I've been trying to teach myself the path integral formulation of quantum field theory and there's a point that's really bugging me: why is the integration measure ##\mathcal{D}\phi(x)## invariant under shifts in the field of the form $$\phi(x)\rightarrow\tilde{\phi}(x)=\phi(x)+\int d^{4}y\Delta(x-y)J(y),\qquad\mathcal{D}\phi(x)\rightarrow\mathcal{D}\tilde{\phi}(x)$$ (where ##\Delta(x-y)## is a Green's function corresponding to the differential operator ##\Box + m^{2}## and ##J(x)## is a source).(adsbygoogle = window.adsbygoogle || []).push({});

Is it simply because the shift term ##\int d^{4}y\Delta(x-y)J(y)## is independent of the field configuration ##\phi(x)## at each spacetime point, or is there more to it than that?

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# Invariance of integration measure under shifts in field

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