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For a generic field theory the path integral measure is defined as, \begin{equation} \mathcal{D}\Phi = \prod_i d\Phi(x_i), \end{equation} where $\Phi$ is a generic field (i.e. it may be scalar, spinor, vector field). Now let us consider a coordinate transformation, \begin{equation} x \to x'. \end{equation}

Now my question is:

In general, HOW to show that the measure remains invariant under the co-ordinate transformation if the co-ordinate transformation be a isometry of the underlying space-time?

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Just a quick comment. This requires that one calculates the Jacobian and sees whether its one or not, sometimes its important to regularize the measure to do this properly. At times the measure is not invariant and can lead to so-called quantum anomalies. See for example arxiv.org/abs/0802.0634 . I hope somebody less lazy than me can give a detailed example. –  Heidar Jul 22 '13 at 19:20
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The first step towards understanding how a measure transforms is to understand how a measure is defined in the first place. The statement "Integral over all field configurations" is quite ill-defined. –  Prahar Jul 22 '13 at 20:56
    
@Heidar Thank you very much. Can you help me more on this topic? I mean I want to know clearly how the measure is defined in path integral, how they transform under any diffeomorphism etc. At quantum level how they transform under the action of quantum group etc. If you can suggest me some references I will be very grateful. –  layman Jul 24 '13 at 15:55
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