Massive Degeneracy and Goldstone Bosons: A Challenge for the Light Cone
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Massive Degeneracy and Goldstone Bosons: A Challenge for the Light Cone ∗ †
Marvin Weinstein
SLAC National Accelerator Laboratory,Stanford, CA, USAE-mail: [email protected]
Wherein it is argued that the light front formalism has problems dealing with Goldstone sym-metries. It is further argued that the notion that in hadron condensates can explain Goldstonephenomena is false.
I. A DEFINITION
You might well ask why I have decided to talk about the meaning of the Goldstone theorem and physics on the LightFront. After all, while important, this is not really a new subject. The answer is simple. I have chosen to give thistalk, rather than talk about new research, because at past Light Cone meetings I have discovered that this subject isnot as well understood by all of the participants as it should be. Therefore, people have not understood the discussionsthat have taken place between Stan Brodsky and me on the question of in hadron condensates and the problems withthe uniqueness of the Light Front vacuum state. I have titled this talk ”Massive Degeneracy and Goldstone Bosons:A Challenge for the Light Cone” to emphasize the importance of this issue for a proper understanding of PCAC andCurrent Algebra.As it turns out, during my recent travels I found that my titles are not always easily understood. So I went backto the Wikipedia for a definition of degeneracy.
This is what I found: • de-gen-er-a-cy [ di?jnn?r?ssee ] noun (plural de-gen-er-a-cies)Definition:
1. bad behavior: immoral, depraved, or corrupt behavior, or an instance of this2. worsened condition: a condition that is worse than normal or worse than before3. worsening of condition: the process of becoming physically, morally, or mentally worse4. quantum physics states of equal energy: the condition of two or more quantum states having the sameenergy.I would love to discuss the first three topics, but alas, I will be discussing the fourth in this talk. The point that Iwant to make in the time allotted to me, is that the successes of PCAC and Current Algebra require us to concludethat in our world the hadron sector is very close to a theory where SU (3) × SU (3) is realized as a Goldstone (otherpeople say spontaneously broken) symmetry. A corollary of this, is that in the limit of an exact Goldstone symmetry,the ground state of the theory is enormously degenerate and this feature of QCD is not apparent in the Light Frontformulation of the theory. This is the feature of QCD that has to be better understood.
II. A PARABLE
What would a general talk be without a parable ? Nothing! So, having said that, I will begin with a modifiedversion of a parable that I believe I first heard from Sydney Coleman.Once upon a time in a universe far, far smaller than ours there lived the famous savant Doctorus E, who wasan expert on practically everything and worked at the famous Crystallus U. One day as the Doctorus was deep in ∗ This work was supported by the U. S. DOE, Contract No. DE-AC02-76SF00515. † Talk given at Light Cone 2010 - LC2010, June 14-18, 2010 , Valencia, Spain thought a student interrupted with a strange observation. He said, ”Doctorus, I have just discovered an amazingthing! The world is translation invariant!” Never at a loss for a response the famous savant replied ”Dummkopff,everybody knows that! Come here and look out the hyper-viewer at what can be seen in the sky. Obviously if you
FIG. 1: Read the pictures left to right and top to bottom move the entire world over by one grelp (an astronomical unit in Crystallus) everything looks the same. We haveknown that forever”.”But no Doctorus”, said the student,”that is not what I meant! I mean that the laws of physics don’t change evenif we translate the world by an arbitrarily small amount”. To which the savant responded ”How could you know that? It would take an infinite amount of energy to move the whole world by an arbitrarily small amount! How could wetest this idea ?””That’s the neat thing Doctorus”, said the student. You don’t have to move the whole world to find out if thereis, what I call, a hidden symmetry . All you have to do is see how much energy it takes to excite an arbitrarily longwavelength excitation. The consequence of the hidden symmetry is that this energy will go to zero as the wavelengthgoes to infinity. In addition, these long wavelength excitations satisfy sum rules constraining their interactions withimperfections””Hmm”, said the savant, ”let me think about this.”As it turned out the young student was correct. In fact, parable aside, translation invariance (by arbitrarily smalltranslations) is realized as a hidden symmetry in a crystal. The excitations are called phonons and these phononshave no mass (i.e., their energy as a function of momentum goes to zero as the momentum vanishes).
