# Asymptotic states in gauge theories

## Main Question or Discussion Point

Moderator's note: This is a sub-thread spun off from https://www.physicsforums.com/threads/is-the-ground-state-energy-of-a-quantum-field-actually-zero.953766/.

... this statement is indeed "modulo surface terms". Particularly I don't see any necessity for the energy-momentum tensor to be symmetric in the realm of special relativity.
I should have said that in certain cases in QFT, we can neglect “surface terms”. For example, the (on-shell) difference between the Belinfante and the canonical expressions for the angular momentum can be written as $$J^{ij}_{Bel} - J^{ij}_{C} = \int d^{3}x \ \partial_{\rho}F^{\rho 0 i j} (x) .$$ Now, by careful analysis using wave packets, one can show that the forward matrix elements of the RHS do vanish. Also, the forward momentum space matrix elements (i.e., the physical quantities in collision processes) of the “surface terms” do vanish if these terms are divergences of local operators.
But, you are absolutely right. As far as I know, the Belinfante expression $$J^{\mu\nu}_{Bel} = \int d^{3}x \left( x^{\mu} \theta^{0 \nu}(x) - x^{\nu} \theta^{0 \mu}(x) \right) , \ \ \ \ \theta^{\mu \nu} = \theta^{\nu \mu} \ ,$$ seems to fail to satisfy all the commutation relations of the Poincare’ algebra (to be more accurate, I should say that I have never been able to establish the correct bracket $\big[i J^{0 j}_{Bel} , J^{0 k}_{Bel} \big]$ using the usual methods, and I don't know if somebody else had).
For gauge fields, of course you can argue with gauge invariance,
Again, one can show that the physical matrix elements of the generators are gauge invariant.
In my opinion, there's neither uniqueness in this split nor is there a really well defined treatment of spin in relativistic hydrodynamics,....
You are right on the "uniqueness" part. I had the pleasure of knowing and working with Elliot Leader on the very same problem (in QCD) for many years. So, I suggest you have a look at his book:
E. Leader, “Spin in Particle Physics”, Cambridge University Press (2001)
You may also find the attached PDF’s useful

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## Answers and Replies

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vanhees71
Gold Member
Thanks a lot! That's a lot food for thought :-)).

I once started to make the effort to show that the canonical-quantization approach without using the abstract construction of the representations of the proper orthochronous Lorentz group a la Wigner indeed leads to the correct local representations on the field operators when the Noether-charge definition of the conserved quantities are used. For free fields this works quite straight-forwardly. I'm not so sure about interacting fields though. I've written this up only in German:

https://th.physik.uni-frankfurt.de/~hees/faq-pdf/qft.pdf

It is not possible to find the exact form of interacting fields, simply because there is no exact solution to the field equations. However, the representation space of (interacting) field operators can be determined uniquely by the general principle of asymptotic completeness: $\mathcal{V}_{in} = \mathcal{H} = \mathcal{V}_{out}$. So, you can apply the method, you described, to asymptotic fields, since any operator can be expanded into series of normal products of (free) asymptotic fields.

A. Neumaier
any operator can be expanded into series of normal products of (free) asymptotic fields.
But this requires a free asymptotic field for each bound state, hence for QCD an inifinite number of fields.

But this requires a free asymptotic field for each bound state, hence for QCD an inifinite number of fields.
See E. Leader paper in PDF #2 above, section V and VI and in particular the subsections VI.D & VI.E.

A. Neumaier
See E. Leader paper in PDF #2 above, section V and VI and in particular the subsections VI.D & VI.E.
I don't think his exposition can be correct. Could you please explain how the claim in eq. (109) in Section VI.D that states in which quarks have definite momentum and gluons have definite transversal momentum are physical is consistent with confinement? $m=0,n=1$ would correspond to a physical single-quark state!??

