Non-Relativistic QFT: Readings & Vacuum Polarization

[arivero]

Any hints for readings on non relativistic quantum field theory? I guess every NRQFT should be renormalisable, because vacuum polarisation is a relativistic effect. But I would like to read about.

 

[nrqed]

Any hints for readings on non relativistic quantum field theory? I guess every NRQFT should be renormalisable, because vacuum polarisation is a relativistic effect. But I would like to read about.

You mean a NRFT in the sense of an effective field theory, I assume.

They are not renormalizable in the "old" sense. They contain an infinite number of interaction. But they are perfectly fine in the sense of effective field theories, that is, if they are used within a certain range of energy.

Take for example the non relativistic effective field theory of QED, called NRQED ( ). All electron/positron states with energies of the order m or above are not included in the theory (here, I mean energy in the nonrelativistic sense i.e. not including the rest mass energy) . All photon states with energies of the order m or above must also be excluded.


So there is no explicit vacuum polarization diagram in NRQED, that is true.
But this effect *must* be included in some way otherwise the theory would be useless (it could not be used to do any realistic calculation!).

Notice that already the simple s chanel process e+e- ->photon -> e+ e- (at tree level) is not explicitly in NRQED. But it must be incorporated somehow. This is done by including a four-fermion vertex.
Now, what about the same process but a one-loop vacuum polatrization bubble? In NRQED this is included as a *renormalization* of the previous four-fermion interaction. That is, to lowest order the four-fermion operator has a coefficient $C_0 \alpha$ if you will (which included the effect of the tree level QED diagram only). Through the usual matching procedure of effective field theories, the one-loop vacuum polarization will change this to $C_0 \alpha + C_1 \alpha^2$.



A very simple and short intro to NRQED is hep-ph/9209266
Two excellent papers on eft's are hep-ph/0506330 and nucl-th/9706029

Simple examples of applications to atomic physics of NRQED are hep-ph/9611313 and hep-ph/9706449. The following gets more technical but the intro part discusses the ideas hep-ph/9608491

The equivalent theory for QCD is called NRQCD.

Patrick

 

[EL]

You mean a NRFT in the sense of an effective field theory, I assume.

By "effective field theory" did you mean field theories where the high energy degree of freedom have been integrated out, and we are left with some effective Lagrangian, e.g. the Heisenberg-Euler Lagrangian?

But why should there be a problem with considering NRQFT's from the beginning? Just start from some Lagrangian which is not relativisticly invariant...
For instance this is the case in many QFT's used in condensed matter physics...

The unification between SR and quantum mechanic must be treated by a QFT, but a QFT need not be relativistic. Ok?

Any hints for readings on non relativistic quantum field theory? I guess every NRQFT should be renormalisable, because vacuum polarisation is a relativistic effect. But I would like to read about.

I would like to read more about this too, and I think there is at least something at the basic conseptual level in Zee's "QFT in a nutshell". But I'd appreciate any good reference too.

 

[CarlB]

These are beautiful papers. I have the heretical belief that methods of physics that simplify computations are likely to be closer to the underlying physics.

This reminds me of the Koide formula in that the simplicity is showing up at the low energy limit. Koide's formula suggests that the leptons are composite particles, and therefore that we can better understand the leptons by better understanding bound states. It also reminds me of the dressed propagator papers and other things like that.

Carl

 

[arivero]

A very simple and short intro to NRQED is hep-ph/9209266
Two excellent papers on eft's are hep-ph/0506330 and nucl-th/9706029

Simple examples of applications to atomic physics of NRQED are hep-ph/9611313 and hep-ph/9706449. The following gets more technical but the intro part discusses the ideas hep-ph/9608491

Awesome papers. And more if you take into account that in 1989 there were no textbooks on modern QFT. No Weinberg's, nor Peskin... Hard times.

Now, I think this NRQFT should be called "semi-relativistic". In fact for some examples it recovers the terms of semi-relativistic quantum mechanics. In a real NR QFT, if such object exists, I do not expect to have a power series on $$\alpha$$, nor any other object whose construction needs of the relativistic constant c.

 

[Careful]

Any hints for readings on non relativistic quantum field theory? I guess every NRQFT should be renormalisable, because vacuum polarisation is a relativistic effect. But I would like to read about.

Euh, vacuum polarization is not a strict relativistic effect, in the non-relativistic version of Barut Self field, one has a non-relativistic counterpart for the vacuum polarization term studied by Wichmann and Kroll. See : Phys Rev A, vol 32 (1985), 3187-3195.

