# How can we derive the results that we know about QM from QED?

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## Summary:

QED describes all physics under certain energy level. So how we derive results that we know from QM from QED?

## Main Question or Discussion Point

Summary: QED describes all physics under certain energy level. So how we derive results that we know from QM from QED?

I want to know how to get quantum mechanics from QED (Quantum Electrodynamics).
In QED we showed that the potential between two electrons goes like 1/r as in quantum mechanics, but it's not enough since there are other computations in advanced quantum mechanics course which I don't know ho to derive from QED. For example, calculating lifetime of transition levels, like 2p to 1s. Since QED describes all the interactions (relevant) I want to know how to derive that. Is there a good book that explains that?

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A. Neumaier
2019 Award
This is done by an approximation method called NRQED.

This is done by an approximation method called NRQED.
Thank you for your response! Do you know a good source (a book or article) to learn it?

A. Neumaier
2019 Award
vanhees71
Gold Member
2019 Award
I don't know, what you are specifically looking for, but there's a very good treatment of "NRQED" in

Weinberg, Lectures on Quantum Mechanics

There the electrons are treated with 1st-quantized non-relativsitic QM coupled to the (quantized) em. field (which of course is never non-relativistic and thus must be treated either as a classical field, which leads to the semiclassical approximation or quantum field, when quantization is important; in the low-energy sector it's particularly important to describe spontaneous emission; everything else is to a good approximation treatable in semiclassical approximation).

I don't know, what you are specifically looking for, but there's a very good treatment of "NRQED" in

Weinberg, Lectures on Quantum Mechanics
I don't think it's NRQED, this is just ordinary QM. What I was looking for is a way to understand how to derive QM from QED.

vanhees71
Gold Member
2019 Award
I see. Then you need to do a formal expansion in powers of $1/c$. In connection with Dirac particles you need to get the right representation to split in "large" and "small" components, in order to systematically neglect the anti-particles degrees of freedom. It's also non-trivial, because you cannot formualte relativistic QT in the "first-quantization formalism" for particles but only as QFT. So I guess, the most promising idea is to derive non-relativistic QFT from relativistic QFT. A bit of googling has brought me to the paper (open access!)

I've not read it yet, but it looks promising.

A. Neumaier
2019 Award
I guess, the most promising idea is to derive non-relativistic QFT from relativistic QFT. A bit of googling has brought me to the paper (open access!)

I've not read it yet, but it looks promising.
Footnote 7: '' We will be concerned only with the free particle. ''
Thus the scope is very limited. Most of the paper is completely useless, talking about what does not work. Of interest are only Sections 6-7, just 4 pages out of 34. And the outcome is free non-relativistic field theory from a free charged relativistic scalar field, not standard QM. Nothing like QM from QED.

True NRQED, as in my references above, is far more useful. But incomplete.

For example, there is nowhere an account that would derive from QED in a clean way the SE for n electrons in a static electric field. There are only very loose roundabout arguments that fall apart when one tries to make them more precise.

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For example, there is nowhere an account that would derive in a clean way the SE for n electrons in a static electric field from QED.
What is SE mean?

A. Neumaier
2019 Award
SE = Schrödinger equation

A. Neumaier
2019 Award
For example, there is nowhere an account that would derive from QED in a clean way the SE for n electrons in a static electric field. There are only very loose roundabout arguments that fall apart when one tries to make them more precise.
For the handwaving approach to the relations between QED and the relativistic multielectronic Hamiltonians used in quantum chemistry see

This is done by an approximation method called NRQED.
I read a lot of articles on NRQED, which explained how to derive the NRQED Lagrangian from symmetries, but do not explain more basic stuff, like what is the free theory Lagrangian, what are the Feynman rules, what is the propagator? it seems like it's obvious stuff they never wrote in the article, but it's far from obvious to me.
The Lagrangian to order $\frac 1 M$ is:
$$\mathcal L =\psi ^\dagger \{ iD_t +\frac {\mathbf D^2} {2M} +c_F e \frac { \mathbf {\sigma} \cdot \mathbf B} {2M}\}\psi$$
Is the Lagrangian in a specific gauge? and How you derive coulomb law from it?
and my original question- calculating lifetime of transition levels, like 2p to 1s, how is it done?

Since I haven't learned anything about QED, I can't follow much of the above. But here's a naive question: there is a physical quantity (I don't recall which) that has famously been measured to some 20 decimal places, and it agrees with QED predictions to nearly the same precision. And this measurement involves electrons bound in atoms, IIRC.

But it also appears that QED is hard to compute for real world problems like actual atoms. So in broad terms, what are the levels of abstraction / simplification that must necessarily intervene between raw QED and being able to calculate that 20 digit number? For example, do they use NRQED for such predictions?

But it also appears that QED is hard to compute for real world problems like actual atoms. So in broad terms, what are the levels of abstraction / simplification that must necessarily intervene between raw QED and being able to calculate that 20 digit number? For example, do they use NRQED for such predictions?
Maybe in some calculation they use NRQED, but those QED predictions are based on QED itself as far as I know. For example, the anomalous magnetic dipole moment is calculated in QED.

A. Neumaier
2019 Award
there is a physical quantity (I don't recall which) that has famously been measured to some 20 decimal places, and it agrees with QED predictions to nearly the same precision.
20 is an exaggeration. The magnetic moment of the electron is known to 12 decimal places relative accuracy.

For its calculation see, e.g. https://www.bipm.org/cc/CODATA-TGFC/Allowed/2015-02/Nio.pdf
It is an exceedingly complex computation.

