Our goal has been to quantize time using the rules that we use to quantize space, and then see what breaks.
The key idea is the laboratory time, quantum time distinction. These represent two ways of looking at same thing. Laboratory time is time as parameter, quantum time is time as coordinate in wave function, relative time the difference. In the limit as dispersion in relative time goes to zero we get standard quantum theory.
Formal development:
Primary approach via Feynman (coordinate) path integrals, with paths varying in time as well as in path: Feynman path integrals.
Schrödinger equation derived from path integrals: Schrödinger equation.
Operator formalism from that; Operator formalism
The Heisenberg (canonical) path integrals from operator formalism: Hamiltonian path integrals.
And then the Feynman path integrals derived from that, closing the loop and demonstrating consistency of all four approaches.
We used Morlet wavelet decomposition to establish convergence and normalization of the path integrals.
Symmetry between time and space established by construction.
Invoked to define paths, Lagrangian.
Invoked to assert midpoint rule for electric potential.
Two ambiguities in the analysis, in the selection of the Lagrangian and in definition of laboratory time.
Argued Lagrangian given is the simplest scalar which:
Produces the correct classical equations of motion
Gives correct Schrödinger equation (Klein-Gordon equation).
With respect to laboratory time, by using Morlet wavelet decomposition can break up any initial wave function into wavelets with well-defined expectations for initial position and momentum. This gives a well-defined classical trajectory and this in turn a well-defined proper time along each trajectory. At the end we add up the various wavelets to get the result.
We made contact with the standard treatment in a number of limits:
Non-relativistic limit: Non-relativistic limit.
With respect to time, temporal quantization is to standard quantum theory as standard quantum theory is to classical mechanics, so we expect to recover standard quantum theory in the semi-classical limit: Semi-classical approximation.
Turning to full quantum case, argue that standard quantum theory is given by the stationary states of the temporal quantization solution: Stationary states. define stationary states in block time, show reduction to standard quantum theory using relative time.
For non-singular potentials, can use time gauge to show reduction from temporal quantization to standard quantum theory: Scattering.
For singular potentials, i.e. for bound states, time gauge is singular. Multi-particle case is a further step, so use some heuristics and analysis of matrix elements to show that the stationary state condition give the Bohr condition for the atomic levels for an arbitrary potential.
Large number of experimental tests available. By interchanging time and a space dimension, we can turn most foundational experiments into a test of temporal quantization.
Most experiments have two halves, a scatterer and a scatteree say. Both have to vary in time or effects of temporal quantization likely to average out.
Look at several sets: