Is Dirac's notation really necessary? One thing that's always bothered me in Dirac's notation is that it assumes that the Hilbert space contains a "continuum basis" of vectors $|x\rangle$, which happen to be eigenvectors of an operator $X$ (which has no eigenvalues, only a continuous spectrum that spans the whole space). Their inner product is distribution-valued, with $\langle x'|x\rangle = \delta(x'-x)$. There's also the cryptic normalization property: $\int |x\rangle\langle x| dx = Id$. According to this question, some of these can't be made rigorous even with something like the concept of "Rigged Hilbert Spaces".
So, is there another approach to quantum mechanics in general that sidesteps this issue entirely, without loss of descriptive power?
 A: The problem, in my view does not lie with the notation as such, but rather with the physical scenario that is being studied. Space-time is continuous infinite. Therefore, one requires a basis that is also continuous and infinite. So, regardless of how one would represent such a basis in terms of a notation, its orthogonality condition would necessarily have to include a Dirac delta function. In the end, one can take one's hat off to people such as Dirac who came up with some mathematical system that makes it possible to do calculations that can lead to predictions, which in turn can be tested in experiments. The mere fact that such predictions often agree with these experimental results, seems to indicate that this way to calculate these quantities using this mathematical formalism must be correct to some extent. It then becomes a challenge to the mathematitians to try and come up with an axiomatic system that can lead to this formalism in a consistent manner. This often implies that one would need to stretch the notions of integrals, vector spaces and such so that the formalism can work in a strict mathematical sense. Whether or not that happens to be the case usually does not stop physicists from using the formalism.
