## Dequantization and deformation

A rig is a ri**n**g without **n**egatives, like the nonnegative integers. You can add and multiply them, multiplication distributes over addition, and you’ve got additive and multiplicative identities 0 and 1.

There’s another rig, called the “rig of costs,” that everyone uses when planning a trip: given two alternative plane tickets from A to B, we chose the least expensive one. We add the cost of a trip from A to B with the cost of a trip from B to C. This one’s denoted

Notice that “addition” here is min, and the additive identity is : . “Multiplication” here is +, and distributes over min:

As described here, one can deform the rig

to the rig

where

like this:

As the deformed rig approaches This is called **Maslov dequantization**; here’s why.

In quantum mechanics, the path a particle takes is governed by **integrating the amplitude**, so the probability amplitude of arriving at point at time is

In classical mechanics, the path a particle takes is governed by the principle of **least action**, so the action cost of arriving at point at time is

where “inf” means “infimum,” i.e. the least element of an infinite set. You get from the complex numbers to the rig by taking

and classical mechanics falls out of quantum mechanics as If you take the derivative of those two equations above with respect to time, you get Schroedinger’s equation from the quantum case and the Hamilton-Jacobi equation from the classical case.

No one’s heard of the latter one, but you can describe a classical system with a wavefunction! Instead of the probability amplitude at a given point, it’s the action cost.

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