I am reading P&S, specifically Chapter 12. I have trouble understanding why the propagator in momentum space (if the following is indeed the propagator in momentum space) in a $\phi^4$ theory in the Euclidean space of the path integral over the configuration of fields over a single momentum shell has the following form $$\hat{\phi}(k)\hat{\phi}(k)=\frac{1}{k^2}(2\pi)^d\delta^{(d)}(k+p)\Theta(k),\tag{12.8}$$ where despite the fact that I can not denote it, a Wick contraction is implied between the two fields on the left hand side. I understand that the $\Theta$ function must be there because we are integrating over a single momentum shell $b\Lambda<|k|<\Lambda$, cf. eq. (12.9), and hence its role is to ensure that the propagator vanishes for momentum values outside that shell. Moreover, I realise that the $\delta$ function is somehow related to the fact that the field takes only real values. Despite the fact that I realise (to a point) why these $\delta$ and $\Theta$ functions are for, I do not seem to comprehend how they appear in the propagator.

Any help will be appreciated.

P.S. The $\Theta$ function is vanishing besides the shell $b\Lambda<|k|<\Lambda$, where it is equal to one, cf. eq. (12.9).

I am reading P&S, specifically Chapter 12. I have trouble understanding why the propagator in momentum space (if the following is indeed the propagator in momentum space) in a $\phi^4$ theory in the Euclidean space of the path integral over the configuration of fields over a single momentum shell has the following form $$\hat{\phi}(k)\hat{\phi}(k)=\frac{1}{k^2}(2\pi)^d\delta^{(d)}(k+p)\Theta(k),\tag{12.8}$$ where despite the fact that I can not denote it, a Wick contraction is implied between the two fields on the left hand side. I understand that the $\Theta$ function must be there because we are integrating over a single momentum shell $b\Lambda<|k|<\Lambda$, cf. eq. (12.9), and hence its role is to ensure that the propagator vanishes for momentum values outside that shell. Moreover, I realise that the $\delta$ function is somehow related to the fact that the field takes only real values. Despite the fact that I realise (to a point) why these $\delta$ and $\Theta$ functions are for, I do not seem to comprehend how they appear in the propagator.

Any help will be appreciated.

P.S. The $\Theta$ function is vanishing besides the shell $b\Lambda<|k|<\Lambda$, where it is equal to one, cf. eq. (12.9).

P.S.#2 I have noticed that there may be a confusion on what I am asking. I am not asking how to derive the free Lagrangian in momentum space (as in Propagator in $\phi^4$ theory). What I am asking is how to read off the propagator from that expression in a way such that Eq. (12.8) from P&S occurs.

I am reading P&S, specifically Chapter 12. I have trouble understanding why the propagator in momentum space (if the following is indeed the propagator in momentum space) in a $\phi^4$ theory in the Euclidean space of the path integral over the configuration of fields over a single momentum shell has the following form (eq. 12.8) $$\hat{\phi}(k)\hat{\phi}(k)=\frac{1}{k^2}(2\pi)^d\delta^{(d)}(k+p)\Theta(k)$$ where despite the fact that I can not denote it, a Wick contraction is implied between the two fields on the left hand side. I understand that the $\Theta$ function must be there because we are integrating over a single momentum shell $b\Lambda<|k|<\Lambda$ and hence its role is to ensure that the propagator vanishes for momentum values outside that shell. Moreover, I realise that the $\delta$ function is somehow related to the fact that the field takes only real values. Despite the fact that I realise (to a point) why these $\delta$ and $\Theta$ functions are for, I do not seem to comprehend how they appear in the propagator.

Any help will be appreciated.

P.S. The $\Theta$ function is vanishing besides the shell $b\Lambda<|k|<\Lambda$, where it is equal to one.

I am reading P&S, specifically Chapter 12. I have trouble understanding why the propagator in momentum space (if the following is indeed the propagator in momentum space) in a $\phi^4$ theory in the Euclidean space of the path integral over the configuration of fields over a single momentum shell has the following form $$\hat{\phi}(k)\hat{\phi}(k)=\frac{1}{k^2}(2\pi)^d\delta^{(d)}(k+p)\Theta(k),\tag{12.8}$$ where despite the fact that I can not denote it, a Wick contraction is implied between the two fields on the left hand side. I understand that the $\Theta$ function must be there because we are integrating over a single momentum shell $b\Lambda<|k|<\Lambda$, cf. eq. (12.9), and hence its role is to ensure that the propagator vanishes for momentum values outside that shell. Moreover, I realise that the $\delta$ function is somehow related to the fact that the field takes only real values. Despite the fact that I realise (to a point) why these $\delta$ and $\Theta$ functions are for, I do not seem to comprehend how they appear in the propagator.

Any help will be appreciated.

P.S. The $\Theta$ function is vanishing besides the shell $b\Lambda<|k|<\Lambda$, where it is equal to one, cf. eq. (12.9).