3. The *-Category of Hilbert Spaces

What is the category of Hilbert spaces? While we have already given an answer, this is actually a tricky question, one that makes many category theorists acutely uncomfortable.

To understand this, we must start by recalling that one use of
categories is to organize discourse about various sorts of
`mathematical objects': groups, rings, vector spaces, topological
spaces, manifolds and so on. Quite commonly these mathematical objects
are sets equipped with extra structure and properties, so let us
restrict attention to this case. Here by **structure** we mean
operations and relations defined on the set in question, while by
**properties** we mean axioms that these operations and relations
are required to satisfy. The division into structure and properties
is evident from the standard form of mathematical definitions such as
``a widget is a set equipped with ... such that ....'' Here the
structures are listed in the first blank, while the properties are
listed in the second.

To build a category of this sort of mathematical object, we must also
define morphisms between these objects. When the objects are *sets equipped with extra structure and properties*, the morphisms are
typically taken to be *functions that preserve the extra structure*.
At the expense of a long digression we could make this completely
precise -- and also more general, since we can also build categories
by equipping not sets but objects of other categories with extra
structure and properties. However, we prefer to illustrate the idea
with an example. We take an example closely related to but subtly
different from the category of Hilbert spaces: the category of
complex vector spaces.

A **complex vector space** is a set equipped with
extra structure consisting of operations called addition

and scalar multiplication

which in turn must have certain extra properties: commutativity and associativity together with the existence of an identity and inverses for addition, associativity and the unit law for scalar multiplication, and distributivity of scalar multiplication over adddition. Given vector spaces and , a

and

for all and . Note that the properties do not enter here. Mathematicians define the category to have complex vector spaces as its objects and linear operators between them as its morphisms.

Now compare the case of Hilbert spaces. A Hilbert space
is a set equipped with all the structure of a complex vector
space but also some more, namely an inner product

Similarly, it has all the properties of a complex vector spaces but also some more: for all and we have the equations

together with the inequality

where equality holds only if ; furthermore, the norm defined by the inner product must be complete. Given Hilbert spaces and , a function that preserves all the structure is thus a linear operator that preserves the inner product:

for all . Such an operator is called an

If we followed the pattern that works for vector spaces and many other mathematical objects, we would thus define the category to have Hilbert spaces as objects and isometries as morphisms. However, this category seems too constricted to suit what physicists actually do with Hilbert spaces: they frequently need operators that aren't isometries! Unitary operators are always isometries, but self-adjoint operators, for example, are not.

The alternative we adopt in this paper is to work with the category
whose objects are Hilbert spaces and the morphisms are bounded
linear operators. However, this leads to a curious puzzle. In a
precise technical sense, the category of finite-dimensional Hilbert
spaces and linear operators between these is *equivalent* to the
category of finite-dimensional complex vector spaces and linear operators.
So, in defining this category, we might as well ignore the inner product
entirely! The puzzle is thus what role, if any, the inner product
plays in this category.

The case of general, possibly infinite-dimensional Hilbert spaces
is subtler, but the puzzle persists. The category of all Hilbert
spaces and bounded linear operators between them is *not*
equivalent to the category of all complex vector spaces and linear
operators. However, it *is* equivalent to the category
of `Hilbertizable' vector spaces -- that is, vector spaces equipped
with a topology coming from *some* Hilbert space structure --
and continuous linear operators between these. So, in defining
this category, what matters is not the inner product but merely
the topology it gives rise to. The point is that bounded linear operators
don't preserve the inner product, just the topology, and a structure
that is not preserved might as well be ignored, as far as the category
is concerned.

My resolution of this puzzle is simple but a bit upsetting to most category theorists. I admit that the inner product is inessential in defining the category of Hilbert spaces and bounded linear operators. However, I insist that it plays a crucial role in making this category into a -category!

What is a -category? It is a category
equipped with a map sending each morphism
to
a morphism
, satisfying

and

To make into a -category we define for any bounded linear operator to be the

We see by this formula that the inner product on both and are required to define the adjoint of .

In fact, we can completely recover the inner product on every
Hilbert space from the -category structure of .
Given a Hilbert space and a vector ,
there is a unique operator
with
. Conversely, any operator from to determines a
unique vector in this way. So, we can think of elements of a
Hilbert space as morphisms from to this Hilbert space. Using
this trick, an easy calculation shows that

The right-hand side is really a linear operator from to , but there is a canonical way to identify such a thing with a complex number. So, everything about inner products is encoded in the -category structure of . Moreover, this way of thinking about the inner product formalizes an old idea of Dirac. The operator is really just Dirac's `ket' , while is the `bra' . Composing a ket with a bra, we get the inner product.

