What is the physical interpretation of self-contraction being divergent?

Question

If we take vacuum expectation value of two scalar field at the same point (2-point correlation function when 2 points coincide) it diverges. what is the physical reason(interpretation)behind this? Also what happens if i take one scalar and one vector field (or any other field other than scalar one)?

Answer 1

This can be explained by the fact that a 'point-like' field is not a well-defined quantity in quantum field theory (or even quantum mechanics) which can be intuitively understood through the Heisenberg uncertainty principle.

Instead, one considers smeared fields $\phi(f) := \int d^nx f(x) \phi(x)$, where $f$ is a real-valued function with compact support. Then, even if we compute the correlation of two completely overlapping fields, the result is finite: $$\langle \phi(f) \phi(f) \rangle = \int d^nx d^ny f(x) f(y) \langle \phi(x) \phi(y)\rangle < \infty, $$ because the pointwise correlation function $\langle \phi(x) \phi(y)\rangle$ is a distribution, i.e., it can have poles (at $x=y$ as you said) but is still integrable with these smearing functions.

There is no need for an interaction, because the correlator just tells you the amount of "overlap" in a certain sense.

If you really want to define the point-wise product of two fields, i.e., $\phi^2(f) := \int \phi(x)^2 f(x)$, you need to renormalize. This is why, for example $\phi^4$-theory has to be renormalized as it contains a 4-fold point-wise product.

For (massive) higher-spin field, the situation remains much the same, because their correlator is related to the one of the scalar field, e.g.: $$\langle A^P_\mu(x) A^P_\nu(y) \rangle \propto (\eta_{\mu\nu} - \partial_\mu \partial_\nu/m^2) \langle \phi(x) \phi(y)\rangle$$ for the massive spin-1 Proca field.

However, the derivative will make the divergence at $x=y$ worse and the need for renormalization all the greater.