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I need to evaluate a path integral which involves a set of fields $X=\left\{ \psi_i \right\}$:

$$ I = \int \prod_i \mathcal{D} \psi_i e^{-S \left[ \left\{ \psi_i \right\} \right] } $$

In order to simplify the treatment I'd want to make a change of variables, and express the integrand as a function of a new set of fields, let's call it $Y=\left\{ \phi_i \right\}$. My problem is: how do I implement the change of variables? Can I extend the multi-dimensional case to the continuum and include the determinant of the Jacobian of the transformation in the integral, i.e.

$$ J_{ij} = \frac{\delta X_i}{\delta Y_j} $$

Moreover I'd want to defined one or more of the $Y_i$ variables as containing some derivatives of the original $X_i$ ones. My intuition leads me to write down the Jacobian in momentum space, where each derivative is then replaced by a $- \mathrm{i} k_\mu$ then evaluate it, also in momentum space. I would also say that as long as the Jacobian does not depends on the fields, it can be taken outside of the integral sign, i.e. it is a multiplicative constant. Is my intuition right?

Any suggestion, or any reference to a textbook is welcomed. In particular I couldn't find in any textbook I own the cases in which one of the redefined fields depends upon the derivative of one the original fields.

I know that this a mathematical question more than a physics question; however I decided to post it here because (usually) physicist are more used to manipulating path integrals.

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1 Answer 1

up vote 6 down vote accepted

There is a well established method to evaluate the Jacobian for path integrals. This most often appears in the literature as a very nice way of understanding anomalies. In a theory with an anomaly, the integrand (action) must be invariant under a certain transformation (classical symmetry), but the symmetry disappears at the quantum level due to the path integral measure not being invariant under the transformation. So the most obvious references I can suggest are those which demonstrate anomalies via the path integral method. Examples include the chiral anomaly and the conformal anomaly. The original idea is due to Fujikawa and the method is named after him. So probably best to start at the Fujikawa method page on Wikipedia. Many good text books also cover this (I know Srednicki does) and then just google "Fujikawa method", "anomaly path integral" etc. Hope that helps get you started.

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Just an addendum: Determinants of path integral Jacobians also play a key role in the Fadeev-Popov method for computing gauge theory correlation functions. –  user1504 Jan 7 '13 at 1:06

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