Circuits of Linear Resistors as a Dagger Compact Category. John C. Baez

Transcription

1 Circuits of Linear Resistors as a Dagger Compact Category John C. Baez Centre for Quantum Technologies National University of Singapore Singapore and Department of Mathematics University of California Riverside CA Brendan Fong Department of Computer Science University of Oxford United Kingdom OX1 3QD baez@math.ucr.edu, brendan.fong@cs.ox.ac.uk September 2013 Abstract There is a dagger-compact category whose morphisms are equivalence classes of electrical circuits made of linear resistors. To construct this category, we begin by recalling Ohm s law and Kirchhoff s laws, and show that we may also understand circuits through the so-called principle of minimum power. We show that a lumped circuit made of linear resistors that is, a circuit of this sort treated as a black box whose inside workings we cannot see amounts mathematically to a Dirichlet form: a finite-dimensional real vector space with a chosen basis and a quadratic form obeying some conditions. There are rules for composing and tensoring Dirichlet forms, which correspond to the operations of composing circuits in series and setting circuits side by side. However, these rules do not give a category, because the would-be identity morphisms are made of wires with zero resistance, which fall outside the Dirichlet form framework. The most elegant solution is to treat Dirichlet forms as a special case of Lagrangian correspondences. This leads to a dagger-compact category of electrical circuits that can include wires with zero resistance. 1 Introduction Control theory concerns itself with the following picture: Two questions present themselves: analysis: given a system, what is the relationship between input and output? synthesis: given an desired relationship between input and output, how do we build a system that produces this relationship. Notions of effort and flow. product equals power displacement flow momentum effort q q p p Mechanics (translation) position velocity momentum force Mechanics (rotation) angle angular velocity angular momentum torque Electronics charge current flux linkage voltage Hydraulics volume flow pressure momentum pressure Thermodynamics entropy entropy flow temperature momentum temperature Chemistry moles molar flow chemical momentum chemical potential 1

2 So, 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: We can study fully general systems, or restrict our attention to linear ones. We can study quantum-mechanical systems, or classical ones. We can study dissipative systems (where energy is not conserved), or conservative ones (where it is). We can study dynamical systems or static ones. We can study open systems or closed ones. Here we study electrical circuits made of linear resistors where the voltages and currents don t depend on time. This amounts to studying linear classical dissipative static open systems. 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. We restrict to resistors to elucidate key ideas. Later work will involve general circuits: by moving from the real numbers to complex numbers we can talk about impedance rather than resistance, and this allows discussion of more general circuits, ones that include inductors and capacitors as well. Principle of minimum power. consider two circuits the same if they behave the same. taking equivalence classes is known as lumping In discussion of large scale systems, Thoma argues for the importance of breaking down large, complex systems into multilevel systems, with the systems divided into local control units whose actions are coordinated by a higher level coordinator system [11]. Compositionality. Spivak: operad of wiring diagrams [24]. We assume familiarity with compact symmetric monoidal categories and their graphical calculi. 2 Circuits of linear resistors The concept of an abstract electrical circuit made of linear resistors is well-known in electrical engineering, but we shall need to formalize it with more precision than usual. The basic idea is that a circuit of linear resistors is a graph whose edges are labelled by positive real numbers called resistances, and whose sets of vertices is equipped with two subsets: the inputs and outputs. This unfolds as follows. A circuit of resistors looks like this: 1Ω 2Ω 1Ω We can consider this a labelled graph, with each resistor an edge of the graph, its resistance its label, and the vertices of the graph the points at which resistors are connected. Composition of circuits involves creating connections between distinct circuits. To do this we first mark points at which connections can be made by denoting some vertices as input and output terminals: inputs 1Ω 2Ω outputs 1Ω Then, given a second circuit, we may choose a relation between the output set of the first and the input set of this second circuit, such as the simple relation of the single output vertex of the circuit 2

3 above with the single input vertex of the circuit below. 1Ω inputs outputs 3Ω We compose the two circuits by identifying output and input vertices according to this relation, giving in this case the composite circuit: inputs 1Ω 2Ω 1Ω outputs 1Ω 3Ω We will formalise these notions of marking terminals and composition of circuits using cospans and push forwards. Treating these cospans as morphisms between finite sets, we shall show that this defines a dagger compact category. 2.1 L-labelled graphs Given a set of labels L, we shall define a category L-Graph of L-labelled directed multigraphs. In this paper we will refer to directed multigraphs simply as graphs. So, define a graph to be a pair of functions s, t : E N where E and N are finite sets. An L-graph is a graph further equipped with a function r : E L. r L E s N We call elements of E edges and elements of N vertices or nodes. We say that the edge e E has source s(e) and target t(e), and also say that e is an edge from s(e) to t(e). Given L-graphs Γ = (E, N, s, t, r) and Γ = (E, N, s, t, r ), a morphism of L-graphs Γ Γ is a pair of functions ɛ, v such that the following diagrams commute: t L r r E ɛ E E ɛ s N v E N s E ɛ t N v E N. t These L-graphs and their morphisms form the category L-Graph. Using results about colimits in the category of sets, it is straightforward to check that this category has finite colimits. We will later make use of the notion of connectedness in graphs, and so we give a definition here. Given two vertices v, w N of a graph, a path from v to w is a finite sequence of vertices v = v 0, v 1,..., v n = w and edges e 1,..., e n such that for each 1 i n, either e i is an edge from v i to v i+1, or an edge from v i+1 to v i. A subset S of the vertices of a graph is connected if for each pair of vertices in S, there is a path from one to the other. A connected component of a graph is a maximal connected subset of its vertices. Note that in the theory of directed graphs the qualifier strongly is commonly used before the word connected in these two definitions. As we never consider any other sort of connectedness, we omit this qualifier. 3

