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)).(adsbygoogle = window.adsbygoogle || []).push({});

Here we have these two formulas of Peskin:

[tex]U\left( t,t_{0}\right)=e^{iH_{0}\left( t-t_{0}\right) }e^{-iH\left( t-t_{0}\right) } [/tex];

[tex]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) } [/tex].

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

[tex]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) }[/tex]

and we cannot transport [tex]e^{iH_{0}\left( t-t_{0}\right) }[/tex] to the left of [tex]\left( H-H_{0}\right)[/tex] so easily to obtain Peskin's result, since, according to my calculations, [tex]\left[ H,H_{0}\right]\neq0[/tex]. 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?

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

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