The issue is addressed in my previous blogs but is important and worth repeating. In Schroedinger quantum mechanics with a time-independent potential, the issue is find the eigenfunctions of the operator . There is a corresponding classical Hamiltonian system. Since the universe is compact, we can apply the results of Vieri Benci to show that the classical system has periodic orbits. The results of James Ralston shows that there are quasimodes of the Schroedinger operator that concentrate around stable periodic orbits of the Hamiltonian systems. Together, therefore we have the general conclusion that the Schroedinger eigenfunctions are probabilistic (because the modulus squared of the the solution to the Schroedinger equation which can be expressed in terms of eigenfunctions is interpreted as a probability) approximations of the periodic orbits of the classical system. Now remember that Schroedinger’s breakthrough was based on moving from classical to a *wave* mechanics in analogy to geometric to wave optics. Conceptually there is the wave aspect to Schroedinger’s mechanics as well as the stochastic aspect. Note that classical systems on a compact universe preserves the wave aspect but the stochastic interpretation is unnecessary. One can think thus of Schroedinger’s equation as a probabilistic convenient approximation to the a ‘real’ classical solution focusing on periodic solutions of the Hamiltonian system. Therefore quantum mechanics can be said to be fooled by randomness: this is not to say that the Schroedinger equation is not a good approximation; indeed, it is a spectacularly successful approximation from the numerical accuracy of various quantities such as energy levels of ‘quantum systems’. However, it is not an exact description of nature.

Previously, I had spoken of the Gutzwiller trace formula as giving us insight into the link of classical and quantum descriptions. The Gutzwiller trace formuls is very important in this respect because it relates to quantization of a classical system that is chaotic. It shows concretely how the above analysis is right in the setting of chaotic classical system as the trace formula picks out the periodic orbits ignoring the others.

Much more detailed analyses are justified but the main ideas are very clear for why quantum mechanics is fooled by randomness in the actual universe which must be compact by arguments I have presented from 2008. The right description of the universe would therefore be expected to be deterministic and classical; my general picture ignoring nuclear forces is that the physical universe is an evolving hypersurface of a stationary 4-sphere (scaled appropriately). Electromagnetism is described by a smooth 1-form normal at every time to the physical universe. Gravitation produces a relation for the Ricci curvature of the hypersurface.

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