Understanding Particles in Lattice QCD: Challenges and Potential Solutions"

[Demystifier]

Can someone explain to me how the concept of particle is defined in lattice QCD?

Here are the reasons why it seems problematic to me:
1) Lattice QCD is based on functional-integral formulation of QFT, which does not contain any operators in the Hilbert space. In particular, it does not contain the particle creation and destruction operators.
2) It is a non-perturbative theory with confinement, which means that one cannot define particles though the LSZ reduction based on assumption that asymptotic states are free particle states (quarks and gluons).

References in which these things are explained would be highly desirable.

Also, if one knows a simpler toy model of a non-perturbative definition of particles in interacting QFT, that might be even more interesting.

 

[tom.stoer]

I can only talk about the quenched approximation. That means that the fermionic determinant is set to one = all fermion loops are suppressed and fermions behave classically. The quarks are "static", only the gluon degrees of freedom are dynamical. So the quarks are defined as static particles.

I do not know how lattice QCD (as of today) goes beyond this quenched approximation.

 

[Demystifier]

I can only talk about the quenched approximation. That means that the fermionic determinant is set to one = all fermion loops are suppressed and fermions behave classically. The quarks are "static", only the gluon degrees of freedom are dynamical. So the quarks are defined as static particles.

I do not know how lattice QCD (as of today) goes beyond this quenched approximation.

Thanks, but it still does not answer my question, not even approximately. What I want is a representation of a 1-particle state (be it quark, gluon, hadron, glueball, ... whatever) as a state in a Hilbert space. How to find such a representation when
1) one deals with path integrals that do not even use the concept of Hilbert spaces, and
2) free particle states are not even a good approximation

Actually, QCD serves here only as an example. What I want to know is how, in general, one can derive the concept of PARTICLE from a theory of quantum FIELDS. It is well known how to do it with PERTURBATIVE CANONICAL quantization of fields, but the example of lattice QCD is interesting because it is neither perturbative nor canonical.

 

[tom.stoer]

OK, understood.

The canonical quantization is definately not limited to the perturbative treatment, but can be defined rigorously even non-perturbatively. It is e.g. possible to write down the full, gauge-fixed QCD Hamiltonian w/o unphysical degrees of freedom (zero norm states, ghosts, ...).

Regarding particle concept in the PI formalism: it's rather simple: every localized and stable (physical) field configuration is a particle. A localized gauge field configuration can e.g. be considered as a glueball.

The problem is how to prepare and/or identify such configurations on the lattice, especially if fermions are taken into account.

 

[Demystifier]

Regarding particle concept in the PI formalism: it's rather simple: every localized and stable (physical) field configuration is a particle. A localized gauge field configuration can e.g. be considered as a glueball.

But such a definition of a particle has nothing to do with the usual Fock-space definition of particle in free QFT:
First, a Fock 1-particle state does NOT need to be localized. (For example, a momentum eigenstate does not have a localized wave function).
Second, a localized configuration does not need to be a Fock 1-particle state. (For example, two bosons with the same localized wave-packet wave function.)

What I want is a generalization of the Fock-space definition of particle in non-perturbative QFT. I don't want a "classical" concept of particle based on local configurations.

 

[tom.stoer]

I think you can't have a "non-perturbative Fock-space particle". It's self-contradictory.

A Fock state is something like a state $$|ABC\ldots\rangle = A^\dagger B^\dagger C^\dagger|0\rangle$$ with A, B, C being field operators creating 1-particle states $$|A00\dots}, |0B0\ldots\rangle, |00C\ldots\rangle$$ which are solutions of the free theory.

If you take a non-perturbative solution of the full theory, a soliton for example, you can try to create it via $$|soliton\rangle = S^\dagger |0\rangle$$. Of course you can now define $$|soliton 1, soliton 2\rangle = S_1^\dagger S_2^\dagger|0\rangle$$ but unfortunately it's no longer a solution of the theory. It may not even be close to a solution!

So even if you are able to define this soliton creation operator, I doubt that you will be able to use it to create reasonable states. I don't even know whether you can span the whole Hilbert space.

Why do you want to throw away the benefit of the PI formalism to be able to deal with real, localized particles instead of plane waves? Look at lattice gauge theory: they are able to visualize localized gauge field configurations; they can calculate their properties; they can calculate hadron masses w/o using Fock states. In the PI formalism the classical concept of a localized field configuration remains valid even after quantization.

