Another doubt in Peskin Schroeder Sec 4.2

[praharmitra]

This doubt is about a text in Peskin Schroeder Pg 86. I reproduce it here.
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$$U(t,t')$$ satisfies the same differential equation (4.18),

$$i \frac{\partial}{\partial t} U(t,t') = H_I(t) U(t,t')$$

but now with the initial condition $$U=1$$ for $$t=t'$$. From this equation you can show that

$$U(t,t') = e^{iH_0(t-t_0)}e^{-iH(t-t')}e^{-iH_0(t'-t_0)}$$

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Here $$H = H_0+H_{int} = H_{KG} + \int d^3x \frac{\lambda}{4!} \phi(\textbf{x})^4$$
and
$$H_I = \int d^3x \frac{\lambda}{4!} \phi_I^4$$.

Can anyone explain how "one can show" the second statement that Peskin Schroeder makes?

 

[praharmitra]

OK, I have been able to prove that the function mentioned above does indeed satisfy the boundary conditions and the differential equation mentioned. So, it is after all a solution to the diffeq. Now, my doubt is this. The original definition of $U(t,t_0)$ was as follows (ref. Page 84)

$$U(t,t_0) = e^{iH_0(t-t_0)}e^{-i H(t-t_0)}$$

Then we found the differential equation that the above satisfies so as to simplify the expression in terms of $\phi_I$. However, if we use the above definition of $U(t,t_0)$, we surely do not reproduce the expression I have written in my first post. What is going wrong here?

 

[dm4b]

I don't have my Peskin and Schroeder in front of me, but IIRC, aren't the U's different because of the pictures being considered in each case (i.e. Schrondinger vs Heisenberg vs Interaction picture)?

Maybe the first takes you from the Schrödinger Picture to the Interaction Picture, or something similar?

Just a guess. I'll try and check P&S later on.

Seems like I struggled with this same thing a ways back

 

[muppet]

The point is (I think) related to their use of a reference time $$t_0$$ which is different from their use of t'.

One defines fields at a fixed time $$t_0$$ in the Schrödinger picture (p83), and all subsequent statements about time evolution have got to make reference to this initial definition in some way, shape or form. The original definition of U relates the interaction picture field at some time t to its original definition back at $$t_0$$. The second seems to relate the field at time t to some arbitrary earlier time, t'. I think the "extra" complex exponential appearing relates the field at t' to that at $$t_0$$. Note that it's the free Hamiltonian $$H_0$$ that's used here, as this is what relates the interaction field configurations at two different times. What I can't work out is precisely what picture field is being related to what, a precise statement of the point of U(t,t') analogous to equation 4.16. I might try and look at this again tomorrow.

 

[strangerep]

praharmitra,

Please try to phrase your question more explicitly. I've read your posts 3 times, and
I still don't know for sure what you're asking.

I can tell you that the stuff in that section of P&S is indeed correct.

I can tell you that $$t_0$$ is the time at which the fields expressed in
Heisenberg picture coincide with the fields expressed in Interaction picture.
See their eq(4.14).

Please be more explicit about which expression(s) and/or equation(s)
you're unable to derive in P&S. I.e., you said:

if we use the above definition of $$U(t,t_0)$$ , we surely do not
reproduce the expression I have written in my first post.

Which expression?
And show a detail calculation why/where you think P&S is wrong.