Elementary example of exact effective-Hamiltonian computation
EElementary example of exact effective-Hamiltonian computation
Stanisław D. Głazek ∗ Institute of Theoretical PhysicsFaculty of Physics, University of Warsaw,Pasteura 5, 02-093 Warsaw, Poland (Dated: December 26, 2020) a r X i v : . [ h e p - t h ] J a n bstract We present an exact computation of effective Hamiltonians for an elementary model obtainedfrom the Yukawa theory by going to the limit of bare fermions being infinitely heavy and barebosons being at rest with respect to the fermions that emit or absorb them. The coupling constantcan be arbitrarily large. The Hamiltonians are computed by solving the differential equation of therenormalization group procedure for effective particles (RGPEP). Physical fermions, defined in themodel as eigenstates of the effective Hamiltonians, are obtained in the form of an effective fermiondressed with a coherent state of effective bosons. The model computation illustrates the method thatcan be used in perturbative computations of effective Hamiltonians for realistic theories. It showsthe mechanism by which the perturbative expansion and Tamm-Dancoff approximation increase inaccuracy along the RGPEP evolution.
I. INTRODUCTION
Complexity of relativistic quantum field theory (QFT) implies a need for approximatecomputational methods. One needs a systematic scheme for improving their accuracy. Thatis the case in computing observables using expansion in powers of a small coupling constant,solving eigenvalue problems using a limited basis in the space of states or using renormal-ization group methods. A combination of all three of these techniques requires a clear-cutpattern to follow. Such pattern can only be provided by an exactly solvable model, becauseone needs the exact solution to unambiguously assess accuracy of the approximate calcula-tions. On the other hand, to obtain an exactly solvable model, one has to simplify a theory.A compromise needs to be struck between simplifying and obtaining a helpful pattern.This article presents a novel, exact renormalization-group computation of effective Hamil-tonians in a model that results from drastic but precisely specified simplifications of QFT,so that one can see the steps that would have to be reconstructed in an analogous computa-tion of the effective Hamiltonians and their spectra in QFT. The presentation is thus quitelimited but it includes enough of the QFT features for addressing the issues of high ordersof perturbation theory, few-body approximations in the Fock space, renormalization-groupimprovements and the form of effective Hamiltonians that change, but not limit the number ∗ [email protected]
2f interacting field quanta.The computation presented in this article concerns a model that is obtained by drasticallysimplifying the Yukawa theory. The simplifications made here partly resemble the onesthat Wilson adopted in formulating his approach to renormalization using a Yukawa-likemodel [1], but they go much further. As a result, the ultraviolet divergences of a local theoryare eliminated at the outset. This is useful because the goal of the presented computation isnot to find the ultraviolet counter terms using the triangle of renormalization [2], as it was inWilson’s case, but to deal with the issue of computation of an effective theory Hamiltonianafter the right counter terms have already removed the divergences. The model computationincludes a pattern of handling terms that are analogous to the finite parts of counter terms.It should be stressed that the history of models that incorporate elements of the Yukawatheory and are helpful in understanding renormalization in QFT beyond the weak-couplingexpansion have a long history [3]. There are exactly solvable models among them, e.g. [4].Also, a model may employ some elements of the Yukawa theory formalism and be exactlysolvable without encountering any need for renormalization. For example, a class of two-levelmodels for a system of a fixed number of fermions, whose Hamiltonians can be written usingbilinear products of fermion creation and annihilation operators, could be solved exactly.One takes advantage of the SU(2) symmetry associated with the two levels [5, 6] or uses thesymmetry of the model’s boson representation [7]. One can even show that such fermionsystems exhibit thermalization when they are weakly coupled to a boson bath [8]. Thisvariety of models that can be solved suggests to the author it should be clearly stated thatthe main purpose here is different. It is to apply a recently formulated renormalization groupequation for Hamiltonians of QFT, to a simplified, one-level model for fermions coupled tobosons in a way that is analogous to the Yukawa theory coupling. In the model case, anexact operator solution to the equation is obtained in the form of a whole family of effectiveHamiltonians that are strictly equivalent. They all act in an infinite dimensional Fock space.Their common spectrum is obtained as a byproduct of the solution to the renormalizationgroup equation. Comparisons and comments concerning the most similar models known tothe author are provided in Sec. VIII.The model used here is defined using the front form (FF) of Hamiltonian dynamics [9]instead of the instant form (IF) used in [1]. It is known that to obtain the Wilson model fromthe front form of Yukawa theory one needs to consider the limit of fermions that are much3eavier than the momentum cutoff parameter [10], say Λ . Here, in addition, also the bosonmass is assumed much greater than Λ . This limit is called the static limit, since bosonsemitted or absorbed by fermions do not move with respect to their source. Further, themodel we use does not include isospin, which leads to a significant simplification: only fourdistinct operators appear in the effective Hamiltonians. This feature will become clear inthe course of computation. Despite these far reaching simplifications, the model interactionHamiltonian changes the number of bosons and the number of Fock components involved inthe dynamics is infinite.To compute the effective Hamiltonians, we use the method called the renormalizationgroup procedure for effective particles (RGPEP). The RGPEP differs conceptually from theWilson renormalization group procedure. Namely, instead of integrating out high-energymodes in the basis of the space of states in which the Hamiltonian acts, one changes the basisin the space of operators to which the Hamiltonian belongs. In other words, the Hamiltonianis seen as an element of the operator space formed by normal-ordered polynomials of barecreation and annihilation operators. The change of basis in the space of such polynomialsis obtained by replacing the bare creation and annihilation operators with the analogousones for the effective quanta of fields, called effective particles, see Sec. III for details. Inperturbation theory, the effective particle operators are polynomials in terms of the bareparticle operators and vice versa [11]. The interactions of effective particles are limited bythe running cutoff Λ that provides an upper bound on the magnitude of the invariant masschange that an interaction can cause. The RGPEP evolution of the computed Hamiltoniansdescribes the variation of their form with the running cutoff.The RGPEP employs the rules of the similarity renormalization group procedure forHamiltonians [12, 13] and takes advantage of the double-commutator feature of Wegner’sflow equation for Hamiltonian matrices [14]. In application to local QFT, the RGPEPhas been recently illustrated in [16], which also includes references to the previous works.However, in all these examples one is forced to use the approximations that are not underprecise control, such as the mentioned earlier weak-coupling expansion [2] or a limitation ona number of virtual particles, called the Tamm-Dancoff (TD) approximation [17, 18]. Theseapproximations obscure the core features of the RGPEP in the context of realistic theories.In contrast, the exact RGPEP computation of effective Hamiltonians