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 $V$ equipped with extra structure consisting of operations called addition

$$+ \colon V \times V \rightarrow V$$

and scalar multiplication

$$\cdot \colon {\mathbb{C}}\times V \rightarrow V ,$$

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 $V$ and $V'$, a linear operator $T \colon V \rightarrow V'$ can be defined as a function preserving all the extra structure. This means that we require

$$T(\psi + \phi) = T(\psi) + T(\phi)$$

and

$$T(c\psi) = c T(\psi)$$

for all $\psi,\phi \in V$ and $c \in {\mathbb{C}}$. Note that the properties do not enter here. Mathematicians define the category ${\rm Vect}$ 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 $H$ is a set equipped with all the structure of a complex vector space but also some more, namely an inner product

$$\langle \cdot, \cdot \rangle \colon H \times H \rightarrow {\mathbb{C}}.$$

Similarly, it has all the properties of a complex vector spaces but also some more: for all $\phi, \psi, \psi' \in H$ and $c \in {\mathbb{C}}$ we have the equations

$$\langle \phi , \psi + \psi' \rangle = \langle \phi, \psi \rangle + \langle \phi, \psi' \rangle,$$

$$\langle \phi, c\psi \rangle = c \langle \phi, \psi \rangle,$$

$$\langle \phi,\psi \rangle = \overline{\langle \psi,\phi \rangle},$$

together with the inequality

$$\langle \psi , \psi \rangle \ge 0$$

where equality holds only if $\psi = 0$; furthermore, the norm defined by the inner product must be complete. Given Hilbert spaces $H$ and $H'$, a function $T \colon H \rightarrow H'$ that preserves all the structure is thus a linear operator that preserves the inner product:

$$\langle T \phi, T \psi \rangle = \langle \phi,\psi \rangle$$

for all $\phi,\psi \in H$. Such an operator is called an isometry.

If we followed the pattern that works for vector spaces and many other mathematical objects, we would thus define the category ${\rm Hilb}$ 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 ${\rm Hilb}$ 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 $\ast$-category!

What is a $\ast$-category? It is a category $C$ equipped with a map sending each morphism $f \colon X \rightarrow Y$ to a morphism $f^\ast \colon Y \rightarrow X$, satisfying

$$1_X^\ast = 1_X ,$$

$$(fg)^\ast = g^\ast f^\ast,$$

and

$$f^{\ast\ast} = f.$$

To make ${\rm Hilb}$ into a $\ast$-category we define $T^\ast$ for any bounded linear operator $T \colon H \rightarrow H'$ to be the adjoint operator $T^\ast H' \rightarrow H$, given by

$$\langle T^\ast \psi,\phi \rangle = \langle \psi, T\phi \rangle .$$

We see by this formula that the inner product on both $H$ and $H'$ are required to define the adjoint of $T$.

In fact, we can completely recover the inner product on every Hilbert space from the $\ast$-category structure of ${\rm Hilb}$. Given a Hilbert space $H$ and a vector $\psi \in H$, there is a unique operator $T_\psi \colon {\mathbb{C}}\rightarrow H$ with $T_\psi(1) = \psi$. Conversely, any operator from ${\mathbb{C}}$ to $H$ determines a unique vector in $H$ this way. So, we can think of elements of a Hilbert space as morphisms from ${\mathbb{C}}$ to this Hilbert space. Using this trick, an easy calculation shows that

$$\langle \phi,\psi \rangle = {T_\phi}^{\!\!\ast}\, T_\psi .$$

The right-hand side is really a linear operator from ${\mathbb{C}}$ to ${\mathbb{C}}$, but there is a canonical way to identify such a thing with a complex number. So, everything about inner products is encoded in the $\ast$-category structure of ${\rm Hilb}$. Moreover, this way of thinking about the inner product formalizes an old idea of Dirac. The operator $T_\psi$ is really just Dirac's `ket' $\vert\psi\rangle$, while ${T_\phi}^{\!\!\ast}$ is the `bra' $\langle \phi \vert$. 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 $\ast$ operation in any $\ast$-category? Most fundamentally, the $\ast$ 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 $\langle T^\ast \phi,\psi \rangle = \langle \phi, T\psi \rangle$ says that $T^\ast$ is the unique operation we can perform on any state $\phi$ so that the transition amplitude from $T\psi$ to $\phi$ is the same as that from $\psi$ to $T^\ast \phi$. So, in a suggestive but loose way, we can say that the process described by $T^\ast$ is some sort of `time-reversed' version of the process described by $T$. If $T$ is unitary, $T^\ast$ is just the inverse of $T$. But, $T^\ast$ makes sense even when $T$ has no inverse!

