I'm stuck on the derivation of the Adler-Bell-Jackiw anomaly. This is discussed on page 666 of Peskin and Schroeder (equation 19.76) or these notes on page 14 (equation 39).

According to these sources, we can evaluate a matrix element as: \begin{equation} \langle x | e^{-\partial^2/M^2} | x \rangle = \displaystyle\lim_{x \to y} \int \frac{\mathrm{d}^4 k}{(2\pi)^4} e^{-ik \cdot (x-y)} e^{k^2/M^2} \end{equation} where $M$ is a regularizer. Maybe this equation is really trivial, but I'm completely lost and would really like some help.

What I've tried so far is inserting momentum eigenstates: \begin{equation} \begin{aligned} \langle x | e^{-\partial^2/M^2} | x \rangle & = \int \frac{\mathrm{d}^4 p}{(2 \pi)^4} \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; \langle x |p \rangle \langle p | e^{-\partial^2/M^2}|k \rangle \langle k | x \rangle \\& = \int \frac{\mathrm{d}^4 p}{(2 \pi)^4} \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; e^{ix \cdot (p-k) } \langle p | e^{-\partial^2/M^2}|k \rangle \\& = \int \frac{\mathrm{d}^4 p}{(2 \pi)^4} \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; e^{ix \cdot (p-k) } e^{k^2/M^2} \delta^4(p-k) \end{aligned} \end{equation} I have no idea if I'm even on the right track, or the above is complete nonsense. Any help is much appreciated.

In response to Adam's comment:

The reason why I wasn't sure about what I've done was the $\lim_{x \to y}$ part. Is there any reason why we would evaluate it as: \begin{equation} \begin{aligned} \displaystyle\lim_{x \to y} \langle y | e^{-\partial^2/M^2} | x \rangle & = \displaystyle\lim_{x \to y} \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; \langle y | e^{-\partial^2/M^2}|k \rangle \langle k | x \rangle \\& = \displaystyle\lim_{x \to y} \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; e^{k^2/M^2} e^{i k \cdot (y-x)} \\& = \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; e^{k^2/M^2} \end{aligned} \end{equation} Whereas we could also evaluate it as: \begin{equation} \begin{aligned} \langle x | e^{-\partial^2/M^2} | x \rangle & = \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; \langle x | e^{-\partial^2/M^2}|k \rangle \langle k | x \rangle \\& = \int \frac{\mathrm{d}^4 k}{(2 \pi)^4} \; e^{k^2/M^2} \end{aligned} \end{equation} which seems to give the same result and is a simpler method. Why do most sources seem to use the first approach rather then the second?

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    $\begingroup$ That looks good to me, though you should look at the matrix element $\lim_{x\to y}\langle x|\cdots|y\rangle$ and you don't need to insert the identity twice. $\endgroup$ – Adam Mar 22 '14 at 0:16
  • $\begingroup$ @Adam thank you! Your method does indeed get the right answer, but it has sparked another question. Could you look at my edit? $\endgroup$ – Hunter Mar 22 '14 at 0:38

You can't remove brackets $|k \rangle\langle k|$ into the second method which you've reprsesented because there is $e^{\frac{k^{2}}{M^{2}}}$ term; you have to make a summation over all $k$, which is completely equivalent to the first method (note also that $|k\rangle $ doesn't coincide with integration variable $k$)

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