III. EXAMPLES OF GOLDSTONE SYMMETRIES
Of course, I wouldn’t be talking about this if the only example of a hidden symmetry was phonons in a crystal. Infact, condensed matter physics is replete with examples of systems that exhibit this phenomenon.For example, ferromagnets are objects that exhibit spontaneous magnetization; i.e., in these systems magneticmoments of atoms align with one another to produce an observable, persistent magnetic field. Since this magneticfield points in a definite direction, it follows that the rotational invariance of the full theory is hidden from us. Themassless excitations (or gapless excitations of this system are called magnons ). Since the manetization could pointin any direction, it follows that the ground state of this system is infinitely degenerate. Similarly, there are anti-ferromagnets, where the spins anti-align, but point in a definite direction.(High temperature superconductors exhibitthis behavior when they are under-doped.) Once again the hidden symmetry is rotational invariance and the masslessexcitations are called spin-waves .Finally, not to be outdone by the world of condensed matter physics, particle physics also has its share of hidden (orGoldstone) symmetries. For the purposes of this discussion I will avoid theories that have the additional complication
FIG. 2: Both the ferromagnet and anti-ferromagnet have rotational invariance as a hidden symmetry. of the Higgs phenomenon taking place and will concentrate on QCD. It is an old story that the only consistentexplanation of the Goldberger-Treiman relation, Adler-Weissberger calculation of the g A /g V sum-rule, the sum rulefor the squares of the masses of the pseudoscalar mesons, Weinberg’s formula for the π − π scattering lengths, Dashen-Weinstein theorem on the slope of the form factors in Kl decay, etc., is that QCD is close to a theory in which SU (2) × SU (2) (and in fact SU (3) × SU (3)) is an exact but hidden , (or Goldstone) symmetry. Moreover, the onlysource of symmetry breaking are the quark mass terms in the Hamiltonian. In other words, if quark masses werezero, then this symmetry would be exact and the π , K and η meson would all have zero mass. They would be, likethe phonons, magnons or spin-waves, the massless excitations associated with the hidden symmetry. By the way, onereason I like calling this sort of symmetry a hidden or Goldstone symmetry, rather than the more popular spontaneouslybroken symmetry is that it avoids having to talk about a ”really broken, spontaneously broken, symmetry” when quarkmasses are non-zero.The key point I want to make in the rest of this talk is that it is genrally true, as is obvious in the case of thecondensed matter examples I talked about, that in order for there to be Goldstone bosons (these massless excitations)there has to be something to wiggle . IV. FORMALITIES
Since I see friends in the audience with a bit of a formal bent, I will spend a few moments giving some formalinsight into the meaning of the Goldstone theorem and what it says about there having to be stuff that wiggles. Amore extensive discussion of some of these issues can be found in my Heidelberg lectures[1] .The story begins with that perennial favorite, Noether’s theorem. You all know the theorem, you learned it ingrade school. It says, that if a Lagrangian has a continuous symmetry then there exists a locally conserved currentassociated with that symmetry; i.e., ∂ µ j µ = 0 . (1)The usual follow up to the proof of this theorem is the observation that as a consequence of current conservationthere is an associated conserved charge Q = Z d x j ( x ) . (2)It is usually argued that a consequence of the current conservation equation that the time derivative of this chargevanishes ∂ Q = Z d x ∂ j ( x ) = − Z d x ~ ∇ · ~j ( ~x ) = 0 , (3)at least if surface terms can be neglected. Then, so the story goes, we have a time independent Hermitian operatorthat can be exponentiated to provide a unitary representation of the symmetry group. It is here that the storybecomes more complicated. As with all things in physics there is often a gotcha .To better understand what the gotcha is, define the operator Q Ω = Z Ω d x j ( x ) , (4)where the integration is over a finite three volume Ω. The good thing about his operator is that it exists and the localnature of the commmutator of the current with local fields guarantees thatlim Ω →∞ [ Q Ω , φ ( ~x )] (5)exists, since once Ω is larger than the intersection of either the past or future light cone of the point ~x with thesurface of integration in Eq.5. This observation tells us that by taking the limit Ω → ∞ in all commutators of Q Ω with all local observables we obtain an automorphism of the space of local observables: the question is whetherthis is an inner automorphism ? In other words, is the a Hermitian operator Q defined on the Hilbert space thatgenerates the same automorphism. If so, it can be exponentiated. Furthermore, if the conserved currents close tothe algebra of a compact Lie group, then so will the conserved Hermitian charges and therefore they will generatea unitary representation of the Lie group on the space of physical states. In that case we say that the conservedcurrents are realized as a Wigner symmetry . The hallmark of a Wigner symmetry is that the states are groupedinto finite dimensional representations of the Lie group (since it is compact) and they all have the same mass; i.e.,they are degenerate. Also, using the Wigner-Eckart theorem, we are able to relate matrix elements of operatorsthat transform as irreducible representations of the group, to one another. Hence, we get