# Structure of Matter in Quantum Field Theory

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This is a topic I've mentioned a few times before. Essentially the structure of matter in quantum gauge field theories is unclear to me. I have no clear question here, just some initial discussion points.

So at the first level, it seems a particle based view of quantum field theory is difficult to maintain. In brief on the Hilbert Spaces of interacting field theories there is no well-defined number operator ##N##. In some texts it's said that states in the interacting theory are a superposition of different particle numbers, but in reality the fact that an ##N## operator doesn't exist means there are no states with a well-defined particle number. You can see this explicitly in Glimm, Osterwalder and Schrader's work on ##\phi^{4}_{3}##, where the number operator for the interacting theory diverges as the cutoff is removed.
In addition to this we have Malament's theorem where he shows that relaitivistic quantum theories don't possess any states with the intuitive property of particles. In essence he shows that there are no local particle creation operators.

Then we have the usual picture of fields and particles being their excitations. Now if particle like states aren't elements of interacting field theory Hilbert spaces, I don't think one can say they are excitations of fields. What one can say is that most field theories can be shown to give rise to states that at late times can be proven to have a well defined number of "clicks" they cause in detectors, with the detectors being represented by specific local observables (details are to be found in the monographs of Araki and Haag, which we can go into, but I don't want to overload the initial post).

At this point we might turn to fields as the fundamental objects. However this remains tricky to me, let's look at gauge theories.

The problem with Gauge theories is that fields carrying the gauge charges cannot be local. If ##\Omega_{\Delta}## is a projector onto states confined to the region ##\Delta## of Minkowski spacetime and ##A## is an observable carrying the gauge charge, then it is always the case that ##\Omega_{\Delta}A \neq 0##. For this reason parameters like ##x## in the quark field ##\psi(x)## are formal, carried over from classical notation, but don't reflect actual localisation at a point.

If one wants ##\psi(x)## to be local, it can only be so as an operator on a Hilbert-Krein space, not on a Hilbert Space of physical states. If one does this there needs to be a condition selecting out the subset of the Hilbert-Krein space containing the physical states (BRST condition in non-rigorous language).

All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.

All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".

So in something like QED the electron and photon fields aren't well-defined local operators and thus actual physical states (which can be localised) can't be associated with them directly. In QCD it is even worse, where the quark and gluon fields, even ignoring this problem, would map one out of the physical Hilbert Space regardless as they carry color charge when all physical states are colorless.

Gauge symmetry itself is formulated in terms of these formal objects ##\psi(x), A_{\mu}(x)## carrying guage charges and most properties of such gauge symmetries (e.g. conserved charges) do not survive quantization. The only aspect of them that does are local Gauss laws (see Nakanishi "Covariant Operator Formalism of Gauge Theories of Quantum Gravity" and Strocchi's book).

So really QCD's physical content to me appears to be there are states which at early/late times make detectors click in a fashion that corresponds to the properties we give nucleons, i.e. late time states will act like particles with nucleon properties. There is a relation between the scattering angles and amplitudes of these nucleon like states that can be encoded in differential forms expressing local Gauss laws.

However the entire structure of quarks, gluons and to some degree even their local fields, seems like a crutch we use so that we have objects that give nice integrals and implement the Gauss law in a way that is easy to use (as Guage "charges") since it reduces it to group theoretic calculations. However the price is that these objects are a long way from the physical content of the theory.

This seems to render statements like "The proton is made of three quarks" into shorthand for "There is a state which at late times has localized properties x,y,z. The restriction the Gauss law imposes on its amplitudes can be modelled perturbatively by decomposing its creation operator into a product of three fields containing charges. Though the states corresponding to these fields are unphysical, lacking even positivity, we have conditions (BRST) to recover what we need for the late time particulate state"

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odietrich, Peter Morgan, protonsarecool and 4 others

All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.

All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".
Where does one find the first statement in this chapter? It seems to contradict the likely fact that in any gauge theory there should be local gauge invariant fields like the trace of the ''square'' of the field strength. This is the basis for the Clay Millennium problem on quantum gauge theory, which asks for a Wightman field theory for the vacuum sector, which should in particular give this field a rigorous definition. See, e.g., https://www.physicsoverflow.org/21846

In QED, the gauge group is abelian and more should hold: The field strength itself and the electron currents should be fields in a corresponding Wightman field theory for the vacuum sector.

bhobba, vanhees71 and dextercioby
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There are certainly local field operators, however looking at QED and QCD (not pure Yang-Mills) how are these connected to the particle dynamics? How do you introduce local field theoretic couplings between the physical matter fields and the physical gauge fields?

