The n-Category Café

How are $Hilb_\mathbb{R}$, $Hilb_\mathbb{C}$ and $Hilb_\mathbb{H}$ related?

Of course the complex HIlbert space $\mathbb{C}^n$ has an underlying real Hilbert space $\mathbb{R}^{2n}$, and the quaternionic Hilbert space $\mathbb{H}^n$ has an underlying complex Hilbert space $\mathbb{C}^{2n}$. But there’s a slightly more sophisticated way to say what’s going on.

Let’s start with the chain of inclusions $$\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} .$$ Thanks to the first inclusion, any complex vector space has an underlying real vector space. In other words: if we have a complex vector space, and we deliberately forget how to do scalar multiplication of vectors by complex numbers, and only remember how to multiply them by real numbers, it becomes a real vector space! Similarly, any quaternionic vector space becomes a complex one, thanks to the second inclusion.

The same is true for Hilbert spaces. To make the underlying real vector space of a complex Hilbert space into a real Hilbert space, we take the real part of the original complex inner product, defined by $$\mathrm{Re}(a + b i) = a .$$ Everyone knows that; a bit less familiar is how the underlying complex vector space of a quaternionic Hilbert space becomes a complex Hilbert space. Here we need to take the complex part of the original quaternionic inner product, defined by $$\mathrm{Co}(a + b i + c j + d k) = a + b i .$$ One can check that these constructions give functors $$Hilb_\mathbb{H} \to Hilb_\mathbb{C} \to Hilb_\mathbb{R} .$$

A bit more formally, we have a commutative triangle of homomorphisms:

$$\begin{aligned} \mathbb{C} &\overset{\quad \beta \quad}{\to} & \mathbb{H} \\ \alpha \uparrow & \nearrow \gamma \\ \mathbb{R} \end{aligned}$$

There is only one choice of the homomorphisms $\alpha$ and $\gamma$. There are many choices of $\beta$, since we can map $i \in \mathbb{C}$ to any square root of $-1$ in the quaternions. However, all the various choices of $g$ are the same up to an automorphism of the quaternions. That is, given two homomorphisms $\beta,\beta' : \mathbb{C} \to \mathbb{H}$, we can always find an automorphism $\theta : \mathbb{H} \to \mathbb{H}$ such that $$\beta' = \theta \circ \beta .$$ So, nothing important depends on the choice of $\beta$. Let us make a choice—say the standard one, with $\beta(i) = i$ — and use that.

Our commutative triangle of homomorphisms gives a commutative triangle of functors:

$$\begin{aligned} Hilb_{\mathbb{C}} &\overset{\quad \beta^*}{\leftarrow} & Hilb_{\mathbb{H}} \\ \alpha^* \downarrow & \swarrow \gamma^* \\ Hilb_{\mathbb{R}} \end{aligned}$$

Now, recall that a functor $F: C \to D$ is faithful if given two morphisms $f,f' : c \to c'$ in $C$, $F(f) = F(f')$ implies that $f = f'$. It is easy to see that the functors $\alpha^*, \beta^*$ and $\gamma^*$ are all faithful. When $F: C \to D$ is faithful, we say that $C$ is faithfully embedded in $D$, and we can think of objects of $C$ as objects of $D$ equipped with extra structure. For our three functors above we can describe this extra structure quite explicitly. This lets us describe Hilbert spaces for a larger normed division algebra in terms of Hilbert spaces for a smaller one. None of this particularly new or difficult: the key ideas are all in Adams’ Lectures on Lie Groups.

First we consider the extra structure possessed by the underlying real Hilbert space of a complex Hilbert space:

Theorem: The functor $$\alpha^* : Hilb_\mathbb{C} \to Hilb_\mathbb{R}$$ is faithful, and $Hilb_\mathbb{C}$ is equivalent to the category where:

• an object is a real Hilbert space $H$ equipped with a unitary operator $J: H \to H$ with $J^2 = -1$.
• a morphism $T : H \to H'$ is a bounded real-linear operator preserving this exta structure: $T J = J' T$.

This extra structure $J$ is often called a complex structure.

Next we consider the extra structure possessed by the underlying complex Hilbert space of a quaternionic Hilbert space. For this we need to generalize the concept of an antiunitary operator. First, given $\mathbb{K}$-vector spaces $V$ and $V'$, we define an antilinear operator $T : V \to V'$ to be a map with $$T(v x + w y) = T(v)x^* + T(w)y^*$$ for all $v,w \in V$ and $x,y \in \mathbb{K}$. Then, given $\mathbb{K}$-Hilbert spaces $H$ and $H'$, we define an antiunitary operator $T: H \to H'$ to be an invertible antilinear operator with $$\langle T v, T w \rangle = \langle w, v \rangle$$ for all $v,w \in H$.

