One vertex interactions in QED - error in the book?

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• Wrichik Basu
In summary, the book "A First Book of QFT" discusses the interaction Lagrangian for QED and the normal-ordered interaction Hamiltonian for a special case of the electron. The authors decompose each field operator into creation and annihilation parts, resulting in 8 terms in the interaction Hamiltonian. The book is correct in its representation of virtual particles in the Feynman diagram, as the particles are off-shell. The change in the use of ##\psi## and ##\bar{\psi}## is correct due to the structure of the field operator and is necessary for the proper orthochronous Poincare group. However, it should be noted that the elementary QED vertex does not depict any matrix elements of
Wrichik Basu
Gold Member
I am studying QFT from the book A First Book of QFT. I am currently in the chapter of QED.

The authors have written down the interaction Lagrangian for QED. Thereafter, they have taken a special case of the electron, and written down the normal-ordered interaction Hamiltonian for this case: $$:\mathcal{H}_{\text{int}}: \ = \ -e:\bar{\psi} \gamma ^\mu \psi A_\mu:,$$ where ##\psi## and ##\bar{\psi}## are the free Dirac field operators, one for the particle and the other for the anit-particle, and ##A_\mu## is the operator of the free EM field.

Then the authors have decomposed each field operator in terms of their creation and annihilation parts, like ##\psi \ = \ \psi_+\ + \ \psi_-## where ##\psi_+## has the annihilation part, and ##\psi_-## has the creation part. Same goes for the other two operators.

If we write each field operator in terms of their creation and annihilation sub-parts, the interaction Hamiltonian has 8 terms. The authors have then said:

As you can see, thay have taken ##\bar{\psi}## for the positron and ##\psi## for the electron. Well and good, I have understood everything so far.

Soon after this, they have written,

How can this be correct? If I follow the previous arguments, ##\psi_+## annihilates an electron, ##\bar{\psi}_-## creates a positron, not an electron, and ##A^\mu_-## creates a photon.

The corresponding Feynman diagram is:

Thereafter they have proceeded to show how none of the eight terms can represent physical processes if all particles are considered on-shell. But nothing about this has been mentioned elsewhere.

I checked out the errata of the book, but they have not written anything about this. Am I correct and the book is wrong, or am I missing something?

The book is right. The diagram represents an interaction of virtual particles only, which are not physical and hence can be off shell.

A. Neumaier said:
The book is right. The diagram represents an interaction of virtual particles only, which are not physical and hence can be off shell.
I understood that they can be off-shell. But tell me one thing: following the arguments of the authors, they first took ##\psi## for the electron and ##\bar{\psi}## for the positron, but soon after that, they took the reverse, that is, ##\psi## for the positron and ##\bar{\psi}## for the electron. Is this change correct because the particles are virtual?

The field operator ##\psi## is made up of annihilation operators for electrons (in front of the positive-frequency modes) and a creation operator for positrons (in front of the negative-frequence modes). Since annihilation and creation operators are adjoint to each other this implies that ##\bar{\psi} = \psi^{\dagger} \gamma^0## it's the other way: ##\bar{\psi}## consists of a creation operator for electrons and an annihilation operator for positrons.

This (and only this!) specific structure of the field operator makes a local realization of the proper orthochronous Poincare group and leads to a Hamilton density fulfilling the microcausality constraint. Together with the demand to have an energy spectrum that is bounded from below, you have to quantize the field as fermions (spin-statistics theorem).

Now the author splits the field operators (analogously for photons, only that there particle and antiparticle are identical and of course bosons) and analyzes the various combinations occurring in the elementary QED vertex. As far as I can see from what you've quoted this is all correct, but you must take it with a grain of salt. Since this elementary vertex doesn't depict any matrix elements of physical processes since all these formal processes given by the split of the field operators are forbidden due to energy-momentum conservation. This means the vertex is just a mathematical notation for the perturbative expansion but nothing directly physical. You can of course build Feynman diagrams from these building blocks which describe physical processes. There the external legs refer to the mode functions, the inner lines to Feynman (in the vacuum QFT we are discussing here identical with the time-ordered) propagators. The vertices stand for ##\mathrm{i} e \gamma^{\mu}##. If a process is allowed by all conservation laws you get S-matrix elements and cross sections from the formalism (in the perturbative approximation to an order in ##\alpha_{\text{em}}## given by the number of vertices in the diagram).

Wrichik Basu
vanhees71 said:
The field operator ψψ\psi is made up of annihilation operators for electrons (in front of the positive-frequency modes) and a creation operator for positrons (in front of the negative-frequence modes).
Yes yes, I just missed that somehow. Thanks.

1. What is the concept of "one vertex interactions" in QED?

One vertex interactions refer to the interactions between a single electron and a single photon in quantum electrodynamics (QED). This interaction is described by a mathematical term known as the Feynman vertex, which represents the probability amplitude for the electron to emit or absorb a photon.

2. How does the book describe the error in the concept of one vertex interactions in QED?

The book states that the Feynman vertex used to describe one vertex interactions in QED contains an error in its calculation. This error arises from the use of an incorrect mathematical formula, which leads to incorrect predictions about the behavior of particles in this interaction.

3. What are the consequences of this error in the book's description of one vertex interactions in QED?

The error in the book's description of one vertex interactions in QED can lead to incorrect predictions about the behavior of particles in this interaction. This can have significant implications for research and experiments in the field of quantum electrodynamics, potentially leading to incorrect conclusions or interpretations.

4. Has this error been corrected in current scientific understanding of one vertex interactions in QED?

Yes, the error in the book's description of one vertex interactions in QED has been corrected in current scientific understanding. The correct mathematical formula for the Feynman vertex has been derived and is now used in calculations and predictions related to one vertex interactions in QED.

5. How does the correction of this error impact our understanding of one vertex interactions in QED?

The correction of this error has improved our understanding of one vertex interactions in QED and has allowed for more accurate predictions and interpretations in this field. It also highlights the importance of continually reviewing and refining scientific theories and calculations to ensure accuracy and progress in our understanding of the natural world.

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