Particles as the quanta of fields

[7777777]

Quantum field theory deals with the quantization of the electro-magnetic field, and finds its
quantum: the photon.

Electric and magnetic fields are classical fields. Can QFT quantize also them, and find their quanta?
It is often said the electrons are field quanta (particles are quanta), meaning that the quanta of electric field are electrons. But electrons have mass, and I think that it is mass which distinguishes field and matter. Or does QFT suggest that fields have mass?

 

[vanhees71]

There is no such thing as an electric or magnetic field, but only one electromagnetic field, and its quanta are called photons. This is the only correct way to define, on a theoretical level, what photons actually are. You must not consider them as being particles in a classical sense. Photons are as different as quanta can be from classical particles. The classical limit of photons are not classical particles but the classical electromagnetic field, described in terms of coherent states of the quantized electromagnetic field.

Electrons are the quanta of a different sort of field, called a Dirac field. Electrons are massive and have spin 1/2, while photons are massless and have spin 1.

 

[7777777]

There is no such thing as an electric or magnetic field, but only one electromagnetic field, and its quanta are called photons.

Do you mean that there are no such thing as electric or magnetic field in QFT?

There are electric and magnetic fields at least in classical field theory described by Maxwell's
equations.

 

[Demystifier]

77777777, electro-magnetic field consists of two parts - electric field and magnetic field. So by quantizing electro-magnetic field via photons, one also quantizes electric field and magnetic field. Photons are quanta of electric field and of magnetic field.

Furthermore, electrons are NOT quanta of electric field. Electrons are quanta of a matter field (technically, a spinor field), which is a massive field. So yes, a field may have a mass.

 

[bhobba]

Vanhees is of course correct.

The justification is found in Quantum Field Theory which is normally only done after an advanced course on QM so you basicaly have to take what was said on faith until your QM is advanced enough.

However I have recently come across a book that can be tackled with QM at the level of say Susskind:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

It is Quantum Field Theory for the Gifted Amateur:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

I am going through it right now. With attention and a bit of time things will be a lot clearer.

Thanks
Bill

 

[bhobba]

There are electric and magnetic fields at least in classical field theory described by Maxwell's equations.

Yes - but it is unified into a single field called the Electromagnetic Field.

The reason is tied up with relativity:
http://richardhaskell.com/files/Special Relativity and Maxwells Equations.pdf

Thanks
Bill

 

[jtbell]

Do you mean that there are no such thing as electric or magnetic field in QFT?

In QED one normally starts with the 4-potential, which has components corresponding to the electric (scalar) potential ##\phi## and magnetic (vector) potential ##\vec A##; not the electric and magnetic fields ##\vec E## and ##\vec B##.

http://en.wikipedia.org/wiki/Electromagnetic_four-potential

 

[Demystifier]

It is Quantum Field Theory for the Gifted Amateur:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

Today I've started reading this book. It seems great!

 

[bhobba]

In QED one normally starts with the 4-potential

Indeed.

And just perhaps to motivate the OP to investigate the detail further that 4 potential is not in fact uniquely defined - it possesses what is called a gauge symmetry - meaning you can transform it in a certain way and physically it makes no difference. The strange thing is it turns out that more or less actually determines EM. Theories like this are called gauge theories

The textbook I suggested before examines this in Chapter 14 where it is shown EM is the simplest such gauge theory, and has profound consequences for the QFT of EM. If it intrigues you - it will explain all.

Just to whet your appetite a bit check out:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

Thanks
Bill

 

[7777777]

Furthermore, electrons are NOT quanta of electric field. Electrons are quanta of a matter field (technically, a spinor field), which is a massive field. So yes, a field may have a mass.

A matter field sounds like a strange "mixture" of matter and field. Is it both matter and field at the same time? Does pure matter exist? Matter without a field. Or pure field? A field without matter?
Matter is the source of fields, so if there is a field without matter, what could have caused it?

 

[dextercioby]

Hehe, nice questions, even though they sound a little philosophical. The only theory that really works, *The Standard Model of Fundamental Particles and Interactions*, is a neat almost mathematical theory in which quantum fields are used to describe matter which is interpreted in form of particles. Interactions between matter fields/particles are also described in terms of quantum fields. Actually 'particles' is a name give to quanta of fields. So there is only quantum field theory. Only fields.

 

[vanhees71]

I guess, I was a bit too brief with my statement concerning the em. Field. Even in classical theory it doesn't make sense to say there is an electric and a magnetic field. For each inertial reference frame there can be defined electric and magnetic field components, but this is a frame dependent statement. Only the em. field as a whole is a physically meaningful quantity. That's why there are no electric and magnetic photons but only photons in qft, which are the quanta of the quantized em. field.

In the Standard Model we have a lot more fields to describe the "matter", the quarks and leptons. They all have spin 1/2 and are thus fermions. Then there are the interactions all described by spin-1 fields in terms of a gauge theory with there corresponding quanta, which are the gluons, photons, and the W and Z bosons. Last but not least there's also the Higgs field and the corresponding Higgs particle of spin 0.

 

[bhobba]

Actually 'particles' is a name give to quanta of fields. So there is only quantum field theory. Only fields.

IMHO, and others as well, if you are just starting out, many of the issues in QM are easier to understand in QFT rather than QM:
https://www.amazon.com/dp/B004ULVG9O/?tag=pfamazon01-20

But I do urge the OP to eventually become acquainted with the real deal as found in the text I mentioned.

I read, and have read, a LOT of books on QM, and the text on QFT for the Gifted Amateur is really very good.

