#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.
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)
@aleksi
Perhaps normal functions are actually a fluke and functions are simply things that you integrate. I mean in reality, you never measure any function at any exact point, you always take some average of values (how large your sensor is, how long it takes to take the reading, etc.). So perhaps it's normal functions that don't exist, and then the Delta function doesn't seem so bad, no?
@rednekulo
In the context of quantum mechanics (and physical reality, which seems to obey quantum mechanics) that may be indeed correct.
But there are many contexts where functions *are* evaluated at exact points, for example when evaluated by (non-quantum) computers or by people.
To me the most reasonable (concise, useful) definition of a function is still a mapping that can be evaluated at any point. So calling things like Delta function "generalized functions" seems appropriate.
@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.