Theorem: The functor $$\beta^* : Hilb_\mathbb{H} \to Hilb_\mathbb{C}$$ is faithful, and $Hilb_\mathbb{H}$ is equivalent to the category where:

• an object is a complex Hilbert space $H$ equipped with an antiunitary operator $J : H \to H$ with $J^2 = -1$;
• a morphism $T : H \to H'$ is a bounded complex-linear operator preserving this extra structure: $T J = J' T$.

This extra structure $J$ is often called a quaternionic structure. We have seen it already in our study of the Three-Fold Way.

Finally, we consider the extra structure possessed by the underlying real Hilbert space of a quaternionic Hilbert space. This can be understood by composing the previous two theorems:

Theorem: The functor $$\gamma^* : Hilb_\mathbb{H} \to Hilb_\mathbb{R}$$ is faithful, and $Hilb_\mathbb{H}$ is equivalent to the category where:

• an object is a real Hilbert space $H$ equipped with two unitary operators $J,K : H \to H$ with $J^2 = K^2 = -1$ and $J K = -K J$;
• a morphism $T: H \to H'$ is a bounded real-linear operator preserving this extra structure: $T J = J' T$ and $T K = K' T$.

This extra structure could also be called a quaternionic structure, as long as we remember that a quaternionic structure on a real Hilbert space is different than one on a complex Hilbert space! Of course if we define $I = JK$, then $I, J,$ and $K$ obey the usual quaternion relations.

The functors discussed so far all have adjoints, which are in fact both left and right adjoints: $$\begin{aligned} Hilb_{\mathbb{C}} &\overset{\quad \beta_*}{\to} & Hilb_{\mathbb{H}} \\ \alpha_* \uparrow & \nearrow \gamma_* \\ Hilb_{\mathbb{R}} \end{aligned}$$ These adjoints can easily be defined using the theory of bimodules. As vector spaces, we have:

$$\begin{array}{ccl} \alpha_* (V) &=& V \otimes {{}_\mathbb{R}\mathbb{C}_{\mathbb{C}}} \\ \beta_*(V) &=& V \otimes {{}_\mathbb{C}\mathbb{H}_{\mathbb{H}}} \\ \gamma_*(V) &=& V \otimes {{}_\mathbb{R}\mathbb{H}_{\mathbb{H}}} \\ \end{array}$$

In the first line here, $V$ is a real vector space, or in other words, a right $\mathbb{R}$-module, while ${{}_\mathbb{R}\mathbb{C}_{\mathbb{C}}}$ denotes $\mathbb{C}$ regarded as a $\mathbb{R}$-$\mathbb{C}$-bimodule. Tensoring these, we obtain a right $\mathbb{C}$-module, which is the desired complex vector space. The other lines work analogously. It is then easy to make all these vector spaces into Hilbert spaces. I can’t resist mentioning that our previous functors can be described in a similar way, just by turning the bimodules around:

$$\begin{array}{ccl} \alpha^*(V) &=& V \otimes {{}_\mathbb{C}\mathbb{C}_{\mathbb{R}}} \\ \beta^*(V) &=& V \otimes {{}_\mathbb{H}\mathbb{H}_{\mathbb{C}}} \\ \gamma^*(V) &=& V \otimes {{}_\mathbb{H}\mathbb{H}_{\mathbb{R}}} \end{array}$$ But instead of digressing into this subject (called Morita theory), all I want to do now is mention that the functors $\alpha_*, \beta_*$ and $\gamma_*$ are also faithful. This lets us describe Hilbert spaces for a smaller normed division algebra in terms of Hilbert spaces for a bigger one!

We begin with the functor $\alpha_*$, which is called complexification:

Theorem: The functor $$\alpha_* : Hilb_\mathbb{R} \to Hilb_\mathbb{C}$$ is faithful, and $Hilb_\mathbb{R}$ is equivalent to the category where:

• an object is a complex Hilbert space $H$ equipped with a antiunitary operator $J : H \to H$ with $J^2 = 1$;
• a morphism $T: H \to H'$ is a bounded complex-linear operator preserving this exta structure: $T J = J' T$.

The extra structure $J$ here is often called a real structure. We have seen it already in our study of the Three-Fold Way.

Next let’s look at the functor from complex to quaternionic Hilbert spaces. It has no name, as far as I know:

Theorem: The functor $$\beta_* : Hilb_\mathbb{C} \to Hilb_\mathbb{H}$$ is faithful, and $Hilb_\mathbb{C}$ is equivalent to the category where:

• an object is a quaternionic Hilbert space $H$ equipped with a unitary operator $J$ with $J^2 = -1$;
• a morphism $T: H \to H'$ is a bounded quaternion-linear operator preserving this extra structure: $T J = J' T$.

