Connection between Dyson's equation and Heisenberg equation of motion

In summary: Also, the second order term in the Dyson expansion is just the first order term in the expansion for the time evolution operator. This is because you can always write the time evolution operator as U = 1 + (i/ħ)Ht + R(t) where R(t) is higher order. So the first order term in the expansion for U is the same as the second order term in the Dyson expansion. Also, the Dyson expansion is just a different way of writing the expansion for the time evolution operator, but with a different interpretation for the terms. In summary, there is a connection between the Dyson equation and the Heisenberg equation, as they both describe how a system evolves in
  • #1
Tanja
43
0
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[/tex] usually solved with the iteration steps [tex] G_{j+1} = G_0 + G_0 V G{j}[/tex]

The Heisenberg equation of motion (with the density as the operator):
[tex] \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?
 
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  • #2
Tanja said:
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[/tex] usually solved with the iteration steps [tex] G_{j+1} = G_0 + G_0 V G{j}[/tex]

The Heisenberg equation of motion (with the density as the operator):
[tex] \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?

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?
 
  • #3
strangerep said:
Are you just talking about how the propagator for an interacting theory is obtained by
perturbative expansion around the free propagator?

Thanks for your reply. Yes, that's what I'm talking about.

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?
 
  • #4
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.
 
  • #5
Tanja said:
Thanks for your reply. Yes, that's what I'm talking about.
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 [itex]H_0 + V[/itex],
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, [itex]U = exp(iHt)[/itex] 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.
 
  • #6
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
 
  • #7
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.
 
  • #8
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.
 

1. What is Dyson's equation and Heisenberg's equation of motion?

Dyson's equation and Heisenberg's equation of motion are two fundamental equations in quantum mechanics that describe the time evolution of a quantum system. Dyson's equation is a perturbative method for calculating the Green's function, which is a mathematical representation of the correlation between particles in a system. Heisenberg's equation of motion describes the evolution of an operator in time, which is used to calculate the expectation value of a physical quantity in a quantum system.

2. What is the connection between Dyson's equation and Heisenberg's equation of motion?

The connection between Dyson's equation and Heisenberg's equation of motion lies in the fact that they both describe the time evolution of a quantum system. Dyson's equation provides a way to calculate the Green's function, which is then used in Heisenberg's equation of motion to calculate the expectation values of operators in a system.

3. How are Dyson's equation and Heisenberg's equation of motion derived?

Dyson's equation and Heisenberg's equation of motion are derived from the principles of quantum mechanics and the equation of motion of a quantum system. Dyson's equation is derived from the Dyson series, which is a perturbative method for calculating the Green's function. Heisenberg's equation of motion is derived from the Schrödinger equation, which describes the time evolution of a quantum system.

4. What is the significance of Dyson's equation and Heisenberg's equation of motion in quantum mechanics?

Dyson's equation and Heisenberg's equation of motion are significant because they provide a mathematical framework for understanding the time evolution of a quantum system. They allow scientists to calculate the correlation between particles in a system and the expectation values of physical quantities. These equations are fundamental to many areas of quantum mechanics, including quantum field theory and condensed matter physics.

5. How are Dyson's equation and Heisenberg's equation of motion used in research and applications?

Dyson's equation and Heisenberg's equation of motion are used extensively in research and applications in the field of quantum mechanics. They are used to calculate the behavior of quantum systems, such as atoms, molecules, and materials. They are also used in the development of quantum technologies, such as quantum computing and quantum cryptography. Additionally, these equations are used in theoretical studies to better understand the fundamental principles of quantum mechanics and its applications to various fields of science and technology.

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