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(Anomalous?) Wick rotation in Fujikawa's method

  1. Aug 27, 2010 #1
    In the derivation of the chiral anomaly using the path integral (Fujikawa's method) one encounters a divergent sum which needs to be regulated. The usual prescription is to insert a factor of [tex] e^\frac{(i\gamma^\mu D_\mu)^2}{M^2} [/tex] while taking [tex] M \rightarrow \infty [/tex].

    This eventually leads to (anomaly part)* [tex] \int \frac{d^4k}{(2\pi)^4} e^{-k^\mu k_\mu} [/tex]. At this point textbooks (and Fujikawa's own paper) glibly Wick rotate the timelike component of k to get a simple convergent integral. However without the Wick rotation, the integral is clearly divergent, and therefore the Wick rotation does not give the same integral.

    I suspect the Wick rotation is somehow part of the regularization, but can anyone justify this?
  2. jcsd
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