4 2.2 Cospans In this paper we are particularly interested in the case when our set L of labels is the set (0, ) of positive real numbers. Here we view each edge as a wire between the source and target vertices, and consider each label as assigning a resistance to the wire, so that the objects of (0, )-Graph may be considered as networks of resistors. Cospans provide the machinery for combining these networks in series. Recall that a cospan from X to Y in a category C is an object C in C with a pair of morphisms f : X C, g : Y C. C X f We shall refer to X and Y as the feet, and C as the apex of the cospan. When such pushforwards exist, cospans may be composed using the pushforward from the common foot: given cospans (C, f : X C, g : Y C) for X to Y and (C, f : Y C, g : Z C ) from Y to Z, their composite cospan is (P, i f : X P, i g : Z P ), where (P, i : C P, i : C P ) is the pushforward of the square: g Y. i P i f C g f C g X Y Z. A map of cospans is a morphism h : C C in C between the apices of two cospans (C, f : X C, g : Y C) and (C, f : X C, g : Y C ) with the same feet, such that f C g X h Y f C g commutes. We will also later discuss spans in a category C. The definitions of these, their composition, and their maps are dual to the corresponding definitions above. 2.3 A category of circuits Following the work of Stay, L-labelled graphs, cospans in L-Graph, and maps of cospans form a compact closed bicategory Cospan(L-Graph) [25]. We shall predominantly concern ourselves with the compact closed subbicategory ResCirc of Cospan((0, )-Graph), which we define as follows. Definition 2.1. The bicategory ResCirc is the full subbicategory of Cospan((0, )-Graph) with objects those (0, )-graphs with no edges. We call the 1-morphisms of ResCirc circuits. It can be shown that every object in Cospan((0, )-Graph) is self-dual, and this implies that ResCirc is again a compact closed bicategory [25]. 4

5 Since (0, )-graphs with no edges are uniquely determined by their vertex sets, an object of ResCirc may simply be thought of as a finite set, while a 1-morphism between finite sets X and Y may be thought of as comprising a (0, )-graph Γ = (E, N, s, t, r) with functions i : X N and o : Y N. We call the subsets N = i(x) and N + = o(y ) the inputs and outputs of the circuit, and think of them as points to which we might attach other circuits. Note that N and N + need not be disjoint. Often the difference between inputs and outputs will not matter, so we also define N = N N +, and collectively call elements of this set terminals. Abusing notation slightly, we also write ResCirc for the 1-categorical decategorification of this bicategory, with objects finite sets, and morphisms isomorphism classes of cospans in (0, )-Graph with feet such objects. This category is also compact closed, with dual objects and their units and counits given by the compactness of the bicategory. The compactness of our category of circuits of linear resistors captures the interchangeability between individual input and output terminals of circuits that is, the fact that we can choose any input terminal to our circuit and consider it instead as an output terminal, and vice versa. Exploring the structure more explicitly, the monoidal product for ResCirc is simply the disjoint union of finite sets on objects, and the disjoint union of graphs for morphisms. The monoidal unit is the empty set. Observe that given a morphism f : X Y in some category C, we may construct the cospan Y X from X to Y, with this cospan an isomorphism if and only if f is, and with inverse f id Y Y id Y Y f Y X in this case. The structural morphisms for the symmetric monoidal structure may be specified by set functions in this way, with for example the associator the cospan induced by the canonical functions (X Y ) Z X (Y Z), and the braiding the cospan induced by the canonical functions X Y Y X. The existence of duals may be seen as follows. Let X be a finite set, and let a : X X X be the function given by taking the coproduct of two identity functions X X, and let! : X be the unique such map. We may take X as its own dual object, with unit and counit given by X X! a and a! X X X X. It is readily verified that these cospans do indeed form a unit and counit for the self-dual object X. 2.4 The dagger structure A dagger functor expresses the idea that the direction of morphisms can be reversed: through a dagger functor each morphism specifies a map from its codomain to its domain, in addition to the map it is from its domain to its codomain. This is true of electrical circuits: if we like we may treat the set of inputs as the set of outputs instead, and the set of outputs as the set of inputs. Recall that a dagger functor is an involutive, contravariant endofunctor that is the identity on objects. That is, given a category C, a dagger functor is a contravariant functor : C C such 5

6 that (A) = A for all objects A ObC and ( (f)) = f for all morphisms f in C. We write f for (f). We say that a morphism f is unitary if the dagger functor takes it to its inverse, that is, if f = f 1. When other structure is present, we prefer this dagger to play nice with it. A dagger symmetric monoidal category is a symmetric monoidal category equipped with a dagger symmetric monoidal functor that is, a dagger functor that coherently preserves the symmetric monoidal structure. Concretely, this requires that the dagger functor preserve the tensor product, and that the associator, unitors, and braiding of the symmetrical monoidal category be unitary. Furthermore, letting L and R be dual objects of a dagger symmetric monoidal category, with monoidal unit I, braiding σ L,R : L R R L, and unit η : I R L and counit ɛ : L R I, we say that L and R are dagger dual if η = σ ɛ. A dagger compact category is a dagger symmetric monoidal category in which every object has a dagger dual. Let : ResCirc ResCirc be the functor that acts as the identity on objects and swaps the feet of cospans, so : i Γ o o Γ i X Y Y X. This is an involutive contravariant functor that is the identity on objects, and hence a dagger functor. A cursory glance at our discussion of the structural morphisms for ResCirc in the previous subsection yields the following proposition. Proposition 2.2. The category ResCirc is a dagger compact category. 3 Equivalences of circuits So far we have modelled our circuits of linear resistors according to their physical form. But another, often more relevant, way to understand a circuit is by its function. To an electric circuit we associate two quantities to each edge: voltage and current. However, circuits are subject to governing laws that imply these quantities must have certain relationships; we are not free to choose voltages and currents as we like. From the perspective of control theory we are particularly interested in the values these quantities take at the terminals, and how altering one value will affect the other values. We shall see that the potentials at the terminals of the circuit determine the flow of current in or out of these terminals. We call two circuits equivalent when they determine the same relationship. Our task in this section is to explore when two circuits are equivalent. We begin by discussing how to find the function of a circuit from its form, advocating in particular the perspective of the principle of minimum power. In the following let X and Y be finite sets, and Γ = (E, N, s, t, r), together with maps i : X N and o : Y N, be a circuit. In particular E will always stand for the edge set of the apex of a circuit, and N for the set of nodes. For a finite set S, we shall write R S for the vector space of functions ψ : S R. Elementary circuit theory is most frequently presented in terms of voltages and currents on the edges or wires of the circuit. These are functions V R E and I R E respectively. We use I for intensity of current, following Ampère. 3.1 Ohm s law and Kirchhoff s laws In 1827 Georg Ohm published a book which included a relation between the voltage and current for circuits made of resistors [16]. At the time, the critical reception was harsh: one contemporary called Ohm s work a web of naked fancies, which can never find the semblance of support from even 6

