# Ground State in Peskin and Schroeder

Summary:
I am having a little confusion regarding the limit taken in Peskin and Schroeders quantum field theory book in ch. 4.
In P&S, it is shown that $$e^{-iHT}\ket{0}=e^{-iH_{0}T}\ket{\Omega}\bra{\Omega}\ket{0}+\sum_{n\neq 0}e^{-iE_nT}\ket{n}\bra{n}\ket{0}$$.
It is then claimed that by letting $$T\to (\infty(1-i\epsilon))$$ that the other terms die off much quicker than $$e^{-iE_0T}$$, but my question is why is this the case? For example, why wouldn't the other terms also die off quicker if we simply sent $$T\to \infty$$ instead? Perhaps there is something about the limit of complex numbers I'm not understanding. Any insight would be appreciated. Thanks.

Last edited:

Nevermind, I think I figured it our. I mistakenly assumed the $$e^{-iE_n T}\to 0$$ as $$T\to \infty$$, but that is not the case, which is why the substitution is needed.

mfb
Mentor
Yes, it's only the small imaginary term in T that leads to an e-x behavior. The real term just leads to oscillation.

vanhees71
Gold Member
It becomes much clearer by "renormalizing" the Hamiltonian such that ##E_0=0##, i.e., the ground state energy eigenvalue is set to 0 by shifting the total energy of the system (represented by the Hamilton operator) by adding a constant such that ##E_0=0##. Then all ##E_n>0##.

Now it is utmost important to add an infinitesimal imaginary part to ##T##, i.e., substituting ##T \rightarrow T-\mathrm{i} \epsilon##. This is crucial for all further developments of the theory to get the correct propagator (in vacuum QFT perturbation theory the time-ordered free-field propagator) and the correct "adiabatic switching" for the LSZ reduction. This is a pretty subtle point and should be very well studied!

It becomes much clearer by "renormalizing" the Hamiltonian such that ##E_0=0##, i.e., the ground state energy eigenvalue is set to 0 by shifting the total energy of the system (represented by the Hamilton operator) by adding a constant such that ##E_0=0##. Then all ##E_n>0##.

Now it is utmost important to add an infinitesimal imaginary part to ##T##, i.e., substituting ##T \rightarrow T-\mathrm{i} \epsilon##. This is crucial for all further developments of the theory to get the correct propagator (in vacuum QFT perturbation theory the time-ordered free-field propagator) and the correct "adiabatic switching" for the LSZ reduction. This is a pretty subtle point and should be very well studied!

Can you perhaps say a bit more about vacuum QFT and adiabatic switching. Recently I saw these terms more frequently and would like to know more about them. Do you know a good reference to read about this subject?

vanhees71
Gold Member
That's one of the points which are very nicely and carefully presented in the classic textbook by Bjorken and Drell (of course the 2nd volume on quantum field theory; the 1st volume is not so much my favorite ;-)).

M91
Do you also happen to know which chapter? I think its either: chapter 16 Vacuum Expectation Values and S-Matrix or, chapter 17 Perturbation Theory

vanhees71