Wheeler's often unconventional vision of nature was grounded in reality through the principle of radical conservatism, which he acquired from Niels Bohr: Be conservative by sticking to well-established physical principles, but probe them by exposing their most radical conclusions.
As relativity and quantum mechanics are arguably the two best confirmed theories we have the dichotomy is troubling.
The obvious lines of attack are to add something (another time dimension perhaps), subtract something (find hidden variables that replace quantum theory) or push harder. The first two are perhaps the most popular, we here try the third. We will quantize time using the same rules as we use for space and see what breaks.
To have a reasonable chance at reaching at definite conclusion we will impose limits on the current investigation: we will only look at single particle case, only work to the first approximation, and take the simplest approach available.
Our guiding principle: the most complete symmetry between time and space.
Laboratory time is defined by Alice using clocks, laser beams, and grad students. It is defined operationally, measured in seconds, clock ticks, cycles of a cesium atom.
The term is used in [Busch-2001] and elsewhere. See also description of coordinate time in [Hilgevoord-1996] [Hilgevoord-1996b].
Definitely manufactured: have to pay the clockminders, laser makers and (sometimes) the graduate students.
We will take this as understood "well enough" for our purposes. (Deeper examination in, for instance, [Helling-2008], [Prati-2008].)
We will refer to laboratory time as .
To get the discussion started, we will use Alice's proper time, an invariant as our laboratory time. Once we have worked up enough structure, we will refine the definition so that Bob – travelling at relativistic speed with respect to Alice – and Alice can agree on what is meant by laboratory time, see below.
Hilgevoord distinguishes between the use of space coordinates as parameter and as operator, we will do the same for time.
Laboratory time is definitely a parameter, not an operator.
The usual wave function is "flat" in time: it represents a well-defined measure of our uncertainty about the particle's position in space, but shows no evidence of any uncertainty in time. This seems very "unquantum-mechanical". Given that any observer, Bob say, going at high velocity with respect to Alice will mix time and space, what to Alice looks like uncertainty only in space to Bob will look like uncertainty in a blend of time and space.
We will therefore extrude Alice's wave function into the time dimension, positing that , at any given instant in laboratory time, is a function of time as well, so that Alice will now be uncertain as to the particle's position in time as well as in space.
Now we ask what happens if Alice generalizes her , defined at laboratory time , to include time:
Reasoning by analogy we project the wave function into the time dimension, as if we were clicking the "extrude" button in a CAD program, turning a three into a four dimensional form.
This extruded wave function represents uncertainty in time, just as it does in space.
If we are to treat time and space symmetrically – our basic assumption – this is almost mandatory. There can be no justification for treating time as flat but space as fuzzy.
As we will see, the properties of this wave function are strongly constrained by time/space symmetry.
The extrusion of the wave function into time may be thought of fluctuations in time, a kind of Zeitbewegung, movement in time (by analogy to Zitterbewegung, fluctuations in space).
[ 1]We will treat quantum time (for which we will use a roman ) the same as any other unmeasured quantum variable, computing its effects by taking expectations over reduced density matrices and the like.
The usual wave function changes shape as laboratory time advances; if it did not it would not be very interesting. The quantum time part of the wave function must evolve as well. At each instant in laboratory time, we expect that will have a slightly different shape, if still centered on the current laboratory time.
We define the relative quantum time as the offset in quantum time from the current value of Alice's laboratory time. If the lab clock says 10 seconds past the hour, the relative quantum time might be 10 attoseconds before or after that.
We define the absolute or block quantum time as:
We expect in general:
The situation is analogous to the use of "center of mass" coordinates. We use center of mass coordinates to subtract off the average value of the space coordinates, letting us focus on the interesting part. And we can use "center of time" coordinates the same way, to focus on what is essential.
As an example, suppose Alice and Bob are travelling by train from Berne to Zurich. They decide to while away the time by doing quantum mechanical experiments. If they are doing a standard double slit experiment, then they will compute x and y and z relative to their current location on the train. But an outside observer, say Eve (mysteriously left behind at the Berne station) will see the space coordinates in the experiment as the sum of the coordinates Alice and Bob are using plus the space coordinates of the train.
The same with time.
We are not inventing a new time dimension or assigning new properties to the existing time dimension. We are merely treating, for purposes of quantum mechanics, time the same as the three space dimensions.
How are we to compute what we expect the wave function at the next instant in laboratory time when we know it at the current instant? We need dynamics.
We will use Feynman path integrals as our defining methodology; we will derive the Schrödinger equation, an operator formatism, and the Hamiltonian path integrals from them.
A path will be defined as a series of wave functions in laboratory time. The set of all possible paths is the set of all possible wave functions at each of all possible laboratory times. The only change we need to make to generalize Feynman path integrals to include quantization in time is to allow our paths to include motion in time as well as in space. In the Feynman path integrals, we change the integral over the paths to include time:
To keep the closest practicable connection with existing treatments of path integrals, we can use as starting set of wave functions Gaussian test functions sharply localized in space. And time. Of this, much much more below.
The laboratory time is like the frames in a movie; not part of the dynamics. When dividing the paths for the path integrals, each step can be thought of as one frame. Laboratory time is time seen as parameter.
Alice's location in time defines laboratory time. If Alice were walking her dog, it might frisk ahead or behind her at any moment. the dog's location is like quantum time: centered on laboratory time, but usually either a bit ahead or a bit behind. Quantum time is time seen as observable.
Alice's own quantum time is an average over the many many quantum times of her particles, amino acids, sugars, water molecules, and so on. Her average quantum time will be almost exactly her laboratory time.
To sum the infinity of paths, we will weight each path using the exponential of its action, defined as the integral of a suitable Lagrangian over the laboratory time from start to finish.
Any scalar Lagrangian that reproduces the classical equations of motion will do; we will argue below the best of these is:
We have to show we can get meaningful limits, agree with existing experimental results, et cetera.
Intuitively we expect effects like pre and post acceleration, interference in time, and so on. Any quantum effect we would look for in space, we might expect to see in time.
In fact most foundational experiments in quantum mechanics (see the perhaps 300 in [Auletta-2000] for instance) have a "flipped" form, where time and a space dimension are interchanged.
Because of the strong constraints implied by time/space symmetry, there are no free parameters in theory, so any one of these could in principle falsify quantum time.
[ 1]Note by quantum time we mean time quantized using rules for space; not little bits of time, like Planck times. Although these are not mutually exclusive possibilities.