You conjecture that the functions Q giving power as a function of the potentials at the exposed wires from some collection of wires and resistors are all non-negative quadratic forms that do not change if you add the same number to each potential. I believe that this conjecture is false.

Consider Q(x, y, z) = (x + y - 2z)^2. If you put in voltages of -1, 1 and 0 you’ll get 0 power, indicating no net current and infinite resistance within the circuit. But voltages of 1, 1 and 0 gives you a power of 4, when I’d expect a power of 0 again from the infinite resistance we saw earlier. This doesn’t seem physically plausible.

In fact let me go farther.

Here is an approach that may lead to a proof and a correct characterization of all possible circuits. Take a circuit with 3 exposed wires. Apply voltages of 1, 0, 0. Let c_12 be the current flowing out of wire 2, and c_13 be the current flowing out of wire 3. Next apply voltages of 0, 1, 0. Let c_21 be the current flowing out of wire 1, and c_23 be the current flowing out of wire 3.

Now I strongly suspect that c_12 and c_21 must be the same. Can this be proven? Does this generalize? In an n exposed wire circuit can it be shown that for any distinct i and j, that in a configuration where wire i is at 1 volt and everything else is at 0, the current c_ij flowing from i to j is the same as the current c_ji flowing from j to i if j is at 1 volt and everything else is 0?

If it can be, then for any circuit at all we can collect all of the measurements of the currents flowing in and out when each of the first n-1 wires are at 1 volt and everything else is at ground. Note that these measured currents are all non-negative. By linearity if we apply any voltages at all we can figure out all of the currents in and out of all of the wires. We can therefore figure out how much work must be applied to keep the voltages there, and therefore how much work is being spent in the circuit. When you calculate it and expand the terms out the work is the sum, over all unordered pairs {i, j} with i not equal to j, of c_ij(x_i - x_j)^2.

An immediate consequence is that all cross terms x_i*x_j must be non-positive. Which tells us that (x+y-2z)^2 can’t work because it has a positive cross term of 2xy.

Note that all polynomials written out that way can easily be realized by real circuits by just connecting wires with appropriate resistances directly between the exposed wires. Each term tells us the current flowing at 1 volt between two exposed wires, which therefore tells us the resistance we need to have. (A current of 0 means no wire.)

I leave finding a nicer characterization of all polynomials of that form to the reader.

## Re: This Week’s Finds in Mathematical Physics (Week 296)

You write that in the category of circuit diagrams (extended from a semicategory),

I have in mind a circuit diagram made to look like the zig-zag relation. That is, it’s the composite of two other diagrams. The first has a straight short-circuit from bottom to top to the left of a bent short-circuit connecting the top back to itself. The second has a bent short-circuit from the bottom back to itself, and this is to the left of a straight short-circuit from the bottom to the top.

Neither of these two diagrams is an identity – indeed, they don’t even seem go from one object back to itself – and yet their composite is the short-circuit. What have I misunderstood?