The rules of quantum mechanics and special relativity are so strict and powerful that it's very hard to build theories that obey both.
Now we could travel anywhere we wanted to go. All a man had to do was to think of what he says and to look where he was going.
In this chapter develop the formal rules for temporal quantization.
Like a snake headed out for its morning rat, we will need to take some twists and turns to get to our objective. Therefore now map out the overall path.
We use Feynman path integrals as the defining formalism. Path integrals are easy to understand and, as we will see, relatively easy to include the time dimension.
Comprehensive treatments of path integral formalism provided in: [Feynman-1965d], [Schulman-1981], [Swanson-1992], [Khandekar-1993], [Kleinert-2004], [Zinn-2005].
We will develop four formalisms:
Feynman path integrals– defining formalism.
Schrödinger equation– derived from Feynman path integral.
Operator formalism– derived from Schrödinger equation.
Hamiltonian path integrals– derived from time evolution of operators.
We will close the loop by deriving the Feynman path integral from the Hamiltonian path integral.
All four approaches make a common use of laboratory time from source to detector; to complete the treatment we need to define the laboratory time in a frame-invariant way as well.
The proper time of the particle will not do; what if we have many particles?
Instead in the last section of this chapterbreak the initial wave function down using Morlet wavelet decomposition; then evolve each part along a geodesic to its destined detector. These geodesics have each a well-defined proper time which will serve as its laboratory time. At the detector, assemble the parts back into one self-consistent whole.
With this done, will argue that the first two requirements,
Well-defined
Manifestly symmetric between time and space
Met.