We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group $SU(N)$. The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits $\sim 2^{2N}$ non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all $N$ these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, $m\ll g_\mathrm{YM}$, the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the $k$-string tension for $N\leq 5$. Our results suggest that the $k$-string tension, $T_k$, is $T_k\sim |m| \sin(πk /N)$ for all $N$. We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for $SU(N\leq5)$ gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of $N$. The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.
We show that $2$d adjoint QCD, an $SU(N)$ gauge theory with one massless adjoint Majorana fermion, has a variety of mixed 't Hooft anomalies. The anomalies are derived using a recent mod $2$ index theorem and its generalization that incorporates 't Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of $N$, are matched by spontaneous chiral symmetry breaking. We find that massless $2$d adjoint QCD confines for $N >2$, except for test charges of $N$-ality $N/2$, which are decon…
We analyze the 2+1 dimensional gauge theory with two fermions in the real adjoint representation with non-zero Chern-Simons level. We propose a new fermion-fermion dualities between strongly-coupled theories and determine the quantum phase using the structure of a `Duality Chain'. We argue that when Chern-Simons level is sufficiently small, the theory in general develops a strongly coupled quantum phase described by an emergent topological field theory. For special cases, our proposal predicts an interesting dynamical scenario with spontaneous breaking of partial 1-form or 0-form global symmet…
We study 2+1 dimensional gauge theories with a Chern-Simons term and a fermion in the adjoint representation. We apply general considerations of symmetries, anomalies, and renormalization group flows to determine the possible phases of the theory as a function of the gauge group, the Chern-Simons level $k$, and the fermion mass. We propose an inherently quantum mechanical phase of adjoint QCD with small enough $k$, where the infrared is described by a certain Topological Quantum Field Theory (TQFT). For a special choice of the mass, the theory has ${\cal N}=1$ supersymmetry. There this TQFT is…
In this paper we study dualities for $\mathcal{N}=2$ gauge theories in three dimensions with matter in the fundamental and adjoint representation. The duality we propose, analogous to mirror symmetry, is obtained starting from $\mathcal{N}=4$ mirror theories and turning on a certain superpotential deformation involving monopole operators. We study the role of emergent symmetries in the dual theory, focusing on the case of models with gauge symmetry $U(2)$ or $SU(2)$. We find that $SU(2)$ adjoint SQCD with one flavor and zero superpotential is dual to SQED with two flavors and three singlets. A…
Using a loop formulation approach of QCD$_2$, we study the potential between two heavy quarks in the presence of adjoint scalar fields, and demonstrate how 't Hooft's planar rule is manifested in this formulation. Based on some physical assumptions, we argue that large adjoint loops ``confined'' inside an external fundamental one give a Casimir type contribution to the potential energy, while the small loops only renormalize the string tension. We also extend the results to the case of massive adjoint fields.
We discuss an exotic phase that adjoint QCD possibly exhibits in the deep infrared (IR). It is a confining phase, with a light spectrum consisting of massless composite fermions. The discrete chiral symmetry is broken, with unbroken continuous chiral symmetry. We argue that it may give a description of the IR of adjoint QCD with three massless Weyl flavors and that it passes all consistency checks known to us.
Two-dimensional QCD with adjoint fermions has many attractive features, yet its single-particle content remains largely unknown. To lay the foundation for a crucially improved approximation of the theory's spectrum, we developed a method to find the basis of eigenstates using the symmetry structure of the asymptotic theory where pair production is disallowed. This method produces complete sets of multi-dimensional harmonic functions for the massless and the massive theory. Previously only part of such a basis was known. The method presented here should be applicable to other theories and has t…
We briefly review the 1+1-dimensional SU(N) gauge theory minimally coupled to an adjoint scalar, based upon a dimensional reduction of 2+1-dimensional pure gauge theory, which approximates the dynamics of the transversely polarized gluon. The lightest glueball states are investigated non-perturbatively using light-front quantisation in the large-N limit a la 't Hooft, and the physical meaning of the results elucidated. Comparison is made with recent lattice Monte Carlo data for 3-dimensional quenched QCD.
