A How can quarks exist if they are confined?

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vanhees71

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Not exactly, since there are contributions from soft photons and electron-positron pairs, and to a smaller extent also of proton-antiproton pairs (if the proton is taken as elementary)
That's an interesting point and almost philosophical again. It boils down to: "What is a proton or electron?" Of course both are first of all concepts to order observations in Nature, and there are different levels of descriptions, all in some way valid at a certain level of accuracy and in some way again invalid if one has a closer look. It's also a question of context, which description is necessary and adequate for describing a certain aspect of natural phenomena.

E.g., a proton (or atomic nucleus) can on the one hand be described as a classical particle if it comes to a useful, however approximate, description of molecules (Born-Oppenheimer approximation) with the electrons binding them together as classical (even static!) electromagnetic fields. It's quite well understood why this approximation works, considering the next precise level of description, namely the molecule with all its "constituents" as non-relativistic quantum particles.

If you look closer at an atomic nucleus you realize it "consists of" protons and neutrons, and you can again ask at different levels of descriptions, how to understand their binding in the nucleus. This reaches (particularly for large nuclei) from a classical fluid-description (Bethe, Weizsächer, Bohr, Wheeler et al) to the nuclear shell model using sophisticated realistic nucleon-nucleon potentials and their derivation from chiral perturbation theory employing renormalization-group methods.

Then you switch again the perspective by going to even higher energies up to deel-inelastic scattering of electrons at a nucleon, and some substructure emerges, described by the "parton model". At even higher energies you resolve also sea quarks and gluons etc. etc.

There's not one answer to "what is a proton" from a theoretical-physics perspective but there's an entire hierarchy of models, each valid within its domain of applicability. What's pretty "stable" across all these levels of description are only a few very fundamental properties like the mass, spin, and various charges.

Even an electron, which is at the level of our knowledge today is still considered an "elementary particle", is not so uniquely described. E.g., an accelerator physicist can treat it usually as a classical point particle or describe it in a continuum-mechanical way (particularly at higher space charges), including some effective way to take into account radiation reaction, which is in principle an unsolved problem in classical electrodynamics. Then the atomic theorist comes far with the idea to use non-relativistic QM or with "relativistic QM" and describe it as a particle with spin 1/2. Then there's of course QFT which is used at higher energies, and particularly if you restrict yourself to situations where QED is sufficient, you can get some way with the perturbative concept of an electron, i.e., a particle with a given mass, spin, and charge which is non-interacting to begin with and then you take into account interactions perturbatively, get into the well-known trouble with divergences and cure them with renormalization (UV) and resummation (IR).

Particularly considering the notorious IR problems, you come to the conclusion that for QED, where you have unscreend and unconfined long-ranged interactions, that the naive perturbative picture and the notion of the corresponding asymptotic free states, is inaccurate, and that one has to use some kinds of other concept. In the picture of the naive perturbation theory you have to "dress the bare electron" with a cloud of soft photons (which we have discussed before in this forum, maybe even in this thread!). This holds even true for the non-realtivistic treatment of "Coulomb scattering" since it's a IR phenomenon.

Concerning the UV problems the answer seems to be Wilson's physical interpretation of the renormalization procedure, i.e., the realization that relativistic QFTs are all effective descriptions at some resolution (or equivalent energy ranges). Whether or not there's a more comprehensive theory from which these effective QFT descriptions can be derived is, as far as I can see, still an open question. At least I don't see a really convincing candidate (given that string theory, as interesting it might be from a mathematical point of view, seems to be completely oversold, because there's no satisfactorial phenomenology derived from it, in contradistinction to the still incomplete concept of QFTs, which are phenomenologically utmost successful).

vanhees71

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Yes, but clearly naive quantization of gauge theories produces states that can't conceivably describe reality (negative-norm or zero-norm) from which one must select out the states that can via some BRST like condition. That's all that is meant here by "physical Hilbert" space, the subset of the states from naive quantization that actually do have sensible properties and are selected out by BRST conditions.

It's not "physical" in some grander or philosophical sense.
Yes sure, there are mathematical problems with (perturbative) gauge models which are solved with some mathematical tools (Faddeev-Popov quantization is more pragmatic, while the operator approach based on BRST is also very illuminating to understand some finer aspects). No matter which mathematical sophistication is necessary, one must not forget that these are all descriptions of nature, not nature itself!

