Point splitting technique in Peskin and Schroeder 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 $$\frac{g^{00}}{2} = \frac{1}{2}.$$
 A: 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:


*

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

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

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


*

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

A: 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.