 

[strangerep]

What I want is a generalization of the Fock-space definition of particle in non-perturbative QFT. I don't want a "classical" concept of particle based on local configurations.

Are you only interested in "particles" at asymptotic times, or for all times?

If the latter, then I'm guessing you want an operator A s.t.
$$[H,A] ~\propto~ A$$
?

 

[genneth]

Particles only make rigorous sense if the particle number operator commutes with the Hamiltonian. However, if the commutator is small, one can still make reasonably sensible statements, ala LSZ and perturbation theory. However, if the commutator is not small, then I would say that is a sign to let go of the picture of particles, because that view will simply make conceptual understanding difficult (even simple states in the system will be a complicated superposition of particles).

Something we often start forgetting as we move up in abstraction: only energy eigenvalues and eigenstates are physical (pedant cutoff: up to usual isomorphisms, etc.); the labels which we attach to them are just labels.

 

[daschaich]

Can someone explain to me how the concept of particle is defined in lattice QCD?

In practice, we work with quark and gluon fields defined across the entire spacetime lattice, with our observables being various correlation functions of these fields.

But it sounds like you're interested in more general considerations, rather than the details of how we perform our calculations. Perhaps you would find relevant the classic 1977 paper by Luscher, "Construction of a selfadjoint, strictly positive transfer matrix for Euclidean lattice gauge theories", http://dx.doi.org/10.1007/BF01614090 or http://ccdb4fs.kek.jp/cgi-bin/img_index?7611148 .

 

[tom.stoer]

I think we should define what "particles" are

- lumps of energy confined to some small region of space, like solitons
- states in a representation of the Poincare group, defined via its mass, spin etc.
- plane wave states in Fock space
- ...

 

[Demystifier]

I think you can't have a "non-perturbative Fock-space particle". It's self-contradictory.

A Fock state is something like a state $$|ABC\ldots\rangle = A^\dagger B^\dagger C^\dagger|0\rangle$$ with A, B, C being field operators creating 1-particle states $$|A00\dots}, |0B0\ldots\rangle, |00C\ldots\rangle$$ which are solutions of the free theory.

If you take a non-perturbative solution of the full theory, a soliton for example, you can try to create it via $$|soliton\rangle = S^\dagger |0\rangle$$. Of course you can now define $$|soliton 1, soliton 2\rangle = S_1^\dagger S_2^\dagger|0\rangle$$ but unfortunately it's no longer a solution of the theory. It may not even be close to a solution!

So even if you are able to define this soliton creation operator, I doubt that you will be able to use it to create reasonable states. I don't even know whether you can span the whole Hilbert space.

Why do you want to throw away the benefit of the PI formalism to be able to deal with real, localized particles instead of plane waves? Look at lattice gauge theory: they are able to visualize localized gauge field configurations; they can calculate their properties; they can calculate hadron masses w/o using Fock states. In the PI formalism the classical concept of a localized field configuration remains valid even after quantization.

Thanks, that's helpfull.

 

[Demystifier]

Particles only make rigorous sense if the particle number operator commutes with the Hamiltonian.

I've seen many times such an argument, but I never liked it. After all, what is so special about the Hamiltonian operator? OK, if the number operator does not commute with the Hamiltonian, then the number of particles is not conserved. But so what? If some quantity is not conserved, it does not mean that this quantity does not make sense.

 

[Demystifier]

I think we should define what "particles" are

- lumps of energy confined to some small region of space, like solitons
- states in a representation of the Poincare group, defined via its mass, spin etc.
- plane wave states in Fock space
- ...

It is a difficult question. But let us, at least for the moment, replace it with a more practical question: How a particle is defined in lattice QCD? It seems that the answer is the following: A hadron particle is the lowest energy state with given values of flavor quantum numbers. Then two such states localized at very different locations (quantum solitons with a negligible overlap) would approximately correspond to a 2-particle state.

 

[tom.stoer]

Fine!

The basic disctinction made is between hadrons and quarks. One can e.g. write down a Hamiltonian for QCD using Fock space quarks and gluons in order to describe localized hadrons.

So essentially one mixes two different concepts of "particles". But this is fine as long as everybody agrees what is meant by "particle".