This shows how adjoints are closely tied to the inner product structure on Hilbert space. But what is the physical significance of the adjoint of an operator, or more generally the operation in any -category? Most fundamentally, the operation gives us a way to `reverse' a morphism even when it is not invertible. If we think of inner products as giving transition amplitudes between states in quantum theory, the equation says that is the unique operation we can perform on any state so that the transition amplitude from to is the same as that from to . So, in a suggestive but loose way, we can say that the process described by is some sort of `time-reversed' version of the process described by . If is unitary, is just the inverse of . But, makes sense even when has no inverse!

This suggestive but loose relation between operations and
time reversal becomes more precise in the case of .
Here the operation really *is* time reversal. More
precisely, given an -dimensional cobordism
,
we let the **adjoint** cobordism
to be
the same manifold, but with the `past' and `future' parts of its
boundary switched, as in Figure 7. It is easy to check
that this makes into a -category.

In a so-called **unitary** topological quantum field theory
(the terminology is a bit unfortunate), we demand that the
functor
preserve the -category
structure in the following sense:

All the TQFTs of interest in physics have this property, and a similar property holds for conformal field theories and other quantum field theories on curved spacetime. This means that in the analogy between general relativity and quantum theory,

Taking this analogy seriously leads us in some interesting directions.
First, since the operation in is given by time
reversal, while operation in is defined using the inner
product, there should be some relation between time reversal and the
inner product in quantum theory! The details remain obscure, at least
to me, but we can make a little progress by pondering the following
equation, which we originally introduced as a `trick' for expressing
inner products in terms of adjoint operators:

An equation this important should not be a mere trick! To try to interpret it, suppose that in some sense the operator describes `the process of preparing the system to be in the state ', while describes the process of `observing the system to be in the state '. Given this, should describe the process of first preparing the system to be in the state and then observing it to be in the state . The above equation then relates this composite process to the transition amplitude . Moreover, we see that `observation' is like a time-reversed version of `preparation'. All this makes a rough intuitive kind of sense. However, these ideas could use a great deal of elaboration and clarification. I mention them here mainly to open an avenue for further thought.

Second, and less speculatively, the equation
sheds some light on the relation between topology change and the
failure of unitarity, mentioned already in Section
2. In any
-category, we may define a morphism
to be **unitary** if
and . For a morphism in
this reduces to the usual definition of unitarity for a linear
operator. One can show that a morphism in is unitary if involves no
topology change, or more precisely, if is diffeomorphic to the
Cartesian product of an interval and some -dimensional manifold.
(The converse is true when
is less than or equal to 3, but it
fails in higher
dimensions.) A TQFT satisfying
maps
unitary morphisms in to unitary morphisms in , so
for TQFTs of this sort, *absence of topology change
implies unitary time evolution*. This fact reinforces a point already
well-known from quantum field theory on curved spacetime, namely that
unitary time evolution is not a built-in feature of quantum theory
but rather the consequence of specific assumptions about the nature
of spacetime [13].

To conclude, it is interesting to contrast and with the more familiar category , whose objects are sets and whose morphisms are functions. There is no way to make into a -category, since there is no way to `reverse' the map from the empty set to the one-element set. So, our intuitions about sets and functions help us very little in understanding -categories. The problem is that the concept of function is based on an intuitive notion of process that is asymmetrical with respect to past and future: a function is a relation such that each element of is related to exactly one element of , but not necessarily vice versa. For better or worse, this built-in `arrow of time' has no place in the basic concepts of quantum theory.

Pondering this, it soon becomes apparent that if we want an easy
example of a -category other than to help build our
intuitions about -categories, we should use not but
, the category of sets and *relations*. In fact, quantum
theory can be seen as a modified version of the theory of relations in
which Boolean algebra has been replaced by the algebra of complex
numbers! To see this, note that a linear operator between two Hilbert
spaces can be described using a matrix of complex numbers as soon as we
pick an orthonormal basis for each. Similarly, a relation between
sets and can be described by a matrix of truth values,
namely the truth values of the propositions where
and . Composition of relations can be defined as matrix
multiplication with `or' and `and' playing the roles of `plus' and
`times'. It easy to check that this is associative and has an
identity morphism for each set, so we obtain a category with
sets as objects and relations as morphisms. Furthermore,
becomes a -category if we define the relation by saying
that if and only if . Just as the matrix for the linear
operator is the conjugate transpose of the matrix for , the
matrix for the relation is the transpose of the matrix for .

So, the category of Hilbert spaces closely resembles the category of relations. The main difference is that binary truth values describing whether or not a transition is possible are replaced by complex numbers describing the amplitude with which a transition occurs. Comparisons between and are fruitful source of intuitions not only about -categories in general but also about the meaning of `matrix mechanics'. For some further explorations along these lines, see the work of Abramsky and Coecke [1].

© 2004 John Baez

baez@math.removethis.ucr.andthis.edu