…the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations mechanics hold good. We will raise this conjecture to the status of a postulate…
I spark, I fizz, for the lady who knows what time it is.
If Alice is in her lab, while Bob is jetting around like a fusion powered mosquito, how are their wave functions related?
Can start with the same four dimensional wave function.
Definitions of kernel differ:
Lagrangian is a scalar is independent of Alice or Bob's coordinate system.
Coordinate systems at each point related by a Lorentz transformation:
So integral transformation at each time step has Jacobian one:
So unaffected.
As noted, Lagrangian already a scalar; we can make the transition to general relativity simply if we wish:
Only change is in step size for Alice or Bob:
However, Alice and Bob's times along the specific paths are different. Example: free particle:
To be sure realistic equations dominated by case where:
So the problem not that immediate, but we would like to handle correctly in any case.
We need a way to discuss laboratory time which is independent of choice of frame, of whether we use Alice's or Bob's frame.
Work in context of general relativity, consider the geodesic from wave function to detector.
We can use the proper time along geodesic from point A to point B , source to detector. This is observer independent. So we replace Alice's laboratory time with geodesic time.
We ignore for now the issue of what happens along spacelike or lightlike trajectories.
We are not quite done, as both initial wave functions and final detectors extend across spacetime.
We need to have a way to deal with their extended character.
In one of Andrew Wyeth's famous Helga paintings, he drew each part of Helga's full length portrait from a perspective chosen for that part (top from top height, torso from torso height, and toes from toe height).
In general relativity we use coordinate patches; here we can use Morlet wavelets in conjunction with coordinate patches.
Break the wave function up into Morlet wavelets in four dimensions, giving sixteen Gaussian test functions.
Using Morlet wavelet analysis, break Alice's wave function up into parts:
Consider the journey of each of these patches/Morlet wavelets to a single point at the detector.
For each wavelet, have well-defined expectations for location:
And momentum:
A well-defined initial position and momentum implies a well-defined classical trajectory. If this intercepts our detector, then we can use the proper time along it. If it misses, we can use the point of closest approach.
Geodesic is now least time path from the average to the detector's coordinate (given by capital D ):
Kernel now:
With geodesic time defined on a per wavelet basis, have wavelet at the detector:
To get full wave, add them all up:
Not ideal, but clearly reasonable.
If significant components of the Morlet wavelets have scale given by s much bigger than the radius of curvature from gravity, this approach breaks down.
Ultimately, one would need a proper theory of quantum gravity. Any predictions based on this analysis would become problematic near a black hole.
Alice and Bob can use same four dimensional wave function; both may use geodesic time for each component of the wave function.
Point of principle established; what is needed for practical work done. Have a well-defined observer-independent method of determining laboratory time.
And therefore have addressed the ambiguity about the spacelike hyperplanes mentioned in the introduction. Wave function independent of observer; geodesic time independent of observer.
Needed for completeness of the argument, also make direct use below in Slits in time.