Simple S matrix example in Coleman's lectures on QFT

In summary, Coleman's QFT lectures introduce equation 7.57, which attempts to calculate the scattering matrix for a Hamiltonian that includes a time-dependent interaction term. However, there are some discrepancies and confusion surrounding the use of the free Hamiltonian, the interaction picture, and the state's time parameter. Coleman's definition of the evolution operator in the interaction picture, U_I, also leads to some inconsistencies. It is important to properly define and use "adiabatic switching" in order to obtain consistent results. A good explanation of this concept can be found in Bjorken and Drell's Quantum Field Theory. It is also noted that taking a step function for the interaction can lead to problems, as shown in a related paper.
  • #1
Glenn Rowe
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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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  • #2
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.
 
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  • #3
vanhees71 said:
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?
 
  • #4
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.
 
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Related to Simple S matrix example in Coleman's lectures on QFT

1. What is an S matrix in the context of Coleman's lectures on QFT?

The S matrix, or scattering matrix, is a mathematical tool used in quantum field theory to calculate the probability of particles interacting and changing their states. It is a fundamental concept in understanding how particles behave and interact in the quantum world.

2. How is the S matrix related to Feynman diagrams?

The S matrix is closely related to Feynman diagrams, which are graphical representations of particle interactions. The S matrix is calculated by summing over all possible Feynman diagrams for a given process.

3. Can you provide a simple example of an S matrix calculation in Coleman's lectures on QFT?

One simple example is the calculation of the S matrix for electron-electron scattering. In this case, the S matrix is given by a single Feynman diagram with two electrons exchanging a photon. The calculation involves integrating over all possible momentum transfers and spin orientations.

4. How does the S matrix account for the uncertainty principle?

The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. The S matrix takes this into account by including a factor that accounts for the probability of particles being at different positions and momenta simultaneously.

5. What is the significance of the S matrix in quantum field theory?

The S matrix is a crucial concept in quantum field theory as it allows us to make predictions about the behavior and interactions of particles. It is also the basis for many advanced calculations and theories in quantum physics, such as the Standard Model.

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