7 the most superficial of observations, and the German Minister of Education said that a professor who preached such heresies was unworthy to teach science [6, 10]. However, a simplified version of his relation is now widely used under the name of Ohm s law. We say that Ohm s law holds if for all edges e E the voltage and current functions of a circuit obey: V (e) = r(e)i(e). Kirchhoff s laws date to Kirchhoff in 1845, generalising Ohm s work. They were in turn generalised into Maxwell s equations a few decades later []. We say Kirchhoff s voltage law holds if there exists φ R N such that V (e) = φ(t(e)) φ(s(e)). We call the function φ a potential, and think of it as assigning an electrical potential to each node in the circuit. The voltage then arises as the differences in potentials between adjacent nodes. If Kirchhoff s voltage law holds for some voltage V, the potential φ is unique only in the trivial case of the empty circuit: when the set of nodes N is empty. Indeed, two potentials define the same voltage function if and only if their difference is constant on each connected component of the graph Γ. We say Kirchhoff s current law holds if for all nonterminal nodes n N \ N we have I(e) = I(e). s(e)=n t(e)=n This is an expression of conservation of charge within the circuit; it says that the total current flowing in or out of any nonterminal node is zero. Even when Kirchhoff s current law is obeyed, terminals need not be sites of zero net current; we call the function ι R N that takes a terminal to the difference between the outward and inward flowing currents, the boundary current for I. ι : N R n I(e) t(e)=n 3.2 The principle of minimum power s(e)=n A boundary potential is also a function in R N, but instead thought of as specifying electric potentials on the terminals of a circuit. As we think of our circuits as open circuits, with the terminals points of interaction with the external world, we shall think of these potentials as variables that are free for us to choose. Using the above three principles Ohm s law, Kirchhoff s voltage law, and Kirchhoff s current law it is possible to show that choosing a boundary potential determines unique voltage and current functions on that circuit. The principle of minimum power gives some insight into how this occurs, by describing a way potentials on the terminals might determine potentials at all nodes. From this, Kirchhoff s voltage law then gives rise to a voltage function on the edges, and Ohm s law gives us a current function too. We shall show, in fact, that a potential satisfies the principle of minimum power for a given boundary potential if and only if this current obeys Kirchhoff s current law. A circuit with current I and voltage V dissipates energy at a rate proportional to the real number P = e E I(e)V (e). I(e), Ohm s law allows us to rewrite I as V/r, while Kirchhoff s voltage law gives us a potential φ such that V (e) can be written as φ(t(e)) φ(s(e)), so for a circuit obeying these two laws the power can also be expressed in terms of this potential. We thus arrive at a functional mapping potentials φ to the power dissipated by the circuit when Ohm s law and Kirchhoff s voltage law are obeyed for φ. 7

8 Definition 3.1. The extended power functional P : R N R of a circuit is the map P (ϕ) = e E 1 ( ) 2. ϕ(t(e)) ϕ(s(e)) r(e) We call this the extended power functional as we shall see that it is defined on potentials that are not compatible with the three governing laws of electric circuits. We shall later restrict the domain of this functional so that it is defined precisely on those potentials that are compatible with the governing laws. Note that this functional does not depend on the directions chosen for the edges of the circuit. This expression lets us formulate the principle of minimum power, which gives us information about the potential φ given its restriction to the boundary of Γ. Call a potential φ R N an extension of a boundary potential ψ R N if φ is equal to ψ when restricted to R N that is, if φ R N = ψ. Definition 3.2. We say a potential φ R N obeys the principle of minimum power for a boundary potential ψ R N if φ minimizes the extended power functional P subject to the constraint that φ is an extension of ψ. As promised, in the presence of Ohm s law and Kirchhoff s voltage law, the principle of minimum power is equivalent to Kirchhoff s current law. Proposition 3.3. Let φ be a potential extending some boundary potential ψ. Then φ obeys the principle of minimum power for ψ if and only if the induced current I(e) = 1 r(e) (φ(t(e)) φ(s(e))) obeys Kirchhoff s current law. Proof. Fixing the potentials at the terminals to be those given by the boundary potential ψ, the power is a nonnegative quadratic function of the potentials at the nonterminals. This implies that an extension φ of ψ minimises P precisely when P ϕ(n) ϕ=φ = 0 for all nonterminals n N \ N. Note that the partial derivative of the power with respect to the potential at n is given by P ϕ(n) = 2 1 ( ) φ(t(e)) φ(s(e)) 2 1 ( ) φ(t(e)) φ(s(e)) ϕ=φ r(e) r(e) t(e)=n s(e)=n = 2 I(e) I(e). t(e)=n s(e)=n Thus φ obeys the principle of minimum power for ψ if and only if s(e)=n I(e) = t(e)=n I(e) for all n N \ N, and so if and only if Kirchhoff s current law holds. 3.3 A Dirichlet problem We remind ourselves that we are in the midst of understanding circuits as objects that define relationships between boundary potentials and boundary currents. This relationship is defined by the stipulation that voltage current pairs on a circuit must obey Ohm s law and Kirchhoff s laws or equivalently, Ohm s law, Kirchhoff s voltage law, and the principle of minimum power. In this subsection we show these conditions imply that for each boundary potential ψ on the circuit there exists a potential φ on the circuit extending ψ, unique up to what may be interpreted as a choice of reference potential on each connected component of the circuit. From this potential φ we can then compute the unique voltage, current, and boundary current functions compatible with the given boundary potential. 8

