#math #quantummechanics
How do mathematicians feel about Dirac's delta function? Got any pointers to gaining a more precise, "mathy" understanding of it?
I've been reading Shankar's book on quantum mechanics (really like it). Dirac's delta function is something that has always bothered me somewhat, since it's not really a function.
@aleksi you mostly consider it as a linear operator, operating on a suitable class of functions (e.g. continuous functions or smooth functions), for which it respects the topology on those functions: operator delta_x applied to function f, gives the number f(x). This is a linear operation in f, continuous with respect to suitable topology on the (vector) space of functions. The theory of distributions, associated to the name of Schwartz, is usually treated in functional analysis books.
Thanks for the pointer to Schwartz distributions! Now that I've read and thought a bit more about it, seems like these distributions are "derivatives" of functions that are not necessarily differentiable. So they only "work" as functions when inside integrals.
For example Dirac's delta function is the derivative of the Heaviside step function. Heaviside function is not differentiable, since it's not continuous.
@aleksi that's right.. the interpretation as functions (rather than operators "dual" to functions where duality is given by integration) can survive if you stretch your intuition about functions.. Gelfand and the Russian tradition, talk of "generalized functions" (there's a book with that title in fact)