For many weeks I've been threatening to bring some serious math into my discussion of electrical circuits. Today I'll finally start: I'll try to use a little symplectic geometry to treat electrical circuits made of linear resistors as morphisms in a dagger-compact category.
But first, here's a great book you should all grab:
1) Jerry Shurman, Geometry of the Quintic, Wiley, New York, 1997. Also available at http://people.reed.edu/~jerry/Quintic/quintic.html
I've recommended this book before. Now Shurman has made it freely available on his website! In 1888, Felix Klein used the icosahedron to solve the general quintic equation:
a x5 + b x4 + c x3 + d x2 + e x + f = 0
Shurman's book explains Klein's ideas in a very nice way, using a bit of modern math to make them easier to understand. It's a wonderful story. For a bit about how it connects to other ideas, see "week230".
Now, on to electrical circuits. Before I get into the math, I want to remind you why it's worth bothering with. Electrical circuits are interesting and important in themselves, but that's not all! They're also great examples of physical systems built from interacting components. As we've seen starting in "week288", there are many types of systems like this. But luckily, there's a big mathematically precise analogy relating a bunch of them:
displacement flow momentum effort q q' p p' Mechanics position velocity momentum force (translation) Mechanics angle angular angular torque (rotation) velocity momentum Electronics charge current flux voltage linkage Hydraulics volume flow pressure pressure momentum Thermodynamics entropy entropy temperature temperature flow momentum Chemistry moles molar chemical chemical flow momentum potentialSo, we can go quite far by picking one kind - say, electrical circuits - and focusing on that. The rest are isomorphic.
Even if we focus on systems of one kind, there are lot of choices left:
In fact all five choices are independent, so we have 32 subjects to study! But in recent weeks I've focused on electrical circuits made of linear resistors where the voltages and currents don't depend on time. This amounts to studying
"Linear", "classical" and "static" are all ways to make our system boring - or at least, easy to understand. But "open" brings category theory into the game, since we can combine two open systems by feeding the outputs of one into the inputs of the other - and this can be seen as composing morphisms. Also, we saw last week that linear classical dissipative static open systems can be understood using the principle of least power!
Now I would like to describe a category that has linear classical dissipative static open systems as morphisms. To make things more concrete, let's think of these systems as electrical circuits made of linear resistors.
But there is a sixth choice to be made! We can treat these circuits either as "distributed" or "lumped".
What do I mean by this? Well, if we treat a circuit as "distributed", we know about every detail of it: for example, we can say it's made of 27 resistors, with particular resistances, hooked up in a certain way. But if we treat it as "lumped", we treat it as a black box with some wires hanging out. We're not allowed to peek inside the box. All we know is what it does, viewed externally.
For example, we could have a fancy circuit like this:
x | o / \ / \ o-----o \ / \ / o | xEach edge has some resistance, as explained in "week294", and the x's mark the input and output. Here's another, simpler, circuit of the same sort:
x | | | xWhen we treat these circuits as "distributed", they're different. Why? Because they look different. But when we treat them as "lumped", they might be the same! Why? Because no matter what resistances we choose for the edges of the fancy circuit, the current through it is proportional to the voltage across it... just like the simple one. So if this constant of proportionality is the same, they count as the same "lumped" circuit.
(In case you're about to object: we're only treating these circuits statically. If you feed a rapidly changing voltage across the two circuits, they will behave differently, since it takes time for changes to propagate. But that's irrelevant here.)
More precisely, let us say that two circuits built from linear resistors count as the same "lumped" circuit if:
they have the same number of inputs, say m,
they have the same number of outputs, say n
the currents on their input and output wires are given by the same function of the electrostatic potentials on those wires, say
f: Rm+n → Rm+n
Since we're looking at linear circuits, the function f will be linear. However, not every linear function f is allowed! To understand which ones are, it's good to use the principle of least power. Here we describe a lumped circuit using a function
Q: Rm+n → R
This gives the power as a function of the potentials at the inputs and outputs. We can recover f by taking the gradient of Q. Since Q is quadratic, its gradient is a linear function of position.
Which functions Q are allowed? Well, this function must be what mathematicians call a "quadratic form": in other words, a homogeneous quadratic polynomial. It must be nonnegative. And, it must not change if we add the same number to each potential.