9 Fix again a circuit with extended power functional P : R N R. Let : R N R N be the operator that maps a potential φ R N to the function N R given by n P ϕ(n). ϕ=φ As we have seen, this function takes potentials to twice the pointwise currents that they induce. We have also seen that a potential φ is compatible with the governing laws of circuits if and only if ( φ ) R N = 0 (1) The operator acts as a discrete analogue of the Laplacian for the graph Γ, so we call this operator the Laplacian of Γ, and say that the equation (1) is a version of Laplace s equation. We then say that the problem of finding an extension φ of some fixed boundary potential ψ that solves this Laplace s equation or, equivalently, the problem of finding a φ that obeys the principle of minimum power for ψ is a discrete version of the Dirichlet problem. As we shall see, this version of the Dirichlet problem always has a solution. However, the solution is not necessarily unique. If we take a solution φ and some α R N that is constant on each connected component and vanishes on the boundary of Γ, it is clear that φ + α is still an extension of ψ and P that ϕ(n) ϕ=φ = P ϕ(n) ϕ=φ+α, so φ + α is another solution. We say that a connected component of a circuit touches the boundary if it contains a vertex in N. Note that such an α must vanish on all connected components touching the boundary. With these preliminaries in hand, we can solve the Dirichlet problem: Proposition 3.4. For any boundary potential ψ R N there exists a potential φ obeying the principle of minimum power for ψ. If we also demand that φ vanish on every connected component of Γ not touching the boundary, then φ is unique. Proof. For existence, observe that the power is a nonnegative quadratic form, the extensions of ψ form an affine subspace of R N, and a nonnegative quadratic form restricted to an affine subspace of a real vector space must reach a minimum somewhere on this subspace. For uniqueness, suppose that both φ and φ obey the principle of minimum power for ψ. Let Then α = φ φ. α N = φ N φ N = ψ ψ = 0, so φ + λα is an extension of ψ for all λ R. This implies that f(λ) := P (φ + λα) is a smooth function attaining its minimum value at both t = 0 and t = 1. In particular, this implies that f (0) = 0. But this means that when writing f as a quadratic, the coefficient of λ must be 0, so we can write f(λ) = 1 ( ) 2 (φ + λα)(t(e)) (φ + λα)(s(e)) r(e) e E = e E = e E = e E 1 r(e) ( (φ(t(e)) φ(s(e)) ) + λ ( α(t(e)) α(s(e)) ) ) 2 1 ( ) 2 φ(t(e)) φ(s(e)) + λ-term + λ 2 r(e) e E 1 ( ) 2 φ(t(e)) φ(s(e)) + λ 2 r(e) e E 1 r(e) 1 ( ) 2 α(t(e)) α(s(e)) r(e) ( α(t(e)) α(s(e)) ) 2. 9

10 Then f(1) f(0) = e E 1 ( ) 2 α(t(e)) α(s(e)) = 0, r(e) so α(t(e)) = α(s(e)) for every edge e E. This implies that α is constant on each connected component of the graph Γ of our circuit. Note that as α N = 0, α vanishes on every connected component of Γ touching the boundary. Thus, if we also require that φ and φ vanish on every connected component of Γ not touching the boundary, then α = φ φ vanishes on all connected components of Γ, and hence is identically zero. Thus φ = φ, and this extra condition ensures a unique solution to the Dirichlet problem. Note also from the proof of the above proposition that: Proposition 3.5. Suppose ψ R N and φ is a potential obeying the principle of minimum power for ψ. Then φ obeys the principle of minimum power for ψ if and only if the difference φ φ is constant on every connected component of Γ and vanishes on every connected component touching the boundary of Γ. Furthermore, φ depends linearly on ψ. Proposition 3.6. Fix ψ R N, and suppose φ R N is the unique potential obeying the principle of minimum power for ψ that vanishes on all connected components of Γ not touching the boundary. Then φ depends linearly on ψ. Proof. Fix ψ, ψ R N, and suppose φ, φ R N obey the principle of minimum power for ψ, ψ respectively, and that both φ and φ vanish on all connected components of Γ not touching the boundary. Then, for all λ R, (φ + λφ ) R N = φ R N + λφ R N = ψ + λψ and ( (φ + λφ )) R N = ( φ) R N + λ( φ ) R N = 0. Thus φ + λφ solves the Dirichlet problem for ψ + λψ, and thus φ depends linearly on ψ. Bamberg and Sternberg [2] describe another way to solve the Dirichlet problem, going back to Weyl [29]. 3.4 Lumped circuits We have seen that boundary potentials determine, essentially uniquely, the value of all the electric properties across the entire circuit. But from the perspective of control theory, this internal structure is irrelevant: we can only access the circuit at its terminals, and hence only need concern ourselves with the relationship between boundary potentials and boundary currents. In this section we streamline our investigations above to state the precise way in which boundary currents depend on boundary potentials. In particular, we shall see that the relationship is completely captured by the functional taking boundary potentials to the minimum power used by any extension of that boundary potential. Furthermore, each such power functional determines a different boundary potential boundary current relationship, and so we can conclude that two circuits are equivalent if and only if they have the same power function. A lumped circuit is an equivalence class of circuits, where two are considered equivalent when the boundary current is the same function of the boundary potential. The idea is that the boundary current and boundary potential are all that can be observed from outside, i.e. by making measurements at the terminals. Restricting our attention to what can be observed by making measurements at the terminals amounts to treating a circuit as a black box : that is, treating its interior as hidden 10

