## Is Higgs alone?

14/03/2015

I am back after the announcement by CERN of the restart of LHC. On May this year we will have also the first collisions. This is great news and we hope for the best and the best here is just the breaking of the Standard Model.

The Higgs in the title is not Professor Higgs but rather the particle carrying his name. The question is a recurring one since the first hints of existence made their appearance at the LHC. The point I would like to make is that the equations of the theory are always solved perturbatively, even if exact solutions exist that provide a mass also if the theory is massless or has a mass term with a wrong sign (Higgs model). All you need is a finite self-interaction term in the equation. So, you will have bad times to recover such exact solutions with perturbation techniques and one keeps on living in the ignorance. If you would like to see the technicalities involved just take a cursory look at Dispersive Wiki.

What is the point? The matter is rather simple. The classical theory has exact massive solutions for the potential in the form $V(\phi)=a\phi^2+b\phi^4$ and this is a general result implying that a scalar self-interacting field gets always a mass (see here and here). Are we entitled to ignore this? Of course no. But today exact solutions have lost their charm and we can get along with them.

For the quantum field theory side what could we say? The theory can be quantized starting with these solutions and I have shown that one gets in this way that these massive particles have higher excited states. These are not bound states (maybe could be correctly interpreted in string theory or in a proper technicolor formulation after bosonization) but rather internal degrees of freedom. It is always the same Higgs particle but with the capability to live in higher excited states. These states are very difficult to observe because higher excited states are also highly depressed and even more hard to see. In the first LHC run they could not be seen for sure. In a sense, it is like Higgs is alone but with the capability to get fatter and present himself in an infinite number of different ways. This is exactly the same for the formulation of the scalar field as originally proposed by Higgs, Englert, Brout, Kibble, Guralnik and Hagen. We just note that this formulation has the advantage to be exactly what one knows from second order phase transitions used by Anderson in his non-relativistic proposal of this same mechanism. The existence of these states appears inescapable whatever is your best choice for the quartic potential of the scalar field.

It is interesting to note that this is also true for the Yang-Mills field theory. The classical equations of this theory display similar solutions that are massive (see here) and whatever is the way you develop your quantum filed theory with such solutions the mass gap is there. The theory entails the existence of massive excitations exactly as the scalar field does. This have been seen in lattice computations (see here). Can we ignore them? Of course no but exact solutions are not our best choice as said above even if we will have hard time to recover them with perturbation theory. Better to wait.

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v2

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N Physics Reports 526 (2013) 93-163 arXiv: 1210.4997v2

## The question of the mass gap

10/09/2014

Some years ago I proposed a set of solutions to the classical Yang-Mills equations displaying a massive behavior. For a massless theory this is somewhat unexpected. After a criticism by Terry Tao I had to admit that, for a generic gauge, such solutions are just asymptotic ones assuming the coupling runs to infinity (see here and here). Although my arguments on Yang-Mills theory were not changed by this, I have found such a conclusion somewhat unsatisfactory. The reason is that if you have classical solutions to Yang-Mills equations that display a mass gap, their quantization cannot change such a conclusion. Rather, one should eventually expect a superimposed quantum spectrum. But working with asymptotic classical solutions can make things somewhat involved. This forced me to choose the gauge to be always Lorenz because in such a case the solutions were exact. Besides, it is a great success for a physicist to find exact solutions to fundamental equations of physics as these yield an immediate idea of what is going on in a theory. Even in such case we would get a conclusive representation of the way the mass gap can form.

Finally, after some years of struggle, I was able to get such a set of exact solutions to the classical Yang-Mills theory displaying a mass gap (see here). Such solutions confirm both the Tao’s argument that an all equal component solution for Yang-Mills equations cannot hold in any gauge and also my original argument that an all equal component solution holds, in a general case, only asymptotically with the coupling running to infinity. But classically, there exist solutions displaying a mass gap that arises from the nonlinearity of the equations of motion. The mass gap goes to zero as the coupling does. Translating this in the quantum realm is straightforward as I showed for the Lorenz (Landau) gauge. I hope all this will help to better elucidate all the physics around strong interactions. My efforts since 2005 went in that direction and are still going on.