A. Neumaier

Science Advisor
I'm genuinely uncertain as to how to interpret them in light of the fact that no quark states exist in the QCD Hilbert space. Without quark states existing in what sense can anything be made of them.
This is why I place currents before particles. A hadron current can be composed of quark (or as you prefer to call it, flavor) currents.

Currents are gauge invariant operator-valued distributions acting on both vacuum representations and representations with finite temperature and density. I believe that quarks as localized particles don't exist (in the sense of cannot be defined in terms of the physical Hilbert space), but that quark currents exist.

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A. Neumaier

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Or for a General Relativistic anology they're as real as Christoffel fields.

I won't get hung up on "real" if people think of it differently. However there is some difference to me between things like the metric and things like Christoffel fields and it's a distinction in the theory not just philosophical.
Could you please explain in more detail which of these GR fields you would say exist in which sense? Measurable is only the gravitational field strength (the Riemann tensor), but neither the metric tensor nor the Christoffel connection.

See also the discussion in the early part of the thread https://www.physicsforums.com/threads/can-the-christoffel-connection-be-observed.948264/

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A. Neumaier

Science Advisor
Quarks have color, but no physical state has color,
QCD quarks have color, but constituent quarks don't. It is important to keep in mind that the meaning of the term quark depends a bit on the context, i.e., on the model within which it is used. The relationship between the different models is not so clear, but the clue to the answer must be found by investigating the relations between these models.

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A. Neumaier

Science Advisor
E.g., a proton (or atomic nucleus) can on the one hand be described as a classical particle if it comes to a useful, however approximate, description of molecules (Born-Oppenheimer approximation) with the electrons binding them together as classical (even static!) electromagnetic fields.
Hmmm, in the Born-Oppenheimer approximation, the proton remains a quantum particle and has its own wave function. Only the interaction between the slow protons and the fast electrons is truncated.
Even an electron, which is at the level of our knowledge today is still considered an "elementary particle", is not so uniquely described.
The moving electrons in a metal are also different objects (quasiparticles) than the (asymptotic) electrons in S-matrix QED. The precise meaning of all subatomic concepts depends on the model within they are used.

A. Neumaier

Science Advisor
these are all descriptions of nature, not nature itself!
Well, everything we say about nature (whether within or without physics, whether classical or quantum) is only a description. Nature just is, and we describe little idealized pieces of it.
In our perception of physical reality there are neither Hilbert and Fock spaces, Lie and other groups in QT, nor configuration and phase spaces, no fiberbundles, Minkowksi and pseudo-Riemannian manifolds in classical physics. These are all description of our perceptions of Nature.
Neither are there coordinates or position vectors, particles, electromagnetic or gravitational fields, etc..
These are all description of our perceptions of Nature.

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vanhees71

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Precisely, and what I wanted to say with this self-evident ideas is that there is no principle difference between the notions of classical and quantum abstract descriptions of nature. Often, and if I remember right also in this thread, people tend to think that the classical description of, say point particles, the "coordinates or position vectors" are "more real" or "more direct" of the entities described by them than in QT, where one has "more abstract", i.e., less familiar mathematical concepts (which we learn about not already early in our school education but later in life). There is, however no such difference in the description. It's only a different level of descriptions valid in different levels of observational accuracy or resolution.

DarMM

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What I'm trying to understand in this thread is how quarks can be absent from the physical Hilbert space and their associated fields ill defined as operators on the physical Hilbert space, with the obvious evidence of the multilocal nature of the proton as mentioned by @A. Neumaier and @bhobba for Deep Inelastic scattering.

DarMM

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Often, and if I remember right also in this thread, people tend to think that the classical description of, say point particles, the "coordinates or position vectors" are "more real" or "more direct" of the entities described by them than in QT
I don't recall that in this thread. The question has more been what way are we to view QCD quarks within the theory.

A. Neumaier

Science Advisor
how quarks can be absent from the physical Hilbert space and their associated fields ill defined as operators on the physical Hilbert space, with the obvious evidence of the multilocal nature of the proton
Nothing forbids the associated currents to be physical (gauge invariant). Only the particle interpretation must be given up.

