Singular external potential in a field theory?

[evilcman]

I have a theoretical question.

When doing canonical quantization, and writing the equation for he time evolution of
operator in Heisenberg picture, we make use of the statement that the external
potential commutes with the field variables
$$[V(\phi), \phi] = 0$$
This is obviously true if the external field has a Taylor series expansion, but, I am wondering about two things.

* In realistic cases is it possible that the external potential is singular?

One thing I can think of is that if we use field theory to describe charge carriers in a metal/semiconductor then an external potential would be the potential of the lattice, which could be singular where the lattice points are. But right now this is just a guess. Is this a good example? Are there others?

* If the potential is singular and does not have a Taylor series expansion(which converges everywhere) then is the commutator still 0? Why?

 

[Bob_for_short]

You probably meant V(x)?

Choosing an interaction external potential is dictated with its source.

If you forget it, you can easily invent a non physical function and obtain severe physical and mathematical difficulties. In other words, wrong modelling.