# Simple S matrix example in Coleman's lectures on QFT

• A
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Simple S matrix example in Coleman's lectures on QFT
In Coleman's QFT lectures, I'm confused by equation 7.57. To give the background, Coleman is trying to calculate the scattering matrix (S matrix) for a situation in which the Hamiltonian is given by
$$H=H_{0}+f\left(t,T,\Delta\right)H_{I}\left(t\right)$$
where ##H_{0}## is the free Hamiltonian, ##H_{I}## is the interaction, and ##f## is a function that turns the interaction on only for a time interval ##T## around ##t=0##. ##\Delta## determines the rate at which the interaction is switched on and off.
Since the interaction is off for times in the distant past and future, the state at these times will be the exact state determined by the free Hamiltonian ##H_{0}##. Coleman calls this state (for the distant past) ##\left|\psi\left(-\infty\right)\right\rangle ^{\text{in}}## and claims that it is given by
$$\left|\psi\left(-\infty\right)\right\rangle ^{\text{in}}=\lim_{t^{\prime}\rightarrow-\infty}e^{iH_{0}t^{\prime}}e^{-iHt^{\prime}}\left|\psi\right\rangle =\lim_{t^{\prime}\rightarrow-\infty}U_{I}\left(0,t^{\prime}\right)\left|\psi\right\rangle$$
where ##U_{I}## is the evolution operator in the interaction picture. He doesn't specify what the state ##\left|\psi\right\rangle## is, but I can't make sense of this equation no matter what I assume about it. Is it the state in the Schrodinger picture or the interaction picture? What time is the state supposed to be at?
If it's the Schrodinger picture (as seems to be the case, as he says this when calculating ##S## in equation 7.59) and the time is ##t=0##, then the ##e^{-iHt^{\prime}}## operator would evolve the state to time ##t^{\prime}##, but then what is the additional ##e^{iH_{0}t^{\prime}}## for?
Finally, how does he get the last equality above? According to Coleman's definition of ##U_{I}## (his equation 7.31) we should have
$$U_{I}\left(t,0\right)=e^{iH_{0}t}e^{-iHt}$$
where the ##t## and the 0 are swapped from its occurrence in the above equation.
Anyone have any thoughts? Thanks.

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2022 Award
I hope Coleman didn't really mean that ##f## is a step function, because then he's generally in big trouble. I don't believe that Coleman really made such a claim. It's really important to do this right and introduce "adiabatic switching" as Gell-Mann and Low did to define the S-matrix in a consistent way. A very good explanation in the QFT context is given in Bjorken and Drell, Quantum Field theory.

• Demystifier
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I hope Coleman didn't really mean that ##f## is a step function, because then he's generally in big trouble.
What exactly goes wrong if one takes a step function?

Gold Member
2022 Award
Have a look at this:

https://arxiv.org/abs/1310.5019

I think this is a nice example underlining the importance of a correct and smooth "adiabatic switching" (both on and off!) in QFT.

I ordered Coleman's book, because this must simply be a gem. Unfortunately it'll take more than 4 weeks to arrive :-(.

I found some other lecture notes from Coleman's QFT lectures online

https://arxiv.org/abs/1110.5013

There it's of course correct and very well discussed, as expected.

• Demystifier and physicsworks