Quantum Mechanics

Quantum Mechanics (QM) is the fundamental theory of modern physics. Thus far, it has successfully been used to describe all processes on scales smaller than molecules. For systems on the scale of molecules, the complexity of interacting processes hampers direct analysis using raw QM: however, analyses via approximations plausible on the scales involved are consistent with the view that all processes fit in with QM. It has been shown that the randomness inherent in quantum mechanics cannot be explained away by any theory of hidden variables – i.e. internal mechanisms with deterministic behaviour, that we simply can't see. I am persuaded that the entire universe is governed by quantum mechanical processes. One stumbling block remains: we have no fully unified understanding of General Relativity (i.e. gravity) alongside QM. The theories which manage to combine the two, to greater or lesser degrees, are known as Quantum Cosmology.

Quantum mechanical discussions in this directory include:

• de Broglie's contribution
• Schrödinger's equation
• Analysis of the energy levels of atoms
• The position observable, considered in the context of a smooth manifold (i.e. a curved universe, as required for general relativity)
• Preliminary thoughts on Quantum Cosmology
• Some ruminations about the effects of observation
• Some related thoughts about the nature of the vacuum

It is my belief that the notation in which modern physics is conducted is old, clunky and in need of revision. In particular, I believe that it is sufficiently clumsy that it is currently hampering the development of better models of the universe and impairing communication between researchers in different fields (c.f. Donald Kennedy's lament here). This belief is, however, only a bunch of hot air until such time as I produce a notation in which it is actually easier to develop such models.

Consequently, while I study QM, I am exploring the mathematical notations which can do what is required. One example of this is my analysis (dating from 1995) of the position observable in QM, given that position on a smooth manifold (the domain in which General Relativity says we must work) is not a linear quantity (i.e. one to which one can apply scaling and addition). This replaces the orthodox description of observables, as Hermitian operators on a complex linear space S, with a description as a measure, on the space of values of the observable, except that the measure maps subsets of this space to projection operators on S, rather than to real probabilities; the projection operators yielded by the measure all commute with one another; and the measure can be used to integrate any well-enough-behaved function f from the observable's space of values to a fixed real linear space U, the integral being a hermitian operator on S whose eigenvalues lie in U. In particular, when the observable's space of values is linear, the identity on it can be used as f and its integral is orthodoxy's hermitian operator.

If a body of given momentum, spin and charge lies entirely within some region bounded by a Kerr event horizon matching those parameters, then we can know nothing, in this outside world, about what is going on inside that region. For a tiny thing like an electron, having it all inside such a tiny region would entail a huge uncertainty of momentum, which was one of the inputs to what region of space-time you examined. Still, it suggests a low-level granularity beyond which one knows one is ignorant of the internals, save only as to their momentum, spin and charge. When we come to stitch together charts of a region of space-time that is an electron, we can expect to get a full description of it without needing to use charts smaller than chunks of would-be event tubes.

I picture an electron as a superposition of solutions to the field equations, each amounting to a chart of a region of space-time that looks like an electron from the outside: all the charts in the superposition give, internally, solutions of the field equations, and the surrounding universe reads them as agreeing on the spin, momentum and charge within the same charted region, at least to within the available uncertainty of the relevant quantities. Such a view of an arbitrary portion of space-time would make for interesting reading, too, I suspect.

See Also

• An analysis of balancing a US penny on its side
• The transactional interpretation of Quantum Mechanics; and a hidden-time variant of it.
• Applying quantum mechanics to game theory changes the nature of even The Prisoners' Dilemma (article provides many fascinating links).
• Causaloid theory.
• Applying Quantum Information
• A relatively accessible account of Bell's Theorem and how naïve assumptions predict negative probabilities.
• An enlightening derivation of quantum mechanics from the elementary postulate that measurement is wont to change the system measured.
• A quantum solution to the arrow-of-time dilemma: entropy can decrease, but only in a process which destroys all evidence of its having happened; for all practical purposes, it thus hasn't happened–!

Written by Eddy.