I don't think his exposition can be correct. Could you please explain how the claim in eq. (109) in Section VI.D that states in which quarks have definite momentum and gluons have definite transversal momentum are physical is consistent with confinement? $m=0,n=1$ would correspond to a physical single-quark state!??
There is no “physical single-quark state”. In his introduction, Leader tells you (throughout this paper “quark” will mean a sum over all flavours of quarks and antiquarks). Also, he uses the by-now standard definitions* for physical states and operators in the covariant quantization of QCD.

(*) See Ref’s [12], [13], [14] and [17] in Leader’s paper.

A. Neumaier
he uses the by-now standard definitions* for physical states and operators in the covariant quantization of QCD.
Kugo and Ojima define a physical state to be one annihilated by $Q_B$ and $Q_C$. Leader claims on p.28 of arXiv:1101.5956v2 (your elliot-2) that ''the states with quarks having definite momentum $P_1,\ldots,P_n$ and transversal gluons having momenta $k_1,\ldots,k_m$ [...] are physical states''. Without any restriction on the number and momenta of the quarks. This is simply wrong.
In his introduction, Leader tells you (throughout this paper “quark” will mean a sum over all flavours of quarks and antiquarks).
This would not restrict the total number of quarks involved in any way. If such a sum were implied in his equation (109), it would be, as he explicitly stated, for definite momenta and hence for fixed (but arbitrary) $n$ and $m$, including the 1-quark case $n=1,m=0$. Maybe the paper is correct in spirit but it is certainly written so sloppily that essential assumptions are missing.

Kugo and Ojima define a physical state to be one annihilated by $Q_B$ and $Q_C$.
So does he. See equations (72) and (73).
Leader claims on p.28 of arXiv:1101.5956v2 (your elliot-2) that ''the states with quarks having definite momentum $P_1,\ldots,P_n$ and transversal gluons having momenta $k_1,\ldots,k_m$ [...] are physical states''.
Read the line just below equation (109).

A. Neumaier
Read the line just below equation (109).
This line just says that Leader's statement follows from commutation rules in the paper by Kugo and Ojima. But since Leader's statement is wrong, this implies that either this line itself or the paper by Kugo and Ojima is also faulty.

Indeed, looking at Kugo and Ojima, one finds that it is not about QCD at all but about SU(2) Higgs/Kibble (with broken symmetry) and pure Yang-Mills only. Hence they say nothing at all about quarks. This means that Leader's claim that his statement about QCD follows from Kugo and Ojima is wrong.

Moreover, Kugo and Ojima claim to have proved in paper III of their sequence (to which the cited paper refers for details about the pure Yang-Mills case) that the asymptotic gluon field is physical. But because of confinement it cannot be physical since it is colored! Thus their analysis is faulty, too. Indeed, they explicitly say that they ignore infrared effects - which make their asymptotic gluon field divergent, hence nonexistent.

Thus everything is based on faulty reasoning.

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All of Leader’s paper (given in the PDF’s) were published in Physical Review. The one in question was published 7 years ago

E. Leader, Phys. Rev., D83, 096012(2011)

So far, no one of the thousands of experts in QCD (around the world) has pointed out that Leader’s paper has any technical mistake. So, if you think otherwise, why don’t you write a letter to Phys. Rev. D. detailing any technical mistake in Leader’s paper? If you are unable to do so, then you should re-examine your (could be poor) understanding of QCD.

The physical content of (covariantly quantized) Yang-Mills theory and the confinement problem in QCD is the subject of chapter 4 in

N. Nakanishi and I. Ojima, “Covariant Operator Formalism of Gauge Theories and Quantum Gravity” World Scientific 1990.

So, have a good study of it, in particular the norm-cancellation mechanism (the so-called quartet mechanism).

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A. Neumaier
Why do you argue from authority rather than by pointing out where my mistakes or misunderstandings are and why my arguments are not coherent (if they aren't)?
All of Leader’s paper (given in the PDF’s) were published in Physical Review.
I published in Physical Review, too. I also know that there is no guarantee that results published there are correct. Thus what you write is not a logical argument against my statements.
you should re-examine your (could be poor) understanding of QCD.
According to the standards of mathematical physics, no one at present understands QCD, since currently there is not even a mathematically consistent definition of what QCD is.
The physical content of (covariantly quantized) Yang-Mills theory and the confinement problem in QCD is the subject of chapter 4
This is a discussion forum, where things should be explained for everyone to check and understand at the indicated (in this case intermediate) level, rather than superficially settled by references to whole book chapters.

