reperiendi

Dequantization and deformation

Posted in Math, Quantum by Mike Stay on 2007 January 31

A rig is a ring without negatives, 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

\displaystyle R_{\min}=(R>0 \cup \{\infty\}, \min, \infty, +, 0).

Notice that “addition” here is min, and the additive identity is \infty: \min(x,\infty)=x. “Multiplication” here is +, and distributes over min:

\displaystyle x+\min(y,z) = \min(x+y, x+z).

As described here, one can deform the rig

\displaystyle (R>0, +, 0, *, 1)

to the rig

\displaystyle R_h=(R>0 \cup \{\infty\}, \oplus_h, \infty, +, 0),

where

\displaystyle u \oplus_h v = -h \log(e^{-u/h} + e^{-v/h}),

like this:

\displaystyle x \mapsto -h \log x.

As h \to 0, the deformed rig approaches R_{\min}. 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 b at time t is

\displaystyle \psi(b,t) = \int_{b_0} \int_{\mbox{paths } \gamma:(b_0, t_0) \to (b,t)}e^{-iS(\gamma)} \psi(b_0, t_0)\, D\gamma\, db_0

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

\displaystyle \psi(b,t) = \inf_{b_0} \inf_{\mbox{paths } \gamma:(b_0, t_0) \to (b,t)}S(\gamma) + \psi(b_0, t_0)

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

\displaystyle z \mapsto -h \log |z|,

and classical mechanics falls out of quantum mechanics as h \to 0. 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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