbert space we can indeed extract a set of states, namely the set of unit vectors modulo phase. However, this is often more misleading than productive, because this process does not define a well-behaved map -- or more precisely, a functor -- from ${\rm Hilb}$ to ${\rm Set}$. In some sense the gap between ${\rm Hilb}$ and ${\rm Set}$ is too great to be usefully bridged by this trick. However, many of the ways in which ${\rm Hilb}$ differs from ${\rm Set}$ are ways in which it resembles $n{\rm Cob}$! This suggests that the interpretation of quantum theory will become easier, not harder, when we finally succeed in merging it with general relativity.

In particular, it is easy to draw pictures of the objects and morphisms of $n{\rm Cob}$, at least for low $n$. Doing so lets us visualize many features of quantum theory. This is not really a new discovery: it is implicit in the theory of Feynman diagrams. Whenever one uses Feynman diagrams in quantum field theory, one is secretly working in some category where the morphisms are graphs with labelled edges and vertices, as shown in Figure 1.

Figure 1: A Feynman diagram
\begin{figure}\vskip 2em

The precise details of the category depend on the quantum field theory in question: the labels for edges correspond to the various particles of the theory, while the labels for vertices correspond to the interactions of the theory. Regardless of the details, categories of this sort share many of the structural features of both $n{\rm Cob}$ and ${\rm Hilb}$. Their resemblance to $n{\rm Cob}$, namely their topological nature, makes them a powerful tool for visualization. On the other hand, their relation to ${\rm Hilb}$ makes them useful in calculations.

Though Feynman diagrams are far from new, the fact that they are morphisms in a category only became appreciated in work on quantum gravity, especially string theory and loop quantum gravity. Both these approaches stretch the Feynman diagram concept in interesting new directions. In string theory, Feynman diagrams are replaced by `string worldsheets': 2-dimensional cobordisms mapped into an ambient spacetime, as shown in Figure 2. Since these cobordisms no longer have definite edges and vertices, there are no labels anymore. This is one sense in which the various particles and interactions are all unified in string theory. The realization that processes in string theory could be described as morphisms in a category was crystallized by Segal's definition of `conformal field theory' [29].

Figure 2: A string worldsheet
\begin{figure}\vskip 2em
\xy0 ;/r.35pc/:
...;''A1'' **\crv{(-8,7) & (-3,5)};

Loop quantum gravity is moving towards a similar picture, though with some important differences. In this approach processes are described by `spin foams'. These are a 2-dimensional generalization of Feynman diagrams built from vertices, edges and faces, as shown in Figure 3. They are not mapped into an ambient spacetime: in this approach spacetime is nothing but the spin foam itself -- or more precisely, a linear combination of spin foams. Particles and interactions are not `unified' in these models, so there are labels on the vertices, edges and faces, which depend on the details of the model in question. The category-theoretic underpinnings of spin foam models were explicit from the very beginning [4], since they were developed after Segal's work on conformal field theory, and also after Atiyah's work on topological quantum field theory [2], which exhibits the analogy between $n{\rm Cob}$ and ${\rm Hilb}$ in its simplest form.

Figure 3: A spin foam
\begin{figure}\vskip 2em

There is not one whit of experimental evidence for either string theory or loop quantum gravity, and both theories have some serious problems, so it might seem premature for philosophers to consider their implications. It indeed makes little sense for philosophers to spend time chasing every short-lived fad in these fast-moving subjects. Instead, what is worthy of reflection is that these two approaches to quantum gravity, while disagreeing heatedly on so many issues [30,31], have so much in common. It suggests that in our attempts to reconcile the quantum-theoretic notions of state and process with the relativistic notions of space and spacetime, we have a limited supply of promising ideas. It is an open question whether these ideas will be up to the task of describing nature. But this actually makes it more urgent, not less, for philosophers to clarify and question these ideas and the implicit assumptions upon which they rest.

Before plunging ahead, let us briefly sketch the contents of this paper. In Section 2 we explain the analogy between $n{\rm Cob}$ and ${\rm Hilb}$ by recalling Atiyah's definition of `topological quantum field theory', or `TQFT' for short. In Section 3, we begin by noting that unlike many familiar categories, neither ${\rm Hilb}$ nor $n{\rm Cob}$ is best regarded as a category whose objects are sets equipped with extra structures and properties, and whose morphisms are functions preserving these extra structures. In particular, operators between Hilbert spaces are not required to preserve the inner product. This raises the question of precisely what role the inner product plays in the category ${\rm Hilb}$. Of course the inner product is crucial in quantum theory, since we use it to compute transition amplitudes between states -- but how does it manifest itself mathematically in the structure of ${\rm Hilb}$? One answer is that it gives a way to `reverse' an operator $T \colon H \rightarrow H'$, obtaining an operator $T^\ast \colon H' \rightarrow H$ called the `adjoint' of $T$ such that

\begin{displaymath}\langle T^\ast \phi,\psi \rangle = \langle \phi, T\psi \rangle \end{displaymath}

for all $\psi \in H$ and $\phi \in H'$. This makes ${\rm Hilb}$ into something called a `$\ast$-category': a category where there is a built-in way to reverse any process. As we shall see, it is easy to compute transition amplitudes using the $\ast$-category structure of ${\rm Hilb}$. The category $n{\rm Cob}$ is also a $\ast$-category, where the adjoint of a spacetime is obtained simply by switching the roles of future and past. On the other hand, ${\rm Set}$ cannot be made into a $\ast$-category. All this suggests that both quantum theory and general relativity will be best understood in terms of categories quite different from the category of sets and functions.

In Section 4 we tackle some of the most puzzling features of quantum theory, namely those concerning joint systems: physical systems composed of two parts. It is in the study of joint systems that one sees the `failure of local realism' that worried Einstein so terribly [14], and was brought into clearer focus by Bell [8]. Here is also where one discovers that one `cannot clone a quantum state' -- a result due to Wooters and Zurek [32] which serves as the basis of quantum cryptography. As explained in Section 4, both these phenomena follow from the failure of the tensor product to be `cartesian' in a certain sense made precise by category theory. In ${\rm Set}$, the usual product of sets is cartesian, and this encapsulates many of our usual intuitions about ordered pairs, like our ability to pick out the components $a$ and $b$ of any pair $(a,b)$, and our ability to `duplicate' any element $a$ to obtain a pair $(a,a)$. The fact that we cannot do these things in ${\rm Hilb}$ is responsible for the failure of local realism and the impossibility of duplicating a quantum state. Here again the category ${\rm Hilb}$ resembles $n{\rm Cob}$ more than ${\rm Set}$. Like ${\rm Hilb}$, the category $n{\rm Cob}$ has a noncartesian tensor product, given by the disjoint union of manifolds. Some of the mystery surrounding joint systems in quantum theory dissipates when one focuses on the analogy to $n{\rm Cob}$ and stops trying to analogize the tensor product of Hilbert spaces to the Cartesian product of sets.

This paper is best read as a followup to my paper `Higher-Dimensional Algebra and Planck-Scale Physics' [5], since it expands on some of the ideas already on touched upon there.

Next: Lessons from Topological Quantum Field Theory Up: Quantum Quandaries

© 2004 John Baez