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Why do you argue from authority
Me needs no authority in field theory, everyone in here knows this about me. You stamped Leader’s paper as “wrong” even before reading it. And talking about authority, Leader himself is an authority on QCD, with 45 years of expertise and impressive records of more than 200 published paper on the theory of strong interactions.

where my mistakes or misunderstandings are and why my arguments are not coherent (if they aren't)?
Asymptotic Completeness, Poincare’ covariance and the fact that (anti)commutation relations of asymptotic fields are c-numbers (Greenberg-Robinson theorem) allow one to determine all of the non-vanishing (anti)commutators of asymptotic fields in terms of the invariant functions $$D(x) = - \Delta (x ; m^{2})|_{m = 0} = - \frac{1}{2 \pi} \epsilon (x^{0}) \delta (x^{2}) ,$$$$E(x) = - \frac{\partial}{\partial m^{2}} \Delta (x ; m^{2})|_{m^{2} = 0} ,$$ and their spacetime derivatives. Now, by direct lengthy computations (using the properties of the invariant functions) you can show that $$[Q_{B} , \vec{P}_{as}(q)] = [Q_{B} , \vec{P}_{as}(g)] = 0,$$ meaning that $P^{i}_{as}(\mbox{quark})$ and $P^{i}_{as}(\mbox{gluon})$ are observables with physical eigen-states in the sense that Fock states can be expanded in terms of these eigen-states in the process of evaluating the quarks and gluons contributions to the Nucleon momentum (see the paragraph after equation (103)).

I published in Physical Review, too.
Then you know that there are sections called “brief reports”, “rapid communications” and/or “letters to editors”. So, if you have found mistakes in Leader’s paper, then you may communicate your findings to Phys.Rev.D. Or, you could tell Leader directly (by Email) that his claim that $[Q_{B} , \vec{P}_{\mbox{as}}(q)] = [Q_{B} , \vec{P}_{\mbox{as}}(g)] = 0$ is "not correct".

I also know that there is no guarantee that results published there are correct.
I did not say that “physical review paper" are necessarily correct. However, mistakes almost always (and here I am only talking about Phys.Rev.D and Phys.Lett.B) get discovered very fast. This is because: (i) usually there are thousands of people working on the same problem, and (ii) all serious researchers around the world read Phys.Rev.D and Phys.Lett.B.
Now, after 7 years in publications, nobody showed that Leader’s paper contains mistakes or said it is wrong. On the contrary, many people admitted to their mistakes that the paper said they did.
This is a discussion forum
Actually, this is exactly the reason why I suggested a reference for you.
where things should be explained for everyone to check and understand at the indicated (in this case intermediate) level, rather than superficially settled by references to whole book chapters.
If I were to write about the physical content of the covariant QCD in here, my post would had been unreasonably long and complicated for “Forum discussions”.

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DarMM
Gold Member
meaning that $P^{i}_{as}(\mbox{quark})$ and $P^{i}_{as}(\mbox{gluon})$ are observables with physical eigen-states in the sense that Fock states can be expanded in terms of these eigen-states in the process of evaluating the quarks and gluons contributions to the Nucleon momentum (see the paragraph after equation (103)).
The Hilbert space of nonperturbative QCD wouldn't be a Fock space though and the physical states couldn't be expanded in terms of quark-gluon states, as those states lie in a Hilbert-Krein space.

The Hilbert space of nonperturbative QCD wouldn't be a Fock space though and the physical states couldn't be expanded in terms of quark-gluon states, as those states lie in a Hilbert-Krein space.
Wrong. The structure of the indefinite-metric Hilbert (Krein) space can be analysed as the Fock space of the asymptotic fields: By assumption $Q_{B}|0 \rangle = Q_{c}|0 \rangle = 0$, i.e., the charges of the BRST-algebra are not broken spontaneously. So, the classification of the representations of the BRST-algebra can be applied to the one-particle asymptotic states and translated into statements about the properties of creation and annihilation operators of asymptotic fields. See sections (4.1.2) & (4.1.3) of Nakanishi and Ojima book.

