Ground State in Peskin and Schroeder

[Diracobama2181]

TL;DR 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.

 

[Diracobama2181]

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]

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]

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!

 

[M91]

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]

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]

I'd say both chapters are worth to be carefully studied. It's a pretty concise formulation of LSZ reduction and the perturbative evaluation of the S-matrix elements.