I suspect that every function Q meeting these three conditions comes from an actual electrical circuit built from resistors. If you know, please tell me!
I don't love the third condition, because it depends heavily on the standard basis of the vector space Rm+n. I hope we can drop this condition if we allow circuits that include an extra kind of circuit element: besides resistors, also "grounds". A "ground" is a place where a wire is connected to the earth, which - by convention - has potential zero.
For example, suppose we have this circuit, with one input connected to a ground, and no outputs:
x | | =The funny little "=" thing is the ground. For this circuit the power is described by a quadratic form Q in one real variable v. If the wire has resistance R > 0, we have
Q(v) = v2 / R
What if we want Q = 0? Well, then we should use a circuit like this, instead:
x | | oIn other words: one input, no outputs, and a wire that just dangles in mid-air instead of being connected to a ground.
Using resistors and grounds, I hope we can build circuits corresponding to arbitrary nonnegative quadratic forms. So, let's try to describe a category where:
How do we compose these morphisms? Using the principle of minimum power! Given morphisms P: U → V and Q: V → W, we define their composite QP: U → W by
QP(u,w) = minv ∈ V P(u,v) + Q(v,w)
It's easy to check that this is associative: it's analogous to matrix multiplication, but with addition replacing the usual multiplication of numbers, and min replacing the usual sum. Indeed, this idea has been widely used to reformulate the principle of least action in classical mechanics as a mutant version of the "matrix mechanics" approach to quantum mechanics:
2) G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, available as arXiv:math/0507014.
3) John Baez, Spans in quantum theory, http://math.ucr.edu/home/baez/span/
But the physics is different now: we are minimizing power rather than action.
There's just one slight glitch. Our would-be category doesn't have identity morphisms! This is easy to check mathematically. Physically, the reason is clear. The identity morphism 1: R → R should correspond to a perfectly conductive wire, like this:
x | | | xThis is also called a "short circuit" - see "week294". But what's the corresponding quadratic form? Well, it doesn't exist. But the idea is that the power used by this circuit would be infinite if the potentials at the two ends were different. So, heuristically, the quadratic form should be
Q(v,w) = +∞(v-w)2
This doesn't really make sense, except as some sort of mysterious limit of the quadratic form for a resistor with resistance R:
Q(v,w) = (v-w)2 / R
as R approaches 0 from above. In other words, the perfectly conductive wire is the limiting case of a resistor.
So, what can we do? There are lots of options. One is to note that perfectly conductive wires don't really exist, and be content with what we've got. Namely: a "semicategory", which is like a category, but without identity morphisms.
Most people don't like semicategories. So, if you're like most people, you'll be relieved to hear that any semicategory can be made into a category by formally throwing in identity morphisms. We don't lose any information this way. Even better, a category arises from a semicategory in this way iff it has this special property: whenever the composite of two morphisms is an identity morphism, both must be identity morphisms. So, semicategories aren't really more general than categories. We can think of them as categories with this extra property!
If we extend our semicategory to a category this way, the result has some nice properties. First, it's a "monoidal category", meaning roughly a category with tensor products:
4) nLab, Monoidal category, http://ncatlab.org/nlab/show/monoidal+category
The tensor product corresponds to setting two circuits side by side:
| | | | -------- ------------ | | | | | | | | -------- ------------ | | | | | |Mathematically, the tensor product of objects V and W is their direct sum V+W, while the tensor product of morphisms Q:V → W and Q':V' → W' is the quadratic form Q+Q' given by:
(Q+Q')(v,v',w,w') = Q(v,w) + Q'(v',w')
Our category also has "duals for morphisms". Intuitively, this means that we can take any circuit Q: V → W built from resistors:
| -------- | Q | -------- | |and reflect it across a horizontal line, switching inputs and outputs like this:
| | -------- | Q† | -------- |to obtain a new circuit Q†: W → V. Mathematically this operation is defined as follows:
Q†(v,w) = Q(w,v)
A category with duals for morphisms is usually called a "dagger-category". It's easy to check that our category is one of those:
5) nLab, Dagger-category, http://ncatlab.org/nlab/show/dagger-category
However, our category has some defects. First of all, there's no morphism corresponding to two perfectly conductive wires that cross like this:
\ / \ / / / \ / \If we had that, we'd get a "symmetric monoidal category":
6) nLab, Symmetric monoidal category, http://ncatlab.org/nlab/show/symmetric+monoidal+category
Our category so far also lacks a morphism corresponding to a perfectly conductive bent wire like this:
| | | | \_/or like this:
_ / \ | | | |If we had these morphisms, obeying the obvious "zig-zag identities":
| _ | _ | | / \ | / \ | | / | = | = | \ | \_/ | | | \_/ | | |then our monoidal category would have "duals for objects", in the sense explained back in "week89".
