Perturbation Theory: Deciphering Missing Lines of Explanation

[vertices]

Again, I am having difficulty deciphering my class notes - in this case there are missing lines of explanation. If we consider a system of particles that approach and interact, the Heisenberg representation of the interacting field is:

\phi(\vec{x} , t) = U^{-1} (t) \phi_{a} (\vec{x} , t) U(t)

(where \phi_a is the free field before the interaction.

Why is it that we can write:

\frac{\partial}{\partial t} \phi_{a}= \frac{\partial}{\partial t} U \phi U^{-1}=[\frac{\partial}{\partial t} UU^{-1},\phi_{a}]+iU[H,\phi]U^{-1}

where the square brackets in the third equality are commutators?

I don't understand where the third expression comes from?

Thanks.

 

[ismaili]

Again, I am having difficulty deciphering my class notes - in this case there are missing lines of explanation. If we consider a system of particles that approach and interact, the Heisenberg representation of the interacting field is:

\phi(\vec{x} , t) = U^{-1} (t) \phi_{a} (\vec{x} , t) U(t)

(where \phi_a is the free field before the interaction.

Why is it that we can write:

\frac{\partial}{\partial t} \phi_{a}= \frac{\partial}{\partial t} U \phi U^{-1}=[\frac{\partial}{\partial t} UU^{-1},\phi_{a}]+iU[H,\phi]U^{-1}

where the square brackets in the third equality are commutators?

I don't understand where the third expression comes from?

Thanks.

<br /> \frac{\partial \phi_a}{\partial t} = <br /> \frac{\partial U}{\partial t} \phi U^{-1} <br /> + U\phi\frac{\partial U^{-1}}{\partial t} <br /> + U\frac{\partial\phi}{\partial t}U^{-1} <br /> ...(*)<br />
where the last term of RHS involves the derivative of time with respect to the field \phi whose equation of motion is well known, the Heisenberg's EoM.

For the first two terms of eq(*), note that,
\frac{\partial U^{-1}}{\partial t} = -U^{-1}\frac{\partial U}{\partial t} U^{-1}<br />
then you will see why they can be grouped into
<br /> \left[ <br /> \frac{\partial U}{\partial t}U^{-1} , \phi_a<br /> \right]<br />

 

[SpectraCat]

\frac{\partial U^{-1}}{\partial t} = -U^{-1}\frac{\partial U}{\partial t} U^{-1}<br />

I do not recall this identity .. can you provide a brief derivation/proof/justification? It seems quite useful ...

 

[meopemuk]

Take time derivative of both sides of the equality

1 = UU^{-1}

Eugene.

 

[SpectraCat]

Take time derivative of both sides of the equality

1 = UU^{-1}

Eugene.

That'll do it ... and it certainly was brief. Thanks!

 

[vertices]

Thank you ever so much ismaili - spent ages trying to see this!