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DarMM
Gold Member
Wrong. The structure of the indefinite-metric Hilbert (Krein) space can be analysed as the Fock space
First the Hilbert Krein space is not a Fock space, which must be formed from Hilbert spaces, but that is a minor detail.

Secondly, I specfically said the physical Hilbert space, which is not a Fock space. The Hilbert space of any interacting theory beyond infrared cutoff ones in 2D are non-Fock.

I'm not talking about the Hilbert Krein space, as it contains non-physical states.

Kugo and Ojima's series of papers from the late-70s explicitly state that gluon and quark fields are non-physical.

For example Section VI of:
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

A. Neumaier
Kugo and Ojima define a physical state to be one annihilated by $Q_B$ and $Q_C$.
So does he. See equations (72) and (73).
But when claiming near (109) that the quark states are physical states Leader only discusses annihilation by $Q_B$. What happens with $Q_C$?
by direct lengthy computations (using the properties of the invariant functions) you can show that $$[Q_{B} , \vec{P}_{as}(q)] = [Q_{B} , \vec{P}_{as}(g)] = 0,$$ meaning that $P^{i}_{as}(\mbox{quark})$ and $P^{i}_{as}(\mbox{gluon})$ are observables with physical eigen-states.
You claim that these observables are physical already because of commutation with $Q_B$ only. But this only proves (72), not (73). Thus your (and Leader's) arguments are incomplete and not conclusive. Maybe this is the cause of our disagreement?

But when claiming near (109) that the quark states are physical states Leader only discusses annihilation by $Q_B$. What happens with $Q_C$?

You claim that these observables are physical already because of commutation with $Q_B$ only. But this only proves (72), not (73). Thus your (and Leader's) arguments are incomplete and not conclusive. Maybe this is the cause of our disagreement?
Again you are making judgement before reading the relevant sections of the paper. In subsection A, after equation (73), Leader presents arguments for the fact that $\vec{P}$ has vanishing ghost number, and he does that for both the canonical as well as the Bellinfante expressions. You can also (if you wish) repeat the previous lengthy calculations replacing $Q_{B}$ by $Q_{c}$ to arrive at $[Q_{c} , \vec{P}_{as}] = 0$ which is consistent with the following commutator of the BRST-algebra $[iQ_{c} , Q_{B}] = Q_{B}$.

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First the Hilbert Krein space is not a Fock space, which must be formed from Hilbert spaces, but that is a minor detail.
Again, I refer you to Nakanishi & Ojima book, section 4.1.3 "Multi-particle structure and asymptotic fields"
where the total state space $\mathcal{V}$ (i.e., Krein space) is analysed as the Fock space of the asymptotic fields.

Secondly, I specfically said the physical Hilbert space, which is not a Fock space. The Hilbert space of any interacting theory beyond infrared cutoff ones in 2D are non-Fock.
I'm not talking about the Hilbert Krein space, as it contains non-physical states.
AGAIN, I ask you to look up section (4.1.2) of the book, where (i) the cohomology group of the BRST charge $Q_{B}$ on $\mathcal{V}$:

$$\mbox{H}(Q_{B}, \mathcal{V}) = \frac{\mbox{Z}(Q_{B} , \mathcal{V})}{\mbox{B}(Q_{B} , \mathcal{V})} = \frac{\mbox{ker}(Q_{B})}{\mbox{im}(Q_{B})} \equiv \frac{\mathcal{V}_{phy}}{Q_{B}\mathcal{V}} ,$$
(ii) the representations of the BRST-algebra and (iii) the inner product structure, are studied in detail.

Kugo and Ojima's series of papers from the late-70s explicitly state that gluon and quark fields are non-physical.
Statements like “gluon and quark fields are physical or non-physical” are meaningless and I don’t understand. What I do understand is the fact that the matrix elements of gluon and quark momentum and angular momentum are physically measurable quantities.

