I had a question about about the integration measure for the path integral after a unitary change of variables. First they consider a 4D spacetime lattice with volume [itex]L^4[/itex]. The measure is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\mathcal{D}\phi = \prod_i d\phi(x_i)

[/tex]

They expand the field variables in a Fourier series [itex]\phi(x_i)=\frac{1}{V}\sum_n e^{-ik_n\cdot x_i}\phi(k_n)[/itex]. My questions are as follows:

1) Why do they consider the real and imaginary parts of [itex]\phi(k_n)[/itex] as independent variables?

2) Why do they re-write the measure as

[tex]

\mathcal{D}\phi(x)=\prod_{k_n^0>0}dRe\phi(k_n)dIm \phi(k_n)

[/tex]

I've never seen a measure re-written like that, I was wondering what allows them to do so.

There's already a thread about this here but I wasn't comfortable bumping a three year old thread, and the response didn't clear up my confusion.

I appreciate any help.

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# Peskin & Schroeder p. 285, change of variables integration measure

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