Proof of validity of the Fundamental Equation.
Hi Russell,
Hope you can follow this! If you have any questions, let me know.
Doctordick said:
The essence of the proof that \vec{\Psi} must satisfy my fundamental equation rests with a proof that the constraints already shown as necessary can be recovered from any solution to that equation: i.e., that any solution to my fundamental equation will satisfy the constraints already laid out and secondly, it must be shown that there exist no solutions satisfying the given constraints which will not be solutions to my fundamental equation.
The first step is to note that, from the definition of \alpha_{ix}, we know that \alpha_{kx}\alpha_{ix}\,=\,-\alpha_{ix}\alpha_{kx} + \delta_{ik}. We then left multiply the fundamental equation by \alpha_{kx} (left multiply means \alpha_{kx} is on the left of the expressions in the equation).
[Well, I just discovered a Latex error in the fundamental equation as shown in message #4 of this thread. I have corrected the error. If anybody sees any errors in anything I say, please point them out to me. I would appreciate it very much and it would do nothing but raise my opinion of whoever noticed it.]
Back to the issue at hand! Since the sum over i in the fundamental equation is over all events (both knowable and unknowable), i=k will occur exactly once in that sum and commuting \alpha_{kx} with the other \alpha and \beta yields a simple sign change. (Note that, since these matrices are not functions of x, \tau or t, \alpha_{kx} commutes with the various partial derivatives.) The result will be:
\frac{\partial}{\partial x_k}\vec{\Psi}-\left\{\sum_i\vec{\alpha_i}\,\cdot\,\vec{\nabla_i}\,+\,<br />
\sum_{i\not=j}\beta_{ij}\delta(\vec{x_i}\,-{\vec{x_j}})\right\}\alpha_{kx}<br />
\vec{\Psi}\,\,=\,\,K\frac{\partial}{\partial t}\alpha_{kx}\vec{\Psi}
<br />
=\,iKm\alpha_{kx}\vec{\Psi}
If the above expression is summed over all k, since \sum_k\alpha_k \vec{\Psi}\equiv0, only the first term survives. Thus we know that, if \vec{\Psi} is a solution to the fundamental equation, it is also a solution to the equation
\sum_k \frac{\partial}{\partial x_k}\vec{\Psi}\,=\,0.
This seems to be quite different from the original constraint. That difference is the source of the "small shift in perspective" which I mentioned in message #4. If \vec{\Psi}_0 is a solution to the above equation, simple substitution will confirm that
\vec{\Psi}_1\,=\,\displaystyle{e^{\frac{i}{n}{\sum_{i=1}^n}\kappa x_i}}\vec{\Psi}_0
is a solution to
<br />
\sum_{i=1}^n \frac{\partial}{\partial x_i}\vec{\Psi}_1\,=\, i \kappa\vec{\Psi}_1\,.
Exactly the same process will pull out the constraint on the \tau axis. The F=0 constraint is zero already so that the result of using the commutation properties of \beta_{ij} to isolate F yields exactly the correct constraint.
Anyone who is familiar with quantum mechanics will recognize this as essentially the mechanism to shift a many particle wave function into a new frame of reference moving with respect to the first. Of course, the partial differential corresponds to the momentum operator of standard quantum mechanics. This whole collection of relationships will be obvious to anyone with a basic education in beginning quantum.
In fact, it is precisely the presumption of shift symmetry [P(x+a) =P(x)] in quantum which is used to establish conservation of momentum. (The only real difference between my development of that constraint and the common physics notion is that my development does not constitute an assumption: it is instead a direct consequence of the arbitrary[/color] labeling of those as yet undefined references we started with.)
Since, in standard beginning quantum mechanics, the \kappa=0 solution of the many particle wave function only exists in the "center of mass" reference frame (when the total momentum of the system is zero), I will
define my fundamental equation as valid only in the "center of mass" reference frame! Just as Newton's equation F=ma is valid only in an inertial frame, my equation is only valid in the frame where the partials with respect to x and \tau summed over all \vec{x_i} vanish.
Finally, general differential with respect to t may need to be a constant but there is no constraint that the constant be related to the left side of the fundamental equation. However, once again, if \vec{\Psi}_0 is a solution with m=0, simple substitution will confirm that
\vec{\Psi}_1\,=\,e^{i Mt}\vec{\Psi}_0
is a solution to for m=M no matter what M may be desired.
The other side of the coin is equally easy to defend. Any solution which fits the four constraints may be adjusted to one which is a solution to the fundamental equation. I have proved that any explanation of anything may be cast into a form which requires my fundamental equation to be valid. Except for the use of the term "The Universe" to represent "an explanation of anything" this is exactly the purpose of Chapter I of "The Foundation of Physical Reality".
[QUOTE="The Foundation of Physical Reality", Chapter I]So, let us review exactly what has been accomplished in this opening chapter. First, I have constructed a mental model of "The Universe". It is admittedly an extremely simple model in that it consists of nothing more than points in a two dimensional Euclidean space who's position in that space is a function of time. It may be a simple model but it holds forth three very important aspects: first, it is very well defined and thus easy to understand; second, it is complete as there exists no communicable concept of reality which is not representable by this model and finally, we have a very specific method of answering any question asked together with the fact that the answer (i.e., the probability of any given answer) must obey an apparently simple equation.[/QUOTE]
At this point I have
defined only thirteen concepts outside of mathematics itself.
-->"mathematics"; a set of logical relationships and definitions understood by enough people to provide decently unambiguous communication.
-->"A"; Whatever it is we wish to explain; the Universe, a problem, an explation…
-->"B"; That finite set of elements of A available to us which our explanation must absolutely explain.
-->"knowable"; elements of B which are elements of A.
-->"C"; A finite collection of sets B; all knowledge which is available to us from which we must create our model. ("C" is "knowable" information).
-->"D"; A finite collection of hypothetical sets analogous to B which are required by our explanation.
-->"unknowable"; elements attached to B which are not elements of A; hypothetical aspects of D.
-->"\vec{x_i}"; an arbitrary numerical label assigned to references to the elements of a given Bj plus those references in D attached to Bj
-->"time" an arbitrary numerical label attached to "Bj" plus the "unknowables" attached to that particular "Bj".
-->"observation"; A collection of references \vec{x_i}(t) which label all the "knowables" and "unknowables" of a particular Bj.
-->"past"; observations available to a test of the explanation.
-->"future"; observations not available to a test of the explanation.
-->"\vec{\Psi}(\vec{x},t)"; an arbitrary mathematical algorithm which will deliver a measure of the expectations for Bj given the associated observation via a normalized inner product with its adjoint {P(Bj)}.
-->"Center of mass coordinate system"; the abstract Euclidian coordinate system where the fundamental equation is valid.
If you can understand and accept the above, then all that is left is to find the solutions to the fundamental equation. Again, as I show you the solutions, I will define further concepts convenient to talking about those solutions.
Have fun -- Dick