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
The 1990’s book of Nakanishi & Ojima is becoming the textbook study of the covariant operator formalism of gauge theories.

A. Neumaier
Statements like “gluon and quark fields are physical or non-physical” are meaningless and I don’t understand.
A local field is physical iff the corresponding smeared field operators map a dense subspace of the physical Hilbert space into itself.
The 1990’s book of Nakanishi & Ojima is becoming the textbook study of the covariant operator formalism of gauge theories.
I currently don't have access to this book, hence cannot use it within the present discussion.
But when claiming near (109) that the quark states are physical states Leader only discusses annihilation by $Q_B$. What happens with $Q_C$?

You claim that these observables are physical already because of commutation with $Q_B$ only. But this only proves (72), not (73). Thus your (and Leader's) arguments are incomplete and not conclusive. Maybe this is the cause of our disagreement?
Again you are making judgement before reading the relevant sections of the paper.
I only judged that the visible argument was incomplete, which is correct. Now you completed it.

If your and Leader's statements are correct then there must be a clash in the terminology as used by Leader and you and the terminology used by mathematical physicists. I am familiar with the latter, so please provide a dictionary so that I am able to understand you.

1. For mathematical physicists, a Fock space is a Hilbert space with positive definite inner product, constructed as a tensor product of sy,mmetrized or antisymmetrized tensor products of 1-particle Hilbert spaces. Is this also your terminology? Or do you perhaps call analogous indefinite product spaces still Fock spaces?

2. For mathematical physicists, asymptotic fields are in 1-1 correspondence with points in the (real, nonnegative) discrete spectrum of the (self-adjoint, positive semidefinite) mass operator of the Poincare representation on the asymptotic state space, or with branch points of its spectrum (in case of theories without a mass gap). Is this also your terminology, or what is the corresponding relation in your terminology?

3. Assuming we agree on point 1 and 2, the asymptotic quark fields of QCD would have to carry a unitary representation of the Poincare group with nonnegative mass (and presumably spin 1/2). These masses would be well-defined parameters of QCD. What is the physical meaning of these masses? Is it current quark masses or constituent quark masses or something else? Or how is the Poincare group represented on the 1-particle Hilbert space corresponding to an asymptotic quark field?

4. Assuming we agree on points 1 and 2, and assuming in addition that QCD has a mass gap (which seems to me universal agreed upon), it should follow from Haag-Ruelle theory that the asymptotic fields satisfy the cluster decomposition principle and hence are not confined. Indeed, points in the discrete spectrum of the mass operator correspond to bound states, whose asymptotic fields satisfy cluster decomposition.

But bound states corresponding to single-quark fields are not observed. So some of your concepts must carry a different content from what I am used to from mathematical physics.

Sorry, I have been very busy lately. I will come back to you very soon.

DarMM
Gold Member
Again, I refer you to Nakanishi & Ojima book, section 4.1.3 "Multi-particle structure and asymptotic fields"
where the total state space V\mathcal{V} (i.e., Krein space) is analysed as the Fock space of the asymptotic fields.
See @A. Neumaier 's point 1 above, Fock spaces have to have Hilbert spaces as their generator, not Krein spaces.

Statements like “gluon and quark fields are physical or non-physical” are meaningless and I don’t understand
It's standard terminology in mathematical physics, see @A. Neumaier 's post.

See @A. Neumaier 's point 1 above, Fock spaces have to have Hilbert spaces as their generator, not Krein spaces.
No, that need not be the case in general. Fock space is the representation space of oscillators algebra, a complex vector space with non-degenerate inner product and unique vacuum annihilated by all annihilation operators. So, even in ordinary QM Fock space can have an indefinite metric. If fact, the quantization of any (oscillator) system with even number of first-class constraints leads to an indefinite metric Fock space. See the text
M. Henneaux & C. Teitelboim, "Quantization of Gauge Systems", (1992) Princeton Uni. Press.
where an infinite number of systems with indefinite-metric Fock spaces are given.

DarMM