in the model describedhere is quite transparent and the result has a clear interpretation in terms of the Fock-space4mage of physical states.The RGPEP equation we solve, see Eq. (15), determines the evolution of Hamiltoniansusing not the cutoff parameter Λ itself, but the parameter that is denoted by t and corre-sponds to Λ − . Thus, t varies from zero for the initial Hamiltonian to infinity for its final,diagonal form, in which all mass-changing interaction terms disappear. The ability to di-agonalize Hamiltonians is the key design feature of the RGPEP equation. Quite generally,the design secures that the first-order solution of the RGPEP evolution equation results invertex form factors whose width in momentum variables varies with t . The width tends toinfinity or some cutoff value when t → and to zero when t → ∞ . In QFT, these formfactors regulate singularities of the local interactions, e.g. see [16], and can be thought of ascorresponding to a finite size of the effective particles. However, in the model solved here thesituation is much simpler because of the static limit. One only obtains a running couplingconstant, denoted by g t , instead of a function of momentum, since the interacting particlesare at rest with respect to each other.The paper is organized in the following way. Section II describes the model Hamiltonian, cf. [3]. The model is derived in the FF of dynamics using the static limit of the Yukawatheory in Sec. II A. It is rewritten in a more familiar energy notation of the IF of dynamics inSec. II B. Section III describes solution of the RGPEP equations. First the equations designis explained in Sec. III A and then the discussion of solutions follows in Sec. III B. Operatorsthat create and annihilate effective particles are derived in Sec. IV, with final formulas inSec. IV A. Exact spectrum of the model Hamiltonian in the Fock space representation isgiven in Sec. V. The issue of approximate computations is addressed in the remaining partof the article. Section VI discusses the weak-coupling expansion. The TD approximation isdiscussed in Sec.VII. Subsequently, Sec. VIII introduces the concept of effective TD Hamil-tonian matrices, including comments and comparisons to related work on similar models.Section IX briefly outlines the ways of comparing the model solution with realistic theories.Section X concludes the article and reviews motivation for studies of QFT using the RGPEP. II. MODEL HAMILTONIAN
The model Hamiltonian we consider is obtained from the Yukawa theory using results ofRef. [10]. In that work, Eq. (2.1) displays the canonical front-form Hamiltonian of Yukawa5heory that has the structure H c = H f + H b + H fb + [other terms] , (1)where H f stands for the Hamiltonian of free fermions of mass m , H f = (cid:88) σ (cid:90) [ p ] m + p ⊥ p + b † pσ b pσ , (2) H b denotes a free Hamiltonian for bosons of mass µ , H b = (cid:88) σ (cid:90) [ p ] µ + p ⊥ p + a † p a p , (3) H fb is the fermion-boson interaction term, H fb = g (cid:88) σ ,σ (cid:90) [ p p p ] δ c.a ¯ u Γ u × b † p σ ( a † p + a p ) b p σ . (4)The bracket [other terms] indicates the terms that disappear in comparison with the firstthree in the limit of the fermion mass m → ∞ . The symbol [ p ] conventionally denotes themeasure d p ⊥ dp + θ ( p + ) / [2 p + (2 π ) ] and δ a.c is π ) δ ( P c − P a ) , where P c and P a denotethe total + and ⊥ momenta of the bare particles that are created and annihilated by theinteraction, respectively. Fermion spinors are denoted by u and u and the matrix Γ is setto 1. The coupling constant is denoted by g . Further model construction steps use only thefirst three terms in Eq. (1). A. The static limit
As a result of steps fully described in [10], the Hamiltonian H = H f + H b + H fb is alteredin a way that leads to a formula resembling Eq. (2.17) in that reference. One considers afermion eigenstate of H that carries an arbitrary momentum P + and P ⊥ and a fixed value ofspin projection on z -axis. The state is a combination of the Fock component with one barefermion in the same spin state and infinitely many Fock components each of which containsone bare fermion and some natural number of bare bosons. The fermion mass eigenvalueis written as M = m + E , where E/m (cid:28) . Every boson kinematic momentum in a Fockcomponent with n bosons is parameterized according to the rule p + n,i = y n,i P + , (5) p ⊥ n,i = y n,i P ⊥ + κ ⊥ n,i , (6)6here < i ≤ n . The corresponding fermion momentum is given by p + n = x n P + , (7) p ⊥ n = x n P ⊥ − κ ⊥ n,i ... − κ ⊥ n,i , (8)where x n = 1 − y n, ... − y n,n . (9)In the absence of bosons, the bare fermion carries the whole P + and P ⊥ . Approximationsdescribed in [10] are based on the conditions that force κ ⊥ n,i (cid:28) m and (cid:80) ni =1 y n,i (cid:28) . Asufficient condition is provided by imposing a cutoff that forces all bosons to only havemomenta relative to the fermion that are negligible in comparison with the fermion mass.The interaction Hamiltonian H fb is supplied with a cutoff form factor, denoted belowby f Λ . One assumes that Λ (cid:28) m . The cutoff function enforces the condition | P c − P a | < Λ (cid:28) m , where P c and P a denote the free total four-momenta of created and annihilatedparticles, respectively. In consequence, all the fractions y i defined above tend to 0 and thefermion fractions x n → . The resulting Hamiltonian that determines the mass eigenvalue M for a physical fermion, see Eqs. (2.16) and (2.17) in [10], takes the form H fermion = mb † b + H b † b , (10)where b denotes annihilation operator for a bare fermion at rest and only one spin projectionon z -axis. The operator H is the boson Hamiltonian associated with the states that containone bare fermion, H = (cid:90) [ q ] (cid:104) (1 / (cid:18) q + + µ + q ⊥ q + (cid:19) a † q a q + gf Λ ( q ) ( a † q + a q ) (cid:105) . (11)The boson momenta in all Fock sectors are identified according to the same relations q + = y n,i m and q ⊥ = κ ⊥ n,i . The half of the round bracket in Eq. (11) equals energy of a boson withmomentum (cid:126)q in which q z = q + − ( µ + q ⊥ ) /q + . It is possible to determine the allowedmomenta for bosons, including the sampling that Wilson adopted, by choosing the function f Λ ( q ) . At this point one can further proceed as in [10] and show that when the initial Yukawatheory includes isospin, the Hamiltonian one obtains in place of H also includes isospin andmatches the model Hamiltonian studied in [1]. In what follows a different path is taken.7he next great simplification step we make here, which was not made in [10], is to assumethat the boson mass µ is also much larger than the cutoff Λ . This assumption implies thatthe bosons cannot move with respect to the fermion that emits or absorbs them. In thelimit Λ /µ → , the Hamiltonian H involves only bosons that are practically at rest relativeto the fermion. One replaces all bosons nearly at rest with respect to the fermion by justone static boson mode, for which q ⊥ = 0 and q + = µ . We consider the model in which µ/m → . Finally, one can allow the static bosons to appear in the model also without afermion. This way one arrives at the Hamiltonian of a model for which the exact effectiveHamiltonians are computed using the RGPEP in the sections that follow. B. Intuitive notation
