nLab matrix mechanics

Classical, quantum and statistical mechanics may all be seen as varieties of matrix mechanics for different rigs. In quantum mechanics we use linear algebra over the ring $\mathbb{C}$ of complex numbers; in classical mechanics everything is formally the same, but we instead use the rig $\mathbb{R}_{min} = \mathbb{R} \cup \{+\infty\}$, where the addition is $min$ and the multiplication is $+$.

Statistical mechanics (or better, ‘thermal statics’) is matrix mechanics over a rig $\mathbb{R}^T$ that depends on the temperature $T$. As $T \to 0$, the rig $\mathbb{R}^T$ reduces to $\mathbb{R}_{min}$ and thermal statics reduces to classical statics, just as quantum dynamics reduces to classical dynamics as Planck’s constant approaches zero.

The dynamics of particles becomes the statics? of strings after Wick rotation.

I suppose matrix mechanics over a distributive lattice could be thought as a kind of minimax (or maximin) calculation - David.

Revised on December 15, 2009 19:49:17 by Toby Bartels (173.60.119.197)