Example of a timeless system

Section 2.5.1 introduces in a difficult problem the idea of a system which requires a background variable as a surrogate for time in order to be analyzed. This problem seems to have been a favorite of Rovelli's for decades. A more complete explanation is provided in the 1990 paper, Quantum mechanics without time: A model and again in the 2008 paper A simple background-independent hamiltonian quantum system.

As tempting as it is to keep working with this model it seems to be an endless activity so I am moving on. Notes so far, copied from the blog.

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The reference is Quantum mechanics without time: A model by Rovelli, 1990, The American Physical Society

This is a fascinating toy model which Rovelli has been playing with for decades. The version that appears in the textbook (Section 2.5.1) is quite challenging to those of us who do not have a strong background, but the paper is more accessible.

The model demonstrates a dynamical system which can be analyzed both classically and as a quantum-mechanical system (yielding different models, of course) in which there cannot be an internal definition of “time” — i.e. one can analyze the system only by introducing another dimension (i.e. crossing the kinematic space with R (carefully)) that one uses as a stand-in for time.

There is much about this paper that is worth further study which is postponed for now. In particular apparently a quite familiar fact to physicists but one the I need to absorb
 * The relationship between lacking a Hamiltonian formulation and not being able to define intrinsic time
 * The art of finding a classical limit of a quantum system
 * The idea of a Hamiltonian that evolves with the extrinsic time
 * The comment that general relativity lacks a Hamiltonian formulation —
 * Techniques for constructing operators, self-adjoint or not, that are useful (and criteria for utility)

Here is a picture that I developed which attempts to represent a single eigen state of this dynamic: