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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.

$$H=H_{0}+f\left(t,T,\Delta\right)H_{I}\left(t\right)$$

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.