It seems reasonable to allow all these circuits made of perfectly conductive wires, even though they correspond to idealized limits of circuits we can actually build. They don't cause any mathematical contradictions. And they should give a very nice category: a symmetric monoidal category with duals for objects and morphisms. Categories of this sort are called "dagger-compact":
7) nLab, Dagger-compact category, http://ncatlab.org/nlab/show/dagger+compact+category
Dagger-compact categories are very important in physics. A classic example is the category of finite-dimensional Hilbert spaces, with linear operators as morphisms, and the usual tensor product of vector spaces. Another example is the category of (n-1)-dimensional compact oriented manifolds, with n-dimensonal cobordisms as morphisms. The interplay between these examples is important in topological quantum field theory. People like Samson Abramsky, Bob Coecke, Chris Heunen, Dusko Pavlovich, Peter Selinger and Jamie Vicary have done a lot to formulate all of quantum mechanics in terms of dagger-compact categories. Here are the fundamental references:
8) Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols, in Proceedings of the 19th IEEE conference on Logic in Computer Science (LICS04), IEEE Computer Science Press (2004). Also available at arXiv:quant-ph/0402130.
9) Peter Selinger, Dagger compact closed categories and completely positive maps, in Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), ENTCS 170 (2007), 139-163. Also available at http://www.mscs.dal.ca/~selinger/papers.html#dagger
You can use dagger-compact categories to clarify what it means, physically, for a complex Hilbert space to be equipped with an orthonormal basis:
10) Bob Coecke, Dusko Pavlovic, and Jamie Vicary, A new description of orthogonal bases, available as arXiv:0810.0812.
You can use them to explain the traditional approach to quantum logic in terms of orthomodular lattices:
11) Chris Heunen, Categorical Quantum Models and Logics, Amsterdam University Press, 2009. Also available at http://www.comlab.ox.ac.uk/people/chris.heunen/about.html
You can even use them to motivate the appearance of complex numbers in quantum mechanics:
12) Jamie Vicary, Completeness and the complex numbers, available as arXiv:0807.2927.
So, if there is a dagger-compact category of electrical circuits, we should find it and study it. I've decided that category theory should not be saved for fancy stuff like the foundations of quantum theory. It can serve as a general language for studying systems made of parts, and we should take full advantage of it!
Let's try. Let's take the category described so far and supplement it with a "cup":
| | | | \_/a "cap":
_ / \ | | | |and a "symmetry":
\ / \ / / / \ / \We could formally throw in these morphisms, just like we threw in identities. But there is a less artificial solution which solves all these problems in one blow. We can take a lesson from symplectic geometry, and notice that nonnegative quadratic forms are a special case of something called "Lagrangian correspondences". These include identity morphisms as well as the cap, cup, and symmetry.
Let me explain! Suppose Q is a quadratic form on a vector space V. Then its differential dQ is a one-form, so it gives an element of V* for each point of V. But since Q is quadratic, its differential depends linearly on the point of Q, so we get a linear map
dQ: V → V*
This is a highbrow formulation of something I already told you in lowbrow way. But now let's go a bit further. The graph of dQ is a linear subspace of the cartesian product V × V*. But V × V* is better than a mere vector space. We can think of it as the cotangent bundle T*V. So, it's a "symplectic" vector space:
13) Wikipedia, Symplectic vector space, http://en.wikipedia.org/wiki/Symplectic_vector_space
Namely, it has a "symplectic structure" - that is, a nondegenerate antisymmetric bilinear form ω given by:
ω((v,f),(v',f')) = f(v') - f'(v)
And it's a general fact that the graph of any quadratic form on V is a "Lagrangian" subspace of T*V = V × V*, meaning a maximal subspace on which ω vanishes.