relations between couplingconstants, transition matrix elements, etc.Suppose, however, that the automorphism defined by the conserved currents is not inner? This can only be thecase if the limit of the operator Q Ω fails to exist as Ω → ∞ . This will happen if there is a massless particle coupledby the current to the vacuum state. When this happens the existence of the conserved currents implies that thesystem has a non-trivial symmetry, but this symmetry is no longer realized by having the states of the theory bundledinto nice finite dimensional irreducible representations of the Lie group. Rather, the consequences of the symmetryare exact low-energy theorems controlling the low energy behavior of the massless particle that is coupled by theconserved currents to the vacuum. Such a symmetry is not immediately obvious to us and for that reason I will adoptColeman’s terminology and refer to it as a hidden or Goldstone symmetry. This is the sort of symmetry realized bythe conserved axial vector currents in QCD in the limit of vanishing quark masses. The Goldstone bosons, i.e. themassless particles coupled to the vacuum state by these currents, are the π, K and η mesons. Some of the consequencesof this symmetry are: the Goldberger-Treiman relation, the Adler-Weissberger realtion, the PCAC self-consistencyconditions, the Dashen-Weinstein theorem on the form factors in K l decay, and Weinberg’s theorem on the behaviorof low energy π − π scattering. This list is by no means complete, I give it only to convince you that there is a greatdeal of experimental evidence that points to the fact that the axial current part of chiral SU (3) × SU (3) is realizedas a hidden or Goldstone symmetry, whereas the symmetry generated by the vector currents is of the Wigner type.Of course, since this talk is about massive degeneracies, I should explicitly point out that the existence of themassless particle created by this current means that the lowest energy state of the theory is enormously degenerate.This is because we can add any number of zero momentum massless particles to the vacuum without increasing theenergy. It is the fact that the light-front formalism insists that the vacuum is the unique lowest energy state thatmakes reconciling the light front treatment of QCD with the real world so problematic. I would also like to point outthat any argument that says in hadron condensates can explain the Goldstone boson phenomenon is simply incorrect,in that it cannot explain this huge vacuum degeneracy. A. Finite Volume Considerations
I now have to say a few words about what happens if I make the spatial volume finite, instead of infinite. Why doI feel compelled to do this? Because all non-perturbative approaches to dealing with QCD involve beginning with asystem in a finite volume and then taking the volume to infinity, and when the volume is finite, the ground state ofthe theory is unique. Where then is the massive degeneracy I spoke of?To clarify this issue, let us return to the example of the ferromagnet. Imagine we make a state that is the tensorproduct of essentially the same norm one spin state on each lattice site, and assume that this state is constructedso that the expectation value of the spin (or magnetization) points in a definite direction. This product state is acontender for the infinite volume magnetized state.If one now applies a rotation to the spins on each site one obtains a new state for which the magnetization pointsin another direction. Now, because the volume is finite the overlap of two such states is given by | Ψ i = N Y i | ψ i i and | Φ i = N Y i | φ i i (6) h Ψ | Φ i = N Y i h ψ i | φ i i . (7)Note, since two non-aligned states of unit length have an overlap whose magnitude is less than one; i.e., || h ψ i | φ i i || < N . Furthermore, for a spin-spin Hamiltonian it is clear that h Ψ | H | Ψ i = h Φ | H | Φ i , (8) h Ψ | H | Φ i ≈ X N → , (9)where X is some number less than unity. Since the different states are not orthogonal (but the are unit length) onecan use them to form an orthonormal basis and diagonalize the Hamiltonian truncated to this space of states. Thesewill be the correct lowest lying eigenstates in the limit of large N and they will be split by an amount that goesto zero as N → ∞ . It is the fact that these states become split by exponentially small amounts as the number ofsites gets large that explains how a finite size ferromagnet seems to form. Clearly, the state in which the ferromagnetpoints in a definite direction is a linear combination of the eigenstates we constructed. Since the splitting betweenthese states is so small, turning on a small magnetic field will put the system into this magnetized state. If this fieldis then turned off adiabatically the different eigenstates will evolve in time with slightly different phases, due to theirenergy differences. However since these energy differences are exponentially small, it will take an exponentially longtime to see the magnetization vanish.The key point of all of this, is not why we can see ferromagnets that have a finite volume. Rather it is that thatthe signal of the infinite degeneracy of the infinite volume limit, is an enormous number of nearly degenerate stateswhose number grows rapidly with increasing volume. This enormous degeneracy is what a light front calculation,done in finite volume, should see. This is what, to the best of my knowledge, isn’t apparent yet. The statement thatthe virtue of the light front is that the vacuum state is empty (and unique) is, in the case of spontaneous symmetrybreaking, a problem, not a virtue. V. WHAT HAPPENS IN THE INSTANT FORMALISM ?