Careful

 

[nrqed]

Awesome papers. And more if you take into account that in 1989 there were no textbooks on modern QFT. No Weinberg's, nor Peskin... Hard times.

I am not sure which ones you are referring to but just in case you are including some of mine, thank you:shy:

The paper which really started nonrelativistic eft's is a paper by Caswell and Lepage (I *think* from 1984) which has several hundreds of citations.

Now, I think this NRQFT should be called "semi-relativistic". In fact for some examples it recovers the terms of semi-relativistic quantum mechanics. In a real NR QFT, if such object exists, I do not expect to have a power series on $$\alpha$$, nor any other object whose construction needs of the relativistic constant c.

I am not sure why you would not expect any power series in alpha. Nonrelativistic quantum mechanics does produce an expansion in powers of alpha. But NRQM breaks down pretty soon because the theory does not treat properly the high energy modes. The solution is to apply renormalization theory to the theory (i.e., to put back the physics due to the high energy modes in the theory using perturbative matching) and the result is an effective field theory.

(similar ideas have been used in different contexts, for example by Luscher and others in lattice gauge theory. In fact, the fundamental ideas originate from the work of Ken Wilson and the others who really presented the modern point of view of renormalization)

 

[arivero]

The paper which really started nonrelativistic eft's is a paper by Caswell and Lepage (I *think* from 1984) which has several hundreds of citations.

It is scanned at KEK:
http://ccdb4fs.kek.jp/cgi-bin/img_index?8504383

I am not sure why you would not expect any power series in alpha.

Because NR stands for "NON relativistic" Alpha appears in physics as the quotient between the minumum possible angular momentum in a quantum theory and the minimum posible angular momentum in a relativistic classical orbit; this is the definition Sommerfeld does, in a German paper long long time ago. So I expect alpha to appear in SEMI-relativistic expansions.

Of course in modern physics alpha is just the dimensionless coupling constant. Ok, leave alpha to live, and then please restore dimensionality of each term in the effective action... In a purely NR theory, I would expect to be able to restore dimensionality by using only powers of h; surely we will need to use both combinations of c and h to set dimensions right, do we? Hapily, the powers of c will appear only as inverse powers; in this case we can claim we have a semirrelativistic expression because in the limit c--->infinity such terms are completely discarded.

Well, for practical purposes, it is only a denomination issue. I see I was thinking on a different issue

In fact, the fundamental ideas originate from the work of Ken Wilson and the others who really presented the modern point of view of renormalization)

I hold some objections against the modern point of view because it drives people to imply that renormalizability of a theory is just an accidental thing. By I liked to read and re-read all these papers of K Wilson (and the one of Wilson and Kogut, of course).

 

[arivero]

. By I liked to read and re-read all these papers of K Wilson (and the one of Wilson and Kogut, of course).

(colateral remark: as a proof I liked the idea, during my own thesis period I did a small scaling play to define contact interactions and I enjoyed it enough to upload it as a preprint http://arxiv.org/abs/hep-th/9411081 )

 

[nrqed]

It is scanned at KEK:
http://ccdb4fs.kek.jp/cgi-bin/img_index?8504383

Thanks! I have the copy that Lepage gave me somewhere but can't find t anywhere.

I hold some objections against the modern point of view because it drives people to imply that renormalizability of a theory is just an accidental thing. By I liked to read and re-read all these papers of K Wilson (and the one of Wilson and Kogut, of course).

I don't think that it is purely an accidental thing.

There is something non-trivial about the fact that the standard model is renormalizable "in the old sense" (in the sense that all infinities can be reabsorbed into a finite number of parameters), as opposed to "renormalizable in the eft sense".

I mean, sure, if one takes any eft and let the scale of "new physics" go to infinity, almost all terms will disappear, leaving only a theory renormalizable in the old sense. However, when one does that for most efts, one ends up with a non-interacting theory. It is non-trivial if one ends up with an interacting theory (like QED or QCD). So even though we can be sure that QED is an eft of something deeper, it is still non-trivial that it is renormalizable in the old sense. It is because of gauge invariance, I believe.

I have to admit that I never saw any of this discussed this way anywhere so that's my personal opinion. But I think that it is non-trivial that one may take the limit of the scale of new physics in an eft to infinity and still be left with an interacting theory which is renormalizable in the old sense. This is different than, say, the four-Fermi model of the weak interaction which was obviously an eft from the very beginning since it is non-renormalizable in the old sense.