Massless Majorana fermions in the adjoint representation of SU(N_c) are expected to screen gauge interactions in 1+1 dimensions, analogous to a similar Higgs phenomena known for 1+1-dimensional U(1) gauge theory with massless fundamental fermions (Schwinger model). Using the light-cone formalism and large-N_c limit, a non-abelian analogue of the Schwinger boson is shown to be responsible for the screening between heavy test charges. This adjoint boson does not exist simply as a physical state, but boundstates are built entirely from this particle.
In this lecture we discuss some exact results for the low-lying spectrum of the Dirac operator in adjoint QCD. In particular, we find an analytical expression for the slope of the average spectral density. These results are obtained by means of a generating function which is an extension of the QCD partition function with fermionic and bosonic ghost quarks. The low-energy limit of this generating function is completely determined by chiral (super-)symmetries. Our results for the slope of the average spectral density are consistent with the results for the scalar susceptibility which can be obt…
We analyze the large $N$ limit of adjoint QCD, an $SU(N)$ gauge theory with $N_f$ flavors of massless adjoint Majorana fermions, compactified on $S^3 \times S^1$. We focus on the weakly-coupled confining small-$S^3$ regime. If the fermions are given periodic boundary conditions on $S^1$, we show that there are large cancellations between bosonic and fermionic contributions to the twisted partition function. These cancellations follow a pattern previously seen in the context of misaligned supersymmetry, and lead to the absence of Hagedorn instabilities for any $S^1$ size $L$, even though the bo…
We consider a dimensional reduction of 3+1 dimensional SU(N) Yang-Mills theory coupled to adjoint fermions to obtain a class of 1+1 dimensional gauge theories. We derive the quantized light-cone Hamiltonian in the light-cone gauge A^+ = 0$ and large-N limit, then solve for the masses, wavefunctions and of the color singlet boson and fermion boundstates. We find that the theory has many exact massless state that are similar to the t'Hooft pion.
We propose to induce QCD by fermions in the adjoint representation of the gauge group SU(N_c) on the lattice. We consider various types of lattice fermions: chiral, Kogut--Susskind and Wilson ones. Using the mean field method we show that a first order large-N phase transition occurs with decreasing fermion mass. We conclude, therefore, that adjoint fermions induce QCD. We draw the same conclusion for the adjoint scalar or fermion models at large number of flavors N_f when they induce a single-plaquette lattice gauge theory. We find an exact strong coupling solution for the adjoint fermion mod…
Extending the investigations about the theory of duals, we analyze duals built up with the aid of discrete symmetry operators. We scrutinize algebraic and physical constraints (encompassing them in a theoretical scope) in order to verify which combination of discrete symmetries may compose a physical and mathematical well-posed spinorial dual. In this scenario, we relate the Lounesto classification with several other spinor classification possibilities, attempting to connect classes and physical constraints. Several possibilities are investigated.
We construct the magnetic dual of QCD with one adjoint Weyl fermion. The dual is a consistent solution of the 't Hooft anomaly matching conditions, allows for flavor decoupling and remarkably constitutes the first nonsupersymmetric dual valid for any number of colors. The dual allows to bound the anomalous dimension of the Dirac fermion mass operator to be less than one in the conformal window.
We consider SU(N) QCD_{1+1} coupled to massless adjoint Majorana fermions, where N is finite but arbitrary. We examine the spectrum for various values of N, paying particular attention to the formation of multi-particle states, which were recently identified by Gross, Hashimoto and Klebanov in the N -> infinity limit of the theory. It is believed that in the limit of vanishing fermion mass, there is a transition from confinement to screening in which string-like states made out of adjoint fermion bits dissociate into stable constituent ``single particles''. In this work, we provide numerical e…
The single-particle content of two-dimensional adjoint QCD remains elusive due to the inability to distinguish single- from multi-particle states. To find a criterion we compare several approximations to the theory. Starting from the asymptotic theory (no pair production, only singular operators), we construct sets of eigenfunctions in the lowest parton sectors of the theory. A perturbative treatment of the omitted operators is performed. We find that multi-particle states are absent if pair-production is disallowed and hints for a double Regge trajectory of single-particle states. We discuss …