This suggestive but loose relation between $\ast$ operations and time reversal becomes more precise in the case of $n{\rm Cob}$. Here the $\ast$ operation really is time reversal. More precisely, given an $n$-dimensional cobordism $M \colon S \rightarrow S'$, we let the adjoint cobordism $M^\ast \colon S' \rightarrow S$ 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 $n{\rm Cob}$ into a $\ast$-category.

Figure 7: A cobordism and its adjoint

In a so-called unitary topological quantum field theory (the terminology is a bit unfortunate), we demand that the functor $Z \colon n{\rm Cob}\rightarrow {\rm Hilb}$ preserve the $\ast$-category structure in the following sense:

$$Z(M^\ast) = Z(M)^\ast .$$

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, the analogue of time reversal is taking the adjoint of an operator between Hilbert spaces. To `reverse' a spacetime $M \colon S \rightarrow S'$ we formally switch the notions of future and past, while to `reverse' a process $T \colon H \rightarrow H'$ we take its adjoint.

Taking this analogy seriously leads us in some interesting directions. First, since the $\ast$ operation in $n{\rm Cob}$ is given by time reversal, while $\ast$ operation in ${\rm Hilb}$ 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:

$$\langle \phi,\psi \rangle = {T_\phi}^{\!\!\ast}\, T_\psi .$$

An equation this important should not be a mere trick! To try to interpret it, suppose that in some sense the operator $T_\psi$ describes `the process of preparing the system to be in the state $\psi$', while ${T_\phi}^{\!\!\ast}$ describes the process of `observing the system to be in the state $\phi$'. Given this, ${T_\phi}^{\!\!\ast} \, T_\psi$ should describe the process of first preparing the system to be in the state $\psi$ and then observing it to be in the state $\phi$. The above equation then relates this composite process to the transition amplitude $\langle \phi, \psi \rangle$. 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 $Z(M^\ast) = Z(M)^\ast$ sheds some light on the relation between topology change and the failure of unitarity, mentioned already in Section 2. In any $\ast$-category, we may define a morphism $f \colon x \rightarrow y$ to be unitary if $f^\ast f = 1_x$ and $ff^\ast = 1_y$. For a morphism in ${\rm Hilb}$ this reduces to the usual definition of unitarity for a linear operator. One can show that a morphism $M$ in $n{\rm Cob}$ is unitary if $M$ involves no topology change, or more precisely, if $M$ is diffeomorphic to the Cartesian product of an interval and some $(n-1)$-dimensional manifold. (The converse is true when $n$ is less than or equal to 3, but it fails in higher dimensions.) A TQFT satisfying $Z(M^\ast) = Z(M)^\ast$ maps unitary morphisms in $n{\rm Cob}$ to unitary morphisms in ${\rm Hilb}$, 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 $n{\rm Cob}$ and ${\rm Hilb}$ with the more familiar category ${\rm Set}$, whose objects are sets and whose morphisms are functions. There is no way to make ${\rm Set}$ into a $\ast$-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 $\ast$-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 $f \colon S \rightarrow S'$ is a relation such that each element of $S$ is related to exactly one element of $S'$, 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 $\ast$-category other than ${\rm Hilb}$ to help build our intuitions about $\ast$-categories, we should use not ${\rm Set}$ but ${\rm Rel}$, 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 $R$ between sets $S$ and $S'$ can be described by a matrix of truth values, namely the truth values of the propositions $yRx$ where $x \in S$ and $y \in S'$. 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 ${\rm Rel}$ with sets as objects and relations as morphisms. Furthermore, ${\rm Rel}$ becomes a $\ast$-category if we define the relation $R^\ast$ by saying that $xR^\ast y$ if and only if $yRx$. Just as the matrix for the linear operator $T^\ast$ is the conjugate transpose of the matrix for $T$, the matrix for the relation $R^\ast$ is the transpose of the matrix for $R$.

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 ${\rm Hilb}$ and ${\rm Rel}$ are fruitful source of intuitions not only about $\ast$-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

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