quantum field theory - What is a point-split?

I encountered the term point-split [1] several times and would like to know what this concept is all about.

From my understanding, a point is splitted by adding $ε$ and $-ε$ to a local point $x$ with $ε → 0$.

1. What is a point-split?


2. Why the blur of locality?


3. Is it a general concept at all?


4. If so, why was it introduced?


5. What is it used for?



6. Does it belong to topology?

Grateful for your answers.

[1] https://arxiv.org/abs/hep-th/9803244

Answer

1. 
   

   A point-splitting procedure is one way to make sense of composite fields in QFT. As an easy example, take a free scalar field $\phi(x)$ in Euclidean signature, in $d$ dimensions. Consider the problem of making sense of the local square $\phi(x)^2$ "inside correlations". The point-splitting amounts to doing so via the limit $$ \phi(x)^2:=\lim_{\epsilon\rightarrow 0}\ \ [\ \phi(x+\epsilon)\phi(x-\epsilon)-\langle\phi(x+\epsilon)\phi(x-\epsilon)\rangle\ ]\ . $$

   
   


2. 
   
   

   Indeed this seems to break locality but one recovers it because $\epsilon$ is sent to $0$. The reason one has to do something like is that correlations diverge at coinciding points. For example $$ \langle\phi(x)\phi(y)\rangle\sim \frac{1}{|x-y|^{d-2}} $$ so you need to subtract a divergent quantity before taking the limit. You also need to introduce the splitting $\epsilon$ to see what you need to subtract.

   
   


3. 
   

   Yes. The basic idea of this point-splitting procedure was introduced by Dirac, Heisenberg, Euler and Valatin (see the references and discussion of the history of this method in Section 1.11 of my article "A Second-Quantized Kolmogorov-Chentsov Theorem"). However, the general framework for understanding this is Wilson's Operator Product Expansion.

   
   


4. 
   

   Already answered in 2.

   
   


5. 
   

   Already answered in 2.

   
   
   


6. 
   

   Not really. This is (hard) analysis, not topology. Although, I should mention that similar constructions appear in compactifications of configuration spaces which topologists and algebraic geometers use. I am not an expert on this, but you can look up "factorization algebras", "Fulton-MacPherson compactification", "wonderful compactifications" for more on this aspect.