Quantum Mechanics and the Fourier-Legendre Transform over a Semiring

Update: see also my more recent notes on this subject.

Consider the two semirings: (R,+,*) and (R',min,+) where R' is R with a positive infinity value added in. The former is just the real numbers added and multiplied in the usual way and in the latter we use the operations min (which gives the lower of its two arguments) and + instead. There are many formal similarities shared by these structures. For example we have the distributive laws a*(b+c) = a*b+a*c and a+min(b,c) = min(a+b,a+c). Zero acts as the identity for + and we have a+0 = a and infinity is the identity for min with min(a,infinity) = a.

It turns out that the former semiring can be viewed as a quantum version of the latter semiring. In particular we can frequently take statements from quantum mechanics and consider them to be statements in a more general semiring rather than over (R,+,*). When we interpret these more general statements in the semiring (R',min,+) they turn out to say things about classical mechanics.

Consider for example the Hamiltonian formulation of classical mechanics. This essentially says that dynamical systems evolve in such a way that the integral of the Lagrangian is minimised. In other words, the integral of the Lagrangian is the min of its value for all possible paths. In quantum mechanics we no longer have systems taking the minimum but in a sense they take all paths. To compute physical quantities we must instead use the Feynman path integral to integrate over all paths. The factor we must integrate is essentially the exponential of the Lagrangian. In classical mechanics we have the min over an infinite set, in quantum mechanics we have the sum (ie. integral) over the same set. See here for a recent paper on this subject.

Another surprising analogy between these two semirings arises we we try to transfer the concept of the Fourier transform to (R,+,*) to (R',min,+). It's not obvious how to interpret the exponential function in (R',min,+) but it turns out that the natural choice is to consider the ordinary linear functions (in the conventional sense) to be the correct analogue. If we then replace the exponentials with linear functions in the definition of the Fourier transform and replace the integral with an infinite min what we end up with is another familiar transform: the Legendre transform. So the Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings.

The analogy carries quite far. In classical mechanics the Legendre transform converts between the Lagrangian and Hamiltonian formulations of classical mechanics. So it changes equations of motion written in terms of (generalised) position into equations written in terms of momentum and vice versa. In quantum mechanics the Fourier transform does much the same thing: the Fourier transform of a wavefunction in space gives the wavefunction in momentum space. Sean Walston has a paper on this. I'm not sure he was aware of the semiring connection when he wrote that.

In summary we have:

(R',min,+)	(R,+,*)
min(a,infinity) = a	a+0 = a
a+0 = a	a*1 = a
a+infinity = infinity	a*0 = 0
Classical	Quantum
integral	minimisation
x -> k*x	x -> exp(k*x)
Legendre Transform	Fourier Transform
Hamiltonian principle of least action	Feynman path integral

I never did understand the Legendre transform. It always seemed like this strange thing plucked out of nowhere. So it's amazing to see that in some sense it is the 'right' thing to study and is as natural as the Fourier transform. Fascinating stuff!