A direct coupling between the field strengths and the electron field will either give chargeless electron fields or be nonlocal.

Certainly it can be done, but it would be more in terms of a nucleon Lagrangian in QCD's case having some sort of Gauss's law encoded in a differential form which is some function of the Nucleon fields.

A far cry from the fields that appear in the usual Lagragian. I doubt the physicality of the fields we typically use, not fields in general.

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Just to say I only have my phone at the moment, I'll give better references when I am back at my library.

There are certainly local field operators, however looking at QED and QCD (not pure Yang-Mills) how are these connected to the particle dynamics? How do you introduce local field theoretic couplings between the physical matter fields and the physical gauge fields?
I believe that, just like particle numbers, whatever couplings are ''introduced'' by a Lagrangian ansatz, these are washed away by renormalization, hence are not rigorously meaningful notions. However the local fields that survive renormalization in a rigorous limit are of course correlated, unlike their free counterparts. This implies interaction.

In electrodynamics one does not need a notion of local charge but only those of a 4D charge current and an electromagnetic field, as these are the only fields that figure classically. Thus the vacuum sector of QED describes everything needed to recover macroscopic electrodynamics. Charged states cannot be found there, of course, but these can be taken to be an unphysical idealization. Rays of charged particles may be viewed as approximate notions that can probably be modelled by appropriate charge current distributions. I haven't seen this done but I don't see any obstacle in doing this.

I understand much less about QCD, but there the matter content should be describable too by local and gauge invariant quark currents. Again the vacuum sector should describe everything of true physical relevance. For example, current-current correlations are among the observables that reveal information testable by experiment. See, e.g., the book ''Quantum chromodynamics: an introduction to the theory of quarks and gluons'' by Yndurain.

DarMM, bhobba, vanhees71 and 1 other person
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I think possibly we are saying similar things, but I've rambled.

The currents exist (I'll have more to say on them later) and are local fields, but their decomposition or expression in terms of quark fields is not directly physically sensical, as the quark fields only exist on an enlarged Hilbert-Krein space.

This to me obscures the physical content of the standard description. Really we should say there are nucleon currents obeying Gauss laws. Quark fields are an unphysical expansion of the physical nuclei fields on a Hilbert-Krein space. So these aren't really quark currents and if we had full nonperturbative control of the theory quarks could be eliminated.

I'm saying something similar to what you say about virtual particles I guess. They're a concept useful for calculations, but not part of the theory's genuine physical content. Same with gluons.

Quark fields are an unphysical expansion of the physical nuclei fields on a Hilbert-Krein space. So these aren't really quark currents
I agree that quark fields themselves are unphysical. But what is your argument that quark currents (the renormalized version of the corr. quadratics in the quark fields) cannot be physical? They should exists in the 6 quark flavors (or, ignoring spin, as a 6x6 matrix valued field), and hence should deserve to be called quark currents, even though the quarks themselves (as fields or particles) are virtual only.

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I think they are physical, but more so that they are really nuclei currents and the decomposition in terms of quark fields is an unphysical convenience. In a proper nonperturbative formulation they would probably be expressed as functions of nuclei fields.

Would this seem reasonable to you?

I think they are physical, but more so that they are really nuclei currents and the decomposition in terms of quark fields is an unphysical convenience. In a proper nonperturbative formulation they would probably be expressed as functions of nuclei fields.

Would this seem reasonable to you?
I guess you mean baryon currents, since we are discussing QCD and not the standard model - nuclei are held together by the weak force.

But there are far too many baryons, and the flavor information is lost.

Thus I think more than just the asymptotic bound state structure, namely that of quark currents, is preserved in the local fields. Other uncharged fields would be expressed in terms of these and glueball fields, and charged fields would emerge as soliton-like fields in other sectors of the theory, not described by Wightman axioms.

