Point-splitting regularization in quantum field theory uses the fact, that UV-divergences occurring in expressions of the type $\left< \phi \left( x \right) \phi \left( x \right) \right>$ can be regulated by writing this as $\left< \phi \left(x+\epsilon \right) \, \phi \left(x \right) \right>$.

My question is now, if you know any scheme which implements the point-splitting technique for polynomials of field operators, i.e. something like $\left< \sum_{n=1}^N a_n \, \phi {\left(x\right)}^n \right>$ or even for general functions of field-operators $\left< f \left( \phi \left(x \right) \right) \right>$ or general functionals $\left< F \left[ \phi \right] \right>$?

Of course I could think of some brute-force method to implement such a scheme, but basically I would like to see a paper, where something like this is presented and used in a reasonable way.


The textbook "String Theory", by Joseph Polchinski, first volume, uses the point-splitting regularization and a minimal subtraction scheme adequate to preserve the conformal symmetry of a $d=2$ Conformal Field Theory (i.e. preserving the fact that the trace of the renormalized energy-momentum tensor $T_{a}\,^{a}$ is zero).

The prerequisite is just the basics of QFT, and path integral. The book itself contain an appendix to fulfill an eventual gap. Is not mathematically rigorous, and very intuitive.

All this is presented in the beginning of chapter 2, section 2.1, and developed further in chapter 3, section 3.4 and 3.6. I recommend to read all the chapter 2 and 3 for the sake of completeness. I think that this two chapters will give a very intuitive picture about it. Through all the this chapters he applies the splitting-point regularization to polynomials of $\phi(x)$ and even derivatives $\partial_a\phi(x)$. Because the theory is $d=2$, $\phi(x)$ does not have dimension, so there is the possibility for operators such as $e^{i\phi(x)}$. Polchinski renormalize polynomials of $e^{i\phi(x)}$ as well.

Just some comments to speed up your reading of this chapters:

  • The source of divergence of $\langle \mathcal{T}\left[\phi(x)\phi(y)\right]\rangle$, as $x\rightarrow y$, is due to the fact that we are taking the time ordering of operators that does not commute for time-like intervals, as you can see here. This means that this divergence is related to the ordering ambiguity of $\phi(x)$ and $\phi(y)$. The time ordering operator $\mathcal{T}$ trade this ordering ambiguity into a divergence.

  • The point splitting will tame this divergence by keeping all the points separated. Then we cancel all the divergences that comes of each pair of $\phi(x)$ and $\phi(y)$ when $x\rightarrow y$. When all the divergences was subtracted, we join all the points back. This will give the renormalized operator, what Polchinski calls 'normal ordered operators'. He call it in that way since, in some theories, one can show that those renormalized operators are the same as put the annihilation operators to the right and the creation operators to the left (the usual normal ordering).

  • In chapter 3 he uses the point-splitting regularization for QFTs in curved spaces (curved world-sheet), preserving the Diffeomorphism. He shows that there is no way to preserve the Weyl-symmetry (i.e. $\left[ {T_a}^{a}\right]_{\text{Rernomalized}} \neq 0$) if $c \neq 0$ and $R \neq 0$, there is an anomaly. In point-splitting regularization, the Ward identities may be violated by the regularization. When there is no subtraction scheme that restore the Ward identity back, there is a anomaly in that theory.

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