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 -dimensional, we are in principle free to choose any
-dimensional manifold to represent space at a given time.
Moreover, given two such manifolds, say and , we are free to
choose any -dimensional manifold-with-boundary, say ,
to represent the portion of spacetime between them, so long as
the boundary of is the union of and . In this situation
we write
, even though is not a function from
to , because we may think of as the process of time passing
from the moment to the moment . Mathematicians call a
**cobordism** from to . For example, in Figure 4
we depict a 2-dimensional manifold going from
a 1-dimensional manifold (a pair of circles) to a 1-dimensional
manifold (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'.

All this has a close analogue in quantum theory. First, just as we can
use any -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 and , respectively, any bounded linear operator
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 is unitary, or at least an isometry.
After all, given a state described as a unit vector , we can
only be sure is a unit vector in if is an isometry.
So, only in this case does 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 and an orthonormal basis
of , we declare that the **relative probability** for a
system prepared in the state to be observed in the state
after undergoing the process is
.
By this, we mean that the probability of observing the system in the
th state divided by the probability of observing it in the th
state is

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 qualify as
`processes', just as we let *all* morphisms in 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 -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 |

-dimensional manifold (space) | Hilbert space (states) |

cobordism between -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, and are
Riemannian manifolds, and
is a **Lorentzian
cobordism** from to : that is, a Lorentzian manifold with
boundary whose metric restricts at the boundary to the metrics on
and . 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 -dimensional manifold
and spacetime is a cobordism between such manifolds. A topological
quantum field theory then consists of a map assigning a Hilbert
space of states to any -manifold and a linear operator
to any cobordism between such manifolds.
This map cannot be arbitrary, though: for starters, it must be a
*functor* from the category of -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 -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
from any object
to any object , a rule for composing morphisms
and
to obtain a morphism
, and for
each object an identity morphism
. These must
satisfy the associative law and the left and right unit
laws and 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 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 we take the objects to be -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 and , obtaining a cobordism , as in Figure 5.

The idea here is that the passage of time
corresponding to followed by the passage of time corresponding to
equals the passage of time corresponding to . This is
analogous to the familiar idea that waiting seconds followed by
waiting seconds is the same as waiting 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:

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

and

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 more seconds, or wait seconds and then wait 0 more seconds, this is the same as waiting seconds.

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

we merely mean that it assigns a Hilbert space of states to any -dimensional manifold and a linear operator to any -dimensional cobordism in such a way that:

- For any -dimensional cobordisms
and
,

- For any -dimensional manifold ,

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 with a category where the objects
are -dimensional Riemannian manifolds and most of the morphisms are
Lorentzian cobordisms between these. In this case a cobordism
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 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 to 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 and , and to be physically well-behaved it must also be compatible with their `-category' structure. We examine these extra structures in the next two sections.

© 2004 John Baez

baez@math.removethis.ucr.andthis.edu