11 from view. So, two circuits give the same lumped circuit when they behave the same as black boxes. First let us check that the boundary current is a function of the boundary potential. For this we introduce an important quadratic form on the space of boundary potentials: Definition 3.7. The (scaled) power functional Q : R N R of a circuit with extended power functional P is given by Q(ψ) = 1 2 min P (φ). φ R N =ψ Proposition 3.4 shows the minimum above exists, so the power functional is well-defined. Up to a factor of 1 2, Q(ψ) is just the power dissipated by the circuit when the boundary voltage is ψ, thanks to the principle of minimum power. The factor of 1 2 simplifies the next proposition, which uses Q to compute the boundary current as a function of the boundary voltage. We will later see that in fact Q(ψ) is a nonnegative quadratic form on R N. Since Q is a smooth real-valued function on R N, its differential dq at any given point ψ R N defines an element of the dual space (R N ), which we denote by dq ψ. In fact, this element is equal to the boundary current ι corresponding to the boundary voltage ψ: Proposition 3.8. Suppose ψ R N. Suppose φ is any extension of ψ minimizing the power. Then dq ψ (R N ) = R N gives the boundary current of the current induced by the potential φ. Proof. Note first that while there may be several choices of φ minimizing the power subject to the constraint that φ R N = ψ, Proposition 3.5 says that the difference between any two choices vanishes on all components touching the boundary of Γ. Thus, these two choices give the same value for the boundary current ι : N R. So, with no loss of generality we may assume φ is the unique choice that vanishes on all components not touching the boundary. Write ι : N R for the extension of ι : N R to N taking value 0 on N \ N. By Proposition 3.6, there is a linear operator sending ψ R N to this choice of φ, and then f : R N R N Q(ψ) = 1 2 P (fψ). 11

12 Given any ψ R N, we thus have dq ψ (ψ ) = d dλ Q(φ + λψ ) λ=0 = 1 d 2 dλ P (f(ψ + λψ )) λ=0 = 1 d 1 ( (f(ψ + λψ ))(t(e)) (f(ψ + λψ ))(s(e)) ) 2 2 dλ r(e) e E e E λ=0 = 1 d 1 ( (fψ(t(e)) fψ(s(e))) + λ(fψ (t(e)) fψ (s(e))) ) 2 2 dλ r(e) = e E 1 r(e) (fψ(t(e)) fψ(s(e)))(fψ (t(e)) fψ (s(e))) = e E I(e)(fψ (t(e)) fψ (s(e))) = n N t(e)=n = ι(n)fψ (n) n N = ι(n)ψ (n). n N This shows that dq ψ = ι, as claimed. I(e) s(e)=n I(e) fψ (n) Note this only depends on Q, which makes no mention of the potentials at nonterminals. This is amazing: the way power depends on boundary potentials completely characterises the way boundary currents depend on boundary potentials. So we can black box according to power usage. 3.5 Equivalent circuits: some examples Example 3.9 (Resistors in series). Resistors are said to be placed in series if they are placed end to end or, more precisely, if they form a path with no self-intersections. It is well known that resistors in series are equivalent to a single resistor with resistance equal to the sum of their resistances. To prove this, consider the following circuit comprising two resistors in series, with input A and output C: r AB B r BC A C Now, the extended power functional P : R {A,B,C} R for this circuit is P (φ) = 1 r AB ( φ(a) φ(b) ) r BC ( φ(b) φ(c) ) 2, while the power functional Q : R {A,C} R is given by minimisation over values of φ(b) = x: Q(ψ) = 1 ( ) 1 2 min ( ) 2 1 ( ) 2 ψ(a) x + x ψ(c). x R r AB r BC Differentiating with respect to x, we see that this minimum occurs when 1 ( ) 1 ( ) x ψ(a) + x ψ(c) = 0, r AB r BC λ=0 12

13 and hence when x is the r-weighted average of ψ(a) and ψ(c): x = r BCψ(A) + r AB ψ(c) r BC + r AB. Substituting this value for x into the expression for Q above and simplifying gives Q(ψ) = This is also the power functional of the circuit 1 r AB + r BC ( ψ(a) ψ(c) ) 2. A r AB + r BC C and so the circuits are equivalent. Example resistors in parallel Example Y- transform planar Proposition Two planar circuits are equivalent if and only if they both reduce to the same circuit after applications of the following five steps: (i) removal of loops (ii) removal of spikes (iii) series resistors substitution (iv) parallel resistors substitution (v) Y- transform Proof. de Verdière, Gitler and Vertigan [28] Normal forms for circuits star, mesh The star-mesh transform Proposition Every network of linear resistors can be normalised to an equivalent mesh on the boundary nodes. Proof. 3.6 Dirichlet forms We have seen that a lumped circuit is completely specified by the power functional Q on the vector space R N. Now we describe which functionals can arise this way. They are the quadratic forms known as Dirichlet forms, and they admit a number of equivalent characterizations. We start with the simplest. A Dirichlet form on S will be a certain sort of quadratic form on R S : Definition Given a finite set S, a Dirichlet form on S is a quadratic form Q : R S R given by the formula Q(ψ) = c ij (ψ i ψ j ) 2 i,j for some nonnegative real numbers c ij. 13

