In my copy of your derivation you expressly exclude the case of xi=xj in the summation over the delta functions. Has that step in the derivation changed?
You are clearly confusing the constraint on the sum, which is i can not equal j, with the values of the arguments indexed by i and j: i.e., xi and xj which can certainly be equal to one another. If one allows i=j in the sum, then the sum is always infinite because it includes the term xi-xi which is exactly zero at all times!
This is elementary stuff Richard!
And if xi=xj is allowed in the rather early step defining F, how do you avoid the resulting infinity that F would then equal?
I don't! That F blows up is of no consequence; all it requires is that the probability that xi=xj is zero which is true if psi is continuous. This also is elementary theoretical physics. I am sure these kind of integrals were done in E&M.
Also PS: You seem to be claimint rigor in the above post. Yet between eqs (2.14) and (2.15) you assume that psi is given by an exponential in time. That is how you go from a wave equation to Schroedinger's equation. Hardly a rigorous step to assume the solution rather than deriving it.
It is exactly rigorous Richard as I am looking at a specific approximate solution to my fundamental equation and I express exactly what approximations I am making. At no point in this particular piece of my presentation do I claim that the solution being obtained is exact!
In fact, one absolutely cannot derive the Schrodinger equation without making approximations because the Schrodinger equation is not correct; it is only approximately correct! If one could derive it without making an approximation then either the derivation is in error or the starting point is wrong. Any physicist knows that Schrodinger's equation is only valid in the non-relativistic case and, if you read on, you will discover that after establishing the definitions of mass, momentum and energy, I show that the approximation I use here is exactly that approximation: i.e., the exponential adjustment to psi immediately below equ(2.15) is exactly the required correction to remove the relativistic mass energy term from the total energy. And that approximation I made turns out to be that the total energy (given by the partial with respect to time) must be approximately constant! That says that the energy of the system being described by Schrodinger's equation (that energy which is changing) must be small compared to the total energy: i.e., the relativistic mass energy of the system must be large compared to the other energy of the system. It must be a non-relativistic situation Richard!!! That is all that approximation amounts to and it absolutely must be made if you are to rigorously derive the Schrodinger equation!
What I have presented is very rigorous -- Dick