The model Hamiltonian introduced in the previous section is rewritten here using anintuitive notation that does not require familiarity with the front form of dynamics andinstead relies on the intuition rooted in the IF of dynamics, H = E f b † b + E b a † a + gE I b † ( a † + a ) b , (12)where E f , E b and E I are the fermion, boson and interaction energy parameters, g is acoupling constant, while b and a are annihilation operators for the static fermion and boson,respectively. These operators and their hermitian conjugates are normalized to obey thestandard (anti)commutation relations, of which the only nonzero ones are { b, b † } = 1 , (13) [ a, a † ] = 1 . (14)The Hamiltonian describes fermions of just one spin state and preserves their number, whichcan only be 0 or 1. The number of bosons is neither specified nor limited and it varies as aresult of interactions.In states with the fermion number equal zero, the interaction vanishes and the spectrummatches the one of a Hamiltonian for free bosons at rest, H b = E b a † a , with eigenvalues E bn = nE b , where n is zero or a natural number. The corresponding normalized eigenstatesof H b are | n (cid:105) = ( n !) − / a † n | (cid:105) . In states with the fermion number equal 1, the Hamiltonianchanges the boson number by 1 and the distribution of bosons needs to be computed.8e apply the RGPEP to this model in the remaining part of this work. This means thatinstead of directly evaluating all of the Hamiltonian eigenvalues and eigenstates in termsof bare quanta, one introduces creation and annihilation operators for effective bosons andfermions and computes the effective Hamiltonians for them. The eigenstates of these effectiveHamiltonians are then found in terms of the basis in the Fock space that is constructedusing the creation operators of the effective particles instead of the bare ones. The exerciseis meant worth carrying out since one can unfold the simplifications used in deriving H ofEq. (12) and look at the dynamics of Yukawa theory anew from the perspective of the modelcomputation. III. COMPUTATION OF THE EFFECTIVE HAMILTONIAN
In the model considered here, the RGPEP equations for a family of renormalized Hamil-tonians, labeled by parameter t , can be written in the operator form, ddt H t = [ G t , H t ] , (15) G t = [ H f + H b , H t ] , (16)where G t is called the generator. The initial condition at t = 0 is provided by H of Eq. (12),which is denoted for that reason as H .Equations (15) and (16) resemble Wegner’s flow equations that describe the evolutionof band-diagonal Hamiltonian matrices as functions of their width on energy scale; thewidth decreases as t increases [14]. There are two differences. One is that Eq. (15) cannotbe represented exactly by finite matrices, because the commutation relations for a and a † cannot. The other one is that the generator G t is a commutator of H t with the sum H f + H b that does not depend on t , cf. [19]. In the Wegner generator, the Hamiltonian matrixis commuted with its diagonal part that varies with t . It should be noted that Eq. (15)is written for the opertaor H t that only contains t -independent creation and annihilationoperators for bare particles, which are replaced by the corresponding t -dependent operatorsfor effective particles in order to obtain the renormalized Hamiltonians H t , see below.9 . Design of Eqs. (15) and (16) Design of Eqs. (15) and (16) originates in the idea that one can consider the Hamiltonianeigenvalue problems in local QFT in terms of some kind of effective quanta instead of thebare ones. The change from bare to effective quanta is motivated by the concept thatthe effective quanta interact in a so much less violent way than the bare quanta do thatthe eigenvalue problem may be convergent in the effective Fock-space basis, even if it doesnot exhibit convergence in the bare Fock-space basis. The appearance of convergence is aconsequence of the vertex factors that emerge in solutions of Eq. (15). Emergence of suchfactors is the feature of double-commutator equations like Eq. (15) with the generator givenby Eq. (16). The model application discussed here shows this feature in a simplified way, seebelow, and demonstrates how convergence in the effective Fock-space basis improves withincrease of t .The key examples of physical elementary particle systems in terms of which one canthink about the design of Eqs. (15) and (16) are hadrons. In QCD, represented in terms ofbare quanta, hadrons are complex mixtures of infinitely many quarks and gluons that areconfined. In the particle tables, most of the known hadrons are classified as bound statesof just a few constituent quarks. The design of the RGPEP equations can be described asaiming at the derivation of a mathematically precise connection between these two picturesof hadrons.The creation and annihilation operators for effective particles are defined using a unitarytransformation of the form q t = U t q U † t , (17)where q stands for the operators a , a † , b or b † , and U † t = T exp (cid:18)(cid:90) t dτ G τ (cid:19) . (18)The symbol T denotes ordering in τ . The Hamiltonian operator H t = U t H t U † t (19)is defined to be the same as the initial one, H t = H , but H t is expressed in terms of theeffective particle operators a t , a † t , b t and b † t instead of the initial operators a , a † , b and b † thatcorrespond to t = 0 . Thus, in H t , the coefficients of products of the effective creation and10nnihilation operators are different from the coefficients of products of the correspondinginitial operators in H . The coefficients in H t contain factors that follow from the double-commutator structure of Eq. (15). These factors are obtained in the process of solvingEq. (15). They emerge in a way similar to the emergence of the band-diagonal matricesfrom the Wegner flow equation.If the model were divergent, as it is the case for bare Hamiltonians in local QFT, H would be supplied with the counter terms that would be computed from the condition thatthe coefficients of effective particle operators in H t for any finite, fixed value of t are notsensitive to the adopted regularization of the divergences. Since the model Hamiltonianof Eq. (12) does not generate divergences, the computation of counter terms to divergentexpressions is not needed and this aspect of local QFT is not illustrated in the model solution.The divergence counter term computation in QFT significantly complicates the RGPEPprocedure with a lot of details that depend on the adopted regularization. These largelyarbitrary details obstruct the conceptual view of the method while the model computationmakes it clear. Counter terms appear in the model computation only in a finite form, whichis analogous to the appearance of the unknown finite parts of the divergence counter termsin local QFT. B. Solution of Eq. (15)
In order to solve Eq. (15), one writes H t = ( E f + δE ft ) b † b + E b a † a + g t E I b † ( a † + a ) b , (20)where the subscript t indicates dependence on that argument. Only four distinct Fock-spaceoperators appear in this formula because no other operators are generated from the initialcondition of Eq. (12). Using a dot to indicate the derivative, one obtains Eq. (15) in theform δ ˙ E ft b † b + ˙ g t E I b † ( a † + a ) b = [ G t , ( E f + δE ft ) b † b + E b a † a + g t E I b † ( a † + a ) b ] , (21) G t = g t E b E I b † ( a † − a ) b . (22)11he generator takes the simple form since the fermion number is conserved by the interaction.Evaluation of the commutator on the right-hand side of Eq. (21) yields δ ˙ E ft b † b + ˙ g t E I b † ( a † + a ) b = − g t E b E I b † ( a † + a ) b − g t E b E I b † b . (23)Equating coefficients in front of the same operators on both sides, one gets δ ˙ E ft = − g t E b E I , (24) ˙ g t = − g t E b . (25)These are ordinary differential equations and solving them leads to the solution of Eq. (15)in the form H t = (cid:2) E f + g t ∆ t (cid:3) b † b + E b a † a + g t E I b † ( a † + a ) b , (26)where g t = ge − E b t , (27) ∆ t = (1 − e E b t ) E I /E b . (28)This result shows that the increase of t from zero to infinity causes the effective fermion-boson coupling constant g t to decrease exponentially fast from its initial value g to zero atthe rate given by an inverse of the boson energy squared. This is the promised suppressionof interactions by the vertex factor. One obtains the vertex factor in this model solely inthe form of a varying coupling constant g t instead a whole form factor that is a functionof momentum and energy transfer between quanta in the vertex. The simplification occursbecause the model contains only static modes for fermions and bosons.The boson energy E b stays constant as a function of t . The fermion energy, E f + g t ∆ t ,evolves from the initial value E f to the final fermion eigenvalue energy E f ∞ = lim t →∞ E f + g t ∆ t = E f − g E I /E b . (29)It seems that E f ∞ may be negative. However, it could only happen outside the range ofapproximations made in the model, where the fermion energy E f is assumed much larger12han the boson energy E b and much larger than the energy change due to the interaction, gE I , while E b and E I are of similar magnitude. Therefore, for any fixed value of g , one onlyconsiders E f much larger that g E I /E b . IV. EFFECTIVE PARTICLES