Having criticized the light front approach because it doesn’t make it easy to address Goldstone symmetries, it wouldbe remiss of me to not argue that this problem is less difficult in the instant formalism. I will now contend that for anon-Abelian gauge theory, such as QCD, the formation of Goldstone bosons is an inescapable property of the strongcoupling limit of the theory.
A. The Schwinger Model
Given the limitations of time, I will begin with a very short discussion of the 1 + 1-dimensional Schwinger model,because it exhibits most of the physics I wish to discuss. After that I will make an even briefer foray into QCD. Thepoint of this, as I have already said, is to show that for these theories it is trivial to argue that the strong couplinglimit of the theory explains why the vacuum state can be very degenerate and support the existence of Goldstonebosons.We begin with the formulation of the lattice version of the Schwinger model in A = 0 gauge. The setting forthe model is a 1-dimensional lattice whose sites are labelled by the integer j . The fermions in this model live onthe sites and are represented by the two-component fermion field ψ j and its conjugate ψ † j . The Abelian gauge fieldof the model, A j , lives on the link joining the pair of lattice sites j and j + 1, The conjugate variable to A j is theelectric field variable E j = ˙ A j and they satisfy the canonical commutation relations [ A j , E j ′ ] = i δ j,j ′ . Since thevariable A appearing in the Schwinger model Lagrangian, it follows that the Maxwell equation coming from varyingthe Lagrangian with respect to A will not be an equation of motion. With these definitions, if we follow the usualprescription, we construct the Hamiltonian of the generic form of the lattice Schwinger model: H = H E + H f , (10)where H E = g X n E n H f = X n,n ′ ( ψ † n ) α K ( n − n ′ ) αβ e − i P n ′− j = n A j ( ψ n ′ ) β , (11)where the kinetic term K ( n − n ′ ) αβ is a two-by-two matrix for each value of n − n ′ , the fermion fields satisfy theanti-commutation relations (cid:8) ( ψ † n ) α , ( ψ n ′ ) β (cid:9) = δ n,n ′ δ α,β . (12)The link fields satisfy the harmonic oscillator commutation relations given above.Now in any number of dimensions, the missing Maxwell equation is just the Gauss law. In one dimension this lawtakes the particularly simple and suggestive form G j = E j +1 − E j − ψ † j ψ j = 0 . (13)The important fact is that although this equation is not one of the Euler-Lagrange, or Heisenberg equations of thetheory, it follows from the specific form of the Hamiltonian that all of the operators G j commute with the Hamiltonian;i.e., [ G j , H ] = 0 ∀ j. (14)From this it follows that, although the theory contains states that do not satisfy the Gauss law, we are free torestrict ourselves to the subspace of states which do, since the Hamiltonian will never take us out of this subspace.Furthermore, since the operators G j are precisely the operators that generate local time-like gauge-transformations,all gauge invariant operators must also commute with the G j ’s. From this discussion we see that by beginning in A = 0 gauge we over quantize the theory, in that canonical manipulations can create more states than we wish;however, thanks to gauge invariance we can select a subspace of states that gives us the theory we are interested in.Obviously, since we have the choice of how to choose our basis states, we see that satisfying the Gauss law will bemost easily done if we work in a basis in which the operators E j and ρ j = ψ † j ψ j are diagonal. Since we are dealingwith fermions, we have the possibility of having four possible fermion states: these correspond to the state having noparticles, one particle of charge −
1, or one anti-particle of charge +1, or finally, one particle and one anti-particle ona single lattice site. Note, if we impose the Gauss condition then specifying the charges on each site completely (upto a constant background field which we will take to be zero) specifies the state. Note, since the Hamiltonian onlycontains the operators e i A j , it can only couple together states in which the electric field changes in absolute value byone unit.At this juncture I am in a position to keep my promise and argue that in the limit g → ∞ the Schwinger modelhas an infinitely degenerate ground-state. Moreover, I will argue that for very large, but finite g the theory has aGoldstone boson that, due to the anomaly, fails to appear as we take the limit g → g P j E j means that when g ≫ E j ) costs a lot of energy. Thus, the ground state of the theory in this limit must be a state where E j = 0 for alllinks. By the Gauss condition this means that the charge on each site must vanish. However, we have already seenthat there are two possible zero charge states for each site. Thus we see that in the large g limit, for a lattice with V sites, there will be 2 V degenerate states with energy zero. All other states will have infinite energy. Now, if we take g large but finite, then we observe that the kinetic term can separate a pair and create a particle and anti-particleon different sites joined by a unit of flux. Since this is a high-energy state the energy denominator appearing inperturbation theory is large and so we are invited to treat the effects of the kinetic term on the ground state bysecond order perturbation theory. However, since the ground is so degenerate, we must do degenerate perturbationtheory, since the different degenerate states are mixed by the kinetic term. This leads us to an effective Hamiltonianwhich is our friend the Heisenberg anti-ferromagnet, and as I already said, this theory has a symmetry that is realizedin the Goldstone mode. Of course, there are no little magnetic spins in this case, rather the role of spin has to dowith the chiral charge of the state which, for each site, is the sum of the particle and anti-particle number on thatsite minus one.Time doesn’t permit me to discuss this model further, especially the interesting story of what happens to theGoldstone mode in the continuum (i.e. g →
0) limit. Since what happens in this case is specific to the anomaly inthe axial current, it presumably is not relevant to the case of the octet of axial currents in QCD. I refer you to mypaper with Kirill Melnikov[2] on the subject for all of the details.