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Other uncharged fields would be expressed in terms of these and glueball fields
in an OPE-fashion

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Okay a nice reference is:
Taichiro Kugo, Izumi Ojima; Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem, Progress of Theoretical Physics Supplement, Volume 66, 1 February 1979, Pages 1–130

In particular in section 5.3 they discuss the physical content of QCD. Now their discussion has certain propositions they can't prove (if they could it would constitute a rigorous construction of Yang-Mills), but under these assumptions the Wightman axioms imply:
$$\mathcal{H}_{phys} = \overline{\mathcal{A}\left(\mathcal{O}\right)\Omega}$$
That is the physical Hilbert space can be constructed from the closure of local hadron fields operating on the vacuum. To be clear ##\mathcal{A}\left(\mathcal{O}\right)## is the local algebra of color singlet local fields, hence this is essentially a Reeh-Schlieder type theorem.

Could you clarify:
But there are far too many baryons, and the flavor information is lost.
Do you mean the flavor information becomes obscured or do you mean it is literally not present?

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I guess you mean baryon currents, since we are discussing QCD and not the standard model - nuclei are held together by the weak force.

But there are far too many baryons, and the flavor information is lost.

Thus I think more than just the asymptotic bound state structure, namely that of quark currents, is preserved in the local fields. Other uncharged fields would be expressed in terms of these and glueball fields, and charged fields would emerge as soliton-like fields in other sectors of the theory, not described by Wightman axioms.
I'm a bit puzzled by these statements... Of course the main force to keep "quarks and gluons" bound (even confined!) in hadrons, which are so far the only asymptotic states of QCD that can be measured, is the strong force. The only "ab-initio way" to understand hadrons from QCD are lattice-QCD calculations. These are Monte-Carlo evaluations of appropriate gauge-invariant correlation functions. One of the key achievements of this approach is a pretty accurate calculation of the hadronic mass spectrum of the empirically known as well as not-yet seen hadrons.

I've no clue what you mean by "the flavor information is lost". Of course to get the hadron spectrum the lattice-QCD calculations involve correlation functions with the right quantum numbers, including flavor, i.e., the corresponding valence-quark flavor.

Another approach to understand hadron phenomenology are of course effective low-energy models. The most important approach in the light-quark sector is chiral perturbation theory (and unitarized versions thereof). This uses the approximate chiral symmetry of QCD in the light-quark sector (flavor SU(2) for up and down or flavor SU(3) for up, down, and strange quarks), which is spontaneously broken in the vacuum and at low temperatures and or (net-baryon) densities. These models are governed by the symmetry properties of corresponding composite fields. Also these involve of course well-defined flavor degrees of freedom. For a nice introduction to this approach, see

https://arxiv.org/abs/nucl-th/9706075

bhobba
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nuclei are held together by the weak force.

I don't think so.

Auto-Didact and bhobba
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Where does one find the first statement in this chapter? It seems to contradict the likely fact that in any gauge theory there should be local gauge invariant fields like the trace of the ''square'' of the field strength. This is the basis for the Clay Millennium problem on quantum gauge theory, which asks for a Wightman field theory for the vacuum sector, which should in particular give this field a rigorous definition. See, e.g., https://www.physicsoverflow.org/21846

In QED, the gauge group is abelian and more should hold: The field strength itself and the electron currents should be fields in a corresponding Wightman field theory for the vacuum sector.
I just read this again and my phrasing was very bad. I meant to say the concept of local field is dubious for the quark and gluon fields and the only versions of them that are well-defined on the physical Hilbert space are loop-like, see Bert Schroer's paper here:
https://arxiv.org/abs/1601.04528

Hadrons however do have a local field theoretic description.

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To summarise, in QCD the real physical description involves a complicated non-Fock space on which we have only hadron operators.

However this leads into my next point. If quarks fields had been defined on the physical Hilbert Space, they would have formed a basis (in the operator theoretic sense) for all other local fields and thus some sense could be given to the notion that "protons are made of quarks, pions are made of quarks".

However since the only physical fields are hadrons what now is the picture of their composition? They seem fundamental. Perhaps there is a basic set of hadron fields that others can be expressed as functions of, but I don't know if there is a unique choice of such.

since the only physical fields are hadrons
You didn't demonstrate this. To say that charged fields are not in the physical Hilbert space does not say anything about which local fields exist in the positive part of the Krein space. For example, glueballs should have local gauge invariant fields associated with them, namely the renormalized trace of the ''square'' of the field strength.

