Differentiation of an exponential with operators (Peskin p.84)

[gremezd]

Does anyone know how to differentiate an exponential, which has an operator in its power? I found it quite a trouble in Peskin's QFT (page 84, formulas (4.17), (4.18)).
Here we have these two formulas of Peskin:

U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }e^{-iH\left( t-t_{0}\right) };
i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }\left( H-H_{0}\right) e^{-iH\left( t-t_{0}\right) }.

I agree with this. However, if we write U\left( t,t_{0}\right) as U\left( t,t_{0}\right)=e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }, then

i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=\left( H-H_{0}\right)e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }

and we cannot transport e^{iH_{0}\left( t-t_{0}\right) } to the left of \left( H-H_{0}\right) so easily to obtain Peskin's result, since, according to my calculations, \left[ H,H_{0}\right]\neq0. Do we have a rule, which explains where to put the operators from the exponential after differentiation, when we have several noncummuting operators in the power of exponential?

 

[nrqed]

Does anyone know how to differentiate an exponential, which has an operator in its power? I found it quite a trouble in Peskin's QFT (page 84, formulas (4.17), (4.18)).
Here we have these two formulas of Peskin:

U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }e^{-iH\left( t-t_{0}\right) };
i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }\left( H-H_{0}\right) e^{-iH\left( t-t_{0}\right) }.

I agree with this. However, if we write U\left( t,t_{0}\right) as U\left( t,t_{0}\right)=e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }, then

But you can't write this since H and H_0 don't commute. e^A e^B = e^{A+B} only when A and B commute. Otherwise you have to use the Campbell-Hausdorf formula.

i\frac{\partial}{\partial t}U\left( t,t_{0}\right)=\left( H-H_{0}\right)e^{i\left( H_{0}-H\right) \left( t-t_{0}\right) }

and we cannot transport e^{iH_{0}\left( t-t_{0}\right) } to the left of \left( H-H_{0}\right) so easily to obtain Peskin's result, since, according to my calculations, \left[ H,H_{0}\right]\neq0. Do we have a rule, which explains where to put the operators from the exponential after differentiation, when we have several noncummuting operators in the power of exponential?

You just differentiate as usual, making sure that you never pass an operator "through" another operator that does not commute with it.

 

[wasia]

Thanks a lot! This has been tormenting me for ages!

 

[gremezd]

Thank you, nrqed, for pointing out my mistake. I appreciate it :)

 

[nrqed]

Thank you, nrqed, for pointing out my mistake. I appreciate it :)

You are very very welcome.

And thank you for posting your question since this apparently helped Wasia too!

Patrick