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v1

## Do quarks grant confinement?

21/07/2014

In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

• Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
• Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
• The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
• The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with $N_f$ the number of quarks (flavours):

From this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal
window of QCD arXiv arXiv: 1405.7340v1

## f0(500) and f0(980) are not tetraquarks

27/06/2014

Last week I have been in Giovinazzo, a really beautiful town near Bari in Italy. I participated at the QCD@Work conference. This conference series is now at the 7th edition and, for me, it was my second attendance. The most striking news I heard was put forward in the first day and represents a striking result indeed. The talk was given by Maurizio Martinelli on behalf of LHCb Collaboration. You can find the result on page 19 and on an arxiv paper . The question of the nature of f0(500) is a vexata quaestio since the first possible observation of this resonance. It entered in the Particle Data Group catalog as f0(600) but was eliminated in the following years. Today its existence is no more questioned and this particle is widely accepted. Also its properties as the mass and the width are known with reasonable precision starting from a fundamental work by Irinel Caprini, Gilberto Colangelo and Heinrich Leutwyler (see here). The longstanding question around this particle and its parent f0(980) was about their nature. It is generally difficult to fix the structure of a resonance in QCD and there is no exception here.

The problem arose from famous papers by Jaffe on 1977 (this one and this one) that using a quark-bag model introduced a low-energy nonet of states made of four quarks each. These papers set the stage for what has been the current understanding of the f0(500) and f0(980) resonances. The nonet is completely filled with all the QCD resonances below 1 GeV and so, it seems to fit the bill excellently.

Someone challenged this kind of paradigm and claimed that f0(500) could not be a tetraquark state (e.g. see here and here but also papers by Wolfgang Ochs and Peter Minkowski disagree with the tetraquark model for these resonances). The answer come out straightforwardly from LHCb collaboration: Both f0(500) and f0(980) are not tetraquark and the original view by Jaffe is no more supported. Indeed, people that know the Nambu-Jona-Lasinio model should know quite well where the f0(500) (or $\sigma$ ) comes from and I would also suggest that this model can also accommodate higher states like f0(980).

I should say that this is a further striking result coming from LHCb Collaboration. Hopefully, this should give important hints to a better understanding of low-energy QCD.

$\overline{B}^0\rightarrow J/ψπ^+π^-$ decays arXiv arXiv: 1404.5673v2
Irinel Caprini, Gilberto Colangelo, & Heinrich Leutwyler (2005). Mass and width of the lowest resonance in QCD Phys.Rev.Lett.96:132001,2006 arXiv: hep-ph/0512364v2
Jaffe, R. (1977). Multiquark hadrons. I. Phenomenology of Q^{2}Q[over ¯]^{2} mesons Physical Review D, 15 (1), 267-280 DOI: 10.1103/PhysRevD.15.267
Jaffe, R. (1977). Multiquark hadrons. II. Methods Physical Review D, 15 (1), 281-289 DOI: 10.1103/PhysRevD.15.281
G. Mennessier, S. Narison, & X. -G. Wang (2010). The sigma and f_0(980) from K_e4+pi-pi, gamma-gamma scatterings, J/psi,
phi to gamma sigma_B and D_s to l nu sigma_B Nucl.Phys.Proc.Suppl.207-208:177-180,2010 arXiv: 1009.3590v1

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

## Nailing down the Yang-Mills problem

22/02/2014

Millennium problems represent a major challenge for physicists and mathematicians. So far, the only one that has been solved was the Poincaré conjecture (now a theorem) by Grisha Perelman. For people working in strong interactions and quantum chromodynamics, the most interesting of such problems is the Yang-Mills mass gap and existence problem. The solutions of this problem would imply a lot of consequences in physics and one of the most important of these is a deep understanding of confinement of quarks inside hadrons. So far, there seems to be no solution to it but things do not stay exactly in this way. A significant number of researchers has performed lattice computations to obtain the propagators of the theory in the full range of energy from infrared to ultraviolet providing us a deep understanding of what is going on here (see Yang-Mills article on Wikipedia). The propagators to be considered are those for  the gluon and the ghost. There has been a significant effort from theoretical physicists in the last twenty years to answer this question. It is not so widely known in the community but it should because the work of this people could be the starting point for a great innovation in physics. In these days, on arxiv a paper by Axel Maas gives a great recount of the situation of these lattice computations (see here). Axel has been an important contributor to this research area and the current understanding of the behavior of the Yang-Mills theory in two dimensions owes a lot to him. In this paper, Axel presents his computations on large volumes for Yang-Mills theory on the lattice in 2, 3 and 4 dimensions in the SU(2) case. These computations are generally performed in the Landau gauge (propagators are gauge dependent quantities) being the most favorable for them. In four dimensions the lattice is $(6\ fm)^4$, not the largest but surely enough for the aims of the paper. Of course, no surprise comes out with respect what people found starting from 2007. The scenario is well settled and is this:

1. The gluon propagator in 3 and 4 dimensions dos not go to zero with momenta but is just finite. In 3 dimensions has a maximum in the infrared reaching its finite value at 0  from below. No such maximum is seen in 4 dimensions. In 2 dimensions the gluon propagator goes to zero with momenta.
2. The ghost propagator behaves like the one of a free massless particle as the momenta are lowered. This is the dominant behavior in 3 and 4 dimensions. In 2 dimensions the ghost propagator is enhanced and goes to infinity faster than in 3 and 4 dimensions.
3. The running coupling in 3 and 4 dimensions is seen to reach zero as the momenta go to zero, reach a maximum at intermediate energies and goes asymptotically to 0 as momenta go to infinity (asymptotic freedom).

Here follows the figure for the gluon propagator

and for the running coupling

There is some concern for people about the running coupling. There is a recurring prejudice in Yang-Mills theory, without any support both theoretical or experimental, that the theory should be not trivial in the infrared. So, the running coupling should not go to zero lowering momenta but reach a finite non-zero value. Of course, a pure Yang-Mills theory in nature does not exist and it is very difficult to get an understanding here. But, in 2 and 3 dimensions, the point is that the gluon propagator is very similar to a free one, the ghost propagator is certainly a free one and then, using the duck test: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck, the theory is really trivial also in the infrared limit. Currently, there are two people in the World that have recognized a duck here:  Axel Weber (see here and here) using renormalization group and me (see here, here and here). Now, claiming to see a duck where all others are pretending to tell a dinosaur does not make you the most popular guy  in the district. But so it goes.

These lattice computations are an important cornerstone in the search for the behavior of a Yang-Mills theory. Whoever aims to present to the World his petty theory for the solution of the Millennium prize must comply with these results showing that his theory is able to reproduce them. Otherwise what he has is just rubbish.

What appears in the sight is also the proof of existence of the theory. Having two trivial fixed points, the theory is Gaussian in these limits exactly as the scalar field theory. A Gaussian theory is the simplest example we know of a quantum field theory that is proven to exist. Could one recover the missing part between the two trivial fixed points as also happens for the scalar theory? In the end, it is possible that a Yang-Mills theory is just the vectorial counterpart of the well-known scalar field, the workhorse of all the scholars in quantum field theory.

Axel Maas (2014). Some more details of minimal-Landau-gauge Yang-Mills propagators arXiv arXiv: 1402.5050v1

Axel Weber (2012). Epsilon expansion for infrared Yang-Mills theory in Landau gauge Phys. Rev. D 85, 125005 arXiv: 1112.1157v2

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

## Ending and consequences of Terry Tao’s criticism

21/09/2013

Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

As generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1

## Return in Paris

15/06/2013

After two years since the last edition, I was back in Paris to participate to the Twelfth Workshop on Non-perturbative Quantum Chromodynamics. The conference is organized by high-energy group at Brown University and held at Institut d’Astrophysique de ParisProfessor Chung-I Tan and Professor Berndt Mueller from Duke University are the organizers. As it also happened in the precedent edition, the workshop was really interesting and rich of ideas for research. The first talk was given by Kostantinos Orginos and was about nuclear physics emerging from lattice computations. This is a matter that I am involved into as a “final user” and so, very near my interests. It is noteworthy to point out how current technology permits  to extract such results from lattice QCD making this a useful tool for the understanding of low-energy phenomenology. With Kostantinos,  his wife Vassiliki Panoussi and sons, we have had a nice social dinner on Tuesday night and I have had an interesting discussion about the current situation of lattice computations. The next speaker was Philippe de Forcrand that is well-known for his works on finite temperature QCD on the lattice.   He showed how the effective Yang-Mills theory at high temperature is surprisingly good with respect to lattice results also lowering temperatures at few times the critical temperature. Another interesting talk was the one by Peter Petreczy about the observables of QCD at finite temperature presenting also the most recent value for the critical temperature. As my readers may know, I computed this value in my recent paper on Physical Review C (see here) properly corrected by the mass gap of Yang-Mills theory. Norberto Scoccola and Daniel Gomez-Dumm showed similar results (see here).

