# Connection between Dyson's equation and Heisenberg equation of motion

Might there be a similarity between Dyson's equation and Heisenberg equation? (It's just a feeling, nothing based on arguments.) Both describe how a system (density matrix or Green's function) behaves in time. Both require knowledge of the intial system at time t=0 and the potential acting on the system.

The Dyson equation:
$$G = G_0 + G_0 V G$$ usually solved with the iteration steps $$G_{j+1} = G_0 + G_0 V G{j}$$

The Heisenberg equation of motion (with the density as the operator):
$$\rho = U^{\dag} \rho_0 U{ [\tex] with [tex] U = e^{\frac{-i}{\hbar} H t[\tex] (in the case of a time independent Hamiltonian). There must be bridge, but I can't find a mathematical transition or a common ground. My knowledge on Green's function is just too limited. Does anybody has an idea what could lead in the right direction? ## Answers and Replies strangerep Science Advisor Might there be a similarity between Dyson's equation and Heisenberg equation? (It's just a feeling, nothing based on arguments.) Both describe how a system (density matrix or Green's function) behaves in time. Both require knowledge of the intial system at time t=0 and the potential acting on the system. The Dyson equation: [tex] G = G_0 + G_0 V G$$ usually solved with the iteration steps $$G_{j+1} = G_0 + G_0 V G{j}$$

The Heisenberg equation of motion (with the density as the operator):
$$\rho = U^{\dag} \rho_0 U{$$ with $$U = e^{\frac{-i}{\hbar} H t$$ (in the case of a time independent Hamiltonian).

There must be bridge, but I can't find a mathematical transition or a common ground. My knowledge on Green's function is just too limited.
Does anybody has an idea what could lead in the right direction?
Are you just talking about how the propagator for an interacting theory is obtained by
perturbative expansion around the free propagator?

If so, this is standard fare in many QFT textbooks, e.g: Greiner & Reinhardt's
"Quantum Electrodynamics" presents this sort of thing at a pedestrian pace.

Or did I misunderstand what you were asking?

Are you just talking about how the propagator for an interacting theory is obtained by
perturbative expansion around the free propagator?

And my question is: Are these two equations connected in any way?

I know that the propagator U is a matrix element of the Green's function: G = <x|U|x'>. But I've never seen a derivation.
Dou you know some online resources treating this topic?

Gokul43201
Staff Emeritus
Gold Member
There is no direct mapping between the two equations - why should there be one?

You will find a derivation to Dyson's equation in Fetter & Walecka, or if you have a strong stomach, in Abrikosov, Gorkov & Dzyaloshinskii.

strangerep
And my question is: Are these two equations connected in any way?
The Dyson recursion equation is what you get when you try to solve the
Heisenberg equation by splitting the Hamiltonian H into $H_0 + V$,
and treat V as a small perturbation.

I know that the propagator U is a matrix element of the Green's function: G = <x|U|x'>.
Actually, $U = exp(iHt)$ is the time evolution operator. The "propagator"
and the "Green's function" are the same thing by different names.

But I've never seen a derivation.
Do you know some online resources treating this topic?
Sorry,... I normally just consult a textbook when I want to check this sort
of thing. I had a quick look in Srednicki's online QFT book, but he
doesn't seem to cover your question explicitly.

I vaguely recall requests on this forum about online QFT books, so maybe
if you search back through other threads you'll find something.

reilly
Dyson's work is done in the Interaction rep. For all practical purposes, that means the time dependence due to the unperturbed, free, Hamiltonian is factored out. This is very similar to the way we solve first order differential equations of the form dW/dt= ivW + F. That is, introduce a new dependent variable S, such that W=Exp(ivt)S. This idea was a key to the Nobel Prizes of Feynman, Schwinger and Tomonaga -- Dyson should have been included.

The Heisenberg equations of motion are directly formed from the role of the Hamiltonian as the generator of displacements in time -- commutators and all that.
This is all explained in any book on QFT, and in many on ordinary QM. Very basic. And of course the two approaches are intimately connected -- they are describing the same thing. Good basic exercise to show the connection.
Regards,
Reilly Atkinson

Thanks strangerep. I found Srednicki's book and it really seems to be good. Anyway it will take me some time to go through it.

Reilly, thank you for the deep insight. I guess, I will be going to the library next week to find a book on QFT.

I think that the book you would be looking for is Fetter and Walecka. Also Mahan's book.

Both of these book derive the Dyson equation starting from perturbation theory using the interaction picture.