Sources of electron-positron field in QFT

[Quetzalcoatl]

H'ya guys! (and gals!)

This is a question that has been bugging me for some time. I can't find an answer online, and neither in the books I posses. So I'll ask the experts:

The source of EM field is the EM charge density (J, the electric currents and charges).

My question is, given that QFT theorizes that electrons are waves in an electron-positron quantum field, let's call that field [latex]\psi[/latex] - what are the sources of [latex]\psi[/latex]? Does it have a source at all? or is the source just zero?

(I'm thinking about this as the analogous of the "divergence" of [latex]\psi[/latex])

Also, is this the case with other 'matter' fields?

Thank you!

[ajb]

In the formalism you add anticommuting sources so that you can write the partition function in terms of the propagator and then the interactions in terms of functional derivatives with respect to these sources. I am not sure if you should think of these anticommuting sources as much more than a mathematical trick. Remember than you have really coupled this source to the free theory.

 

 

They way you are thinking is that these sources should be the Noether current (densities) for some symmetry. I have not seen anything like that listed as a symmetry of the Dirac action. See if you can cook this up using Noether's theorem. Maybe something goes wrong and you just cannot think in terms of some symmetry, I don't know I have never looked into it.

 

Please keep us informed if you make any progress.

 

 

p.s. It may pay to first think about the real scalar field, you have less signs to worry about and one source.

Edited October 6, 2014 by ajb

[elfmotat]

The electron field doesn't have a source like the photon (electromagnetic) field does because electron number is conserved (which is why charge is conserved!) while photon number isn't, as a consequence of U(1) symmetry. If you create an electron then you must destroy one somewhere else, and vice versa. You can create as many photons as you want.

 

Something semi-related is that if you get a large number of photons together in coherent states, you get a macroscopic classical electromagnetic field. There's no macroscopic classical version of the electron field because electrons obey the Pauli exclusion principle, which means that you can't get a large collection of electrons in coherent states.

[elfmotat]

In the formalism you add anticommuting sources so that you can write the partition function in terms of the propagator and then the interactions in terms of functional derivatives with respect to these sources. I am not sure if you should think of these anticommuting sources as much more than a mathematical trick. Remember than you have really coupled this source to the free theory.

 

Indeed it's just a clever trick. In canonical QFT it's actually an unnecessary trick. The propagator is just defined as the amplitude for a particle to be created at one point and destroyed at another from the vacuum: [math]iS(x-y)= \langle 0| \, T \left \{ \psi (y) \bar{\psi} (x) \right \} |0 \rangle[/math]. If you expand the field into Fourier modes and go through a tedious calculation you get the propagator. It's the same propagator that you get by introducing anti-commuting "sources" into the Lagrangian.

Edited October 7, 2014 by elfmotat

[ajb]

I also realised that these sources cannot be something to do with symmetries as Noether's theorem will give us a current [math]J^{\mu}[/math] associated with any continuous symmetry. So the form of the sources is just wrong and there is no candidate for this symmetry.

 

 

So as elfmotat and I agree, the sources are just a nice trick in the path integral formalism.

 

 

(It has been a while since I actually performed any QFT calculations )

[Quetzalcoatl]

ajb, elfmotat:

Thank you for giving me a precise answer! I would like to delve deeper, and so I realize I need more math (you always need more math )

I'm currently reading (for the 3rd time) Theodore Frankel's great book, "The Geometry of Physics", which has a lot of fiber/vector bundle, exterior algebra and manifolds stuff. I suspect this is just the beginning. The book itself goes more into math than physics though. I'm eager to apply that math. Can you recommend books that have to do with Quantum Field Theory? I've studied GR and QM, but never got very deep into QFT. I've been watching Dr. Susskind's lectures online, but I feel like the last piece of the puzzle to understanding the standard model is still missing.

I am trying to compare gravity to EM and to other fields and draw analogies. I find it helps imagine things, and is more intuitive. For example, in the context of gravity we have the Lorentz symmetries, in EM we have the gauge symmetry (of the EM potential), which is similar, but in the fiber dimensions (U(1)). Those symmetries manifest as a conservation law, such as space-time momentum conservation in the case of gravity. In EM does it give us charge conservation? Anyways, I'm really interested in finding more good books in the field of QFT.

I also see this forum as a way to learn from others, as I almost don't know any theoretical physicists.

Thanks again!

[elfmotat]

 

ajb, elfmotat:

Thank you for giving me a precise answer! I would like to delve deeper, and so I realize I need more math (you always need more math )

I'm currently reading (for the 3rd time) Theodore Frankel's great book, "The Geometry of Physics", which has a lot of fiber/vector bundle, exterior algebra and manifolds stuff. I suspect this is just the beginning. The book itself goes more into math than physics though. I'm eager to apply that math. Can you recommend books that have to do with Quantum Field Theory? I've studied GR and QM, but never got very deep into QFT. I've been watching Dr. Susskind's lectures online, but I feel like the last piece of the puzzle to understanding the standard model is still missing.

 

I'd recommend reading Griffiths' Introduction to Elementary Particles before anything else. It gives a general overview of the standard model without much of the theoretical basis. That may sound like a bad thing, but it's probably a good idea to get familiar with the notation and general concepts (Feynman diagrams, etc.) before jumping straight into QFT.

 

For full-fledged QFT, the easiest introductory textbook I've come across is Klauber's. It takes the canonical quantization approach instead of the path integral formalism, so it will seem more familiar to you. He provides the first few chapters for free on his website: http://quantumfieldtheory.info/ .

