David asked how the wavefunction of a quantum particle reduces to the ‘wavefunction of a classical particle’ — whatever that is — in the $\hbar \to 0$ limit.

I think I finally figured it out. The answer is pretty simple: a ‘classical wavefunction’ is a real-valued function $\psi$ on spacetime. We can think of the number $\psi(t,x)$ as the least possible action for a particle to reach the point $x$ at time $t$. We then evolve this in time following the principle of least action. It’s sort of obvious how — but before I describe that, let me bring in a handy analogy.

We globe-trotting academics are all familiar with the problem of minimizing the cost of travelling from here to there. So, the principle of least action becomes more intuitive if we call it the principle of ‘least cost’.

In these terms, you should think of $\psi(t_0,x_0)$ as the price you have to pay to start a trip at the point $x_0$ at time $t_0$. The ‘startup cost’, as it were.

Given this, we define $\psi(t_1,x_1)$ to be the least cost of getting to the point $x_1$ by the time $t_1$.

It’s easy to compute this function if we know the price of any path that takes us from the point $x_0$ at time $t_0$ to the point $x_1$ at time $t_1$.

To compute $\psi(t_1,x_1)$, we first take the startup cost $\psi(t_0,x_0)$ for each point $x_0$. Then we add the cheapest price of a path from $x_0$ at time $t_0$ to $x_1$ at time $t_1$. Then we minimize this over all such paths and all points $x_0$.

Note this is *exactly* like the path integral at the top right of this week’s notes, but with the ‘cost’ or action $S$ replacing the exponentiated action $exp(iS/\hbar)$, addition replacing multiplication, and minimization replacing integration. This is just what we’d expect: classical mechanics is just like quantum mechanics, but with the rig $\mathbb{R}^{min}$ replacing the ring of complex numbers!

Starting from what I’ve said, we can write down a differential equation describing the time derivative of the wavefunction $\psi(t,x)$. In the quantum case this is called Schrödinger’s equation; in the classical case it’s called the Hamilton–Jacobi equation.

If you try to work out how the quantum wavefunction reduces to the classical one in the $\hbar \to 0$ limit, you need to see the ring $\mathbb{C}$, or at least $\mathbb{R}$, as a one-parameter deformation of the rig $\mathbb{R}^{min}$. The formulas for doing this involve exponentials, which is why Urs mentioned logarithms in his answer to your question.

It’s all very pretty — and it’s a bit shocking that more people don’t discuss this stuff. So, maybe it’s my duty to figure it out and explain it thoroughly in this course.

## Re: Quantization and Cohomology (Week 10)

The notes say that Schrödinger

But didn’t de Broglie say that first?