Is it possible to have a field f([pard](x)) - fractal or otherwise - where [pard](x) are discrete Dirac delta functions, and f interrelates the various magnitudes of those singularities as a conventional field would for points over a continuum?
selfAdjoint
Sep10-03, 05:39 PM
I think the phrase "various magnitudes of all the singularities" is a contradiction in terms. a δ doesn't have a value until it's integrated, as in ∫f(x)δ(x-a)dx = f(a).
Loren Booda
Sep10-03, 07:19 PM
selfAdjoint,
Couldn't local Dirac singularities in my proposed field represent also various magnitudes obeying their point-by-point (distribution) normalization through overall integration?
selfAdjoint
Sep11-03, 11:08 AM
Loren, go back and look at the integral I posted, notice that f(x) in there. It could be anything. What the Dirac δ does is to pick out a particular value of any function that you integrate. Dirac modelled it on the finite case of a vector like (0,1,0). If you inner multiply that by any arbitrary vector (a,b,c) you get
(a,b,c)(0,1,0) = 0*a + 1*b + 0*c = b
(so it picks out the second component, and if you used a 1 in a different place you would pick out a different component. Now in QM math the integral of the product of two "functions" is an inner product in the algebra of those functions, so δ(x-a) in the integral picks out the "a-value component" of the function.