Re: Parity Eotvos Experiment-Discrete Noether

Posted by Uncle aAl on May 14, 2003 at 17:41
(68.5.243.16)

Re: Parity Eotvos Experiment-Discrete Noether (Pasti)

John Baez has no problem with a fully discretized Noether's theorem.

Arbitrary cellular automata mostly don't have Lagrangians. (If you have a Lagrangian you have a Hamiltonian, and vice-versa.) Noether's theorem considered in the context of algebraic geometry instead of differential geometry allows purely discrete variables. A theory of dynamical systems wherein time and the phase space are discrete still allows all the tools of ordinary classical mechanics to apply - calculus, differential forms, the Euler-Lagrange equations, Noether's theorem, etc.

A variant of Noether’s Theorem implies that exact conservation of discrete angular momentum must enforce asymptotic continuous angular isotropy as one looks at processes some level above the scale of the cellular array. Similarly, absolute microscopic conservation of discrete units of momentum must enforce asymptotic continuous translational symmetry. Conservation of discrete units of energy does the same for asymptotic continuous time symmetry.

Discrete Time-Step Noether Theorem
Discrete Noether Theorem
Discrete Time Noether Theorem

Interesting discussion!

The parity pair P3121 P3221 crystallographic space groups that comprise contrasted solid sphere tellurium test masses in the parity Eotvos experiment are wholly tolerant of a small fraction of compositional and structural imperfections including a crystallographer's mosaicity. Ditto very slight macroscopic inertial moment mismatches (impact is no worse than the square of the fractional relative mismatch). This is explicitly calculated here and down from there.

More to to the point, we benefit from a random dispersion of infrequent flaws! The tellurium lattice has two sets of C2-axes,

1) A C2-axis lays in the ab-plane, exactly midway between the a- and b-axes and passing through the origin, also passing through a Te atom. Additional copies of this C2-axis are generated by the 31 screw: they are rotated by 120 and 240 degrees and are at heights (1/3)c and (2/3)c above the ab-plane.

2) A C2-axis lays in the bc-plane, parallel to the b-axis and at a height of (1/6)c above the ab-plane as a perpendicular bisector of a Te-Te bond. Additional copies of this C2 axis are generated by the 31M/sub> screw: they are rotated by 120 and 240 degrees and are at heights (1/2)c and (5/6)c above the ab-plane.

A little random crud breaks the C2-axes. While two minor rotational correspondences don't severely impact CHI (rigorous computation in Petitjean's QCM), we'd like them reduced. Machining tolerences also mute the C2-axes. 1017 imperfections in 1022 atoms are good things. They still leave 1022 atoms in their correct places.

--
Uncle Al
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)


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