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I'm reading section 9.2 of Peskin and Schroeder, specifically where they begin the explicit computation of the two point correlation function for a free scalar field using the path integral (p.285). They use a finite (size $L$ along each dimension) square lattice (lattice spacing $\epsilon$ along each dimension) to do this, then take the continuum limit in the end.

They begin by using the discrete Fourier transform to do a change of variables $$\phi(x_{i})=\frac{1}{V}\sum_{n}e^{-ik_{n}\cdot x_i}\phi(k_n)$$$$\phi(x_{i})=\frac{1}{V}\sum_{n}e^{-ik_{n}\cdot x_i}\phi(k_n)\tag{9.21}$$ where $V=L^4$, $k^\mu_n=2\pi n^\mu/L $ and the four sums are over all integers such that $|k^{\mu}_n|<\pi/\epsilon$, so they are finite sums. I think this also means that the range of the summation indices includes the integer zero. More precisely (I think), the lattice is assumed to have $(2M+1)$ points and each of the four sums runs from $-M$ to $M$. Here is where my confusion comes in. Since the $\phi(x_i)$ are real then $\phi^*(k_n)=\phi(-k_n)$ so P&S then consider the real and imaginary parts of the $\phi(k_n)$ with $k_n^0>0$ as the independent variables in the path integral. My question is what about all the $\phi(k_n)$ that have $k_n^0=0$? it seems to me there should be $d\phi(k_n)$ measures in the path integral that have $k_n^0=0$ but the restriction they state just ignores them which does not make sense to me. I'm not sure where I'm going wrong with my reasoning so any clarification is appreciated.

I'm reading section 9.2 of Peskin and Schroeder, specifically where they begin the explicit computation of the two point correlation function for a free scalar field using the path integral (p.285). They use a finite (size $L$ along each dimension) square lattice (lattice spacing $\epsilon$ along each dimension) to do this, then take the continuum limit in the end.

They begin by using the discrete Fourier transform to do a change of variables $$\phi(x_{i})=\frac{1}{V}\sum_{n}e^{-ik_{n}\cdot x_i}\phi(k_n)$$ where $V=L^4$, $k^\mu_n=2\pi n^\mu/L $ and the four sums are over all integers such that $|k^{\mu}_n|<\pi/\epsilon$, so they are finite sums. I think this also means that the range of the summation indices includes the integer zero. More precisely (I think), the lattice is assumed to have $(2M+1)$ points and each of the four sums runs from $-M$ to $M$. Here is where my confusion comes in. Since the $\phi(x_i)$ are real then $\phi^*(k_n)=\phi(-k_n)$ so P&S then consider the real and imaginary parts of the $\phi(k_n)$ with $k_n^0>0$ as the independent variables in the path integral. My question is what about all the $\phi(k_n)$ that have $k_n^0=0$? it seems to me there should be $d\phi(k_n)$ measures in the path integral that have $k_n^0=0$ but the restriction they state just ignores them which does not make sense to me. I'm not sure where I'm going wrong with my reasoning so any clarification is appreciated.

I'm reading section 9.2 of Peskin and Schroeder, specifically where they begin the explicit computation of the two point correlation function for a free scalar field using the path integral (p.285). They use a finite (size $L$ along each dimension) square lattice (lattice spacing $\epsilon$ along each dimension) to do this, then take the continuum limit in the end.

They begin by using the discrete Fourier transform to do a change of variables $$\phi(x_{i})=\frac{1}{V}\sum_{n}e^{-ik_{n}\cdot x_i}\phi(k_n)\tag{9.21}$$ where $V=L^4$, $k^\mu_n=2\pi n^\mu/L $ and the four sums are over all integers such that $|k^{\mu}_n|<\pi/\epsilon$, so they are finite sums. I think this also means that the range of the summation indices includes the integer zero. More precisely (I think), the lattice is assumed to have $(2M+1)$ points and each of the four sums runs from $-M$ to $M$. Here is where my confusion comes in. Since the $\phi(x_i)$ are real then $\phi^*(k_n)=\phi(-k_n)$ so P&S then consider the real and imaginary parts of the $\phi(k_n)$ with $k_n^0>0$ as the independent variables in the path integral. My question is what about all the $\phi(k_n)$ that have $k_n^0=0$? it seems to me there should be $d\phi(k_n)$ measures in the path integral that have $k_n^0=0$ but the restriction they state just ignores them which does not make sense to me. I'm not sure where I'm going wrong with my reasoning so any clarification is appreciated.

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Peskin and Schroeder path integral discretization

I'm reading section 9.2 of Peskin and Schroeder, specifically where they begin the explicit computation of the two point correlation function for a free scalar field using the path integral (p.285). They use a finite (size $L$ along each dimension) square lattice (lattice spacing $\epsilon$ along each dimension) to do this, then take the continuum limit in the end.

They begin by using the discrete Fourier transform to do a change of variables $$\phi(x_{i})=\frac{1}{V}\sum_{n}e^{-ik_{n}\cdot x_i}\phi(k_n)$$ where $V=L^4$, $k^\mu_n=2\pi n^\mu/L $ and the four sums are over all integers such that $|k^{\mu}_n|<\pi/\epsilon$, so they are finite sums. I think this also means that the range of the summation indices includes the integer zero. More precisely (I think), the lattice is assumed to have $(2M+1)$ points and each of the four sums runs from $-M$ to $M$. Here is where my confusion comes in. Since the $\phi(x_i)$ are real then $\phi^*(k_n)=\phi(-k_n)$ so P&S then consider the real and imaginary parts of the $\phi(k_n)$ with $k_n^0>0$ as the independent variables in the path integral. My question is what about all the $\phi(k_n)$ that have $k_n^0=0$? it seems to me there should be $d\phi(k_n)$ measures in the path integral that have $k_n^0=0$ but the restriction they state just ignores them which does not make sense to me. I'm not sure where I'm going wrong with my reasoning so any clarification is appreciated.