B. What About QCD ?
At this point I can only give the briefest summary of what happens in QCD. If one works in the correspondingversion of A = 0 gauge, the story parallels that of the Schwinger model. Once again, gauge invariance requires thatin the large coupling limit the color charge on every site must vanish in order to avoid having non-vanishing flux onany link. Thus, each site can have as many q ¯ q or qqq states as are allowed by the exclusion principle. If we simplyfocus on the possible mesons that there are a large number of zero energy states on each site. As in the case of theSchwinger model we do degenerate second order perturbation theory to understand what happens when we turn onthe kinetic terms.The result is that we obtain a frustrated SU (12) × SU (12) anti-ferromagnet. The frustration is due to the presenceof next nearest neighbor hopping terms. The same terms break the global symmetry to chiral SU (3) × SU (3). It isstraightforward to show that, in this system, the vector SU (3) symmetry is realized in Wigner fashion, meaning thatthere are degenerate SU (3) multiplets of particles with non-vanishing mass, but the axial part of the symmetry isrealized in Goldstone mode. The massless multiplet of mesons are the π , K and η mesons. Another bonus is that inthis limit we get the good predictions of the ratio of magnetic moments obtained in the old SU (6) symmetry scheme,but none of the bad predictions. A complete discussion of this approach appears in paper by myself, Sid Drell, HelenQuinn and Ben Svetitsky[3]-[4]. VI. THE CHALLENGE
This talk can be summarized as follows: • Exact symmetries can be realized in Wigner or Goldstone mode. • When a symmetry is realized in Wigner mode the states of the theory form degenerate irreducible representationsof the symmetry group and the lowest energy state is unique. • When a symmetry is realized in Goldstone mode the lowest energy state of the theory is infinitely degenerate,the states of the theory do not form irreducible representations of the symmetry group and there are masslessparticles coupled by the conserved currents to any one of the possible ground states. • In finite volume the signal of a Goldstone realization of a symmetry is that the number of nearly degeneratestates grows rapidly with increasing volume and the gap between these states shrinks exponentially with thevolume. • The existence of a condensate such as the magnetization, for a ferromagnet, or the staggered magnetization foran anti-ferromagnet, signals a Goldstone symmetry. This is because this condensate transforms non-triviallyunder the symmetry transformations and so its existence implies the ground state isn’t unique. • PCAC means that the pion, kaon and eta are would be Goldstone bosons of the theory where the quark massesare set to zero. This interpretation is overwhelmingly supported by experimental data. This means that theseparticles are really the wiggling of the order parameter or condensate. • Finally, in order for the Goldstone particle to exist there has to be something to wiggle every place where theparticle can exist.
This means that the condensate that is the order parameter for this Goldstone symmetrycannot be confined to the interior of hadrons.
Thus, to reiterate, the challenge for the Light Front is to show how the formalism gives rise to this sort of patternof degeneracy when the physical volume of space becomes large. [1] M. Weinstein, Springer Tracts Mod. Phys. , 32 (1971).[2] K. Melnikov and M. Weinstein, Phys. Rev. D , 094504 (2000) [arXiv:hep-lat/0004016].[3] B. Svetitsky, S. D. Drell, H. R. Quinn and M. Weinstein, Phys. Rev. D , 490 (1980).[4] M. Weinstein, S. D. Drell, H. R. Quinn and B. Svetitsky, Phys. Rev. D22