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
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
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.
One of the rare, really thoughtful posts on Slashdot.
I find this intensely moving.
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.