2. Lessons from Topological Quantum Field Theory

Thanks to the influence of general relativity, there is a large body of theoretical physics that does not presume a fixed topology for space or spacetime. The idea is that after having assumed that spacetime is $n$-dimensional, we are in principle free to choose any $(n-1)$-dimensional manifold to represent space at a given time. Moreover, given two such manifolds, say $S$ and $S'$, we are free to choose any $n$-dimensional manifold-with-boundary, say $M$, to represent the portion of spacetime between them, so long as the boundary of $M$ is the union of $S$ and $S'$. In this situation we write $M \colon S \rightarrow S'$, even though $M$ is not a function from $S$ to $S'$, because we may think of $M$ as the process of time passing from the moment $S$ to the moment $S'$. Mathematicians call $M$ a cobordism from $S$ to $S'$. For example, in Figure 4 we depict a 2-dimensional manifold $M$ going from a 1-dimensional manifold $S$ (a pair of circles) to a 1-dimensional manifold $S'$ (a single circle). Physically, this cobordism represents a process in which two separate spaces collide to form a single one! This is an example of `topology change'.

Figure 4: A cobordism

All this has a close analogue in quantum theory. First, just as we can use any $(n-1)$-manifold to represent space, we can use any Hilbert space to describe the states of some quantum system. Second, just as we can use any cobordism to represent a spacetime going from one space to another, we can use any operator to describe a process taking states of one system to states of another. More precisely, given systems whose states are described using the Hilbert spaces $H$ and $H'$, respectively, any bounded linear operator $T \colon H \rightarrow H'$ describes a process that carries states of the first system to states of the second. We are most comfortable with this idea when the operator $T$ is unitary, or at least an isometry. After all, given a state described as a unit vector $\psi \in H$, we can only be sure $T\psi$ is a unit vector in $H'$ if $T$ is an isometry. So, only in this case does $T$ define a function from the set of states of the first system to the set of states of the second. However, the interpretation of linear operators as processes makes sense more generally. One way to make this interpretation precise is as follows: given a unit vector $\psi \in H$ and an orthonormal basis $\phi_i$ of $H'$, we declare that the relative probability for a system prepared in the state $\psi$ to be observed in the state $\phi_i$ after undergoing the process $T$ is $\vert\langle \phi_i, T\psi \rangle\vert^2$. By this, we mean that the probability of observing the system in the $i$th state divided by the probability of observing it in the $j$th state is

$$\frac{\vert\langle \phi_i, T\psi \rangle\vert^2} {\vert\langle \phi_j, T\psi \rangle\vert^2} .$$

The use of nonunitary operators to describe quantum processes is not new. For example, projection operators have long been used to describe processes like sending a photon through a polarizing filter. However, these examples traditionally arise when we treat part of the system (e.g. the measuring apparatus) classically. It is often assumed that at a fundamental level, the laws of nature in quantum theory describe time evolution using unitary operators. But as we shall see in Section 3, this assumption should be dropped in theories where the topology of space can change. In such theories we should let all the morphisms in ${\rm Hilb}$ qualify as `processes', just as we let all morphisms in $n{\rm Cob}$ qualify as spacetimes.

Having clarified this delicate point, we are now in a position to clearly see a structural analogy between general relativity and quantum theory, in which $(n-1)$-dimensional manifolds representing space are analogous to Hilbert spaces, while cobordisms describing spacetime are analogous to operators. Indulging in some lofty rhetoric, we might say that space and state are aspects of being, while spacetime and process are aspects of becoming. We summarize this analogy in Table 1.

GENERAL RELATIVITY	QUANTUM THEORY
$(n-1)$-dimensional manifold (space)	Hilbert space (states)
cobordism between $(n-1)$-dimensional manifolds (spacetime)	operator between Hilbert spaces (process)
composition of cobordisms	composition of operators
identity cobordism	identity operator

Table 1: Analogy between general relativity and quantum theory

This analogy becomes more than mere rhetoric when applied to topological quantum field theory [5]. In quantum field theory on curved spacetime, space and spacetime are not just manifolds: they come with fixed `background metrics' that allow us to measure distances and times. In this context, $S$ and $S'$ are Riemannian manifolds, and $M \colon S \rightarrow S'$ is a Lorentzian cobordism from $S$ to $S'$: that is, a Lorentzian manifold with boundary whose metric restricts at the boundary to the metrics on $S$ and $S'$. However, topological quantum field theories are an attempt to do background-free physics, so in this context we drop the background metrics: we merely assume that space is an $(n-1)$-dimensional manifold and spacetime is a cobordism between such manifolds. A topological quantum field theory then consists of a map $Z$ assigning a Hilbert space of states $Z(S)$ to any $(n-1)$-manifold $S$ and a linear operator $Z(M) \colon Z(S) \rightarrow Z(S')$ to any cobordism between such manifolds. This map cannot be arbitrary, though: for starters, it must be a functor from the category of $n$-dimensional cobordisms to the category of Hilbert spaces. This is a great example of how every sufficiently good analogy is yearning to become a functor.

However, we are getting a bit ahead of ourselves. Before we can talk about functors, we must talk about categories. What is the category of $n$-dimensional cobordisms, and what is the category of Hilbert spaces? The answers to these questions will allow us to make the analogy in Table 1 much more precise.

First, recall that a category consists of a collection of objects, a collection of morphisms $f \colon A \rightarrow B$ from any object $A$ to any object $B$, a rule for composing morphisms $f \colon A \rightarrow B$ and $g \colon B \rightarrow C$ to obtain a morphism $gf \colon A \rightarrow C$, and for each object $A$ an identity morphism $1_A \colon A \rightarrow A$. These must satisfy the associative law $f(gh) = (fg)h$ and the left and right unit laws $1_A f = f$ and $f 1_A = f$ whenever these composites are defined. In many cases, the objects of a category are best thought of as sets equipped with extra structure, while the morphisms are functions preserving the extra structure. However, this is true neither for the category of Hilbert spaces nor for the category of cobordisms.