14 Note that we may assume without loss of generality that c ii = 0 and c ij = c ji ; we do this henceforth. Any Dirichlet form is nonnegative: Q(ψ) 0 for all ψ R S. However, not all nonnegative quadratic forms are Dirichlet forms. For example, if S = {1, 2}: Q(ψ) = (ψ 1 + ψ 2 ) 2 is not a Dirichlet form. In fact, the concept of Dirichlet form is vastly more general: such quadratic forms are studied not just on finite-dimensional vector spaces R S but on L 2 of any measure space. When this measure space is just a finite set, the concept of Dirichlet form reduces to the definition above. For a thorough introduction Dirichlet forms, see the text by Fukushima [8]. For a fun tour of the underlying ideas, see the paper by Doyle and Snell [7]. The following characterizations of Dirichlet forms help illuminate the concept: Proposition Given a finite set S and a quadratic form Q : R S equivalent: R, the following are (i) Q is a Dirichlet form. (ii) Q(φ) Q(φ ) whenever φ i φ j φ i φ j for all i, j. (iii) Q(φ) = 0 whenever φ i is independent of i, and Q obeys the Markov property: Q(φ) Q(ψ) when ψ i = min(φ i, 1). Proof. See Fukushima [8]. It is crucial for us that the property of being a Dirichlet form is preserved under minimising over linear subspaces of the domain generated by subsets of the given basis. Proposition If Q : R S+T R is Dirichlet, then is Dirichlet. min Q(, ν) : R S R ν R T Proof. We first note that min ν R S Q(, ν) is a quadratic form. Again, min ν R T Q(, ν) is welldefined as a nonnegative quadratic form also attains its minimum on an affine subspace of its domain. Furthermore min ν R T Q(, ν) is itself a quadratic form, as the partial derivatives of Q are linear, and hence the points at with these minima are attained depend linearly on the argument of min ν R T Q(, ν). Now by Proposition 3.15, Q(φ) Q(φ ) whenever φ i φ j φ i φ j for all i, j S + T. In particular, this implies min ν R T Q(ψ, ν) min ν R T Q(ψ, ν) whenever ψ i ψ j ψ i ψ j for all i, j S. Using Proposition 3.15 again then implies that min ν R T Q(, ν) is a Dirichlet form. Corollary Let Q : R N R be the power functional for some circuit. Then Q is a Dirichlet form. Proof. The extended power functional P is a Dirichlet form, and writing R N = R N R N\ N allows us to write 1 Q( ) = min φ 0 R N\ N 2 P (, φ 0). The converse is also true: simply construct the circuit with set of vertices X and an edge of resistance 1 c ij between any i, j X such that the term c ij (ψ i ψ j ) appears in the Dirichlet form. This gives: 14

15 Proposition A function Q is the power functional for some circuit if and only if Q is a Dirichlet form. Note that this is an expression of the star-mesh transform every circuit is equivalent to a mesh on its terminals. We may interpret the proof of Proposition 3.16 as showing that intermediate potentials at minima depend linearly on boundary potentials, in fact a weighted average, and that substituting these into a quadratic form still gives quadratic form. 3.7 Summary In summary, in this section we have shown the existence of a surjective function { } { } circuits with inputs I, outputs O, and I O = Dirichlet forms on I + O mapping two circuits to the same Dirichlet form if and only if they are functionally equivalent. 4 A category of lumped circuits Open circuits can be connected with one another to create new circuits, determining a new relationship. We show that this new relationship is a function only of the current-voltage relationships determined by the composite circuits, and hence does not depend on the internal structure of the composite circuits. This allows us to speak of a composition rule for current-voltage relationships. In this section we define a category of lumped circuits. This is a dagger compact category. Sign convention on current: current flows from high potentials to low potentials. 4.1 Composition of Dirichlet forms We begin with a naïve attempt to construct a category where the morphisms are lumped circuits. This naive attempt doesn t quite work, because it doesn t include identity morphisms. However, it points in the right direction. Recall that lumped circuits, or equivalence classes of circuits, are in one-to-one correspondence with Dirichlet forms. We define a composition rule for Dirichlet forms that reflects composition of circuits. Given finite sets S and T, let S + T denote their disjoint union. Let D(S, T ) be the set of Dirichlet forms on R S+T. There is a way to compose these Dirichlet forms : D(T, U) D(S, T ) D(S, U) defined as follows. Given Q D(S, T ) and R D(T, U), let (R Q)(γ, α) = min β R T Q(γ, β) + R(β, α), where α R S, γ R U. This is consistent with circuits: the power used by the entire circuit is just the sum of the power used by its parts. It is immediate from Proposition 3.16 that this composition rule is well-defined: the composite of two Dirichlet forms is again a Dirichlet form. Moreover, this composition is associative: (P Q) R = P (Q R). However, there is typically no Dirichlet form 1 S D(S, S) playing the role of the identity for this composition. A category without identity morphisms is called a semicategory, so we see Proposition 4.1. There is a semicategory where: the objects are finite sets, 15

