Richard,
I am surprised and gratified by your response. I am always willing to talk to anyone on a rational level.
***** Richard:
Is that a typo. If xi never equals xj then the Dirac delta function is always zero.
*****
I think I understand your confusion; it appears that you misunderstand the meaning of the indices. The data consists of a set of numbers. The index refers to which number in the set we are talking about. Thus the statement that the sum is over i not equal to j says that the pair of numbers taken from that data set to be used in the Dirac delta function shall never be the same piece of data. It does not mean that two different members of the data set can not have the same numerical value. That is to say it is certainly possible that xi=xj without requiring i=j.
***** Richard:
You first introduce the delta function in eq(1.22)
where you equate F to a sum over Delta(xi-xj) and with the sum excluding i=j, so that F=0 as we know it should.
*****
Here again, you are missing the point. The fact that i=j has been excluded does not force F to zero. The situation is rather the other way around, the rule that F is set to zero forces the exclusion of xi=xj for when that case occurs the Dirac delta function blows up. In fact if you think about it for a moment, you should realize that i=j has to be excluded from the sum as, if that term is allowed, F cannot possibly have any value except infinity as xi always equals xi.
***** Richard:
I still cannot understand how you go from the xi values, which are values of your data, to x which is a spatial coordinate.
*****
The xi values are just plotted on the x axis. My data are displayed as a bunch of points on the x axis!
***** Richard:
The exponentials I talk about are solutions of the constraint equations for psi given by eq(1.25). The time dependence is exponential
psi=exp(imt)
If there were just one data point xi (n=1) then
psi=exp(ik*xi)
In eq(1.29) you also express the variation of psi away from the center of mass as an exponential. That is why I expected a wave equation.
*****
These comments lead me to believe that you are not familiar with the standard proof of conservation of energy and momentum as presented in quantum mechanics. (Or perhaps the confusion engendered above is the source of the problem.) Eq(1.29) is exactly the standard change in the wave function generated by changing the value of the total momentum of the collection of entities being represented by the wave function. In my presentation, these constraints (those not including the one with the Dirac delta function) are arrived at by the requirement that a constant may be added to all my data without removing the possibility of modeling the data. This is, in essence, identical to the standard proposition that the origin of the coordinate system used to represent the data cannot have an influence on the answer to the problem.
Returning for a moment to set of equations listed as eq(1.25), the first is exactly the standard expression of conservation of momentum. My presentation is not exactly the same as the standard but the mathematical content is completely equivalent. The second (on the upper right) is exactly the same factor, except it is expressed in the hypothetical tau axis direction, and the third (on the lower left) is the same effect as seen from the perspective of the time variable (the standard proof of conservation of energy in standard quantum mechanics). These three equations have long been known by anyone familiar with quantum theory of particle interactions. As is eq(1.29)! Now, that any possible rule may be forced by the Dirac delta function sum (on the lower right) is original with me.
But, other than that, no competent physicist would argue with eq(1.25) nor with the fundamental nature of my deduction of them. The only real difference here is that I do it before the fact of identifying the differentials with momentum and energy and they do it after the fact.
Note here that, in my presentation, I have not defined the terms momentum, mass or energy. That is one reason why I make the comment about defining the center of mass coordinate system. That coordinate system is defined in my presentation by the fact that the constants displayed in eq(1.25) vanish; whereas, in the standard presentation, that same coordinate system is defined by the fact that the total momentum vanishes (which I clearly cannot do as momentum has not yet been defined in my presentation).
***** Richard:
I am puzzled by eq(2.22) where you say "the probability distribution of the datum represented by xk is given by"
Here you go back to xk as a value of the data rather than a spatial coordinate. What gives?
*****
The only thing I can get from this is that your problem is nothing more than a confusion of the notation I think I have explained at the beginning of this post.
My picture is as follows: I have this knowable data which I have divided into separate sets which I have called observations. I picture these various sets as having been selected from a continuous set at different times (that is a mental model of the circumstances). In my mental model of this data, these observations are not independent. I picture them as the same set of data at a different time: i.e., in my mental model of the universe, this knowable data is a set of numbers which change over time and I am trying to discover the rules which will explain that change.
Notice two very important things here. The fact that I see these observations as being observations of the same data is a complete figment of my imagination. Also, that this data changes continuously into the data seen in a later observation is also a complete figment of my imagination. All I really have to work with is the fact of the knowable data itself. That is, the fact of the observation is taken as real but the idea that any of this information is real when you are not looking is taken to be a complete figment of my imagination.
The unknowable data is that data required to make that figment seem real! In other words, to justify the model I am in the process of creating here. The unknowable data is there to give explanation to the change in knowable data from observation to observation.
So my data is being viewed as a collection of points plotted in an x tau space. The data and the points in the mathematical space are exactly the same thing; they don’t go back and forth in interpretation at all.
***** Richard:
Don't go. Just work with me. You do not strike me as thin skinned.
*****
I am not very thin skinned. I just get very disappointed when I can’t see any light at the end of the tunnel.
Have fun -- Dick |