On Tuesday it was the ultrarelativistic Heavy-ion collision session. This was particularly interesting and involved the talks of two friends of mine: Marco Ruggieri and Salvatore Plumari. In this area of research there is a really interesting and hot debated situation. On the other side, there is plenty of experimental results from RHIC and LHC. The session chair was Jean-Yves Ollitrault. He put the foundations to the current understanding of the quark-gluon plasma through a hydrodynamic approximation. What is observed in the experiments is the production of a flow of particles in a transverse direction named elliptic flow. This is a clear evidence of existence for the quark-gluon plasma. Marco and Salvatore work in the group of Vincenzo Greco at University of Catania in Italy. The idea they based their work on is to derive the hydrodynamic equations from a kinetic description as the one provided by the Boltzmann equation. This approach opens up the scene to the possibility to derive such an equation and the full description of the quark-gluon plasma starting directly from QCD and fixing the collisional integral of the kinetic equation. Of course, one should understand the applicability conditions but my take is that, being the running coupling going to zero due to asymptotic freedom, a quark-gluon plasma should have scarce multi-collision effects. On the other side, this is a charged plasma but lives for a very small time. This means that this approach can prove to be really successful. One of the open questions is if, going at higher energies, a state called “color glass condensate” should form and this is a matter of a hot debate in the community. This is creating some tension that is reminiscent of the story I recounted about Landau gauge propagators for pure Yang-Mills theory (see here). A color glass condensate gives an increasing lower bound on the viscosity to entropy ratio by a factor 2 with respect to $1/4\pi$, also computed from string theory, and appears less efficient with respect to observed elliptic flow at RHIC (see here). This kind of wars is often unproductive in physics and science at large as it slows down progress and good works could turn out unpublished. In situations like this, researchers should have eyes wide open and open minds granting all the contenders to be fairly listened waiting for experiments or careful lattice computations to say the last word. This should teach the history of Landau gauge propagators and also by looking back to history of physics. Otherwise we will stay on a silly forever war  where we are only able to prove to the rest of mankind that nothing has been learned from the past.

On Wednesday the session was dedicated to AdS/CFT, Holography, and Scattering. There was the talk of Carl Bender that is currently working on PT quantum mechanics. He is the pioneer of strong perturbation for quantum systems and quantum field theory. I often cited his work that has been a source of inspiration. David Dudal also spoke and discussed a holographic model for the analysis of strong ion collisions and the effect of the huge magnetic field generated. He gets results reminiscent of the Nambu-Jona-Lasinio model.  David is one of the proponents of the Refined Gribov-Zwanzinger model (see here). This is a real successful approach to the understanding of Landau gauge propagators and fits quite well with my results in the deep infrared behavior of a Yang-Mills theory as I also pointed out in my talk (see below).

It was a great workshop and I have been very happy to be there also this year. I hope people at Brown University will repeat this again. Thanks a lot!

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

Ollitrault, J. (1992). Anisotropy as a signature of transverse collective flow Physical Review D, 46 (1), 229-245 DOI: 10.1103/PhysRevD.46.229

M. Ruggieri, F. Scardina, S. Plumari, & V. Greco (2013). Elliptic Flow from Nonequilibrium Color Glass Condensate Initial
Conditions arXiv arXiv: 1303.3178v1

David Dudal, John Gracey, Silvio Paolo Sorella, Nele Vandersickel, & Henri Verschelde (2008). A refinement of the Gribov-Zwanziger approach in the Landau gauge:
infrared propagators in harmony with the lattice results Phys.Rev.D78:065047,2008 arXiv: 0806.4348v2

Lieb, E., & Simon, B. (1973). Thomas-Fermi Theory Revisited Physical Review Letters, 31 (11), 681-683 DOI: 10.1103/PhysRevLett.31.681

Lieb, E., & Simon, B. (1977). The Thomas-Fermi theory of atoms, molecules and solids Advances in Mathematics, 23 (1), 22-116 DOI: 10.1016/0001-8708(77)90108-6

Marco Frasca (2006). Thermodynamic Limit and Decoherence: Rigorous Results Journal of Physics: Conference Series 67 (2007) 012026 arXiv: quant-ph/0611024v1