16 a morphism from T to S is a Dirichlet form Q D(S, T ). composition of morphisms is given by (R Q)(γ, α) = min β R T Q(γ, β) + R(β, α). We would like to make this into a category. The easy way is to formally adjoin identity morphisms; this trick works for any semicategory. This amounts to introducing some circuits that contains wires with zero resistance. However, we obtain a better category if we include more morphisms: more circuits having wires with zero resistance. As the expression for the extended power functional includes the reciprocals of resistances, such circuits cannot be expressed within the framework we have developed thus far. Indeed, for these idealised circuits there is no function taking boundary potentials to boundary currents: the lack of resistance is thought as as implying that any difference in potentials at the boundary induces infinite currents. To accommodate this idea, we introduce sympletic vector spaces and their Lagrangian subspaces. 4.2 Symplectic vector spaces Inspired by the principle of least action of classical mechanics in analogy with the principle of minimum power, we turn to symplectic methods to discuss lumped circuits. A symplectic vector space allows a more invariant perspective, viewing voltages and currents in the same vector space. The word symplectic is a calque of complex, introduced by Weyl []. A circuit made up of wires of positive resistance defines a function from boundary potentials to boundary currents. A wire of zero resistance, however, does not define a function: the principle of minimum power is obeyed as long as the potentials at the two ends of the wire are equal. More generally, we may thus think of circuits as specifying a set of allowed voltage-current pairs, or as a relation between boundary potentials and boundary currents. This set forms what is called a Lagrangian subspace, and is given by the graph of the differential of the power functional. More generally, Lagrangian submanifolds graph derivatives of smooth functions: they describe the point evaluated and the tangent to that point within the same space. The material in this section is all known, and follows without great difficulty from the definitions. To keep this section brief we omit proofs. See any introduction to symplectic spaces, such as Cimasoni and Turaev [5] or Piccione and Tausk [17] for details. Definition 4.2. Given a vector space V over a field F, a symplectic form ω : V V F on V is a antisymmetric nondegenerate bilinear form. That is, a symplectic form ω is a function V V F that obeys (i) bilinearity: for all λ F and all u, v V we have ω(λu, v) = ω(u, λv) = λω(u, v); (ii) total isotropy: for all v V we have ω(v, v) = 0; and (iii) nondegeneracy: given v V, ω(u, v) = 0 for all u V if and only if u = 0. A symplectic vector space (V, ω) is a vector space V equipped with a symplectic form ω. Given symplectic vector spaces (V 1, ω 1 ), (V 2, ω 2 ), a symplectic map is a linear map f : (V 1, ω 1 ) (V 2, ω 2 ) such that ω 2 (f(u), f(v)) = ω 1 (u, v) for all u, v V 1. A symplectomorphism is a symplectic map that is also an isomorphism. A symplectic basis for a symplectic vector space (V, ω) is a basis {p 1,..., p n, q 1,..., q n } such that ω(p i, p j ) = ω(q i, q j ) = 0 for all 1 i, j n, and ω(p i, q j ) = δ ij for all 1 i, j n, where δ ij is the Dirac delta, equal to 1 when i = j, and 0 otherwise. A symplectomorphism maps symplectic bases to symplectic bases, and conversely: any map that takes a symplectic basis to another symplectic basis is a symplectomorphism. 16

17 Example 4.3 (The symplectic vector space generated by a finite set). Given a finite set S, we consider the vector space R S R S a symplectic vector space, with symplectic form ω((v, φ), (v, φ )) = φ (v) φ(v ). Let {v s } s S be the basis of R S consisting of the functions S R mapping s to 1 and all other elements of s to 0, and let {φ s } s S R S be the dual basis. Then {(v s, 0), (0, φ s )} s S forms a symplectic basis for R s R S. There are a two common ways we will build symplectic spaces from other symplectic spaces: conjugation and summation. Given a symplectic form ω, we may define its conjugate sympletic form ω = ω, and write the conjugate symplectic space (V, ω) as V. Given two symplectic vector spaces (U, ν), (V, ω), we consider the direct sum U V a sympletic vector space with the symplectic form ν + ω. This is not a product in the category of symplectic vector spaces and symplectic maps. The symplectic form gives a map from a symplectic space to its dual space. We study these dual elements, and the symplectic form itself, through their kernels. Given a subspace S of V, we define its complement S = {v V ω(v, s) = 0 for all s S}. Note that this construction obeys the following identities, where S and T are subspaces of V : dim S + dim S = dim V (S ) = S (S + T ) = S T (S T ) = S + T. Given a symplectic vector space V = U U, the subspace U has the property of being a maximal subspace such that the symplectic form restricts to a zero form on U. Subspaces with this property are known as Lagrangian subspaces, and may be realised as the image of U under symplectic isometries V V. Lagrangian subspaces may also be viewed as the subspaces that correspond to possible (point, tangent vector) pairs for quadratic forms on V. They may also mostly be viewed as graphs of self-adjoint maps U U. Definition 4.4. Let S be a linear subspace of a symplectic vector space (V, ω). We say that S is isotropic if ω S S = 0, and that S is coisotropic if S is isotropic. A subspace is Lagrangian if it is both isotropic and coisotropic. Lagrangian subspaces are also known as Lagrangian correspondences and canonical relations. Note that a subspace S is isotropic if and only if S S. This fact helps with the following characterisations of Lagrangian subspaces. Proposition 4.5. Given a subspace L V of a symplectic vector space (V, ω), the following are equivalent: (i) L is Lagrangian. (ii) L is maximally isotropic. (iii) L is minimally coisotropic. (iv) L = L. (v) L is isotropic and dim L = 1 2 dim V. 17

18 From this proposition it follows easily that the direct sum of two Lagrangian subspaces in Lagrangian in the sum of their ambient spaces. An advantage of isotropy is that there is a good way to take a quotient of a symplectic vector space by an isotropic subspace that is, there is a way to put a natural symplectic structure on the quotient space. Proposition 4.6. Let S be an isotropic subspace of a symplectic vector space (V, ω). Then S /S is a symplectic vector space with symplectic form ω (v + S, u + S) = ω(v, u). Proof. The function ω is a well-defined due to the isotropy of S by definition adding any pair (s, s ) of elements of S does not change the value of ω(v + s, u + s ). This function is then immediately a symplectic form as the defining properties are immediately implied true by the fact they are true for ω. 4.3 Lagrangian subspaces from Dirichlet forms We are interested in these symplectic structures as Dirichlet forms on S give rise to Lagrangian subspaces of R S R S. In this subsection we show how these subspaces are constructed. More generally, we show that there is a one-to-one correspondence between Lagrangian subspaces and quadratic forms. Proposition 4.7. Let S be a finite set. Given a quadratic form Q on R S, the subspace L Q = { (v, dq v ) v R S} R S R S, where dq v R S is the value of differential dq of Q at v R S, is Lagrangian. Moreover, this construction gives a one-to-one correspondence } { {quadratic forms on R S Lagrangian subspaces of R S R S } with trivial intersection with R S {0} R S R S. Proof. The symplectic structure on R S R S and our notation for it is that given in Example 4.3. Note that for all s, t S, 2 Q v s v t = dq vs (v t ) = dq vt (v s ), so dq v (u) = dq u (v) for all u, v R S. Thus L Q is Lagrangian: for all u, v R S ω ( (u, dq u ), (v, dq v ) ) = dq v (u) dq u (v) = 0. Observe that the only element of L Q of the form (v, 0), where v R S, is (0, 0). Thus L Q has trivial intersection with R S {0} considered as a subspace of R S R S. This construction forms the leftward direction of the above correspondence. For the rightward direction, suppose that L is a Lagrangian subspace of R S R S such that L (R S {0}) = {(0, 0)}. Then for each v R S, there exists a unique α v R S such that (v, α v ) L. Indeed, if α v and α v were distinct elements of R S with this property, then by linearity (0, α v α v) would be a nonzero element of L R S {0}, contradicting the hypothesis about trivial intersection. Existence then follows from the fact that L has dimension equal to that of R S, since L is Lagrangian. We thus can define a function, indeed a linear map, α : R S R S. This defines a bilinear form Q(u, v) = α u (v) on R S R S. But L is Lagrangian, so ω ( (u, α u ), (v, α v ) ) = α v (u) α u (v) = 0. Thus Q(, ) is a symmetric bilinear form, and so Q(v) = Q(v, v) defines a quadratic form on R S. As dq v (v) = Q(v) for any quadratic form, these two constructions are inverses, and so give the claimed one-to-one correspondence. 18

