Dirac's Equation vs. QFT

[LarryS]

I’m attempting to learn QFT on my own and would like to get an idea of just how much I still do not know.

Consider a system consisting only of electrons and for the purpose of this question, pretend that particle creation and annihilation never occur.

QUESTION: Would Dirac’s famous relativistic equation for spin ½ particles be sufficient for modeling this system or would we still need QFT?

Thanks in advance.

 

[PeterDonis]

Would Dirac’s famous relativistic equation for spin ½ particles be sufficient for modeling this system

Do you mean the Dirac equation derived from the Lagrangian including just the kinetic term ##\bar{\psi} \left( \gamma^\mu \partial_\mu + m \right) \psi##? If so, the answer is yes, but the system in question will be utterly boring and unrealistic, because there is no way for the electrons to interact, so your solutions are just a bunch of free electron states that never change.

If, OTOH, you want a system where the electrons can interact, then the answer is no, because as soon as you include the interaction (the simplest way is just to add the ##\bar{\psi} \gamma^\mu A_\mu \psi## interaction term to the Lagrangian), you no longer have just electrons, you have electrons and the EM field, and then you will end up with QFT when you try to make everything consistent.

 

[A. Neumaier]

Dirac's equation already involves positrons, hence your scenario is somewhat questionable

 

[LarryS]

Dirac's equation already involves positrons, hence your scenario is somewhat questionable

Ok, I forgot about the "Dirac Sea". If I make the system consist of electrons and positrons, then I believe the question is still reasonable. I'm trying to get a feel as to what is in QFT besides creation and annihilation.

 

[LarryS]

Do you mean the Dirac equation derived from the Lagrangian including just the kinetic term ##\bar{\psi} \left( \gamma^\mu \partial_\mu + m \right) \psi##? If so, the answer is yes, but the system in question will be utterly boring and unrealistic, because there is no way for the electrons to interact, so your solutions are just a bunch of free electron states that never change.

If, OTOH, you want a system where the electrons can interact, then the answer is no, because as soon as you include the interaction (the simplest way is just to add the ##\bar{\psi} \gamma^\mu A_\mu \psi## interaction term to the Lagrangian), you no longer have just electrons, you have electrons and the EM field, and then you will end up with QFT when you try to make everything consistent.

Is it possible to include the covariant derivative in the Lagrangian so the EM field interaction between the electrons (and positrons) would be accounted for but the electrons and positrons themselves would remain particles instead of existing as quantized fields in a QFT?

 

[protonsarecool]

You can use the covariant derivative in the Dirac equation and treat it as a "single particle" equation with coupling to a classical EM field. If you solve eg. the hydrogen problem this way, you get several relativistic corrections that you would not get out of using the Schroedinger equation (including spin). This works since you have effectively a single-particle problem coupled to a classical field.

If you really want to describe multiple electrons, as you indicated in the OP, this approach makes no sense anymore, and you should use genuine QFT techniques.

 

[A. Neumaier]

but the electrons and positrons themselves would remain particles instead of existing as quantized fields in a QFT?

The Hilbert space of an interacting relativistic QFT is no longer a Fock space (this is Haag's theorem), so the number operator is ill-defined in the interacting theory. It is lost during the renormalization procedure.

There is, however, a book by Eugene Stefanovich called https://www.researchgate.net/profile/Eugene_Stefanovich/publication/2172727_Relativistic_Quantum_Dynamics_A_non-traditional_perspective_on_space_time_particles_fields_and_action-at-a-distance/links/554865df0cf26a7bf4dabf15.pdf, an ugly hack representing a minority position, that tries to preserve the particle structure (which by Haag's theory is impossible nonperturbatively). To be equivalent with observation he still needs to start with the standard field theory (QED). But then he transforms it in perturbation theory - by means of a dressing transformation that breaks manifest Lorentz invariance - into a representation in which particles and antiparticles figure. The result is therefore observer-dependent at any finite order of perturbation theory, with the result that there appear artifacts with seeming violation of causality.