Point splitting technique in Peskin and Schroeder

Question

One of the cornerstones of point splitting technique of calculating chiral anomaly (Peskin and Schroeder 19.1, p.655) is a symmetric limit $\epsilon \rightarrow 0$. And this is the point that I don't get. Is it really possible to take such a limit? For example, consider the expression $$ \text{symm}\,\text{lim}_{\epsilon \rightarrow 0} \Bigl\{\frac{\epsilon^{\mu}\epsilon^{\nu}}{\epsilon^2}\Bigr\} = \frac{g^{\mu \nu}}{d} \tag{19.23} $$ in $d=2$ spacetime dimensions. Let $\mu = \nu = 0$. Then $$ \frac{\epsilon^{0}\epsilon^{0}}{\epsilon^2} = \frac{1}{1-(\frac{\epsilon^1}{\epsilon^0})^2}. $$ But the latter expression either greater than 1 or less than 0 and so can't be equal to $$\frac{g^{00}}{2} = \frac{1}{2}.$$

Answer 1

The symmetric limit (19.23)

$$S^{\mu\nu}~:=~ \text{symm}\,\lim_{\epsilon \rightarrow 0} \left\{\frac{\epsilon^{\mu}\epsilon^{\nu}}{\epsilon^2}\right\}, \qquad \epsilon^2~:=~\epsilon^{\mu}g_{\mu\nu}\epsilon^{\nu},$$ should be thought of as a regularization prescription. It is part of a symmetric point splitting regularization scheme, cf. Ref. 1. The traditional notion of limit $$\lim_{\epsilon \rightarrow 0} \left\{\frac{\epsilon^{\mu}\epsilon^{\nu}}{\epsilon^2}\right\}$$ does not exist. The prescription $$S^{\mu\nu}~:=~\frac{g^{\mu\nu}}{d}$$ is motivated by three things:

1. The prescription $S^{\mu\nu}$ can only depend on the metric.

2. $S^{\mu\nu}$ should be a (2,0) tensor wrt. Lorentz transformations.

3. If we contract $S^{\mu\nu}$ with $g_{\mu\nu}$, the result should be $1$.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT; Section 19.1, p. 655.

Answer 2

Same logic used in (7.87) in Peskin itself: $l^{\mu}l^{\nu} \rightarrow \frac{1}{d}l^2 g^{\mu\nu}$.

Ref. Peskin & Schroeder, Section 7.5, p. 251.

Answer 3

Taking the 'symmetric limit' means averaging the expression over the $S^{d-1}$ sphere surrounding $\epsilon^\mu=0$ and then taking the size of the sphere to zero. That should give you the expression you're looking for.