This result is less well-known than the previous ones, so let me sketch a proof:

Proof: Suppose $H$ is a quaternionic Hilbert space equipped with a unitary operator $J$ with $J^2 = -1$. Then $J$ makes $H$ into a right module over the complex numbers, and this action of $\mathbb{C}$ commutes with the action of $\mathbb{H}$, so $H$ becomes a right module over the tensor product of $\mathbb{C}$ and $\mathbb{H}$, considered as algebras over $\mathbb{R}$. But this is isomorphic to the algebra of $2 \times 2$ complex matrices. The matrix $$\left(\begin{aligned} 1 & 0 \\ 0 & 0 \end{aligned} \right)$$ projects $H$ down to a complex Hilbert subspace $H_\mathbb{C}$ whose complex dimension matches the quaternionic dimension of $H$. By applying arbitrary $2 \times 2$ complex matrices to guys in this subspace we get back everything in $H$, so $\beta_* H_\mathbb{C}$ is naturally isomorphic to $H$.   █

Composing $\alpha_*$ and $\beta_*$, we obtain the functor from real to quaternionic Hilbert spaces. I’ve seen this called quaternification — or occasionally ‘quaternization’, but that means something else in chemistry!

Theorem: The functor $$\gamma_* : Hilb_\mathbb{R} \to Hilb_\mathbb{H}$$ is faithful, and $Hilb_\mathbb{R}$ is equivalent to the category where:

• an object is a quaternionic Hilbert space $H$ equipped with two unitary operators $J, K$ with $J^2 = K^2 = -1$ and $J K = -K J$.
• a morphism $T: H \to H'$ is a bounded quaternion-linear operator preserving this extra structure: $T J = J' T$ and $T K = K' T$.

Again, let me sketch a proof:

Proof: The operators $J, K$ and $I = J K$ give make $H$ into a left $\mathbb{H}$-module. Since this action of $\mathbb{H}$ commutes with the existing right $\mathbb{H}$-module structure, $H$ becomes a module over the tensor product of $\mathbb{H}$ and $\mathbb{H}^{op} \cong \mathbb{H}$, considered as algebras over $\mathbb{R}$. But this tensor product is isomorphic to the algebra of $4 \times 4$ real matrices! So, the matrix $$\left(\begin{aligned} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{aligned}\right)$$ projects $H$ down to a real Hilbert subspace $H_\mathbb{R}$ whose real dimension matches the quaternionic dimension of $H$. By applying arbitrary $4 \times 4$ real matrices to guys in this subspace we get back everything in $H$, so $\gamma_* H_\mathbb{R}$ is naturally isomorphic to $H$.   █

Finally, it is worth noting that some of the six functors we have described have additional nice properties:

• The categories $Hilb_\mathbb{R}$ and $Hilb_\mathbb{C}$ are symmetric monoidal categories, meaning roughly that they have well-behaved tensor products. The complexification functor $\alpha^*: Hilb_\mathbb{R} \to Hilb_\mathbb{C}$ is a symmetric monoidal functor, meaning roughly that it preserves tensor products.
• The categories $Hilb_\mathbb{R}, Hilb_\mathbb{C}$ and $Hilb_\mathbb{H}$ are dagger-categories, meaning roughly that any morphism $T : H \to H'$ has a Hilbert space adjoint $T^\dagger : H' \to H$ such that $$\langle T v, w \rangle = \langle v, T^\dagger w \rangle$$ for all $v \in H$, $w \in H'$. All six functors preserve this dagger operation.
• For $Hilb_\mathbb{R}$ and $Hilb_\mathbb{C}$, the dagger structure interacts nicely with the tensor product, making these categories into dagger-compact categories, and the functor $\alpha^*$ is compatible with this as well.

For precise definitions of the terms here, click on the links. I’ve spent a lot of time trying explain how these concepts unify physics with topology and other subjects:

• John Baez and Aaron Lauda, A prehistory of $n$-categorical physics, to appear in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson, Cambridge U. Press.
• John Baez and Mike Stay, M. Stay, Physics, topology, logic and computation: a Rosetta Stone, in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics 813, Springer, Berlin, 2000, pp. 95–174.

For more on dagger-categories see also these papers:

• Samson Abramsky, Abstract scalars, loops, and free traced and strongly compact closed categories, in Proceedings of CALCO 2005, Lecture Notes in Computer Science 3629, Springer, Berlin, 2005, 1–31.
• Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols.
• Peter Selinger, Dagger compact closed categories and completely positive maps, Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Elsevier, 2007, pp. 139–163.

and this wonderful book:

• Bob Coecke, editor, New Stuctures for Physics, Lecture Notes in Physics 813, Springer, Berlin, 2000, pp. 95–174.

The three-fold way is best appreciated with the help of these category-theoretic ideas. A more $n$-categorical treatment of symplectic and orthogonal structures can be found in my old paper on 2-Hilbert spaces:

• John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125–189.

If I’m feeling exceptionally energetic I may say more about the $n$-categorical aspects, Morita theory, and so on. More likely, I’ll wrap up the story next time by saying how the Three-Fold Way solves some of the problems of real and quaternionic quantum theory.