Do you see what I mean?

Patrick

 

[arivero]

Hey, you have done a very deep post... and coincidentally, it is the post #1000 you do in physicsforums. Congratulations

There is something non-trivial about the fact that the standard model is renormalizable "in the old sense"

That is my view too. But reading Weinberg's or other modern, post Wilsonian books, one is driven to believe that it is just a consequence of being "renormalizable in the eft sense". Also from some lectures of Lepage it seems to transpire this consequence

It is non-trivial if one ends up with an interacting theory (like QED or QCD). So even though we can be sure that QED is an eft of something deeper, it is still non-trivial that it is renormalizable in the old sense. It is because of gauge invariance, I believe.

I have to admit that I never saw any of this discussed this way anywhere so that's my personal opinion.

I think that one of the goals of the analysis of fixed points in the modern renormalisation group and trajectories between them was to isolate the amazing non-triviality of gauge theories. And probably the hope of GUT modellers was to uplift this observation into some kind of uniqueness statement. But given all the rage on String Theory during the last decades, I thought that this line of thinking had become lost, or abandoned.

 

[selfAdjoint]

On the presence and Sommerfeld definition of alpha as entailing relativity, isn't using it as an unexplained constant in the very spirit of renormalization group thinking? That is, the high energy physics "within" alpha is factored out and replaced with a counter term, just the numeric value of alpha.

 

[arivero]

On the presence and Sommerfeld definition of alpha as entailing relativity, isn't using it as an unexplained constant in the very spirit of renormalization group thinking? That is, the high energy physics "within" alpha is factored out and replaced with a counter term, just the numeric value of alpha.

Hmm I do not see it. Perhaps you could expand a little here.

Note Sommerfeld was not using more high anergy than a bit of relativity for the calculation of Bohr orbits. It is known that for a classical relativistic particle in a 1/r^2 force the coupling constant, say K, equals angular momentum times orbital speed. If the orbital speed can not be higher than c, then the angular momentum can not be smaller than K/c. On the other hand, quantum mechanics a la Bohr has a minimum angular momentum h. Sommerfeld constant was defined as the quotient between both quantities.

I suposse one could say that alpha marks the scaling between two different scales; for instance one could say that in the interacction of particles of mass m we have an interplay between mc^2 and alpha mc^2.

 

[nrqed]

Hey, you have done a very deep post... and coincidentally, it is the post #1000 you do in physicsforums. Congratulations

Thank you!

That is my view too. But reading Weinberg's or other modern, post Wilsonian books, one is driven to believe that it is just a consequence of being "renormalizable in the eft sense". Also from some lectures of Lepage it seems to transpire this consequence

You are right, and it has always bothered me a bit that there was not more discussion on the reason for those non-trivial efts renormalizable in the old sense (QED, QCD, etc). I think that gauge invariance and chiral symmetry have a special role in this. In any case, this is in stark contrast with the four-Fermi model of the weak interaction or with GR.
There must be some paper out there that discusses that in more depth and pedagogically.

I think that one of the goals of the analysis of fixed points in the modern renormalisation group and trajectories between them was to isolate the amazing non-triviality of gauge theories. And probably the hope of GUT modellers was to uplift this observation into some kind of uniqueness statement. But given all the rage on String Theory during the last decades, I thought that this line of thinking had become lost, or abandoned.

I agree with you. I unfortunately never really learn renormalization group flows and all that stuff. I don't have a deep understanding and that's a big deficiency in my background.


Thanks for your comments!

Patrick

 

[arivero]

You are right, and it has always bothered me a bit that there was not more discussion on the reason for those non-trivial efts renormalizable in the old sense (QED, QCD, etc).

At least Huang, in page 338 of his other book, makes an attempt. But it already starts in a no very promising way:

"... the case of QED remains a puzzle. This is ironic, for perturbative renormalization scores its greatest triumph in QED, and yet the fixed point structure is not clear."

 

[arivero]

I have taken some time to browse across Lepage's TASI89 lecture (hep-ph/0506330) and sometimes it results overoptimistic or misleading about how effective the effective theories are. In section 3.1 it uses to the precision of (g-2) for the electron to conclude than the scale for new physics beyond U(1) should be "probably larger than a TeV". Well, it can perhaps to be forgiven that Z0 and W+ are less than 100GeV, as it is only about one order of magnitude. But it also misses the muon and the tau, which are no doubt "new physics in electrodynamics".