The Three-Fold Way (Part 5)
Posted by John Baez
You can now see the paper these blog entries are based on:
- John Baez, Division algebras and quantum theory.
But the blog entries have more jokes! So far, I’ve explained how certain complex representations of groups can be seen as arising from real or quaternionic representations. This gives a sense in which ordinary complex quantum theory subsumes the real and quaternionic theories. But there’s also a sense in which all three theories have equal priority. This idea can be seen already at the level of Hilbert spaces, even before group representations enter the game.
For this we need to think about categories of Hilbert spaces. As usual, let $\mathbb{K}$ be either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Now, let $Hilb_\mathbb{K}$ be the category where:
- an object is a $\mathbb{K}$-Hilbert space;
- a morphism is a bounded $\mathbb{K}$-linear operator.
I want to show you how any one of the categories $Hilb_\mathbb{R}$, $Hilb_\mathbb{C}$ and $Hilb_\mathbb{H}$ can be embedded in any other. This means that a Hilbert space over any one of the three normed division algebras can be seen as Hilbert space over any other, equipped with some extra structure!
So if you ask which is fundamental: real, complex or quaternionic quantum theory, there’s a certain sense in which the answer is: take your pick!
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: $T J = J' T$.
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.
Re: The Three-Fold Way (Part 5)
It seems to me that there is a typo on top of page 28 of the arXiv paper:
Instead of Hilb_R ⊗ Hilb_R → Hilb_C
it should say Hilb_R ⊗ Hilb_C → Hilb_C.