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.

$$\frac{\partial \phi_a}{\partial t} = \frac{\partial U}{\partial t} \phi U^{-1} + U\phi\frac{\partial U^{-1}}{\partial t} + U\frac{\partial\phi}{\partial t}U^{-1} ...(*)$$
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,
$$\frac{\partial U^{-1}}{\partial t} = -U^{-1}\frac{\partial U}{\partial t} U^{-1}$$
then you will see why they can be grouped into
$$\left[ \frac{\partial U}{\partial t}U^{-1} , \phi_a \right]$$

 

[SpectraCat]

$$\frac{\partial U^{-1}}{\partial t} = -U^{-1}\frac{\partial U}{\partial t} U^{-1}$$

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!