- #1
eoghan
- 207
- 7
- TL;DR Summary
- Justify why the fields die off at infinite time, and why the interaction vacuum can be exchanged with the free vacuum.
Hi everyone,
In his book "Quantum field theory and the standard model", Schwartz derives the position-space Feynman rules starting from the Schwinger-Dyson formula (section 7.1.1). I have two questions about his derivation.
1) As a first step, he rewrites the correlation function as
$$
\langle\Omega\vert\phi_1\phi_2\vert\Omega\rangle = i\int d^4x (\Box_xD_{x1})\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle = i\int d^4x D_{x1}\Box_x\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle
$$
where ##D_{x1}## is the Feynman propagator, such that ##\Box_xD_{x1}=-i\delta_{x1}##
In the last step, he integrated by parts supposing that the term ##D_{x1}\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle## disappears on the boundary of the integration domain.
However, previously while deriving the LSZ formula (section 6.1, just before Eq 6.9), he notes "we will obviously have to be careful about boundary conditions at ##t=\pm\infty##. However, we can safely assume that the fields die off at ##\vec x=\pm\infty##, allowing us to integrate by parts in ##\vec x##". Shouldn't this apply also for the present derivation? I mean, how can we justify that ##D_{x1}\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle## dies off also at the time boundary?
2) In computing the two points correlation function in the presence of interaction, Schwartz notices that ##\langle\Omega\vert\phi_1\phi_2\vert\Omega\rangle## contains a term ##g^2\langle\Omega\vert\phi^2_x\phi^2_y\vert\Omega\rangle## (Eq 7.19). Since we are interested only in order ##g^2##, he says that we should use the free field result for ##\langle\Omega\vert\phi^2_x\phi^2_y\vert\Omega\rangle##. This makes sense, but in the free field, the empty state is ##\vert 0\rangle\neq \vert\Omega\rangle##. And indeed, later on he shows that between ##\vert\Omega\rangle## and ##\vert 0\rangle## there is a factor proportional to the exponential of the interaction potential (Eq 7.53 and following).
Of course, his strategy of considering the interacting and free vacuum equal is correct, because the final result is correct. But I do not see how it can be justified.
In his book "Quantum field theory and the standard model", Schwartz derives the position-space Feynman rules starting from the Schwinger-Dyson formula (section 7.1.1). I have two questions about his derivation.
1) As a first step, he rewrites the correlation function as
$$
\langle\Omega\vert\phi_1\phi_2\vert\Omega\rangle = i\int d^4x (\Box_xD_{x1})\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle = i\int d^4x D_{x1}\Box_x\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle
$$
where ##D_{x1}## is the Feynman propagator, such that ##\Box_xD_{x1}=-i\delta_{x1}##
In the last step, he integrated by parts supposing that the term ##D_{x1}\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle## disappears on the boundary of the integration domain.
However, previously while deriving the LSZ formula (section 6.1, just before Eq 6.9), he notes "we will obviously have to be careful about boundary conditions at ##t=\pm\infty##. However, we can safely assume that the fields die off at ##\vec x=\pm\infty##, allowing us to integrate by parts in ##\vec x##". Shouldn't this apply also for the present derivation? I mean, how can we justify that ##D_{x1}\langle\Omega\vert\phi_x\phi_2\vert\Omega\rangle## dies off also at the time boundary?
2) In computing the two points correlation function in the presence of interaction, Schwartz notices that ##\langle\Omega\vert\phi_1\phi_2\vert\Omega\rangle## contains a term ##g^2\langle\Omega\vert\phi^2_x\phi^2_y\vert\Omega\rangle## (Eq 7.19). Since we are interested only in order ##g^2##, he says that we should use the free field result for ##\langle\Omega\vert\phi^2_x\phi^2_y\vert\Omega\rangle##. This makes sense, but in the free field, the empty state is ##\vert 0\rangle\neq \vert\Omega\rangle##. And indeed, later on he shows that between ##\vert\Omega\rangle## and ##\vert 0\rangle## there is a factor proportional to the exponential of the interaction potential (Eq 7.53 and following).
Of course, his strategy of considering the interacting and free vacuum equal is correct, because the final result is correct. But I do not see how it can be justified.