 

[Demystifier]

One additional question: Is meson (or barion) in lattice QCD allways determined by a 2 (or 3) point function, i.e., a correlation function of 2 (or 3) quark fields?

If yes, then the claim that meson (or barion) contains 2 (or 3) quarks is justified.

 

[Demystifier]

Are you only interested in "particles" at asymptotic times, or for all times?

If the latter, then I'm guessing you want an operator A s.t.
$$[H,A] ~\propto~ A$$
?

Excellent, I love it! It seems to be exactly what I wanted (but was unable to spell it explicitly). And I don't see a reason why such A wouldn't exist for ANY H.

Do you know any paper in which such A was constructed explicitly for a nontrivial interacting theory?

 

[tom.stoer]

afaik large-N QCD deals with an approx. like $$\bar{q}Xq ~ <\bar{q}Xq> + \hat{Q} + \ldots$$ where the first operator is bilinear in the quarks with some Dirac and color matrix X, the vev is some condensate and Q is a meson fluctuation operator. One can show that in an appropriate limit the meson fluctions couple only weakly, so this should be related to the above mentioned case; at least in 1+1 dim. QCD the Hamiltonian can be rewritten in terms of these weakly coupled meson fluctuations.

 

[daschaich]

One additional question: Is meson (or barion) in lattice QCD allways determined by a 2 (or 3) point function, i.e., a correlation function of 2 (or 3) quark fields?

If yes, then the claim that meson (or barion) contains 2 (or 3) quarks is justified.

By "two-point function", we refer to the propagation of a hadron (either a meson or a baryon) from one point of spacetime to another. A "three-point function" adds another operator to the correlation function, to probe for example the form factors of this hadron.

But to address what you mean as opposed to what you say, the mesonic creation and annihilation operators always involve two quark fields, while the baryonic operators involve three. There would be no way to obtain the appropriate quantum numbers otherwise.

 

[Demystifier]

By "two-point function", we refer to the propagation of a hadron (either a meson or a baryon) from one point of spacetime to another. A "three-point function" adds another operator to the correlation function, to probe for example the form factors of this hadron.

But to address what you mean as opposed to what you say, the mesonic creation and annihilation operators always involve two quark fields, while the baryonic operators involve three. There would be no way to obtain the appropriate quantum numbers otherwise.

Thanks for this clarification. Are these two quark fields (defining a mesonic operator) fields at the SAME spacetime point? If yes, does it lead to UV divergences on a lattice? (I guess not, because in a continuum Dirac delta(0) is infinite, but on the lattice Cronecker delta(0) is finite.)

 

[Demystifier]

Let me further develop the idea of strangerep. Assume that all eigenstates of H are known. These eigenstates can be labeled by some discrete (as well as some continuous) labels. One of these discrete labels (or a combination of them) has values n=0,1,2,3 ..., where n=0 corresponds to the ground state. Then we can say that the number of particles in a given eigenstate is simply the number n.

The problem, of course, with such a definition of particles is that it is highly non-unique, unless some additional criteria is posed on the choice of the appropriate discrete label n. Perhaps it was not such a good idea as it seemed to me at first.

 

[daschaich]

On the lattice, everything is discrete. That's why we use it!

Since we are concerned with hadrons, our "n=0,1,2,3 ..., where n=0 corresponds to the ground state" refers to hadronic excitations rather than particle number. (Though sometimes multi-hadron "scattering states" rear their ugly heads.)

Back in January, Georg wrote a bit about extracting excited states on the lattice.

 

[Demystifier]

Back in January, Georg wrote a bit about extracting excited states on the lattice.

Very illuminating, especially the first paragraph. Thanks!

Let me explain to myself few points which are not explained there:
If O are the 1-hadron composite operators, then psi_n are 1-hadron wave functions. By direct products of 1-hadron wave functions you can construct many-hadron wave functions (assuming that different hadrons do not interact with each other).

However, in this way you can calculate only |psi_n|^2, not the phase of psi_n. Is there a way to overcome this problem as well?

 

[genneth]

I've seen many times such an argument, but I never liked it. After all, what is so special about the Hamiltonian operator? OK, if the number operator does not commute with the Hamiltonian, then the number of particles is not conserved. But so what? If some quantity is not conserved, it does not mean that this quantity does not make sense.

