Learning QFT from Peskin and Schroeder

[Son Goku]

I'm still pretty new to QFT, so forgive me if I have made a ridiculous mistake.

I've been learning QFT from Peskin and Schroeder mostly but decided to read Ryder recently and I have just come across an amazing result (in my opinion) in Chapter 3.

Ryder basically shows that the electromagnetic field and its coupling to the complex Klein Gordon Field "arises naturally by demanding invariance of the action under gauge transformation of the second kind in the internal space of the complex Klein Gordon field".

My question is, does this also hold for the Dirac Field?

By making similar demands on the Dirac Field can one derive the electromagnetic field and its coupling to the Dirac Field?

(Hopefully I haven't written this in a difficult to understand manner)

 

[Meir Achuz]

I'm still pretty new to QFT, so forgive me if I have made a ridiculous mistake.

I've been learning QFT from Peskin and Schroeder mostly but decided to read Ryder recently and I have just come across an amazing result (in my opinion) in Chapter 3.

Ryder basically shows that the electromagnetic field and its coupling to the complex Klein Gordon Field "arises naturally by demanding invariance of the action under gauge transformation of the second kind in the internal space of the complex Klein Gordon field".

My question is, does this also hold for the Dirac Field?

By making similar demands on the Dirac Field can one derive the electromagnetic field and its coupling to the Dirac Field?

(Hopefully I haven't written this in a difficult to understand manner)

Yes for Dirac. In order to have Local Gauge Invariance for the Dirac field,
\partial_\mu\rightarrow\partial_\mu-eA^\mu is required, just as for the KG field.

 

[Meir Achuz]

In fact, that shows that classical EM theory is required by LGI in QM,
and that all truly elementary particles (other than the gauge particles themselves) must have some kind of charge. It also shows why charge is intrinsicly dimensionless.

 

[Son Goku]

Thanks. That is an unbelievably powerful result.
I still can't get over the idea that electromagnetism can be "derived".

Thanks again.

 

[selfAdjoint]

In fact, that shows that classical EM theory is required by LGI in QM,
and that all truly elementary particles (other than the gauge particles themselves) must have some kind of charge. It also shows why charge is intrinsicly dimensionless.

I agree with Son Goku; this is a marvelous result! Why isn't it better known, broadcast to the hills as one of the triumphs of field theory? You could write a whole popular book around this one topic (with intro; I'm thinking of something which motivates an unrigorous notion of partial derivative, etc.).

 

[Meir Achuz]

I agree with Son Goku; this is a marvelous result! Why isn't it better known, broadcast to the hills as one of the triumphs of field theory? You could write a whole popular book around this one topic (with intro; I'm thinking of something which motivates an unrigorous notion of partial derivative, etc.).

This is done in the last chapter of "Classical Electromagnetism" by Franklin.

 

[selfAdjoint]

This is done in the last chapter of "Classical Electromagnetism" by Franklin.

Thanks! I just ordered a used copy of it online, because I want very much to study the argument used.

 

[Daverz]

I believe the same argument is made in A. O. Barut, Electrodynamics and the Classical Theory of Fields and Particles, section IV.3.