Solution for the operator H t in Eq. (26) is transformed into the Hamiltonian for effectiveparticles using the operator U t according to Eq. (19). The result is H t = (cid:0) E f + g t ∆ t (cid:1) b † t b t + E b a † t a t + g t E I b † t ( a † t + a t ) b t , (30)where b t and a t are given by Eq. (17). Knowing G t in Eq. (22), one obtains from Eq. (18)that U † t = e c t b † ( a † − a ) b = 1 + (cid:104) e c t ( a † − a ) − (cid:105) b † b , (31)where c t = ( g − g t ) E I /E b . (32)Therefore, a t = a (1 − b † t b t ) + b † t a b t , (33) b t = e c t ( a † − a ) b . (34)Analogous formulas hold for creation operators a † t and b † t , obtained by hermitian conjugation. A. Effective particle operators
It is visible in Eq. (33) that the effective boson operators a t are equivalent to the bareones in the subspace of Fock space without effective fermions, for in that case b t ≡ . In thesubspace that contains one effective fermion, one has b † t b t ≡ and is left with a t = b † t a b t , (35)and a corresponding relation for a † t . Evaluation yields a t = ( a + c t ) b † b , (36)13nd a † t is obtained by conjugation.In summary, the annihilation operator for effective fermion, b t , is given by Eq. (34), andthe annihilation operator for an effective boson is a t = a + c t b † b , (37)where c t is given by Eq. (32). The corresponding creation operators are obtained by hermi-tian conjugation. Using these results, one can check by a direct calculation that the effectiveHamiltonian H t of Eq. (30) is equal to the initial Hamiltonian H = H of Eq. (12). V. EXACT SPECTRUM IN THE FOCK SPACE
One observes that there are three ways of seeking the model Hamiltonian spectrum. Inthe first way, one uses the Hamiltonian expressed in terms of the initial particle operatorsthat correspond to t = 0 . In the second way, one uses the Hamiltonian expressed in termsof effective particle operators for some finite value of the RGPEP parameter t . The thirdway is reduced to inspection of the effective Hamiltonian with t = ∞ . The respective formsof one and the same Hamiltonian H = H of Eq. (12) are H = E f b † b + E b a † a + gE I b † ( a † + a ) b , (38) H t = (cid:0) E f + g t ∆ t (cid:1) b † t b t + E b a † t a t + g t E I b † t ( a † t + a t ) b t , (39) H ∞ = E fermion b †∞ b ∞ + E b a †∞ a ∞ . (40)where ∆ t is given in Eq. (28). Taking into account the commutation relations that theoperators with t = ∞ obey, one sees that the eigenvalues are E fermion = lim t →∞ (cid:0) E f + g t ∆ t (cid:1) (41) = E f − g E I /E b , (42) E n bosons = nE b , (43) E fermion+ n bosons = E fermion + nE b , (44)14nd the corresponding normalized eigenstates are | fermion (cid:105) = b †∞ | (cid:105) , (45) | n bosons (cid:105) = 1 √ n ! a † n ∞ | (cid:105) , (46) | fermion + n bosons (cid:105) = 1 √ n ! a † n ∞ b †∞ | (cid:105) , (47)where | (cid:105) denotes the model Hamiltonian ground state that contains no physical particles.According to Eqs. (33) and (34), b ∞ = e g ( E I /E b ) ( a † − a ) b , (48) a ∞ = a + g ( E I /E b ) b † b . (49)A physical fermion state is composed of the bare fermion and a coherent state of bosons.Since a † − a = a † t − a t = a †∞ − a ∞ , one can speak of the coherent state of bare as well aseffective or physical bosons. The n -boson eigenstates without a fermion are the same as ifthe interaction were absent. VI. WEAK-COUPLING EXPANSION
In the weak-coupling expansion one hopes to gain some insight concerning solutions of atheory assuming that the coupling constant in the interaction terms is a very small number.After evaluating some quantity of interest using expansion in powers of an infinitesimalcoupling, one can check how large the coupling would have to be for the result to matchdata. Then there comes the question of how large the remaining terms in the expansion are.In the model with the coupling constant g not very small, such procedure is not viable asan approximation method for obtaining eigenstates of the Hamiltonian H in terms of bareparticle operators that appear in its form H . This form corresponds to the Yukawa theoryexpressed in terms of bare degrees of freedom. Although the fermion eigenvalue E f ∞ is justa quadratic function of g and one might hope that an expansion up to terms order g maybe sufficient, the fermion eigenstate contains terms with all powers of the product g timesthe bare boson creation operator acting on the vacuum state.Quite different situation is encountered when one uses the Hamiltonian in its form H t with E b t sufficiently large for g t of Eq. (27) to be small. The eigenstates without a fermionare just free effective bosons created by a † t from the vacuum state. The eigenstates with15 fermion are given by b † t | (cid:105) plus admixtures of effective bosons that are created from thefermion state with strength g t instead of g . One sees in Eq. (27) that g t can be small forarbitrarily large g when t is made sufficiently large. In that case, the fermion state can beapproximated well by using the expansion in powers of g t .The mechanism described above can be illustrated by the perturbative expansion up tosecond-order for the fermion energy eigenvalue and the corresponding eigenstate. In general,a perturbative expansion is obtained by writing | ψ (cid:105) = ( ψ + ψ + ψ + ... ) b † t | (cid:105) + ( ψ + ψ + ψ + ... ) a † t b † t | (cid:105) + ( ψ + ψ + ψ + ... ) a † t a † t b † t | (cid:105) + ... , (50)where ψ mn ∼ g nt . The eigenvalue problem reads H t | ψ (cid:105) = ( E + E + E + ... ) | ψ (cid:105) , (51)where E n is of order g nt . Assuming that the dominant coefficient in front of b † t | (cid:105) is ψ oforder 1, one can limit the effective Fock-space expansion to only three terms: one effectivefermiom, one effective fermion and one effective boson, and one effective fermion and twoeffective bosons. Coefficients of the components with more effective particles are of order g nt with n > . By projecting both sides of Eq. (51) on these three basis states, one obtainsthree equations. Projection on the component b † t | (cid:105) yields (cid:0) E f + g t ∆ t − E − E − E − ... (cid:1) × ( ψ + ψ + ψ + ... )+ g t E I ( ψ + ψ + ψ + ... ) . (52)Projection on a † t b † t | (cid:105) leads to (cid:0) E f + g t ∆ t − E − E − E − ... (cid:1) × ( ψ + ψ + ψ + ... )+ E b ( ψ + ψ + ψ + ... )+ g t E I ( ψ + 2 ψ + ψ + 2 ψ + ... ) . (53)16rojection on a † t a † t b † t | (cid:105) produces (cid:0) E f + g t ∆ t − E − E − E − ... (cid:1) × ( ψ + ψ + ψ + ... )+ 4 E b ( ψ + ψ + ψ + ... )+ 2 g t E I [ ψ + ψ + ψ + ... ] . (54)Each of these equations contains terms proportional to powers of g t . Equating coefficientsof 1, g t and g t on both sides of these equations, one arrives at a set of 9 equations that mustbe satisfied simultaneously. Assuming that ψ = 1 , setting ψ = ψ = 0 and introducingthe normalization factor N , one obtains E = E f − E b (cid:16) g t e E b t E I (cid:17) , (55) | ψ (cid:105) = N (cid:20) b † t | (cid:105) − E b g t E I a † t b † t | (cid:105) + 12 (cid:18) E b g t E I (cid:19) a † t a † t b † t | (cid:105) (cid:35) . (56)The term ( g t E I ) /E b in the effective fermion energy in Eq. (39) cancels the second-orderself-interaction term that results from emission and absorption of an effective boson. Thus,even though in the model the fermion self-interaction is finite, the term ( g t E I ) /E b in theeffective fermion energy term in the Hamiltonian H t appears in the role of a finite part ofthe fermion self-interaction counter term when the parameter t tends to zero and its inverseplays the role of a cutoff Λ . The finite part is positive, vanishes when Λ → or t → ∞ and implies that in that limit the effective fermion energy in H t approaches the physicalfermion eigenvalue E fermion . One could replace the effective fermion energy term in H t bythe eigenvalue and ignore the self-interaction effects. VII. THE TAMM-DANCOFF APPROXIMATION
