Is the divergence of the propagator between same two spacetime points a pathology?

Question

This question is same as the question here which is not answered yet.

By Poincare invariance, the propagator of any field theory $G(x,y)$ must be a function of $|x-y|$ and by dimensional arguments $$G(x,y)=\langle 0|\phi(x)\phi(y)|0\rangle\sim\frac{1}{|x-y|^2}\tag{1}$$ which diverges as $x\to y$. Is this divergence a pathology of the theory? Is it an indication of taking some unphysical limit? If yes, why is the limit $x\to y$ unphysical? Does it necessarily mean continuum picture of spacetime in quantum field theory has to be given up?

Answer 1

That is not the real-space propagator. For a massless field in $d=4$ Minkowski space-time that obeys a standard wave equation in real space the Green's functions are given by: $$G(x,y) = \frac{1}{2\pi} \delta(\tau^2) \Theta(x^0-y^0) + A + \frac{B}{2\pi} \delta(\tau^2) \operatorname{sign}(x^0-y^0),$$ where $\tau^2 = \left[x^0 - y^0\right]^2 - \left[\vec{x} - \vec{y}\right]^2$ is the standard proper time between events at $x$ and $y$.

The first term is the standard retarded Green's function from E&M, and the other two terms satisfy the homogeneous wave equation ($[\partial_0^2 - \nabla^2] f(x) = 0$) and are there to accommodate different boundary conditions.

The first term can be constructed as the zero mass limit of the retarded propagator for a massive field. The second term is an obvious solution to the homogeneous equation. The third term is the combination of retarded and advanced propagators in such a way that it produces a solution to the homogeneous equation. This is the most general function that obeys: $$\left[\partial_0^2 - \nabla^2\right] G(x,y) = \pm \delta(x-y),$$ and obeys all of the same continuous symmetries as the d'Alembert operator (space-time translations, rotations, space parity, and boosts - time reflection is often broken by boundary conditions). Note that the $\pm$ on the right hand side is there because I'm too lazy to put in the work to ensure the proper sign.

All of this is perfectly sensible and well behaved when $G(x,y)$ is treated only as the kernel of a linear operator. Where QFT runs into trouble is that $G(x,y)$ is treated as an ordinary function. In other words, it's like having a matrix $M$ where all of the powers of $M$, $M^n$, are perfectly well defined and behave well when they are applied to the relevant vectors. QFT relies on calculations analogous to $M_2 = \left[M_{ij}^2\right]$, i.e. squaring of the individual matrix elements. What does that even mean?

Regardless of how we interpret treating $G$ as a function, it is the need to do so that requires the introduction of regularization and renormalization to get results that make sense.

Answer 2

This is indeed a pathology for the computation of some quantities, such as any quantities derived from the stress energy tensor

$$T_{\mu\nu} = \lim_{x\to y} D_{\mu\nu} G(x,y)$$

For some differential operator $D_{\mu\nu}$. This can be easily renormalized, though, as it can easily be shown that in the coincidence limit $x = y$, the divergence term will be a constant (or depend on the curvature of the manifold, if in curved space), and this is then just a renormalization of the cosmological constant, or the gravitational constant as well in curved space.