5
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
2
$\begingroup$

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 $k$)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.