The idea of the TD approximation [17, 18] is to limit the Hamiltonian eigenvalue problemto a subspace of the Fock space defined by a limit on the number of virtual particles. Oneassumes that the eigenstate components with more particles than the limiting number havea small probability and can be neglected in the first approximation. Such approach wasalso proposed in the context of solving QCD in the front form of Hamiltonian dynamics,17sing the idea that a suitable renormalization group algorithm, including the Fock-sectordependent counter terms, could be used to identify the dominant features of the dynamicsas the limit on the number of particles is increased. Subsequently, one could attempt tocompute corrections to the dominant picture using the methods of perturbative expansionand successive approximations [20], including some form of the coupling coherence [21].In case of the RGPEP, the key feature that influences the accuracy of the TD type ofapproach to realistic theories is that instead of the bare, original field quanta one limits thenumber of the effective quanta. The idea is illustrated using Fig. 1. It shows plots of theexpected number of virtual effective bosons in the physical fermion state as a function ofthe RGPEP scale parameter t . The plotted value is defined by (cid:104) N t (cid:105) = (cid:104) fermion | a † t a t | fermion (cid:105) , (57)where the fermion state is given in Eq. (45). Using Eqs. (37) and (48) one obtains (cid:104) N t (cid:105) = g t ( E I /E b ) , (58)which for E I = E b yields the expected number of virtual effective bosons in a physicalfermion, (cid:104) N t (cid:105) = g t = g e E b ( t − t ) . (59)The coupling constant g is the value that g t takes when t = t . We set the value of t to E − b , since this value of the running cutoff corresponds in magnitude to the energy changeassociated with emission or absorption of just one boson. The value of g is arbitrary.To provide examples of the numbers involved, three values of the coupling constant g arearbitrarily selected: 2, 1 and 1/2. The three curves shown in Fig. 1 correspond to thesethree values of g . The number of virtual bosons in a physical fermion strongly depends onthe value of g and these three values are sufficient to illustrate the dependence. Each of thechosen values corresponds to a different value of the bare coupling constant g in Eq. (12), g = eg .It is visible in Fig. 1 that approximations of the TD type with just one or two virtualbosons do not apply in terms of the bare particles if the coupling constant g is not suffi-ciently small. For example, if g = 2 , the expectation value (cid:104) N (cid:105) is almost 30. However,when t grows, the expectation value (cid:104) N t (cid:105) decreases. In the model, where one possesses the18 IG. 1. Expectation value of the number of effective-bosons, see Eq. (57), in the physical fermioneigenstate of Eq. (45) as a function of the RGPEP scale parameter t . The three curves correspondto the three values 2, 1 and 1/2 of the coupling constant g in Eq. (59), defined as the effectivecoupling constant g t for t equal t = 1 /E b , assuming that the free boson energy E b equals thefermion-boson interaction energy parameter E I in the model Hamiltonian, see Sec. VII. It is visiblethat the TD approximation becomes increasingly accurate when t increases, since (cid:104) N t (cid:105) decreasesexponentially fast with increase of t . exact solution to the RGPEP equation, Fig. 1 shows that for t exceeding E − b the effec-tive interaction vertex suppression factor can become so small that the strength of the barecoupling constant is overcome and the TD approximation represents the physical fermionaccurately in terms of a small number of the corresponding virtual effective particles. VIII. TD HAMILTONIAN MATRICES
If the coupling constant g t is sufficiently small and the parameter t large enough for theRGPEP form factors to suppress the interaction terms in H t that change the number ofeffective particles, then the TD approximation may be valid. In that case one can definethe effective Hamiltonian matrices that describe the dynamics in terms of a limited numberof effective Fock-space basis states. We call them the effective TD Hamiltonian matrices, orjust TD matrices, denoted by H TD t . One can compute them following the pattern illustratedbelow in terms of our model.Consider the Hamiltonian H t in which the effective, particle number-changing interactionterm is weak enough to expect that the TD approximation is reasonable. Suppose one is19nterested in an approximate computation of observables for a physical fermion. In themodel, the physical fermion is known exactly. It is represented by the state | fermion (cid:105) inEq. (45). However, in an approximate calculation in a realistic theory a physical fermionstate would not be known exactly.Suppose one expects that the physical fermion state is dominated by the basis state b † t | (cid:105) ,while the basis state a † t b † t | (cid:105) provides the leading correction. Still smaller corrections involvethe basis states a † nt b † t | (cid:105) with n > . To describe the physical fermion state using the TDapproximation, one computes the matrix H TD t that acts on the coordinates of states in thesubspace of the Fock space that is spanned by the basis states with one effective fermion anda limited number of effective bosons. If instead of the physical fermion one were interested inthe properties of states | fermion + n b bosons (cid:105) , one would first compute Hamiltonian matrix H TD t that acts on the coordinates in the subspace spanned by the effective basis states a † n b − t b † t | (cid:105) , a † n b t b † t | (cid:105) and a † n b +1 t b † t | (cid:105) . Corrections would follow from enlarging the matrixto include coordinates in directions of basis states with n b ± effective bosons, etc. A simple illustration of the TD approximation is obtained in case of the physical fermionand the assumption that the matrix H TD t only acts on the two-dimensional vectors ofcoordinates in the Fock-subspace spanned by the basis states b † t | (cid:105) and a † t b † t | (cid:105) . The firstapproximation is obtained by writing | fermion TD (cid:105) = x t b † t | (cid:105) + x t a † t b † t | (cid:105) + 1 √ x t a † t b † t | (cid:105) . (60)Then one observes that the physical fermion eigenvalue probem has the form H t | ψ (cid:105) = E TD | ψ (cid:105) , (61) | ψ (cid:105) = | fermion TD (cid:105) + | n b > (cid:105) , (62)where | n b > (cid:105) stands for all components with more effective bosons than 2. Projecting thisequation on the same range of components that appears in Eq. (60), one obtains the matrix20quation h , h , h , h , h , h , h , x t x t x t + h ,n b > = E TD x t x t x t , (63)where the matrix elements are h m,n = 1 √ m ! n ! (cid:104) | b t a mt H t a † nt b † t | (cid:105) (64)with ≤ m, n ≤ . The TD approximation amounts to setting h ,n b > = 0 . The effectiveTD Hamiltonian matrix is defined by H TD t mn = h m,n . (65)Its eigenvalue problem reads (cid:88) n =0 H TD t mn x n = E T D x m . (66)One has H TD t 22 = E f + g t ∆ t + 2 E b , (67) H TD t 21 = H TD 12 t = √ g t E I , (68) H TD t 11 = E f + g t ∆ t + E b , (69) H TD t 10 = H TD 01 t = g t E I , (70) H TD t 00 = E f + g t ∆ t . (71)The eigenvalues E written in the form E = E f + g t ∆ t + xE b obey the equation (2 − x )(1 − x ) x + α (2 − x ) = 0 , (72)where α = ( g t E I /E b ) . If α were zero due to g t = 0 , the three eigenvalues E n = E f + g t ∆ t + x n E b with x n = n would correspond to a free effective fermion and n free bosons. Assumingthat x = n + yα and neglecting higher powers of α one obtains E n = E fermion + nE b for n equal 0 or 1, as expected