User:UniversalExplanation/appendix2

Appendix 2: Universality of the F=0 Constraint
A simple way to understand the fact that there must exist a set D which will constrain B to any set conceivable through the Dirac delta function is to imagine the x $$\tau$$ plane as being masked by the particular set B under consideration and let D consist of all the remaining points in the x $$\tau$$ plane. It should be clear that D will constrain B to exactly the set of elements of A in B if the constraint on the complete collection of points is that no point can appear twice.

Since the Dirac delta function is defined by

$$\int_a^b\delta(x-c)dx = \left\{\begin{array}{c}0 \text{ for c not in interval } [a,b]\\1 \text{ for c in the interval } [a,b]\end{array} \right. $$ ,

and the interval over which x = c is exactly zero, the value of the Dirac delta function at x = c must be infinite. It follows that, if the two arguments of any term of the sum

$$F = \sum_{i\neq j}\delta(\vec x_i - \vec x_j) = 0$$

are identical, the sum is explicitly infinite. Thus it is that the only case which satisfies the constraint F=0 occurs when no point in the plane appears twice. This proves a D exists for any possible collection of elements in B.

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