I don't think so.
Oh, sorry. Of course, nuclei are held together by meson exchange, which is the strong force.

bhobba
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You didn't demonstrate this. To say that charged fields are not in the physical Hilbert space does not say anything about which local fields exist in the positive part of the Krein space. For example, glueballs should have local gauge invariant fields associated with them, namely the renormalized trace of the ''square'' of the field strength.
Sorry, hadron and glueballs then. More so the point is that what in the conventional way of writing the theory are composite fields are in fact fundamental and that this seems to me to cause a funny detail with the matter, i.e. hadron, fields in that without quarks they aren't all reducible to being "composed" of a unique common set of fields.

Peter Morgan
I'm a bit puzzled by these statements. [...]
I've no clue what you mean by "the flavor information is lost".
We are discussing here what can (probably) be said rigorously about QED and QCD in Minkowski space. Thus Euclidean lattice calculations do not help.
Do you mean the flavor information becomes obscured or do you mean it is literally not present?
What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.

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We are discussing here what can (probably) be said rigorously about QED and QCD in Minkowski space. Thus Euclidean lattice calculations do not help.

What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.
I agree, but is this not correct, i.e. since quark and gluon fields are an unphysical expansion of the physical (hadron and glueball) fields on a Hilbert-Krein space, then in a formulation of QCD that used no unphysical concepts we would simply have a hadron "zoo" with all hadrons on equal footing and thus is this not the actual picture QCD presents of the world?

I agree, but is this not correct, i.e. since quark and gluon fields are an unphysical expansion of the physical (hadron and glueball) fields on a Hilbert-Krein space, then in a formulation of QCD that used no unphysical concepts we would simply have a hadron "zoo" with all hadrons on equal footing and thus is this not the actual picture QCD presents of the world?
That depends on which fields are local and gauge invariant. I believe that there may be local, gauge invariant current fields for all quark flavors, though I haven't checked yet whether this is likely to hold - too many other things to do now, at the beginning of our winter term. It is only at the level of asymptotic fields that we necessarily have the hadron zoo, with nuclear democracy between all bound states.

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We are discussing here what can (probably) be said rigorously about QED and QCD in Minkowski space. Thus Euclidean lattice calculations do not help.

What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.
Well, this is the fundamental difference between physics and math. Physics works through observations first, and of course in the early 60ies one had no clue about quarks. This came only later when Gell-Mann discovered the possibility to order hadrons in terms of the approximate SU(3) flavor symmetry. Only somewhat later with the discovery of Bjorken Scaling at SLAC and Feynman's parton model came the notion that quarks are real constituents. The so far final description is QCD with the discovery of asymptotic freedom in the early 70ies and the "confinement conjecture". What's observable in QFTs are the asymptotic free states, and no colored "objects" are asymptotic free according to confinement. As I said, the only way to investigate this (not yet fully solved problem) of confinement is through lattice-gauge theory, which allows to study well certain aspects of it only, among them the hadron spectrum and in its "thermal" version the equation of state of strongly interacting matter. Of course, this is not mathematically rigorous, but physics is not about a mathematically rigorous formulation of QFT but of the application of unfortunately in some aspects not well-defined models based on QFT ideas. Euclidean lattice QCD may not help in finding a mathematically rigorous description of QFT (although, who knows in which way such a mathematically satisfiable version of non-Abelian gauge theories may be formulated in the future), but allows to answer many physically relevant questions, and it's very confirming for the phenomenologist that QCD doesn't only work in the perturbative regime of deep inelastic scattering but also in the low-energy non-perturbative regime of hadron physics, at least as far as it is accessible with lattice-QCD methods.

I still do not get what you are after concerning flavor. That's a well-defined quantum number, according to which all observable particles are described, including flavor in the sense of the electroweak sector, which is closely related with flavor in strong-interaction physics. Admittedly it's a bit hidden, and that's why it took a while to get the idea of quark (and today also neutrino) mixing right.

king vitamin
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That depends on which fields are local and gauge invariant. I believe that there may be local, gauge invariant current fields for all quark flavors, though I haven't checked yet whether this is likely to hold - too many other things to do now, at the beginning of our winter term. It is only at the level of asymptotic fields that we necessarily have the hadron zoo, with nuclear democracy between all bound states.
No worries, I'll try to get current thinking on that. Assuming it were true let's say, would your view then be that QCD's fundamental objects are flavour current fields and glueball fields? Hadrons then being reducible (in some operator theoretic sense) to the flavour current fields.