on the basis of the exact solution given in Eq. (44) and Eqs. (45)21r (47), respectively. Higher order terms in the expansion of x in powers of α can be usedto compare the TD approximation with the weak-coupling expansion. Next level of the TDapproximation would be obtained by introducing the component a † t b † t | (cid:105) / √ and neglecting h ,n b > .The coordinates x n in realistic theories would not be just numbers but unknown functionsof only n relative momentum variables and discrete quantum numbers of fermions andbosons. The number of momentum arguments would be the same in the non-relativistic andrelativistic theories because the total momentum of the eigenstates drop out from the TDmatrix problem and the eigenvalues E are solely the masses squared of the physical systems.One can use Eq. (63), ignoring h ,n b > , to evaluate the Fock-space coordinate x t in termsof the coordinates x t and x t using a fully non-perturbtive Gaussian elimination or theso-called R operation, the latter when either the boson energy E b is large [22] or g t is small.The Gaussian elimination yields x t = 1 E T D − h , h , x t , (73)which can be put into the remaining two equations. The result is h , + h , E TD − h , h , h , h , h , x t x t (74) = E TD x t x t . (75)This is the TD matrix eigenvalue problem including the fermion self-interaction in thefermion-boson component. Note that the eigenvalue E T D appears on both sides of theproblem, which requires a non-perturbative matching of its left-hand side value with itsvalue on the right-hand side. If instead of the Gaussian elimination one used the operation R [22] in expansion up to second power of the coupling constant g t , including the pertur-bative orthogonality and normalization corrections, then the eigenvalue E T D in Eq. (74) onthe left-hand side would be replaced by E f + E b . Even though in this case the left-hand sidematrix would only contain terms of order up to g t , the eigenstates would depend on g t ina way specific to the particular TD approximation. The issue would then be what changesoccur when one attempts to improve the approximation by including more effective particlesor higher powers of g t in the TD Hamiltonian matrices. An example of a phenomenological22tudy based on the hypothesis that gauge-bosons obtain an effective mass is presented in [23]in the case of description of baryons using heavy-flavor QCD.Examples of perturbative and TD approximations described above and in Secs. VI andVII in the exactly solvable model can be used in assessing convergence of similar approxima-tions in more complex cases. Consider the numerical studies of eigenvalue problems for TDHamiltonian matrices obtained using bare quanta in the Yukawa and Yukawa-like theories,such as reported in [24, 25] and references therein. The same theories can be considered inthe limit of fermion and boson masses much larger than the cutoff parameters, irrespective ofthe form of regularization. One can limit numerical calculations, where the quantum degreesof freedom are discrete, to a single mode for all quanta involved, precisely as it is done herein the Yukawa theory to obtain our model Hamiltonian. In that setup, the TD Hamiltonianmatrices one would obtain would resemble the ones in our model. One can compare theaccuracy and convergence measures adopted in [24, 25] with exact results shown in Fig. 1.In our model case, the bare coupling constant g = eg , see Eq. (27), determines theexpectation value for the number of bare bosons, (cid:104) N (cid:105) , in the exact fermion eigenstate. For g small, a small (cid:104) N (cid:105) is expected. However, for g order √ π ∼ . , which corresponds tothe conventional coupling constant g / (4 π ) ∼ , Fig. 1 shows that the expected number ofbare bosons exceeds 30. It is stated in [24, 25] that in theories considered there one achievesconvergences using TD matrices with 3 or 4 bosons for quite large coupling constants. Itwould hence be of interest to find out what mechanism is at work by which the inclusion ofadditional interactions and motion of bare bosons with respect to bare fermions improves theconvergence so significantly. Convergence for the electromagnetic form factors may be lessindicative of the number of bosons needed because the contributions of the Fock componentswith n constituents at large momentum transfers may quickly decrease with n [26].The fact that the TD matrix eigenvalue problems are particularly suitable as a tool forseeking approximate solutions to QFT in the FF of Hamiltonian dynamics originates in thespecial circumstance that the momentum component p + is conserved by the interactions andcannot be negative, in a sharp distinction from the momentum component p z in the IF ofdynamics, which can have both signs. As a result, discretization of momenta in a box on afront divides the available total momentum P + of an eigenstate of a FF Hamiltonian into adefinite number of pieces, say K . Each Fock-space constituent of an eigenstate must carrya natural number of units P + /K . Therefore, the number of constituents is limited from23bove by K , which ties the maximal number of constituents in the TD approximation tothe resolution of momentum discretization, K . This is the basis of the so-called discretizedlight-cone quantization (DLCQ) [27–29].The DLCQ method has been applied to the Yukawa theory [30]. Divergences were reg-ulated using the Pauli-Villars method that introduces additional massive fields. To obtainsolvable models, the masses of quanta of the additional fields were set equal to those of thecorresponding physical ones [31]. Similar DLCQ computations were also carried out in asolvable model that closely resembles the Yukawa theory of heavy fermions [32, 33]. Thatmodel was used in [32] to introduce the concept of “clothed” particles. The clothed particlewas defined using an exact solution for a state of a single particle. An analogous solutionwas recovered using DLCQ . In these examples, the DLCQ methods were found useful forconstructing low-mass states in which the mean number of bare constituents was small.As resulting from simplifications of one and the same Yukawa theory, the applications ofthe DLCQ mentioned above allow one to pin point basic features by which the RGPEP andDLCQ approaches differ. These features are visible in the three Eqs. (38), (39) and (40) ofSec. V. They display three distinct operator forms of the same Hamiltonian that acts in themodel Fock space.Equation (38) corresponds to the initial, one might say, canonical Hamiltonian of a theorywithout counter terms. This operator provides the starting point for the RGPEP, which isset up at the scale parameter t = 0 . Since the model is ultraviolet finite, no ultravioletdivergences need to be countered.Equation (39) displays the same Hamiltonian written in terms of creation and annihi-lation operators for the effective particles that correspond to an arbitrary positive valueof the finite scale parameter t , as described in Sec. IV. The formula displays an effectivefermion-boson interaction term and a fermion self-interaction term. The Hamiltonian hasthe universal form of a polynomial function of effective particle operators. The polynomialcoefficients and operators are the computed functions of t . The self-interaction term vanishesat t = 0 because there are no counter terms needed in the initial Hamiltonian. If instead theinitial Hamiltonian led to divergences in any term of H t for any finite t , one would computethe counter terms at t = 0 by demanding that the divergences in H t are eliminated. Themodel is too simple to illustrate in detail what is done in the RGPEP regarding computationof counter terms when the initial Hamiltonian is divergent. However, detailed perturbative24llustrations are available in an asymptotically free example of a scalar theory in 5+1 dimen-sions [34] and in a general derivation of formulas for relativistic Hamiltonians of effectiveparticles in QFT [35]. Here it is only noted that in the presence of divergent self-interactions,the self-interaction