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

[naele]

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 $L^4$. The measure is
$$\mathcal{D}\phi = \prod_i d\phi(x_i)$$

They expand the field variables in a Fourier series $\phi(x_i)=\frac{1}{V}\sum_n e^{-ik_n\cdot x_i}\phi(k_n)$. My questions are as follows:
1) Why do they consider the real and imaginary parts of $\phi(k_n)$ as independent variables?
2) Why do they re-write the measure as
$$\mathcal{D}\phi(x)=\prod_{k_n^0>0}dRe\phi(k_n)dIm \phi(k_n)$$

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.

 

[Chopin]

Bear in mind that this is a functional measure. That means that what they're trying to do is consider every possible value that $\phi(k)$ could take on at position $k_i$. Since the Fourier transform requires $\phi$ to be complex, we need a way to parameterize all of the complex plane. We can do this by defining $\phi$ in terms of two real numbers $a$ and $b$, by setting $\phi(k) = a(k) + i b(k)$, and integrating both of them over the entire real line, leading to an integration measure of $da\:db$. Writing $d Re\phi\:d Im\phi$ is just another way of saying the same thing.

 

[naele]

I think I understand that part now, thanks. I do have a problem still with the change of variables from $\phi(x_i)\to\phi(k_n)$. I might be missing something, but there would presumably be a factor of $V^{1/n}$ from the 1/V factor in the Fourier series expansion. And then when I transform from $\phi(k_n)\phi^*(k_n)\to Re \phi(k_n)Im \phi(k_n)$ I get a Jacobian that's not equal to 1.

 

[Chopin]

For your first question, I think the answer is that we're just dealing with the measure, not the full integral. So the 1/V will probably show up in the full integral expression.

As for the Jacobian of the measure, that one may have been my fault--I think you need to define the parameterization as $\phi = \frac{a + ib}{\sqrt{2}}$ or something like that in order for the Jacobian to work out correctly.

 

[naele]

Well the reason I thought there would be a factor is because, unless I'm doing something wrong, I thought the jacobian for $\phi(x_i)\to\phi(k_n)$ is $\frac{1}{V}e^{-ik_n\cdot x_i}$. Although now that I think about it, they do say that it is a unitary transformation, so presumably the Jacobian would have unit modulus, but I'm having difficulties checking that.