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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Nice exposition about the math driving evolution

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

One of the rare, really thoughtful posts on Slashdot.

Powers of 10 toward the black hole in the center of the galaxy

Posted in Astronomy, Fun links, Math by Mike Stay on 2007 January 30

http://www.cs.indiana.edu/%7Ehanson/Movies/blackhole.mov

Also see Hanson’s other visualizations here:
http://www.cs.indiana.edu/~hanson/
and this 3-d projection of the 5-d Calabi-Yau manifold:
http://www.bathsheba.com/crystal/calabiyau/

Great site on the inner complexities of a piano

Posted in Fun links by Mike Stay on 2007 January 26

Exile

Posted in Poetry by Mike Stay on 2007 January 24

I find this intensely moving.

Exile
by Douglas Stay

In a strange, distant land, by decree of the king
I journey down roads that are foreign.
The faces are stone when I’m so far from home,
And the words and the fields seem barren.
An echo of home in the silence of stars
In the breeze as it blows in the morning
My memory fails as I tell myself tales
Hearing ravens cry their shrill warning.
I awake in the night hearing voices I love
Drift away like the leaves in November
And the flavor of bread as these far paths I tread
Recalls meals from the edge of remember.
Oh! The banners we made in a hurry that day
Seem to fly in the corners of vision.
Passing shadows at night play tricks on my sight
Till the star of the morning has risen.
I long for the day when the years pass away,
And my time in exile has ended.
I will make my way home, to my fortress of stone
And the walls that so long I defended.