term would include a free, finite part of the corresponding counter term.That part would be adjusted by comparison with experiment and may be constrained bydemands of symmetry that the resulting theory is meant to posses.Equation (40) is an expression of the same model Hamiltonian in terms of the operatorsthat create physical states from the vacuum state. A state of a single physical particleis an eigenstate of the Hamiltonian. The formula (40) is obtained in the limit t → ∞ .The effective creation and annihilation operators labeled by ∞ correspond to the physicalparticles of the model. Generally, H ∞ that comes out of solving the RGPEP equationcould involve mixing of eigenstates within degenerate multiplets that in addition to theHamiltonian eigenvalues are labeled by the eigenvalues of other operators that commutewith H t , such as a component of the angular momentum, spin, isospin or a similar quantity.Our model Hamiltonian form of Eq. (40) corresponds to both the concept of “clothed”particles in [32] and the DLCQ solutions for single physical particle states. It is visiblein Eq. (40) that the model of Eq. (12) is too simple to produce interactions between theeffective particles that correspond to t = ∞ and match physical ones as single-particle states.It is worth stressing that the effective particles for t → ∞ do not have to correspond to thephysical ones. This is important for considerations that involve the concept of confinement,see below and Sec. X.It is now clear that the RGPEP and DLCQ computations discussed above differ signif-icantly. The RGPEP produces a whole family of equivalent effective Hamiltonians. TheDLCQ does not produce such a family. It does label Hamiltonian matrices with the res-olution K and the transverse momentum cutoff, introduced by the Pauli-Villars masses.However, these are the regularization parameters. The resolution K and the Pauli-Villarsmasses are meant to be sent to infinity in order to obtain solutions of a theory. They arenot the finite parameters analogous to the RGPEP t on which the physical quantities do notdepend, see Sec. V. Each member of the family labeled by t is expressed using a differentchoice of degrees of freedom in one and the same theory, which means using different creationand annihilation operators, or different quantum field operators that are built from them.Solving the TD Hamiltonian matrix eigenvalue problems in terms of bare quanta for which25 = 0 may be very difficult numerically because of involvement of many basis states in thedynamics, as is illustrated in the model by Fig. 1. An effective Hamiltonian with a finite t that is adjusted to the scale of the physical quantity of interest is dominated by the effectivebasis states of a similar scale. An approximate but accurate description of the quantity ofinterest is simpler to achieve that way than by keeping all bare basis states in a computationthat requires handling of all variables of the theory up to the cutoffs. The model exampleillustrates this feature solely in terms of the magnitude of the effective coupling constantthat decreases as t increases and thus weakens the coupling between different effective Fockcomponents that correspond to the parameter t . In contrast, the DLCQ approach attemptsto solve the theory directly in terms of the degrees of freedom present in the quantum Hamil-tonian in its initial form, analogous to Eq. (38), i.e. , the one that is obtained by quantizationof a local theory.Our model example makes it also clear that the concept of “clothed” particles mentionedabove differs from the RGPEP concept of scale-dependent effective particles. The “clothed”particles approach is based on writing a Hamiltonian in terms of operators associated withthe physical particles instead of the bare ones. In the RGPEP language, the idea is toreplace the gradual evolution from t = 0 to t = ∞ by a single jump. Such replacement isnot available in any closed form in complex theories for which one does not have any exactsolutions. Notably, in case of confinement the required physical particles are not supposed toexist. The issue is relevant to the ultimate DLCQ limit K → ∞ that appears to be related toquestions concerning the vacuum, which is assumed to carry p + = 0 . The RGPEP approachis conceptually different from the “clothed” particle approach. Its equations can be solvedfor effective operators making various guesses or approximations and the resulting effectiveparticles do not have to be identified with any physical, individually observable objects.The effective Hamiltonians H t can be studied in terms of their predictions for quantitiesaccessible experimentally. For an example of such attempt in heavy-flavor QCD, see [23].Another basic feature that distinguishes the RGPEP example from the DLCQ examplesmentioned above is that the number of quantum degrees of freedom stays the same in theeffective theory for all values of t , including the canonical theory at t = 0 . However, theinteraction terms in H t evolve with t as the RGPEP Eq. (15) dictates. If the initial theorywere divergent, the ultraviolet counter terms would be computed in the process of solvingEq. (15) and they would be inserted in the initial condition at t = 0 . They would thus not26e constructed by adding degrees of freedom like in the Pauli-Villars approach. Instead, thedemand on the RGPEP evolution that for finite t it yields finite effective Hamiltonians H t would be used to determine the missing counter terms in H .Finally, it should be observed that in the non-relativistic contexts of condensed matterphysics, addressed broadly in [15], as well as in nuclear physics theory developed in [36]and elsewhere, similar Wegner-like equations and corresponding TD Hamiltonian matrixeigenvalue problems appear that resemble the ones obtained by applying the RGPEP to themodel Hamiltonian of Eq. (12) or other model Hamiltonians of analogous nature, cf. [37].According to the rule that the same equations have the same solutions, no matter whattheir interpretation is, and in view of the discussion of this section, it becomes clear that theRGPEP concept of effective particles developed in particle physics and explicitly illustratedusing the elementary Eq. (39), can be introduced in the other branches of physical theory aswell. For example, one can attempt to introduce a whole family of scale-dependent effectiveelectron operators that include phonon operators in a model of a condensed-matter mediumor effective nucleon operators that include meson operators in a model of a nucleus. IX. MODEL SOLUTION AND REALISTIC THEORIES
The model solution illustrates the structure, function and purpose of the RGPEP in thecontext where no divergences appear. The concept of counter terms only shows up throughthe cancellation of the fermion self-interaction energy, due to emission and absorption ofbosons, against the effective fermion energy in the eigenvalue problem for the Hamiltonian H t . The terms that cancel out are finite. The pattern is analogous to the cancellationbetween the finite parts of counter terms and self-interactions in realistic theories.The model solution illustrates the weakening of effective interactions solely in terms ofthe coupling constant that decreases as the RGPEP evolution parameter t grows. Thisweakening corresponds to the weakening obtained in terms of the vertex form factors inrealistic theories. The model running-coupling constant corresponds to the vertex formfactor for the specific value of its argument, corresponding to the invariant mass changecaused by the interaction. Emergence of the RGPEP vertex form factors in the Yukawatheory is described in [38]. Analogous appearance of the vertex form factors in the Abeliangauge theory is shown in [16]. The RGPEP form factors that emerge in the third-order27omputation of the effective vertices in a non-Abelian theory is provided in [39].Extension of the model solution that would include the motion of bosons with respectto fermions and hence produce the associated vertex form factors, would be of great value.As