Multilocality in the plane orthogonal to the flow direction is all that needs to be explained, I think, and currents do that.

vanhees71

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What I'm trying to understand in this thread is how quarks can be absent from the physical Hilbert space and their associated fields ill defined as operators on the physical Hilbert space, with the obvious evidence of the multilocal nature of the proton as mentioned by @A. Neumaier and @bhobba for Deep Inelastic scattering.
Quarks carry a color charge (of the fundamental representation) and thus are gauge dependent and thus can't be observables. The same holds for gluons which carry a color charge (of the adjoint representation) and thus also can't be observables. Only gauge-independent expectation values and gauge-invariant S-matrix element lead to a sensible definition of observables within the theory. Physically that's named "confinement", i.e., all color charges are confined in color-neutral objects which are (putatively) observable. Today these we know this are the baryons and the mesons (or both together the hadrons) and maybe some more "exotic" states like tetraquarks (maybe the XYZ states in the charm sector are such guys, but maybe they are rather "meson molecules", which is still under both theoretical and experimental debate). Then there should also be glue balls, i.e., color-neutral bound states of gluons.

Formally quarks and gluons are the quantum fields of the theory, and they can be used to build observables, e.g., $\bar{u} \gamma_5 d$. This carries the quantum numbers of the negative pion.

DarMM

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Quarks carry a color charge (of the fundamental representation) and thus are gauge dependent and thus can't be observables. The same holds for gluons which carry a color charge (of the adjoint representation) and thus also can't be observables. Only gauge-independent expectation values and gauge-invariant S-matrix element lead to a sensible definition of observables within the theory. Physically that's named "confinement", i.e., all color charges are confined in color-neutral objects which are (putatively) observable. Today these we know this are the baryons and the mesons (or both together the hadrons) and maybe some more "exotic" states like tetraquarks (maybe the XYZ states in the charm sector are such guys, but maybe they are rather "meson molecules", which is still under both theoretical and experimental debate). Then there should also be glue balls, i.e., color-neutral bound states of gluons.

Formally quarks and gluons are the quantum fields of the theory, and they can be used to build observables, e.g., $\bar{u} \gamma_5 d$. This carries the quantum numbers of the negative pion.
That's a good summary and don't take this as rude (I don't mean it that way), but I know all this. What I'm getting at is that it is odd that you can build operators for physical states out of seemingly unphysical fields acting on a state space that isn't a Hilbert Space and yet those unphysical fields seem to be echoed in certain scattering experiments. I think something like @A. Neumaier's comment is what is needed.

Basically quarks seem to unnecessary in a nonperturbative setting on one level (all states are colorless, quark states don't exist in a Hilbert space and hence one might be tempted to construct a purely hadronic Lagrangian directly on a hardonic-glueball Hilbert space), but seemingly a useful concept in other cases, e.g. Deep Inelastic Scattering.

A. Neumaier

Science Advisor
and yet those unphysical fields seem to be echoed in certain scattering experiments.
Formally quarks and gluons are the quantum fields of the theory, and they can be used to build observables, e.g., $\bar{u} \gamma_5 d$. This carries the quantum numbers of the negative pion.
In particular, they can be used to build gauge invariant and hence in principle observable currents $j_q=\bar{q} \gamma q$ (trace over colors) for each of the six quarks $q$. Thus while individual (colored) quarks are unphysical, the six (colorless) quark flows are physical.

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A. Neumaier

Science Advisor
In particular, they can be used to build gauge invariant and hence observable currents $j_q=\bar{q} \gamma q$ (trace over colors) for each of the six quarks $q$. Thus while individual quarks are unphysical, the six quark flows are physical.
In addition, there are physical flavor-changing currents $j_q=\bar{q} \gamma q'$ with two different quarks $q$ and $q'$.

andrew s 1905

The details of this discission are well beyond me, nevertheless I am trying to extract what you mean by "real" and "physical".

Would I be right to conclude that you mean that the entity under discussion is real and or physical if it is a coherent element of the model (rather than an artefact of the mathematical procedure used to solve it) and or that it needs to be an observable (at least in principle)?
Thanks Andrew

A. Neumaier

Science Advisor
the entity under discussion is real and or physical if it is a coherent element of the model
The usage in the present thread is: A quantum field is called physical, or real (i.e., observable in principle) if it is realized as a (densely defined, distribution valued) operator on a Hilbert space with positive definite inner product.