But, there are Lagrangian subspaces of T*V that are not the graphs of quadratic forms. There are also "limits" of graphs of quadratic forms - precisely the sort of thing we want now! After all, every circuit made of perfectly conductive wires can be thought of as a limit of circuits made of resistors.
So, we can try a category where:
Remember that an element of V describes the potentials on the input wires of our circuit, while W does the same job for the output wires. An element of V+W describes the potentials on input and output wires. Currents live in the dual vector space, so an element of T*(V+W) describes the potentials and currents on input and output wires. The Lagrangian subspace describes the potentials and currents that are allowed by our circuit.
We can also change perspective and say:
Here an element of T*V describes the potentials and currents on the input wires, while T*W does the same job for the output wires. As before, the Lagrangian subspace of T*V × T*W describes the potentials and currents that are allowed by our circuit. But now we can think of it as relation between T*V and T*W. This makes it clear how to compose morphisms: we compose them according to the usual method for composing relations.
This perspective will be familiar to symplectic geometers who know about "Lagrangian correspondences", also known as "canonical relations". We're studying a special case of those, namely the linear case. If you want to learn more, try:
14) Alan Weinstein, Symplectic categories, available as arXiv:0911.4133.
Following in part some (unpublished) ideas of the author, Guillemin and Sternberg observed that the linear canonical relations (i.e., lagrangian subspaces of products of symplectic vector spaces) could be considered as the morphisms of a category, and they constructed a partial quantization of this category (in which lagrangian subspaces are enhanced by half-densities). The automorphism groups in this category are the linear symplectic groups, and the restriction of the Guillemin-Sternberg quantization to each such group is a metaplectic representation. On the other hand, the quantization of certain compositions of canonical relations leads to ill-defined operations at the quantum level, such as the evaluation of a delta "function" at its singular point, or the multiplication of delta functions.
Here's the reference to Guillemin and Sternberg:
15) Victor Guillemin and Shlomo Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), 915-955.
I learned symplectic geometry from Guillemin in grad school, so I'm happy to see it being applied to resistors! And the discussion of quantization suggests a way to understand resistors quantum-mechanically. In fact there's a bit of literature on this subject already:
16) Michel H. Devoret, Quantum fluctuations in electrical circuits, in Quantum Fluctuations, eds. S. Reynaud, E. Giacobino and J. Zinn-Justin, Elsevier, 1997. Also available at http://qulab.eng.yale.edu/documents/reprints/Houches_fluctuations.pdf
But for now, the classical theory is interesting enough. I guess I need to start by checking my claim:
Claim: there is a dagger-compact category where:
- An object is a finite-dimensional real vector space.
- A morphism S: V → W is a Lagrangian subspace of T*V × T*W.
- We compose morphisms using composition of relations.
- The tensor product is given by direct sum.
- The symmetry is the obvious thing.
- The dagger of a subspace of T*V × T*W is the corresponding subspace of T*W × T*V.
This category is mathematically elegant, but "too big", because its morphisms include a lot more than Lagrangian subspaces coming from nonnegative quadratic forms, and limits of these. There's a well-known topology on the set of Lagrangian subspaces of a symplectic vector space, so the concept of limit is well-defined here. If we restrict attention to Lagrangian subspaces coming from nonnegative quadratic forms, and limits of these, do we get a subcategory? It might seem obvious - but shockingly, composition is not continuous with respect to this well-known topology! Weinstein gives a counterexample. So, there's something nontrivial to check.
If we do get a subcategory, will it still be a dagger-compact category? Yes, I think so, because it contains the cup
| | | | \_/and cap:
_ / \ | | | |and symmetry:
\ / \ / / / \ / \So, this would be a very nice thing.
I thank James Dolan, Peter Selinger, Alan Weinstein and Simon Willerton for helping me figure out these ideas.
Addendum: I thank Mikael Vejdemo-Johansson for catching a typo.
For more discussion visit the n-Category Café.
Reality has been around since long before you showed up. Don't go calling it nasty names like "bizarre" or "incredible". The universe was propagating complex amplitudes through configuration space for ten billion years before life ever emerged on Earth. Quantum physics is not "weird". You are weird. - Eliezer Yudkowsky
© 2010 John Baez