pointed out earlier, the RGPEP vertex form factors are expected to be important in thederivation of effective quark and gluon dynamics in QCD. However, given the complexityof QCD, one might attempt to first undo some of the model simplifications made here andtackle the problem of applying the RGPEP to the Yukawa theory. To be specific, one mayaim at a comprehensive resolution of the paradox that concerns interactions of nucleonswith pions, and perhaps also other mesons. Namely, the exchange of just one pion betweennucleons yields the Yukawa potential in second-order perturbtion theory, but the couplingconstant one needs to introduce in order to match the phenomenology is so large that thestandard perturbation theory with local interactions cannot be valid. Perhaps the largecoupling corresponds not to a canonical Yukawa theory with t = 0 but to the effectivetheory in which t corresponds to the pion mass scale and the vertex form factors make theinteraction effectively quite weak by suppressing it outside the small momentum transferrange that corresponds to the pion exchange.The model solution includes a coherent state of bosons around a fermion. One could askif the pattern exhibited by the model could be followed for the purpose of explaining if theeffective Yukawa theory could describe the pion cloud around nucleons.Since the RGPEP suppression of interactions corresponds to the vertex form factors, itmakes sense to ask if any theory that introduces vertex form factors of some width mightcorrespond to an effective one in the sense of the RGPEP for a width parameter t matchingthe form factor scale. The example of particular interest is provided by the Nambu andJona-Lasinio model [40] that to the author’s best knowledge was never analyzed using theRGPEP. X. CONCLUSION
The main import of the elementary model study is that it illustrates how the RGPEPworks in an exactly solvable model. However, the realistic theories are much more complexthan the model and one cannot predict on the model basis if the RGPEP can fully providethe means that are required for unambiguous identification of the corresponding effective28amiltonians in complex theories. To find out what can be achieved in that matter, onewould have to focus on the direct application of the RGPEP in terms of perturbative expan-sions and TD approximations to the complex theories. In that context the model solution isof value because such approximate methods quickly get quite convoluted in realistic theories.The value is that one can use the model as a pattern to follow and to consult with whencalculations get hard to see through. The key example of a barrier to break is to solve theRGPEP equation up to the fourth order of perturbation theory in QCD and derive the corre-sponding TD Hamiltonian matrices. Perhaps this is the way to obtain the constituent-quarkpicture of hadrons from QCD.The case of quarks in QCD is particularly pressing even though one can also try to usethe RGPEP for addressing theoretical issues of the Standard Model as a whole. The idea ofconstructing effective quarks dates back to early years of current algebra [41]. As far as theauthor knows it is not fully realized till today, while the particle data tables [42] continue toclassify hadrons mostly in terms of just two or three quark constituents. States that containtwo more quarks are being added in the same spirit of constituents. QCD suggests insteadthat hadrons are built from practically unlimited numbers of quarks, antiquarks and gluonsof canonical theory. Despite the great progress of lattice gauge theory, Gell-Mann’s opinionfrom twenty years ago [43] appears still valid: “The mathematical consequences of QCD havestill not been properly extracted, and so, although most of us are persuaded that it is thecorrect theory of hadronic phenomena, a really convincing proof still requires more work. Itmay be that it would be helpful to have some more satisfactory method of truncating thetheory, say by means of collective coordinates, than is provided by the brute-force latticegauge theory approximation!”The author’s opinion is that the basic difficulty to overcome before one can addressprecise phenomenology that involves fast moving and strongly interacting hadrons, is tofirst somehow gain control of the ground state of the theory. The reason is that all particlestates one considers are meant to be created by action of operators on that special state.Such control is also desired concerning spontaneous breaking of symmetries. In the FF ofHamiltonian dynamics the vacuum problem is formulated in a different way than in theIF dynamics, e.g. see [44]. The condition p + > for all quanta with finite momenta andnon-zero masses can be compared with the condition that the vacuum state carries zeromomentum. The vacuum state should also be invariant with respect to a change of an29nertial frame of reference. This may be a large change, such as to the infinite momentframe used in the parton model. Somehow the vacuum state is limited to states akin tothose with p + = 0 , sometimes called the FF zero modes.The vacuum problem of QCD has a long history, stimulated by the concepts of quarkand gluon condensates and posing questions in cosmology. To gain a perspective, one canconsult the works [45–47]. The leading condensates can be simply incorporated in the FFversion of QCD sum rules [48] using the condition p + > (cid:15) + for all non-vacuum modes whilethe vacuum modes must have p + < (cid:15) + . The constant (cid:15) + is treated as infinitesimal. If oneassumed that the states with momenta p + < (cid:15) + were absent, one could even think that thecosmological vacuum problem may be resolved [49]. However, the dynamics of modes with p + < (cid:15) + is singular and to the author’s best knowledge it is not understood.Of course, the exact computation of effective Hamiltonians for the model of Eq. (12) isonly relevant to the vacuum issue because the computation is used to illustrate the RGPEP.The point is that the vacuum problem in the FF Hamiltonians can be turned into a renor-malization group issue according to [2]. Namely, the counter terms to the cutoff (cid:15) + → are expected to mimic vacuum effects and one hopes to finesse dynamical effects due tothe latter that way. The idea is presented in [2] using the FF power counting and originalsimilarity renormalization group procedure [12, 13]. However, the number and complexityof terms one obtains turns out difficult to handle using the similarity procedure. With theRGPEP the situation is different because one does not need to directly address the multitudeof matrix elements of many complex operators that involve initially unknown functions ofmany momentum variables. Therefore, one can focus instead on behavior of coefficients inpolynomial functions of creation and annihilation operators for effective particles. Moreover,the RGPEP equation in QCD that corresponds to Eq. (15) in our model discussion securesinvariance of the Hamiltonians H t with respect to seven kinematical Poincaré symmetries,leaving only three that are dynamical and need to be renormalized. Consequently, insteadof the cutoff p + > (cid:15) + on the absolute momenta p + , one can use a dimensionless cutoff x > (cid:15) on the ratio x = p +1 /p +2 that momenta of particles 1 and 2 involved in an interaction termcan form.Exact non-perturbative solutions of the RGPEP equation in QFT as complex as QCDare not currently foreseeable. However, one can study the terms that emerge in perturbativeexpansion using asymptotic freedom, known in the FF effective particle Hamiltonians to the30owest order only [39]. The fourth-order calculation mentioned earlier is of interest becausethis is the first place where the running coupling appears in the effective interaction termsand increases with t . General fourth-order RGPEP formulas are available in [35]. As longas the effective coupling constant is not too large, one can use the perturbative expansion tolearn what kinds of terms arise. 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