What Are the Limitations of Multiplying Delta Functions in Quantum Mechanics?

[cjellison]

Hi, I am not seeking a "complete" treatment of distribution functions (like Gelfand or Schwartz). However, I would like some discussion in regards to multiplying delta functions together---especially in QM.

From the little I have discovered, distributions do not form an algebra, and thus, one cannot "legally" multiply delta functions together. However, we do this all the time:

$$\delta(\mathbf{r}-\mathbf{r}_0) = \delta(x-x_0)\delta(y-y_0)\delta(z-z_0)$$

Also, I was wondering about a text that discussed in detail the problem with "complete" sets of eigenfunctions in an infinite-dimensional space ($L_2$) and normalizing to a delta function rather than to 1. I guess I am looking for better explanations than the typical "it just works" explanation. Do I need von Neumann's book?

Thanks.

 

[DarMM]

The theory of multiplying distributions is very much under-developed and is one of the reasons we lack a rigorous definition of Quantum Field Theory. There are several proposals but no full theory. One can define multiple algebras, the trick is one which preserves the properties you want.

For your second question have a look at https://arxiv.org/abs/quant-ph/0502053