 

David Tong's lecture notes and video lectures are good as well, though he doesn't go into nearly the detail that Klauber does. He also uses the canonical quantization approach. You can find the notes here: http://www.damtp.cam.ac.uk/user/tong/qft.html , and the videos here: https://www.youtube.com/watch?v=8yplCob7_Ck&list=PL1C5310BB35555A1C .

 

More advanced treatments of QFT usually make use of the path integral formalism though, so you'll probably want to eventually learn that. Zee's book is probably the easiest introduction to path integrals, though he can be a bit hand-wavy (which I personally can't stand). The first chapter is available for free: http://pup.princeton.edu/chapters/s7573.pdf . He also has a video lecture series, though it doesn't go into any great depth: https://www.youtube.com/watch?v=_AZdvtf6hPU&list=PLPtYfNT-VhvlB7kwjoHTqkHmhDibbl6Dr&index=1 .

 

For a more formal and advanced treatment of QFT that goes into much more detail, you'll probably want Peskin and Schroeder's book. It goes through both the canonical (at the beginning of the book) and path integral (for ~ the latter 2/3 of the book) approaches. I wouldn't recommend it to a first-timer though.

 

 

 

I am trying to compare gravity to EM and to other fields and draw analogies. I find it helps imagine things, and is more intuitive. For example, in the context of gravity we have the Lorentz symmetries, in EM we have the gauge symmetry (of the EM potential), which is similar, but in the fiber dimensions (U(1)). Those symmetries manifest as a conservation law, such as space-time momentum conservation in the case of gravity. In EM does it give us charge conservation? Anyways, I'm really interested in finding more good books in the field of QFT.

I also see this forum as a way to learn from others, as I almost don't know any theoretical physicists.

Thanks again!

 

Yes, gauge symmetry gives rise to charge conservation.

[ajb]

I like Ryder's book 'Quantum field theory'. It is quite explicit which is very useful when first leaning the subject. I also like Kaku's book 'Quantum Field Theory: A Modern Introduction', but it is harder to read and far less explicit. It does however give you some feeling for some more advanced topics.

 

Other books I have also used include Nash 'Relativistic Quantum Fields', which is an old book now but it covers renormalisation very well; and Ramond 'Field Theory : A Modern Primer'.

 

From there you may need books on specific topics depending on what you are interested in.

[Sensei]

 

Yes, gauge symmetry gives rise to charge conservation.

 

Reverse.

Data from experimental observations showed conservation of charge, so any developed by human field theory must obey it (otherwise they would be immediately rejected because of disagreement with experimental data)..

Edited October 29, 2014 by Sensei

[elfmotat]

 

Reverse.

Data from experimental observations showed conservation of charge, so any developed by human field theory must obey it (otherwise they would be immediately rejected because of disagreement with experimental data)..

 

Are you being pedantic on purpose?

[Sensei]

 

Are you being pedantic on purpose?

 

Being pedantic is benefit for scientist.

[ajb]

Reverse.

Data from experimental observations showed conservation of charge, so any developed by human field theory must obey it (otherwise they would be immediately rejected because of disagreement with experimental data)..

The point is that we know mathematically how symmetries are related to conserved charges. So, if we have some conserved charge then it is related to some group via symmetries. In converse, any theory we construct that has symmetries will have conserved charges associated with that symmetry.

[Quetzalcoatl]

Thanks for your suggestions. I'll definitely check those out.

 

 

 

Reverse.

Data from experimental observations showed conservation of charge, so any developed by human field theory must obey it (otherwise they would be immediately rejected because of disagreement with experimental data)..

 

Couldn't it be that symmetries and charge conservation are the same phenomenon in some very abstract way? And then both the statement and it's reverse are true in an if-and-only-if kind of way?

[ajb]

Couldn't it be that symmetries and charge conservation are the same phenomenon in some very abstract way? And then both the statement and it's reverse are true in an if-and-only-if kind of way?

Noether's theorem is what you are looking for; http://en.wikipedia.org/wiki/Noether%27s_theorem

[Quetzalcoatl]

Noether's theorem is what you are looking for; http://en.wikipedia.org/wiki/Noether%27s_theorem

 

I read about Noether's theorem, but I meant finding analogies not just in the way of symmetries, but in everything. I was really impressed for example with the idea of applying Noether's theorem to the fiber coordinates, and the relation that the U(1) symmetry has to the EM potential, namely that the potential can be shifted by a gradient of any function.

 

So:

 

"A' --> A + dF" is related to the rate of change of the phase of the electron's spinor field, and that relates to the charge of electron field being conserved.

I'm not sure that is accurate. I'm still wrapping my head around it. I think this was attributed to Weyl/Weil (the first part) and the rest of course to Emmy Noether. This made me wonder if the gravitational potential in the form of the christoffel symbols (levi-civita connection), has a symmetry, that is in a similar way equivalent to the conservation of its charge - the energy/momentum. Is gravitation a gauge field? I know the Lorentz transformations are such a gauge transformation, but what is the analog of the "dF" for gravity?

Edited October 31, 2014 by Quetzalcoatl

[ajb]

All gauge theories are related to principal fibre bundles.

 

People do argue if gravity is technically a gauge theory or nor. However, the principal fibre bundle here is the frame bundle of a pseudo-Riemannian manifold. It is the principal bundle associated with the tangent bundle. Because you have metric you can define orthonormal vectors and so reduce the structure group to the Lorentz group. The gauge theory is then a choice of local coordinates on the frame bundle.

 

I will have to think a little harder to explicitly answer your question. Maybe you could look for an answer by googling 'Poincare gauge theory of gravity'

[Quetzalcoatl]

Thanks ajb. I'll get those books you and others are recommending, and see how far down the rabbit hole I can get