I still do not get what you are after concerning flavor. That's a well-defined quantum number, according to which all observable particles are described, including flavor in the sense of the electroweak sector, which is closely related with flavor in strong-interaction physics. Admittedly it's a bit hidden, and that's why it took a while to get the idea of quark (and today also neutrino) mixing right.
As I understand it, QCD has quarks of 6 flavors. What is the flavor of a proton or a neutron, or a kaon? You can describe it only as a tensor product of flavors.
But in a description of local hadron fields that DarMM is after, one would have an elementary field for each hadron, and hence for proton, neutron, kaon,.... How would you recover mathematically the flavor description from a description where these are elementary fields (without assuming anything about quarks)? I think it cannot be done.

Assuming it were true let's say, would your view then be that QCD's fundamental objects are flavour current fields and glueball fields? Hadrons then being reducible (in some operator theoretic sense) to the flavour current fields.
Yes. Each meson and baryon current would be built from tensor products of two and three flavor currents, respectively, by some kind of OPE construction.

Note that currents are more fundamental than asymptotic fields, in the sense that they should have a meaning also at finite times.

Peter Morgan and DarMM
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As I understand it, QCD has quarks of 6 flavors. What is the flavor of a proton or a neutron, or a kaon? You can describe it only as a tensor product of flavors.
But in a description of local hadron fields that DarMM is after, one would have an elementary field for each hadron, and hence for proton, neutron, kaon,.... How would you recover mathematically the flavor description from a description where these are elementary fields (without assuming anything about quarks)? I think it cannot be done.
To be clear I'm not pursuing a hadron description, just genuinely confused. Also regarding the flavour description, does the fundamental rep of the flavour group have to physically exist (i.e. be embodied in quarks) for it to be recoverable? I might be wrong, but the hadron fields would still embody other reps.

To be clear I'm not pursuing a hadron description, just genuinely confused. Also regarding the flavour description, does the fundamental rep of the flavour group have to physically exist (i.e. be embodied in quarks) for it to be recoverable? I might be wrong, but the hadron fields would still embody other reps.
On currents, the flavor group acts in the adjoint representation, and this should be a (heavily broken) dynamical symmetry of the vacuum sector of QCD. The fundamental representation would belong to the (in a Wightman QFT) nonexisting quarks. Thus representation theory allows flavor currents without quarks.

Most of the presumably infinitely many meson and baryon currents probably cannot be assigned to representations of the flavor U(6), since the problem is the field analogue of classifying the spectral lines of molecules in terms of their dynamical symmetry groups. This can be done approximately for the lowest multiplets of a weakly broken symmetry, but not for sufficiently excited states - which are in the chaotic regime.

DarMM
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I see what you mean the underlying U(6) symmetry would be almost irrecoverably hidden inside relations between the hadron field amplitudes.

I'll try to find stuff on the flavour currents then.

Thus representation theory allows flavor currents without quarks.
One can read Feynman diagrams for processes involving quarks as intuitive pictures of (in pure QCD) conserved flow lines of the corresponding currents. This requires no quarks to exist.

Maybe it even has a proper hydrodynamical interpretation. Indeed, there is a well-studied hydrodynamic approximation of QFT based on the CTP (closed time path) formalism and the 1PI effective action. See, e.g., the QFT book by Calzetta and Hu. @vanhees71, do you have references where this is used to study QCD?

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As I understand it, QCD has quarks of 6 flavors. What is the flavor of a proton or a neutron, or a kaon? You can describe it only as a tensor product of flavors.
But in a description of local hadron fields that DarMM is after, one would have an elementary field for each hadron, and hence for proton, neutron, kaon,.... How would you recover mathematically the flavor description from a description where these are elementary fields (without assuming anything about quarks)? I think it cannot be done.
It depends a bit on which kind of flavor you mean. If you have pure QCD there are only the mass eigenstates of the quarks involved, and these are called flavors (in the sense of the strong interaction, which is on the other hand flavor blind, i.e., all the quarks have precisely the same strong interaction governed by the fact that they are color-SU(3) triplets, and the anti-quarks anti-triplets; since the color symmetry is gauged, this uniquely defines the strong interaction via "gluon exchange", where the gluons are the gauge bosons of the strong interaction, necessarily transforming according to the adjoint representation). A proton thus has flavor "uud", a neutron "ddu", and a kaon ##\bar{u} s##, ##\bar{d} s##, ##d \bar{s}##, or ##u \bar{s}## depending on whether you have a negatively charged anti-kaon, a neutral anti-kaon, a neutral kaon, or a positive kaon. Usually the quantum numbers stated are for the light quarks isospin (in the SU(2) model) or isospin and hypercharge (in the SU(3) model and also generally for all 6 flavors although there it doesn't make too much sense anymore to group them in SU(N) models). With the electric charge related to the corresponding quantum numbers by ##Q=T_3+Y/2##. The hypercharge is given by the sum over the flavor-quantum numbers "strangeness, charmness, bottomess, topness" and baryon number.

