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AnssiH · 05-31-2009

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Originally Posted by Doctordick

The actual problem here is that the fundamental equation is no longer valid (we are simply no longer in the rest frame of the original object and our altered $\vec{\Psi}$ is thus no longer a solution to the correct equation). What we have is the fact that our mental model of reality must include the fundamental symmetry that all solutions, in the absence of outside influence, must transform to valid solutions to the correct equation in the center of mass system of any collection of data. This symmetry appears to imposes a major constraint on the character of the possible solutions $\vec{\Psi}$ . In reality,it does not as the scale invariant nature of our mental model provides a straight forward resolution of the difficulty.

There was a tricky sentence in there, and just to be absolutely sure I interpreted it correctly, did you mean to say:

What we have is the fact that our mental model of reality must include the fundamental symmetry that all solutions, in the absence of outside influence, must transform to valid solutions to the "fundamental equation" in the center of mass of a system of any collection of data.

At least that would make sense to me.

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It turns out that we are quite lucky in that the consequences of the above symmetry have already been completely worked out long ago by others. Notice that, if one ignores the Dirac delta function (as it has no spacial extension) my fundamental equation is a simple linear wave equation in four dimensions with wave solutions of fixed velocity.

Once again due to my lacking math knowledge, I'm unable to see that clearly. That it's "a linear wave equation with wave solutions of fixed velocity".

I googled "linear wave equation" and came up with a lot of stuff that looks partially familiar but thought maybe it's best if you just point me out to the correct direction.

At any rate, I have no problems with taking that on faith for now, as I figure the important bit is that the elements are expected to travel at fixed velocity.

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The constraint spoken of above is exactly the same constraint placed on the conventional Euclidean mental model of the universe by the fixed speed of light in Maxwell's equations. As we all know, if we constrain ourselves to linear scale changes, it turns out that there exists one very simple (and unique) relativistic transformation which maintains a given fixed velocity for all reference frames moving with constant velocity with respect to one another.

Indeed, Lorentz transformation.

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The velocity in our four dimensional “wave equation” is fixed by the value of K in our representation. (Notice that, in my derivation of Schrödinger's equation, I set $c=\frac{1}{K\sqrt{2}}$ .) For the moment (since K is actually a totally open parameter) I will set this constant velocity to v_?.

Yup.

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In order to solve for the required transformation, consider uniform motion in the x direction (remember, we are still actually working in a four dimensional representation so x can be in any direction (though I will not really worry about tau as in the final analysis any dependence on tau will be integrated out anyway so tau is, in some sense special; particularly as it is a figment of our imagination created solely to allow representation of multiple occurrences of valid elemental entities). In the following picture, the tau axis is not shown. We just can't really show four orthogonal axes in a conventional picture. In this case, tau is simply another axis orthogonal to x and obeys exactly the same relationships as do the y and/or z axes: i.e., $\tau'=\tau$ .

We need to have a formula for translating coordinate points in the first frame, $(x,y,z,\tau)$ , into the identical points represented in the second frame, which have to be $(x',y',z',\tau')$ in a way which continues the validity of the fundamental equation. In order to do that, I will use the fact that the fundamental equation is (sans interactions) a wave equation where the wave velocity, v_? is constant; thus, we can use an opening circumstance where (at t=0), $\Psi$ , the wave function of an object consisting of a single element (i.e., all interactions with the rest of the universe are being ignored), consists of a spike at the origin in both frames and is zero elsewhere (that means we are starting with the origins of both frames of reference exactly aligned origins).

Yup.

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Anyone familiar with wave equations understands that the solution here is quite simple, $\Psi(t)$ is thereafter a spike at r=tv_? (where r is the radius of a four dimensional sphere centered on the origin) and zero elsewhere from then on. (Think of a flashbulb going off at the moment the origins of the two coordinate systems are exactly in the same point and then picture the sphere of light expanding at the speed of light.) The fact that our case is a four dimensional sphere is only of passing significance here, as we are still speaking of uniform radial expansion: i.e., the radius to that pulse of probability must be given by $r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}$ . Please notice that this means that once a scale is set for one coordinate, it is likewise set for all the others (otherwise we wouldn't have a sphere).

Not really familiar with wave equations but still that all seems to be trivially true.

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Thus the wave function is non zero only on the surface of a sphere expanding at a specific velocity (which I am calling v_? for the time being). What is important here is that this must be true in both frames (if it is not true in the primed frame, the non-zero portion of $\Psi(t')$ will not be on the surface of an expanding sphere). That is, both frames must yield exactly the same probability distribution; it is the two frames of reference which are different, not the probability of finding that elemental entity.

First, it is quite easy to show that the transformations in y, z and $\tau$ are trivial as they must always line up exactly with the same points on the unprimed axes (an entity not moving in one of those directions in the unprimed coordinate system can not be moving in those directions in the primed coordinate system): i.e., y'=y, z'=z and $\tau'=\tau$ (the scale of these coordinates must be identical). The only problems occur with the x axis and t. Note that, in my picture (though I can produce x, t diagrams) t is not an axis of my coordinate system; it is instead, a parameter of evolution, a distinctly different concept. It should be clear to the reader that there exists no way to guarantee that t in the primed coordinate system is identical to t used in the unprimed coordinate system (before we can discuss that issue one must first explain how time is to be determined).

Yup, it all seems clear up to this point...

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Nevertheless, it is fairly easy to show that the transformation from one coordinate system to the other can be no more complex than $x'=\alpha x -\beta t$ and $t'=\gamma x -\delta t$ .

For those who don't believe that, consider the terms in a power series expansion of some supposed arbitrary function. The constant terms of that power series can be dropped as they move the primed origin at all times even t=0 where we have already defined it to be in exactly the same position as the unprimed coordinate system.

...but I understand almost nothing of the above

I suspect alphas and betas refer to something different than they do inside the fundamental equation, and that $\gamma$ is the Lorentz factor...? Also don't know what to make of the $\delta t$ . I have no idea what's a "power series". Needless to say, I am quite lost once again

I think I should stop here until I understand that step properly.

-Anssi

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