Maybe in some calculation they use NRQED, but those QED predictions are based on QED itself as far as I know.
Older work was done in NRQED, for example https://arxiv.org/abs/hep-ph/9512327
The Lagrangian to order $\frac 1 M$
https://arxiv.org/pdf/1503.07216.pdf

How you derive coulomb law from it?
This is not easy, but can be done. The Coulomb law and interaction in QED are based on two different physical principles, this is why they look so dissimilar.

The Coulomb law is a direct interaction between two charges, thus it is proportional to the square of the charges: $e^2$.

On the other hand, QED or NRQED model assumes that the elementary interaction occurs between a charge and the electromagnetic field (or a photon). So, QED interaction describes this elementary process (e.g., an electron emits or absorbs a photon) which is proportional to $e$. Then, according to this model, the interaction between charges occurs in two stages: one charge emits a photon ($\propto e$), then the other charge absorbs the photon ($\propto e$). The full interaction requires these two events, hence the $e^2$ factor appears. Of course, everybody denies that the Coulomb force is created by the exchange of virtual photons, but the two-stage character of this force is also evident from the fact that the Coulomb scattering appears only in the 2nd order of the QED S-matrix.

Interestingly, the primary QED process "an electron emits or absorbs a photon" is kind of unphysical, because it violates conservation laws: a free electron can never emit (or absorb) a free photon. In addition, this virtual process is the source of nasty ultraviolet divergences. So, one may wonder, why do we need this non-observable intermediate step in our theory? Perhaps we can exclude this unphysical piece and keep only realistic charge-to-charge interactions in our Hamiltonian? It appears that such a reformulation of the QED interaction can be done, so that the S-matrix is not affected, which means that all those 12-digit accurate predictions remain intact. As a bonus we get a new QED Hamiltonian, which has only physical interactions. The leading term is the Coulomb potential plus there are multiple corrections: relativistic, radiative, bremsstrahlung, etc. Another bonus is that the resulting theory is free of ultraviolet divergences.

The original idea of this "direct interaction" approach to QFT was proposed in

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378.

You can find more modern works in this direction by searching references to this groundbreaking paper.

Eugene.

DarMM
Gold Member
So, one may wonder, why do we need this non-observable intermediate step in our theory?
The "step" refers to $\bar{\psi}\gamma^{\mu}\psi A_{\mu}$.
It's present because that is simply the interaction between the fields. Then since it appears in the Hamiltonian/Action one will have it in perturbative calculations due to it being brought down by Wick's theorem or Gaussian integration.
However interpreting it as "the primary QED process" requires one to be reading Feynman diagrams as literal processes.
It's simply a term in the Hamiltonian and no more an unobservable intermediate step/process than $x^4$ is in the anharmonic oscillator Hamiltonian.

Another bonus is that the resulting theory is free of ultraviolet divergences
Ultraviolet divergences are simply a feature of multiplying operator valued distributions, where only on a particular Hilbert $H_{phys}$ will Wick ordering define the correct multiple. In other words $:\bar{\psi}\gamma^{\mu}\psi A_{\mu}:$ is divergence free provided the Wick ordering is done with respect to the physical vacuum in the true Hilbert space. Unfortunately this is a complex problem as the the true Hilbert space is determined by the form of the Hamiltonian.

Thus we are forced to work in Fock space and remove divergences order by order. If one attempts the clothing transformation one does effectively the same thing by defining the clothing transformation order by order. If you remain in Fock space these things have to occur. The clothed formalism is no more divergence free than the usual approach to QFT.

Of course it is an early exercise in QFT texts to rederive the Coulomb potential from QED.

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vanhees71
Gold Member
2019 Award
This is not easy, but can be done. The Coulomb law and interaction in QED are based on two different physical principles, this is why they look so dissimilar.

The Coulomb law is a direct interaction between two charges, thus it is proportional to the square of the charges: $e^2$.
The usual treatment of atomic physics within non-relativistic QM for the "matter part", i.e., with the electrons and atomic nuclei treated in 1st quantization non-relativistic QM and the quantized electromagnetic field starts from the Coulomb gauge, using an interaction picture with $\hat{H}_0$ consisting of the free-photon piece (i.e., transversely polarized em. waves with 2 polarization degrees of freedom) and the matter particles including the interaction of the charges via the Coulomb interaction potential and $\hat{H}_I$ as the rest describing the interaction of the charges with the quantized electromagnetic field.

A very good treatment of this approximation scheme, valid for atoms with not too large $Z$, can be found in
S. Weinberg, Lectures on Quantum mechanics, Cambridge University Press

The same theory can be formulated of course also using non-relativistic QM in the 2nd-quantization formalism. You can even include spin using the quantized "Pauli equation". All this is then also suitable for condensed-matter calculations.

The same scheme can of course also be used within full QED with the full quantized Dirac field. How to get the hydrogen spectrum and radiation corrections (Lamb shift), you can find in

S. Weinberg, The quantum theory of fields, vol. 1, Cambridge University Press

Demystifier
I've not read it yet, but it looks promising.
I've just finished reading it yesterday, it's really great.

A. Neumaier
2019 Award
I've just finished reading it yesterday, it's really great.
See my post #8 in this thread. It is only about deriving the free nonrelativistic field (collection of particles) from the free relativistic complex scalar field, and shows that when properly done, one actually gets two free fields (electrons and positrons). Nothing deep.

The really problematic case in the case with interactions....

Demystifier
It is only about deriving the free nonrelativistic field (collection of particles) from the free relativistic complex scalar field, and shows that when properly done, one actually gets two free fields (electrons and positrons). Nothing deep.
Well, it's true that the paper is only about free particles, but it contains much more results than you outlined above.

A. Neumaier
2019 Award
Well, it's true that the paper is only about free particles, but it contains much more results than you outlined above.
Well, most of the paper is about how to do things in the wrong way. Not really illuminating. The few pages that show how to do it right are the interesting ones.

vanhees71