Thanks
Bill

 

[7777777]

I guess, I was a bit too brief with my statement concerning the em. Field. Even in classical theory it doesn't make sense to say there is an electric and a magnetic field. For each inertial reference frame there can be defined electric and magnetic field components, but this is a frame dependent statement. Only the em. field as a whole is a physically meaningful quantity. That's why there are no electric and magnetic photons but only photons in qft, which are the quanta of the quantized em. field.

Maxwell unified the electric and magnetic fields into equations implying that light propagates as electromagnetic waves. Before Maxwell the electric field was described by Coulomb's law.
Maybe the quantization of the electric field should be based on Coulomb's law, but I don't
know if anyone has tried to do it.

 

[Demystifier]

Maybe the quantization of the electric field should be based on Coulomb's law, but I don't
know if anyone has tried to do it.

It seems impossible to do it. The Coulomb's law does not involve a time derivative of the field, which means that the theory is not dynamical. Consequently, one cannot define canonical formalism involving a Hamiltonian and a momentum, which implies that one cannot quantize the system by the usual methods of quantization.

 

[ChrisVer]

What do you mean based on Coulomb's law? Coulomb's law exists in Maxwell equations for the electrostatic case... however this "static" gives a hint about what is happening in a different, moving frame...things change and you also get a magnetic field. So you are bound to talk about electromagnetism instead if you want to have a frame independent picture. That is just Special Relativity and that's to be expected for a Quantum field theory (where by definition you are dealing with relativistic quantum mechanics / many particle systems).

Now for the QED itself, it does contain the Coulomb interaction (as a potential) as a low energy limit (non-relativistic limit).

Also the Coulomb's law is not by definition correct. It works fine for single charges, but when you have many particles this can be a mess. In this case, in general, since you are working with charge densities, the coulomb interaction appears as a term (large distances /or low energies) in a multipole expansion.

 

[7777777]

What do you mean based on Coulomb's law? Coulomb's law exists in Maxwell equations for the electrostatic case... however this "static" gives a hint about what is happening in a different, moving frame...things change and you also get a magnetic field. So you are bound to talk about electromagnetism instead if you want to have a frame independent picture. That is just Special Relativity and that's to be expected for a Quantum field theory (where by definition you are dealing with relativistic quantum mechanics / many particle systems).

I mean the static electric field whose source is a charge, an electron. To describe the electric field by Coulomb's law, only a charge is needed, there is no magnetic field. In this way it is possible
to avoid talking about electro-magnetic field.

Now for the QED itself, it does contain the Coulomb interaction (as a potential) as a low energy limit (non-relativistic limit).

Also the Coulomb's law is not by definition correct. It works fine for single charges, but when you have many particles this can be a mess. In this case, in general, since you are working with charge densities, the coulomb interaction appears as a term (large distances /or low energies) in a multipole expansion.

Perhaps a single charge is enough. How to understand the electric field created by an electron?
Does the field have mass? Is the field the same as the charge of an electron, then it could be massless, a charge field, a charge density field. The electric field strength becomes infinite at short distances to the electron, that is the problem of the Coulomb law. I have read the history of Dirac & co and how difficult it was for them to develop a satisfactory QED, there were problems with infinities of electron's mass and charge. They found a solution: polarization of vacuum.

 

[ChrisVer]

I mean the static electric field whose source is a charge, an electron. To describe the electric field by Coulomb's law, only a charge is needed, there is no magnetic field. In this way it is possible
to avoid talking about electro-magnetic field.

But something that is static in a ref frame, will be moving in another. That's what I tried to explain. That's why Special relativity is unifying elec. and magn. into electromagnetism...

Perhaps a single charge is enough. How to understand the electric field created by an electron?

When you start dealing with quantum field theories, you find out that you don't have a single electron, but many particles. That's why I wrote in the previous post the Relativistic QM as quantum mechanics of many particles system. Eg a first indicator of this is that an electron is coming as a solution together with a positron in Dirac's equation.
Not to say that a natural space to work in is the Fock space.

Does the field have mass? Is the field the same as the charge of an electron, then it could be massless, a charge field, a charge density field. The electric field strength becomes infinite at short distances to the electron, that is the problem of the Coulomb law. I have read the history of Dirac & co and how difficult it was for them to develop a satisfactory QED, there were problems with infinities of electron's mass and charge. They found a solution: polarization of vacuum.

Which field? the electromagnetic field or the electron's? The electromagnetic field is massless. The electron is not massless. The bare charge of electron is Q=e... This is however getting corrections from higher order feynman diagrams and thus it doesn't appear always as e.
The infinities are dealt through the renormalization. In the QED renormalization there still exists the Landau Pole, but the last appears in so large energies that of course we don't expect the standard model to be predictive at them.

 

[Nugatory]

I mean the static electric field whose source is a charge, an electron. To describe the electric field by Coulomb's law, only a charge is needed, there is no magnetic field. In this way it is possible
to avoid talking about electro-magnetic field.

It is not possible if the electron is in motion relative to the observer - magnetic effects wil appear. And as we expect our physical laws to work whether the observer is moving relative to the system being observed or not, we don't have the option of ignoring these effects.

Thus, Coulomb's Law is incomplete - it applies only in the special case of stationary charges and test particles, and is a very limited special case of Maxwell's equations which ought to be your starting point (google will find them quickly).

We're straying far from the original question, so I suggest that you try some of the references that other posters in this thread have suggested. After you've looked at them (this one from Bhobba would be a good start) you should feel free to come back with some more focused questions.

 

[Nugatory]

I'm going to close this thread now, not because anything is particularly wrong with it but just because it seems to be reaching a point of diminishing returns. Anyone who feels otherwise or wants to contribute further, PM me.