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.

 

[sir-pinski]

This is a great theoretical question! In field theory, the external potential is typically considered to be a classical background field that does not interact with the quantum fields. This means that the potential is fixed and does not change with time, and therefore it commutes with the field operators.

In realistic cases, it is possible for the external potential to be singular. Your example of the lattice potential in a metal or semiconductor is a good one. Another example could be a point charge in electromagnetism, where the electric potential is singular at the location of the charge. In these cases, the potential is still considered to be a classical background field and does not interact with the quantum fields, so it still commutes with the field operators.

The reason for this is that, in field theory, we are dealing with operators that act on a Hilbert space of quantum states. The commutator represents the fundamental quantum mechanical uncertainty between two operators. In the case of a singular external potential, the commutator is still zero because the potential is a fixed classical background and does not introduce any additional uncertainty in the system.

In summary, the commutator between the external potential and the field operators will always be zero, even if the potential is singular, as long as it is a classical background field that does not interact with the quantum fields. I hope this helps to clarify your question!