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Posted by Richard D. Stafford, Ph.D. on July 16, 2002 21:43:19 UTC

This post is in reference to

posted by Richard David Yannopoluos-Ruguist otherwise known as Yanniru. It seems he is tired of my rudeness! I am tired of his incompetent analysis. In particular, I am tired of his pretending to be a competent physicist.

I am of the opinion that anyone who understands the purpose of indices and college calculus has enough mathematics to comprehend the illogic of his comments and I invite anyone with such rudimentary education to take a careful look at his complaints. Nothing here is beyond your ability to understand!

I am writing this for the benefit of those who think, as I am completely unconcerned with Yanniru; he has shown himself as intellectually incompetent. Anyone who wishes to see the equations and text Yanniru is referring to is invited to visit my site:

Now, to the issue at hand!

Yanniru: In your paper at eq. 1.18 you explicitly set F =0.

What I said was that there must exist a function F such that for all valid collections of data (data which obey the rules) the statement F = 0 is capable of enforcing the rule. Central to this statement is the fact that one is working with an open ended collection of imaginary entities: see Aurino's point # i in

Which exactly expresses this issue!

My statement is not at all the same as explicitly setting F=0: i.e., the rule can be represented by F=0 is not equivalent to the statement that F=0. Only someone totally unfamiliar with the subtleties of mathematics would consider the two statements equivalent. Using the roots of a function as a constraint is a common technique used in advanced mathematical physics; if Yanniru were familiar with such work, he would be familiar with the technique and not confused by the path I presented.

Yanniru: Then in eq. 1.22 you set F equal a summation over delta functions and still set it equal to zero.

Ok, so here I did not make myself as clear as I might have. I presumed the reader would be intelligent enough to realize that I meant that the "rule" F=0 could be so written. Again, there is a big difference between saying F is zero and saying F is zero for valid data! If one includes the probability of observing a particular set of data in the analysis, then the situation where F is not equal to zero is no problem (so long as the probability of seeing such a situation is zero). This kind of logical analysis is common to many advanced physics problems and should not engender difficulties with anyone competent in advanced mathematical physics: ergo, Yanniru is not competent!

Yanniru: You can do this because the summation over j and i explicitly excludes i=j. Those are the indicies I referred to.
This statement is a succinct demonstration that Yanniru has no comprehension at all of what is going on. True, the equation absolutely cannot be true if i=j as, if that were the case, the summation would include a term which would have the argument x sub i minus x sub i which is explicitly zero no matter what value x sub i has. That being the case, that term would be infinite for any and all values of x sub i and there could exist no x sub i which would yield a non infinite result. For that reason, the sum cannot include any terms where the indices are the same. This has absolutely nothing to do with values over which the arguments may range. The indices are indicators as to which argument we are referring to. Only a person completely ignorant of the use of indices would confuse the two issues.

Yanniru: In eq. 1.23 you indicate that the delta function would equal infinity if i=j.
No, I do not! Look at the equation once! The argument is x sub n minus x sub i : i.e., the indices are explicitly different (one is n and the other is i). We are talking about two different arguments here! What I say is that F becomes infinite if x sub n equals x sub i for any i at all. The possibility that i equals n cannot occur because the range of i does not include n. If you go to the paragraph immediately above equation 1.23 you will see the line which explicitly states that we are considering the set of arguments x sub 1 through x sub (n-1). I conclude that Yanniru either did not read the document or did understand what was being said. Given his performance on the other issues I lean towards the latter as he is clearly confusing the value of arguments with the value of the indices which indicate which argument we are talking about.

Yanniru:, and in the following paragraph you elaborate that the infinity must be avoided.
Certainly it must be avoided! If the rule is F=0, (since all terms are positive definite) any term going to infinity certainly violates the rule! Note that, in this case we are talking about the ability of the expression F=0 to constrain the data, not about the value of F. Again, Yanniru's confusion of the two issues leads met to believe he has no experience with any mathematical procedures which depend on these kinds of issues. Any graduate student familiar with theoretical physics should be familiar with this basic concept; ergo, Yanniru has never spent much time in the field of mathematical physics.

Yanniru: Therefore when you later integrate you cannot integrate over the infinity to get a non-zero value.
This line is a complete non-sequitur. Stating that the rule to be enforced is F=0 is not equivalent to stating that F=0. Yanniru has apparently completely failed to comprehend the importance of expression 1.20; I can only presume that he was unable to comprehend the significance of the presentation surrounding that expression. In view of the fact that such analysis is common to advanced mathematical physics, I conclude Yanniru is ignorant of the field.

What is important (and understandable by any of you) is that F=0 for any valid set of data and P (the probability of seeing a particular set of data) is zero for any invalid set of data (i.e., no correct calculation can result in "invalid data"). It follows that the product FP=0 for absolutely any conceivable set of data (valid or not). Thus equation 1.27 is true for any arguments at all and has absolutely nothing to do with whether the arguments are valid data or not. If the arguments are invalid data, then the relations expressed in equation 1.27 will force the magnitude of Psi to zero and the constraint F=0 need not be true! The equation produces the "either or situation": either F=0 (the arguments constitute valid data) or Psi is zero (the probability of seeing the data is zero).

Yanniru: But in actaulity, your whole result is because you got a non-zero value when you integrated over the delta functions, which you yourself constrain to always equal zero.
No, I do not constrain them to equal zero, I constrain the product between them and the magnitude of Psi to be zero; quite a different statement. What bothers me is that anyone familiar with mathematical physics and the use of the Dirac delta function would be right at home with such expressions. Yanniru's complete failure to comprehend the difference (which I do not believe is beyond any of you) implies he has had nothing but a cursory introduction to such work, if that.

Yanniru: For more examples, in eq. 1.27 the middle term containing betas is just the same F function, which we know equals zero. Take out the beta term and you lose all your results.
Apparently, once again I have exceeded Yanniru's comprehension of mathematical physics. I distinctly get the impression that he has no idea what so ever of the meaning of the expressions he is looking at.

Yanniru: Same thing in eq. 2.1 which is your general equation. Then in eq. 2.5 you even call it the term f again, but small f this time. Still equals zero anyway, but the delta functions are gone from faulty integration over them. This f function ultimately becomes the potential function V in Schrodinger's eq 2.17. So what tyou really derived is the wave function, which of course follow from the shift symmetry.
Just more of the same! If he cannot comprehend equation 1.27 he is intellectually lost. He has decided that the integral over the Dirac delta function is zero and thus the term may be neglected. I take this to be an indicator that Yanniru has never done any serious work with the Dirac delta function.

Yes, the integral over the product of the Dirac delta function and Psi is indeed zero (for any specific value of the argument not being integrated over). But that is not a significant fact when compared to the probability that the argument not being integrated over has a specific value . Since we are working with a continuous variable, the integral over all possible values of that argument (the sum of the probabilities of all values) must be unity and the probability of any particular argument must be zero (because the number of possibilities is infinite). Persons experienced in mathematical physics are not bothered by such circumstances. Yanniru's analysis is equivalent to trying to do calculus from the arithmetic perspective. It is only a question of understanding higher mathematics.

Again, anyone familiar with advanced mathematical physics would be familiar with such things. So I can only conclude Yanniru is lying about his training. It is not necessary to understand the field to understand the illogic of Yanniru's comments as he makes use of none of those issues. All of his comments are on a High school level. You should all be capable of understanding this issue and should not surrender your opinion to his claimed authority.

Have fun -- Dick

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