In the category ${\rm Hilb}$ we take the objects to be Hilbert spaces and the morphisms to be bounded linear operators. Composition and identity operators are defined as usual. Hilbert spaces are indeed sets equipped with extra structure, but bounded linear operators do not preserve all this extra structure: in particular, they need not preserve the inner product. This may seem like a fine point, but it is important, and we shall explore its significance in detail in Section 3.

In the category $n{\rm Cob}$ we take the objects to be $(n-1)$-dimensional manifolds and the morphisms to be cobordisms between these. (For technical reasons mathematicians usually assume both to be compact and oriented.) Here the morphisms are not functions at all! Nonetheless we can `compose' two cobordisms $M \colon S \rightarrow S'$ and $M' \colon S' \rightarrow S''$, obtaining a cobordism $M' M \colon S \rightarrow S''$, as in Figure 5.

Figure 5: Composition of cobordisms

The idea here is that the passage of time corresponding to $M$ followed by the passage of time corresponding to $M'$ equals the passage of time corresponding to $M'M$. This is analogous to the familiar idea that waiting $t$ seconds followed by waiting $t'$ seconds is the same as waiting $t'+t$ seconds. The big difference is that in topological quantum field theory we cannot measure time in seconds, because there is no background metric available to let us count the passage of time. We can only keep track of topology change. Just as ordinary addition is associative, composition of cobordisms satisfies the associative law:

$$(M'' M') M = M'' (M' M) .$$

Furthermore, for any $(n-1)$-dimensional manifold $S$ representing space, there is a cobordism $1_S \colon S \rightarrow S$ called the `identity' cobordism, which represents a passage of time during which the topology of space stays constant. For example, when $S$ is a circle, the identity cobordism $1_S$ is a cylinder, as shown in Figure 6. In general, the identity cobordism $1_S$ has the property that

$$1_S M = M$$

and

$$M 1_S = M$$

whenever these composites are defined. These properties say that an identity cobordism is analogous to waiting 0 seconds: if you wait 0 seconds and then wait $t$ more seconds, or wait $t$ seconds and then wait 0 more seconds, this is the same as waiting $t$ seconds.

Figure 6: An identity cobordism

A functor between categories is a map sending objects to objects and morphisms to morphisms, preserving composition and identities. Thus, in saying that a topological quantum field theory is a functor

$$Z \colon n{\rm Cob}\rightarrow {\rm Hilb},$$

we merely mean that it assigns a Hilbert space of states $Z(S)$ to any $(n-1)$-dimensional manifold $S$ and a linear operator $Z(M) \colon Z(S) \rightarrow Z(S')$ to any $n$-dimensional cobordism $M \colon S \rightarrow S'$ in such a way that:

• For any $n$-dimensional cobordisms $M \colon S \rightarrow S'$ and $M' \colon S' \rightarrow S''$,
  
  $$Z(M'M) = Z(M')Z(M) .$$
  
  

• For any $(n-1)$-dimensional manifold $S$,
  
  $$Z(1_S) = 1_{Z(S)} .$$
  
  

Both these axioms make sense if one ponders them a bit. The first says that the passage of time corresponding to the cobordism $M$ followed by the passage of time corresponding to $M'$ has the same effect on a state as the combined passage of time corresponding to $M'M$. The second says that a passage of time in which no topology change occurs has no effect at all on the state of the universe. This seems paradoxical at first, since it seems we regularly observe things happening even in the absence of topology change. However, this paradox is easily resolved: a topological quantum field theory describes a world without local degrees of freedom. In such a world, nothing local happens, so the state of the universe can only change when the topology of space itself changes.

Unless elementary particles are wormhole ends or some other sort of topological phenomenon, it seems our own world is quite unlike this. Thus, we hasten to reassure the reader that this peculiarity of topological quantum field theory is not crucial to our overall point, which is the analogy between categories describing space and spacetime and those describing quantum states and processes. If we were doing quantum field theory on curved spacetime, we would replace $n{\rm Cob}$ with a category where the objects are $n$-dimensional Riemannian manifolds and most of the morphisms are Lorentzian cobordisms between these. In this case a cobordism $M \colon S \rightarrow S'$ has not just a topology but also a geometry, so we can use cylinder-shaped cobordisms of different `lengths' to describe time evolution for different amounts of time. The identity morphism is then described by a cylinder of `length zero'. This degenerate cylinder is not really a Lorentzian cobordism, which leads to some technical complications. However, Segal showed how to get around these in his axioms for a conformal field theory [29]. There are some further technical complications arising from the fact that except in low dimensions, we need to use the C*-algebraic approach to quantum theory, instead of the Hilbert space approach [13]. Here the category ${\rm Hilb}$ should be replaced by one where the objects are C$^*$-algebras and the morphisms are completely positive maps between their duals [15].

Setting aside these nuances, our main point is that treating a TQFT as a functor from $n{\rm Cob}$ to ${\rm Hilb}$ is a way of making very precise some of the analogies between general relativity and quantum theory. However, we can go further! A TQFT is more than just a functor. It must also be compatible with the `monoidal category' structure of $n{\rm Cob}$ and ${\rm Hilb}$, and to be physically well-behaved it must also be compatible with their `$\ast$-category' structure. We examine these extra structures in the next two sections.

© 2004 John Baez
baez@math.removethis.ucr.andthis.edu

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