19 In particular, every Dirichlet form defines a Lagrangian subspace. We call the Lagrangian subspaces arising from Dirichlet forms in this way Dirichlet Lagrangian subspaces. Also note that the differential of a Dirichlet form is a doubly infinitesimally stochastic operator. For more details, see []. 4.4 Lagrangian subspaces as morphisms Recall that a relation between sets X and Y is a subset R of their product X Y. Furthermore, given relations R X Y and S Y Z, there Lagrangian subspaces can be thought of as morphisms: there is a way to compose them. Definition 4.8. A Lagrangian relation L : V 1 V 2 is a Lagrangian subspace L of V 1 V 2. A Lagrangian relation L V 1 V 2 comes equipped with projections L V 1 and L V 2. We may thus consider the Lagrangian relation also as defining a span L V 1 V 2 Composition of Lagrangian relations is given by composition of spans. We think of it as matching up output currents of the first relation with input currents of the second. By default Q gives boundary currents that are outward looking. We conjugate the first symplectic form to produce inward looking currents. Let V 1, V 2, V 3 be symplectic vector spaces, L V 1 V 2 be a Lagrangian relation L : V 1 V 2, and L V 2 V 3 be a Lagrangian relation L : V 2 V 3. As V 1 V 2 are P L L V 1 V 2 V 3 As V 1 V 3 is the categorical product of the symplectic vector spaces V 1 and V 3, there exists a unique linear map π : P V 1 V 3 such that commutes. We write L L = Im π V 1 V 3.. Proposition 4.9. The composite of two Lagrangian relations is again a Lagrangian relation. That is, L L is a Lagrangian subspace of V 1 V 3. Lemma L V Lagrangian, S L V isotropic, then L/S S /S Lagrangian. Proof. As L is isotropic and the symplectic form on S /S is given by ω (v +S, u+s) = ω(v, u), L/S is immediately isotropic. Recall that, by Proposition 4.5, an isotropic subspace S of a symplectic vector space V is Lagrangian if and only if dim S = 1 2 dim V. Also recall that for any subspace dim S + dim S = dim V. Thus dim(l/s) = dim L dim S = 1 2 dim V dim S = 1 2 (dim S + dim S ) dim S = 1 2 (dim S dim S) = 1 2 dim(s /S). Thus L/S is in fact Lagrangian in dim(s /S). 19

20 Lemma L V Lagrangian, S V isotropic, then (L S ) + S V Lagrangian. Proof. This is immediate from the way taking the symplectic complement interacts with sums and intersections: ((L S )+S) = (L S ) S = (L +(S ) ) S = (L+S) S = (L S )+(S S ) = (L S )+S. Since (L S ) + S is its own complement, it is Lagrangian. Combining these two lemmas gives: Corollary L V Lagrangian, S V isotropic, then ((L S ) + S)/S S /S Lagrangian. Proof of Proposition 4.9. Let Then = {(0, v, v, 0) v V 2 } V 1 V 2 V 2 V 3. = {(v 1, v, v, v 3 ) v V 2, v i V i } V 1 V 2 V 2 V 3. Note L L = ((L 1 L 2 ) ) + )/, and / = V 1 V 3. Thus L L is Lagrangian in V 1 V 3 by Corollary Note that this composition is associative. 4.5 Wires of zero resistance This solves our identity problems. identity id : V V, id = {(v, v) v V } V V unitors λ = {(0, v, v)} ρ = {(v, 0, v)} associators α = {(u, v, w, u, v, w)} swaps σ : U V V U, σ = {(u, v, v, u) u U, v V } U V V U. Note that these all come from symplectomorphisms between the domain and codomain. From this viewpoint it is immediate that all the necessary diagrams commute, and so we have a symmetric monoidal category. The monoidal product of two morphisms is just given by the direct sum of symplectic vector spaces. The compactness structure does not come from functions. cup η : {0} V V, η = {(0, v, v) v V } {0} V V = {0} V V cap ɛ : V V {0}, ɛ = {(v, v, 0) v V } V V {0}. This gives duals for each object, so we have a compact category. More generally we have spiders µ m,n that connect m inputs with n outputs via wires of zero resistance. 4.6 The dagger compact category of Lagrangian linear relations Let Lagr be the category with objects symplectic vector spaces and morphisms Lagrangian relations, with composition defined as above. We have seen that this composition is associative, and the diagonal Lagrangian subspace of V V acts as an identity morphism for each symplectic vector space V. Furthermore, it is symmetric monoidal. This is a subspace of the category of relations on symplectic vector spaces, and has a forgetful functor to the category of linear relations on vector spaces, which has in turn a forgetful functor to the category of relations on sets. 20