This defines uniquely "flavor quantum numbers" to all hadrons. E.g., the nucleons (proton and neutron) have isospin ##T_3=\pm 1/2## and hypercharge ##Y=1/2##.

In the context of weak interactions within the Standard Model one has to see that there's a somewhat different notion of flavor. The flavor states are then those with definite weak isospin and hypercharge. By convention the lefthanded part of the uplike-quarks (up, charm, top) mass eigenstates also have good wiso-spin quantum numbers ##T_{W3}=1/2## (the right-handed parts have ##T_{W3}=0##), while the lefthanded (righthanded) part of the downlike-quarks (down, strange, bottom) are different from their mass eigenstates and are connected to the mass eigenstates by the unitary CKM mixing matrix. For details see my "Lecture Week Transpararencies" (section on QFD):

https://th.physik.uni-frankfurt.de/~hees/hqm-lectweek14/ebernburg14-1.pdf

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One can read Feynman diagrams for processes involving quarks as intuitive pictures of (in pure QCD) conserved flow lines of the corresponding currents. This requires no quarks to exist.

Maybe it even has a proper hydrodynamical interpretation. Indeed, there is a well-studied hydrodynamic approximation of QFT based on the CTP (closed time path) formalism and the 1PI effective action. See, e.g., the QFT book by Calzetta and Hu. @vanhees71, do you have references where this is used to study QCD?
Hydro is the "bread-and-butter model" to describe the hot and dense fireball created in ultrarelativistic heavy-ion collisions. The picture is that the "heavily Lorentz contracted" nuclei run through each other and leave behind a hot and dense fireball consisting of "quasiparticles" that are (at high enough collision energies) parton like (i.e., quark and gluon like) in the early phase (starting after a astonishingly short "formation time" of less than 1 fm/c after the collision), undergoing a cross over to a hot and dense hadron resonance gas (which coincides closely with the socalled "chemical freezeout" at a temperature of around 150-160 MeV). Then also the elastic interaction rates cease after the fireball has become dilute and cold enough and also thermal freezeout occurs (at around 90-100 MeV for the most central collisions) after which the hadrons stream freely to the detectors. From the formation time on until the thermal freezeout most observations in the low-transverse-momentum range are very well described by relativistic (viscous) hydrodynamics, i.e., local thermal equilibrium. For a recent review, see, e.g.,

https://arxiv.org/abs/0708.2433

For a systematic derivation from transport theory, see e.g.,

https://arxiv.org/abs/1202.4551

To finally make the link with QFT, see e.g.,

https://arxiv.org/abs/0808.0715

dextercioby and A. Neumaier
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I believe that, just like particle numbers, whatever couplings are ''introduced'' by a Lagrangian ansatz, these are washed away by renormalization, hence are not rigorously meaningful notions. However the local fields that survive renormalization in a rigorous limit are of course correlated, unlike their free counterparts. This implies interaction.
I was thinking over this and I don't fully understand this part.

In constructive models of the Yukawa interaction for example the couplings explicitly exist in the continuum limit. Similarly for all constructive models. Why exactly are they not rigorously meaningful notions?

I was thinking over this and I don't fully understand this part.

In constructive models of the Yukawa interaction for example the couplings explicitly exist in the continuum limit. Similarly for all constructive models. Why exactly are they not rigorously meaningful notions?
Of course in any rigorous construction, there is for every coupling constant some renormalized equivalent. But I don't know of any rigorous definition. How would you define a coupling rigorously, just given some Wightman quantum field theory - i.e., without reference to a particular method of constructing it?