Fleeting thoughts: The Value of Symmetry, or on Emmy Noether and her "Invariante Variationsprobleme"

Albert Einstein and Emmy Noether: the radiant Yang and the dark Yin of modern physics. Both are Jews, refugees in the USA, victimized by the Nazis. But how different are their destinies! Physics and mathematics need symmetry as well as the symmetry between the two.

Noether’s "Invariante Variationsprobleme" allows these symmetries to be restored. The first theorem establishes the symmetry of a set (even infinite) of parameters of the continuous transformations of a Lie group and the Lagrangian of them. Its functional is action. Any symmetry of that transformation corresponds to a law of conservation.   One can think of two groups (this of transformation and this of conservation) and their Cartesian product, which is commutative in the case. For example, conservation of energy is per a unit of time and it implies its opposite counterpart: conservation of time per a unit of energy (mass), which can be interpreted as a cyclic and teleological return of action into itself. Any factorization of action to physical quantities implies a law of conservation in relation to the group of transformation of the rest parameters. What is conserved is always action: Any law of conservation is derivative from its and relative to the parameters. Then the counterpart of action is a dimensionless physical quantity like entropy or information.

The second theorem develop the conservation of action even where the parameters are variable and thus the conserved is only action rather than any multiplier of it.

The symmetry of action and information (entropy), which are conserved, can be interpreted as a corresponding symmetry of physics and mathematics and embodied into the destinies of Einstein and Noether: The equilibrium between them must be restored. The Chinese Taiji, the ultimate limit, through which Yang transforms into Yin, and vice versa, too, can aid to elucidate these interrelations.