Derivation of Doctor Dicks Fundamental Equation -- Part II
There exist a couple of subtle aspects of the model so far described. Of very great significance is the fact that the goal was to create a model which will model any explanation of A obtained from C. The specific mapping of the labels for the elements of C are part of the model and not a given aspect of the phenomena to be modeled: i.e., not at all part of A[/color]. If follows that the \vec{\Psi}(\vec{x},t) yielded by the model cannot be a function of that mapping procedure: i.e., all possible mappings must end up yielding exactly the same probability algorithm (the \vec{\Psi}(\vec{x},t) must yield results consistent with the actual distributions of the elements of B in C independent of the chosen mappings). This fact can be used to prove that \vec{\Psi} must satisfy some very simple partial differential relations.
The process yields three orthogonal differential constraints on \vec{\Psi} in the three dimensional representational space defined by the x, \tau and t axes of the model (if you need clarification on this issue, let me know).
<br />
\sum_i^n \frac{\partial}{\partial x_i}\vec{\Psi}\,=\, i \kappa_x \vec{\Psi}\,\,,\,\,\sum_i^n \frac{\partial}{\partial\tau_i}\vec{\Psi}\,=\, i\kappa_{\tau}\vec{\Psi}\,\,and\,\,\frac{\partial}{\partial t}\vec{\Psi}\,=\, im\vec{\Psi}
The final aspect of the model is to actually design a universal rule which is capable of yielding the distributions of the elements of B in C for every possibility. In this regard, it is quite easy to prove that, for any B (any distribution of points in the (x,\tau) plane) there exists a corresponding set D (a second distribution of points in the (x,\tau) plane), which, under the simple constraint that no two points can be the same, will constrain the distribution of B in C to exactly that distribution, no matter what that distribution might be (if you need clarification on this issue, let me know). The constraint that no two points can be the same is easily enforced by requiring:
<br />
F \, = \, \sum_{i\not= j}\delta (\vec{x_i}-{\vec{x_j}})\, =\, 0.
where \vec{x_k} is defined to be the vector in the x, \tau space defined by (x_k,\tau_k) and \delta represents the Dirac delta function. This express constraint on the elements of B can be converted into an express constraint on \vec{\Psi} by noting that the proper constraint on \vec{\Psi} is that that \vec{\Psi} must vanish whenever the above constraint on the elements is invalid; i.e., when F is not equal to zero, \vec{\Psi} must be zero. Thus the product of the two must always be zero and the correct constraint on \vec{\Psi} is given by:
<br />
\sum_{i\not= j}\delta (\vec{x_i}-{\vec{x_j}}) \vec{\Psi}\, =\, 0.
These four independent constraints on \vec{\Psi} may be expressed in a very succinct form through the use of some very simple well known mathematical tricks.
If one defines a set of anti commuting matrices as follows:
[\alpha_{ix}\, , \,\alpha_{jx}]\,\,{\equiv}\,\,\alpha_{ix}\alpha_{jx} \, + \, \alpha_{jx}\alpha_{ix}\,=\,\delta_{ij}
[\alpha_{i\tau}\, , \,\alpha_{j\tau}]\,=\,\delta_{ij}
[\beta_{ij}\, , \, \beta_{kl}] \, = \, \delta_{ik} \delta_{jk}
[\alpha_{ix}\, , \, \beta_{kl}] \, = \, [\alpha_{i\tau}\, , \, \beta_{kl}]\, = 0\, ,\, \, \mbox{ where } \,\, \delta_{ij}\, = \,\left\{ \begin{array}{ll}<br />
0 & \mbox{if i\noy=j}\\<br />
1 & \mbox{if i=j}.\end{array} \right. <br />
and defines two expressions ( \vec{\alpha_i} = \alpha_{ix}\hat{x} + \alpha_{i\tau}\hat{\tau} and \vec{\nabla_i} \,=\,\frac{\partial}{\partial x_i}\hat{x}\,+\,\frac{\partial}{\partial\tau_i}\hat{\tau} ), a small shift in perspective will allow the four constraints on \vec{\Psi} to be written in a single equation as follows:
<br />
\left\{\sum_i\vec{\alpha_i}\\,\dot\,\vec{\nabla_i}\,+\,<br />
\sum_{i\not=j}\beta_{ij}\delta(\vec{x_i}\,-{\vec{x_j}})\right\}<br />
\vec{\Psi}\,\,=\,\,K\frac{\partial}{\partial t}\vec{\Psi}\,=<br />
\,iKm\vec{\Psi}
constrained by the requirement that
<br />
\sum_i\vec{\alpha_i}\,\vec{\Psi}\,=\,\sum_{i\not=j}\bata_{ij}\,\vec{\Psi}\,<br />
=\,0.
It follows that all explanations of anything may be directly modeled by a set of points in an (x,\tau) space moving through a t dimension and required to obey the fundamental equation given above. The probability of any particular set of elements in B being given by
<br />
P(\vec{x},t) = \vec{\Psi}^{\dagger}(\vec{x},t)\cdot\vec{\Psi}(\vec{x},t)dv<br />
And I have thus successfully created a model of all possible explanations of A consistent with C. The fact that no constraints of any kind were placed on A implies the solution has consequences of great significance to every science under study.
If anyone out there has the gray matter to follow the derivation and is interested in discussing the issues it raises, I am ready to discuss the following issues:
1. Possible errors in my derivation (I will have to leave finding them to you as I am not aware of any errors).
2. How to obtain solutions to the equation and the solutions that I have found.
3. The philosophical implications as to what impact the discovery has on one's mental image of reality.
4. Interpretation of the equation: just what the various parts of the derivation mean.
:zzz: