Why |phi|² ?, Simplicity ?

The wave function is a probability amplitude.
So far so good.
This means that it doesn't give us the probability directly and it can be a complex number.
Taking the square modulus of it, gives rise to interference. That's very good.

But why the square, why not the fourth power or any even function, like cosh(|phi|)?

Simplicity is the winner, but why?

Michel

------- WARNING: Unofficial hypothetical "Steve" musings follow: ;) -------------

A photon is viewed as composed of two quarks. It makes sense that a photon is an interaction between (at least) two objects, because it represents a detectable interaction between two objects.

If they both can travel within specific paths, then the probability of finding an interaction between them would be the square of the probability of either quark existing in that location, as an interaction requires two objects to occur so the interaction between two objects would occur proportional to the density of probability(A) * probability(B). For a single photon, envisioned as an interaction between two points, the paths of A and B would be mutual and so probability(A) = probability(B).

This could apply for more than two objects though. If you were to consider the relative densities of various objects in space, the underlying average number of objects could simply contribute to the uncorrelated "background noise" or virtual particle product.

For the 3-D space we understand, it takes repetitive and localizable disparities in probabilities for us to recognize something as an object. If space was smiply composed of 2 classes of objects but with a uniform distributation, there would be no way to localize either class and the interactions would appear chaotic and unpredictable, but instead if measurements occur in a relative, differential manner, as appears the case, then a local increase in the density of one particle, can be localized (you could also consider this to represent the absense of a complimentary "anti-particle", so one or the other exists, except when the density is uniform then the existance of both or equivalently neither could be observed).

Anyway, in this case the probability of observing the interaction of any of these two classes, A and B, of particles, using differential measuerments (which comprise most every physical scientific measurement I can think of), would be proportional to local_density(A)*local_density(B) and either adjusted by comparing the ratio or difference to the average_density(A)*average_density(B), depending on the scales over which measurements are calibrated.

So let's say two classes of particles A and B had an average density in a volume of 50% each.

The background rate of interaction (assuming constant uncorrelated motion) would be:

You could view these as 4 possible interactions:

density(A)*density(B) = .5*.5 = .25
density(anti_A)*density(B) = (1-.5)*.5 = .25
density(A)*density(anti_B) = .5*(1-.5) = .25
density(anti_A)*density(anti_B) = (1-.5)*(1-.5) = .25

If such probabilities existed on large scales in a very uniform and diffused manner, then these would appear similar to background radiation or virtual particles.

Now if a local increase occured setting A at 60% of the density and B at 40% at the density, then the new probabilities become:

density(A)*density(B) = .6*.4 = .24
density(anti_A)*density(B) = (1-.6)*.4 = .16
density(A)*density(anti_B) = .5*(1-.5) = .36
density(anti_A)*density(anti_B) = (1-.5)*(1-.5) = .24

So this would physically appear as differential changes of:

AB -.01 (a 1% local density of anti_AB)
!AB -.09 (a 9% local density of anti_!AB)
A!B +.11 (no anti particle)
!A!B -.01 (a 1% local density of anti_!A!B)

Anyway, the fact that photons have diffused wave characteristics, indicates more than one, likely 2 or more particles are involved in a spacially separated manner.

Let's say for example, that a photon represented an orbital between two points - the emitter and detector. An electron is also viewed as being composed of two quarks, similar (or identical) to a photon.

If we assume all real observations occur due to interactions or exchanges of information, then an individual fundamental particle could never be seen in isolation (isolated quarks are never observed, simply because it takes at least a second one to "see" it and exchange information).

So an electron could be viewed as a continual interaction between two quarks. This alone is timeless, just as a photon is, because two objects seeing only each other never convey new information and so no external references for time are available. (On the other hand, a mass doesn't exist in such a timeless state because each of the 3 quarks can observer various sequences of the other two quarks and construct a perception of a typical "spacetime" from this).

There's no evidence that photons exist in isolation - they may only be a communication between two points, and bidirectional effects have been observed for photons as well.

So imagine instead that an "electron", or general localized interaction between two quarks near at atom (you couldn't witness an electron in any context of "spacetime" without a third quark to make such measurements anyway so it could make sense that electron densities appear higher near a "mass"), can instead share an orbit of one of its quarks with a distant atom. In this case, one quark may drift off, leaving the remaining quark rather isolated and unobservable, meanwhile another such drifting quark comes in and replaces it. The original quark that drifted off has characteristics of the atomic orbital it occupied prior to this (discrete frequency spectra of emitted light from an atom) while the new quark drifting in would likely need to have some compatibility with the new shell it joins (there may be a size component and extra energy is effectively "chopped off").

Anyway, in this way the exchange of information via a photon is timeless, in the sense that an isolated quark doesn't experience time in itself. When it interacts with a second quark, it carries whatever information it contained from the last interaction (for an atomic orbital, likely related to the energy compatible in an exchang with the second quark in its prior orbital).

Now consider that these motions could be very chaotic. For us to envision the path of a photon, we need two stable references in space. If a quark diffused rather randomly, any single emission of a quark, without an exchange between two points would be diffused and appear as potentially a combination of background radiation and vacuum fluctuations.

But for our physical measurements, we're looking at local probabilities that are constrained to be non-uniform (we need something to grab onto and some other object we can see in order to imagine a path between them). In other words, if we're making detections at one end of a path and can already "see" the other, then this obviously requires a statistically significant pathway to exist between them, and so you'd obviously expect such exchanges would occur more frequently between these objects almost by definition.

There does seem to be a bit more than simple random motion necessary in order to create observed wavelengths. Some possibilities could be:

1) Only one of the two quark classes comprising a photon is visible. So an exchange only appears visible in one direction, though potentially an invisible exchange occurs the other direction.

2) The universe is largely cyclic and repetitive (unlikely entirely) and though motions might appear rather random, destructive collisions can occur and this creates characteristics where motions are biased toward only sustainable interactions and a probabilistic bias against two particles colliding destructively exists. (Imagine two quarks with identical, but complimentary characteristics combining and annihalting each other, such may have been common in the past but followed a sort of exponential decay, or "half-life" to such events).

There could be a way this energy is "recycled" randomly into some other form and the process repeated, so a continual possibility for such events could remain in either case, though with the recycling, it would likely be very constant.

3) If there truly existed the equivalent of an orbital between both endpoints in a photon exchange, this could give various wavelength characteristics, possibly related to a ratio of dimensions for an elliptical orbit.

4) The apparent wave characteristics could exist as an "aliasing" with an underlying quantized dimensional grid, for a discrete universe. In this case (which I personally think could be promising), the wave characteristics would be indicative of a mismatch between our Euclidean space and an underlying Hamiltonian space. Some distances we imagine as physically existing, don't actually exist for specific wavelengths as they physically can't fit to that distance. So we might observer two objects being some distance apart, but we aren't looking at distances relative to that specific wavelength, but instead using an approximate average. If you were to actually view the universe using a specific wavelength as reference for distances, the picture would change and some objects could be effectively invisible to each other, depending on the distance.

I could probably think of a few more possibilities, but that might be enough to wet some tastebuds.

Have fun.

By Aether Wave theory (AWT) the probability function corresponds the energy density profile of quantum wave and the mass density profile of Aether at the place of quantum wave pocket. By such a way, the energy density of standing wave can be computed as the square power of amplitude (see the DHTML applet for MS IE browser).



This post has been edited by Zephir on Sep 2 2006, 08:16 AM