Doubt in Sec 4.2 of Peskin Schroeder-Perturbative Expansion of Correlation Functions

praharmitra

Can someone explain to me how the authors got the second equation of eq (4.19), Page 84, of Peskin Schroeder.

The equation is:

[tex]
H_I(t) = e^{iH_0(t-t_0)}(H_{\text{int}}) e^{-iH_0(t-t_0)} = \int d^3x \frac{\lambda}{4!} \phi_I(t,\textbf{x})^4
[/tex]
where
[tex]
H_{\text{int}} = \int d^3x \frac{\lambda}{4!} \phi^4(\textbf{x})
[/tex]
I do not understand how the second part of this eq is equal to the third. Please explain.

matonski

Since [itex] e^{iH_0(t-t_0)}e^{-iH_0(t-t_0)} = 1[/itex], you can insert it between each factor of [itex]\phi[/itex]:

[tex]
e^{iH_0(t-t_0)}\phi^4 e^{-iH_0(t-t_0)}
= \left[ e^{iH_0(t-t_0)}\phi e^{-iH_0(t-t_0)} \right]
\left[ e^{iH_0(t-t_0)}\phi e^{-iH_0(t-t_0)} \right]
\left[ e^{iH_0(t-t_0)}\phi e^{-iH_0(t-t_0)} \right]
\left[ e^{iH_0(t-t_0)}\phi e^{-iH_0(t-t_0)} \right]
=\phi_I^4
[/tex]

praharmitra

Of course that would be the natural thing to do. However, [tex]e^{iH_0(t-t_0)}\phi e^{-iH_0(t-t_0)}[/tex] is a vague statement since you have not examined the arguments of [tex]\phi[/tex].

In our current context of [tex]H_{int}[/tex], the argument is [tex]\phi(\textbf{x})[/tex]. However, the definition of [tex]\phi_I[/tex] is

[tex]
\phi_I(t,\textbf{x}) = e^{iH_0(t-t_0)} \phi(t_0,\textbf{x}) e^{-iH_0(t-t_0)}
[/tex]
where [tex]\phi(t_0,\textbf{x}) = e^{iHt_0}\phi(\textbf{x})e^{-iHt_0} [/tex]. This surely does not reproduce the same result that has been written.

What am I doing wrong?

muppet

[tex]\phi(t_0,\mathbf{x}) = \phi(\mathbf{x})[/tex].

Schroedinger picture operators are defined at some reference time [tex]t_0[/tex].