It seems that for the time being we shouldn’t consider the possibility of there existing any particular reference frames in which the fundamental equation is valid in and gives different predictions then in any other reference frame and just consider it just as valid in any reference frame as at the time being. I think that it may just be confusing the issue of when the fundamental equation is valid and where the Lorenz transformation comes from. I think that we have been doing this anyhow.
Quote:
Originally Posted by Doctordick
I am not sure of exactly what you mean by “we are interested in”. We are interested in any transformation which can be defined and exactly how the defined transformation should be performed and what the meaning of the transformed solution is (that depends very much on exactly what kind of transformation one is talking about).
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Unless I am missing something I think that the only transformations that you have defined are the those resulting from multiplying
by
and
(where i is the square root of -1 and k is a real number) ( also I am only considering the one dimensional case here) performing these multiplications results in the addition of momentum mass or energy to the explanation.
The problem is that if a subset of the universe exists in which the fundamental equation can be considered valid in then the fundamental equation must be valid in both frames but the energy momentum and mass of the elements won’t agree in both frames.
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Originally Posted by Doctordick
If you examine the deduction of my fundamental equation, you will discover that a very important step in that deduction is the proof that, no matter what patterns the valid elements may have, there always exists a set of hypothesized elements such that the rule  will constrain the valid elements to exactly those required patterns. When one makes the approximations required to deduce Schrödinger's equation (doing all the required integration), that self same rule ends up being a function of “x”. Thus the meaning of the original proof is simply that, under the approximations made, there always exists some function of x which will require the behavior of any specific element to obey Schrödinger's equation where V(x) is that function. Is it possible that V(x) could be so complex that we can no longer use Newtonian mechanics? That depends on what you mean by “use Newtonian mechanics”. If you mean, “so complex that we can not solve the problem”; sure, there are a lot of Newtonian problems so complex that solution has eluded the scientific community for centuries. But can that be taken as evidence that Newtonian mechanics is false (false in the sense that even if the approximations used are valid, Newtonian mechanics is still false)? I think not. |
The sense that I keep thinking of, that I wonder if Newtonian mechanics would be useful in, is if the approximations made are not good approximations (that is if we can’t really ignore the influence of the rest of the universe). Seeing as we can’t solve the problem directly and the only explanation that I know of to compare it to is my experience I see no way to know one way or the other. Although it seems that if we look at the fundamental equation over small enough changes in the axis’s then this can be considered a good approximation or is this even an issue?
How I am understanding this is that if we look at the equation over a sufficiently small change in t then Newtonian mechanics will approximate the fundamental equation. And over those changes in t there will exist objects that can be considered universes onto themselves. That is, they have a rest frame in which the explanation of the object in its rest frame is just as valid as the explanation in the rest frame of the universe.
The problem is that when the Schrödinger equation is considered the change from the rest frame of the universe to the rest frame of just the object will result in changing the energy and momentum of the explanation so that the fundamental equation is no longer valid.
Now we want a transformation that transforms the measurements in one frame to any other frame. This would allow us to take the measurements taken in another frame and transform them to the measurements that they would be if we made them in our reference frame. The fundamental equation without the Dirac delta function is a wave that is expanding at a constant rate. This is given by the equation
now this equation must be valid in any reference frame that the fundamental equation is valid in. Considering the scale invariance of the fundamental equation and that any transformation must be invariant under any shift in the origin the only possible transformation is of the form
using these equations and solving for the necessary transformation we arrive at the Lorenz transformations as the only possible transformation.
In order for this to work it requires that the value of
is the same in both frames but the actual value of
is actually defined by the measures of length and time. Your clock will allow us to define time by counting the oscillations of the oscillator but this still requires that we define distance. In order for us to define distance to be the same in both frames we would have to use a property that is the same in both frames. The problem is that there is no property that is the same in both frames. Any property is a property of what we are explaining not a property of what the explanation must obey. So that all that we can do is use the same procedure to construct a unit of measure (The Schrödinger equation lets us do this?). If both explanations are valid then the procedure must be a scale of the procedure that the other observer is using. That scale is the very thing that we found to be the Lorenz transform.
Quote:
Originally Posted by Doctordick
It seems to me that you are confusing two very different issues here. Deriving the Schrödinger equation and showing that Newtonian mechanics is an approximation to the fundamental equation has nothing to do with any scale issues. Objects are defined to be collections of elements which remain in a coherent structure over a sufficiently long time to be thought of as individual entities. You should understand that one of the major approximations necessary to be made is that the energy of the entities must be approximately given by  . That means that the kinetic energy (the energy of motion) must be small compared to  : i.e., the elements going to make up that object cannot have net relativistic velocities with respect to the rest frame of the object. This is a fundamental constraint on Schrödinger's equation. That further means that V(x) cannot be so large to generate relativistic Newtonian velocities. The net result is that the collection of elements going to make up objects (such as my mirror assembly) cannot have relativistic velocities relative to one another. It follows that the directions of the individual elements making up the objects are all on essentially parallel paths, moving at v ?. Any deviation from those parallel paths must be small. |
But won’t the oscillator always be considered to move at
as it has zero momentum in the
direction or is it not considered part of the mirror assembly and relativistic effects wont effect it as it has a fixed speed after distance and time have been defined. This seems to be the case.
Quote:
Originally Posted by Doctordick
I get the impression that you are trying to confuse yourself. There are three steps involved here and you need to understand each of those steps before proceeding to the next. First there is the deduction of my equation (which has to do with explaining an arbitrary past; that is why my first step is to define time). Second, is the proof that Schrödinger's equation is an approximation to that equation. And, third, is the demonstration that the picture requires the same relativistic transformations as does Maxwell's equation. Mixing and mushing with these three different issues is a procedure just crying to confuse you.
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I’m not sure that I follow exactly what the purpose in the first part of your previous post is. It looks like you were beginning to lay out some of the considerations for the first step in your deduction although I’m not sure I understand why, unless there is something about it that has some kind of influence on the current topic that you are trying to point out or you are suggesting that I take and go back and look at the original deduction. If that is the case, I think that it is best if I use your latest topic “What I believe an explanation is!” Thread for any questions as they come up in the thread as I have read all of the “what can we know of reality” thread as well as parts of the “Is time just an illusion?” thread, although it was quite some time ago and I can’t say how well I understand it while it at least made sense when I read it, and the “what can we know of reality” thread is beginning to get slightly confused in the topic originally meant to be disused as well as quite long. If this is the case just say so.