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vanhees71

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I've never seen a "distribution valued operator" in my everyday life nor by any experimentalists with their refined equipment to extend our senses to the "quantum realm". I've once visited CERN, looking at the accelerator and at 2 of the main experiments (CMS and ALICE). This is what's real, but the outcome of this setup is described very well by the Standard Model (among other things indeed using distribution valued operators on a Hilbert (?) space, or however you defined the Hilbert-like space used to formulate the theory in terms of more rigorous mathematical formulations), and in my opinion that and only that is what makes the SM "realistic". It's the only sense the word "realistic" has for me as a physicist: A theory/model is realistic if and only if it describes successfully the outcome of observations in the real world. E.g., string theory is "not realistic", because it doesn't describe anything observable (yet). This may change in the future when string theorists come up with some real-world testable result, which then is checked by experimentalists with real-world equipment in the lab (or in astronomical observatories on Earth or in space).

A. Neumaier

Science Advisor
It's the only sense the word "realistic" has for me as a physicist
But I described the sense how ''realistic'' is used in mathematical physics and in this thread. For example, $\Phi^4$ theory in 2 space-time dimensions is realistic in the above sense, while it is clearly not realistic in your sense.
the Standard Model (among other things indeed using distribution valued operators on a Hilbert (?) space, or however you defined the Hilbert-like space used to formulate the theory in terms of more rigorous mathematical formulations)
The operator formulation of the standard model is in terms of a Krein space (topological vector space with an indefinite inner product), hence is not immediately physical in the sense of mathematical physics.

vanhees71

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I see. Well, at the end mathematical physics becomes physics when it makes predictions testable by observation. Of course, as $\phi^4$ theory in 1+1 dimensions, it can be of high value to understand the mathematics of the theory, and as such it's of course well worth studying, but mathematics has no aim at all to be "realistic" in any sense, and that makes it so useful in applications too, because it provides a concise language for all kinds of theories and models aiming at the description of "reality".

A. Neumaier

Science Advisor
mathematics has no aim at all to be "realistic" in any sense,
But mathematics uses all sorts of everyday expressions (such as groups, fields, rings, equivalence, truth) to denote specific formal objects or relations, and pins them down by giving precise definitions. Thus these terms can be used in an unambiguous way and convey definite information. This is what makes it the right tool for precision in all sciences. This is also the way the term ''physical field'' is used in the present discussion.

However, the relation of the precisely defined terms to the loose everyday notion described by the same word may be very weak.

vanhees71

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Of course, I agree with that notion of math completely.

DarMM

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@A. Neumaier has already covered this, but just to respond.
I've never seen a "distribution valued operator" in my everyday life nor by any experimentalists with their refined equipment to extend our senses to the "quantum realm".
"Physical field" means an operator valued distribution which (after smearing) acts on the Hilbert Space. This is sort of the bare minimum needed for a field to have correlation functions and thus to be observable in some form (i.e. correspond to detector clicks). It's just a term you tend to see in non-perturbative studies of field theories.

Some of the fields one can have in QFT don't produce physical fields in this sense. An example would be the ghost fields, which don't act on the Hilbert Space (or more strictly aren't endomorphisms on it) and so aren't really connected to observables, i.e. it would be possible to completely eliminate them in an alternate way of formulating the theory.

Hence the distinction, one kind are associated with observable quantities, the other are not and can be written out of the theory.

The surprising thing (and the focus of this thread) is that nonperturbatively quark fields are the same as ghost fields in this sense. However it is confusing, because many observations (e.g. Deep Inelastic Scattering) look most natural in terms of quarks.

vanhees71

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In the pertubative theory the ghost fields cancel contributions of other unphysical field-degees of freedom. They are necessary to organize the perturbative calculation of observable predictions of the theory.

DarMM

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In the pertubative theory the ghost fields cancel contributions of other unphysical field-degees of freedom. They are necessary to organize the perturbative calculation of observable predictions of the theory.
Yes, perturbatively.

Nonperturbatively in some formulations, e.g. lattice gauge theory, they don't even show up.

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"How can quarks exist if they are confined?"

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