I think this is actually the heart of the matter --- returning to your original goal of looking for particles in general, rather than just QCD.

You are completely correct in that the number operator always makes sense. After all, we can define it! That pretty much defines "makes sense". However, just because something makes sense doesn't mean that it is a helpful quantity to consider. This is perhaps an easier thing to accept if you have a condensed matter background. The game is never "what is the exact description", but rather "what is the *simplest* description". If something doesn't commute with the Hamiltonian, then its evolution with time is massively complex, and then even if you had an exact solution, you would have understood almost nothing.

Let's consider QED. There, the Gaussian sector gives you number, charge, momenta, etc. The interaction doesn't commute with number, but only weakly --- for a given initial number, the change is slow and gradual. It might not be a perfect quantum number, but one can still picture Feynman diagrams rather naively and see them as describing important physical processes.

Now in QCD this is no longer true. The number operator has a complex evolution, and also very quickly. But this is simply a sign that one should be trying to label the solutions differently --- so number operators of baryons, mesons, etc. rather than the quarks and gluons. Now, of course, we don't really know how to do this rigorously... I think you might like to look into things like chiral perturbation theory, which I think (in principle at least) would give you what you're looking for.

I don't know what your background is, but I seriously recommend (and to all HEP) some condensed matter. Most importantly, the concept of particle has been vigorously debated and frustrated over. After all, unlike in relativistic quantum fields, sometimes the total lack of translational symmetry completely destroys attempts such as identifying representations of the Poincaré group! One should definitely try and understand superconductivity, because it's a demonstration of creating new bound particles and then completely switching the description to be in terms of those, and also fractional/integer quantum hall effects, where one finds fractionalisation of electrons and also find them fundamentally hard to pin down (localised energy? hah!) but yet find them indispensable in creating a physical picture. The goal is to get a broader view of quantum fields, not restricted by the rather rigid rules that HEP tends to run in. A truly eye opening moment is realising that the "vacuum" is just a material ground state, and has nothing special going for it at all...

 

[Demystifier]

Genneth, you are making good points!

You will understand my position if I tell you that my background is in high energy physics and that most of my research is in the FOUNDATIONS of quantum theory. The latter means that I am more interested in "what is the exact description", than in "what is the simplest description".

The paper that perhaps gives the best representation of my interests and way of thinking is this:
http://xxx.lanl.gov/abs/quant-ph/0609163 [Found.Phys.37:1563-1611,2007]
This is, in fact, my most readed and downloaded (although not most cited) paper.

 

[Demystifier]

If some quantity is not conserved, it does not mean that this quantity does not make sense.

In fact, from a practical point of view, one could even argue that non-conserved quantities make MORE sense than conserved ones. Namely, all practically interesting physics is contained in CHANGES in nature, such as scattering. A change can only be described by a quantity which is NOT conserved.

 

[genneth]

... The latter means that I am more interested in "what is the exact description", than in "what is the simplest description".

This I can completely understand. I would qualify it however, by drawing emphasis to the word "description". Of course, we understand "description" mathematically --- something only qualifies if it is a proper, predictive theory. However, if we have two such available descriptions, then it is a matter of taste. Let me draw from an example:

In superconductivity, the toy model one works with is that of a large number (crucial!) of electrons $$\psi$$, with an attraction (4-point vertex) if they meet (delta potential). The slick method is to work in a path integral formulation. The first step is to introduce an auxilary bosonic field $$\phi$$, which has interactions $$\psi\psi\phi^\dagger$$ (+ h.c.), which then replaces the 4-point vertex. The trick here is to notice that $$\phi$$ may be integrated out to reproduce the original action --- this is called a Hubbard-Stratonovich transform, but is really just undo-ing a Gaussian integral. After doing this, we notice that in terms of the original electrons, we now have a quadratic action! This means we can do the integral, which gives some suitably complicated answer, *outside* of the exponential. This may then by lifted back in by taking a log, and now expresses self-interactions of the auxiliary field $$\phi$$. The key thing to note is that all this is *exact*. No approximations, nothing lost. We completely traded electrons for a new bosonic field (representing pairs of electrons), and found a new description of the system. The two are mathematically identical, but one allows you to understand normal metals easily, the other superconductors easily. Perhaps it would be helpful to meditate on the question of where the electrons are in the bosonic description...?

 

[Demystifier]

Good example, genneth!