In this paper we complete the derivations of finite volume partition functions for QCD using random matrix theories by calculating the effective low-energy partition function for three-dimensional QCD in the adjoint representation from a random matrix theory with the same global symmetries. As expected, this case corresponds to Dyson index $β=4$, that is, the Dirac operator can be written in terms of real quaternions. After discussing the issue of defining Majorana fermions in Euclidean space, the actual matrix model calculation turns out to be simple. We find that the symmetry breaking patter…
The twisted space-time reduced model of large $N$ QCD with various flavours of adjoint Wilson fermions is constructed applying the symmetric twist boundary conditions with flux $k$. The models with one and two flavours show distinctive behaviours. For the two flavor case, the string tension, calculated at $N=289$, approaches zero as we decrease the quark mass in a way consistent with the theory being governed by an infrared fixed point. In contrast, the string tension for the case of a single adjoint Wilson fermion remains finite as the quark mass decreases to zero, supporting that this is a c…
Two dimensional QCD coupled to fermions in the adjoint representation of the gauge group $SU(N)$, a useful toy model of QCD strings, is supersymmetric for a certain ratio of quark mass and gauge coupling constant. Here we study the theory in the vicinity of the supersymmetric point; in particular we exhibit the algebraic structure of the model and show that the mass splittings as one moves away from the supersymmetric point obey a universal relation of the form ${M_i}^2(B)-{M_i}^2(F)=M_iδm+O(δm^3)$. We discuss the connection of this relation to string and quark model expectations and verify it…
We study $2d$ QCD coupled to fermions in the adjoint representation of the gauge group $SU(N)$ at large $N$, and its relation to string theory. It is shown that the model undergoes a deconfinement transition at a finite temperature (analogous to the Hagedorn transition in string theory), with certain winding modes in the Euclidean time direction turning tachyonic at high temperature. The theory is supersymmetric for a certain ratio of quark mass and gauge coupling. For other values of that ratio, supersymmetry is softly broken. The spectrum of bound states contains an infinite number of approx…
This is a review of recent developments regarding the Skyrmion sector of higher representation QCD. Ordinary QCD is a SU(n) gauge theory with n_f Dirac quarks in the fundamental representation. Changing the representation of quarks leads to different and interesting theories, which are not as well studied as ordinary QCD. In order to be able to have a consistent asymptotically free large n limit, we must limit ourselves to three cases: two-index representation (symmetric or anti-symmetric) and adjoint representation. Skyrmions of the low-energy effective Lagrangian shall be the main subject of…
We propose gauging matrix models of string theory to eliminate unwanted non-singlet states. To this end we perform a discretised light-cone quantisation of large N gauge theory in 1+1 dimensions, with scalar or fermionic matter fields transforming in the adjoint representation of SU(N). The entire spectrum consists of bosonic and fermionic closed-string excitations, which are free as N tends to infinity. We analyze the general features of such bound states as a function of the cut-off and the gauge coupling, obtaining good convergence for the case of adjoint fermions. We discuss possible exten…
We consider an 'electric' $U(N)$ level $k$ QCD$_3$ theory with one adjoint Majorana fermion. Inspired by brane dynamics, we suggest that for $k \ge N/2$ the massive $m<0$ theory, in the vicinity of the supersymmetric point, admits a $U(k-\frac{N}{2})_{-(\frac{1}{2}k+\frac{3}{4}N),-(k+\frac{N}{2})}$ 'magnetic' dual with one adjoint Majorana fermion. The magnetic theory flows in the IR to a topological $U(k-\frac{N}{2})_{-N,-(k+\frac{N}{2})}$ pure Chern-Simons theory in agreement with the dynamics of the electric theory. When $k<N/2$ the magnetic dual is $U(\frac{N}{2}-k)_{\frac{1}{2}k+\frac{3}{…
The phase diagram of SU(N) gauge theories with fermions in an arbitrary representation R can be calculated on finite volume manifolds such as S^1 x S^3. When S^3 is small a perturbative analysis is possible and the weak-coupling analogue of the pure Yang-Mills theory confinement-deconfinement transition is accessible in the large N limit. We calculate the large N phase diagram of adjoint QCD [SU(N) gauge theory with adjoint fermions] where periodic boundary conditions are applied to fermions on S^1 such that the confined phase is favored for light enough adjoint fermion mass. We calculate the …
We present an analytic approach to solving 1+1 dimensional QCD with an adjoint Majorana fermion. In the UV this theory is described by a trivial CFT containing free fermions. The quasi-primary operators of this CFT lead to a discrete basis of states which is useful for diagonalizing the Hamiltonian of the full strongly interacting theory. Working at large-$N$, we find that the decoupling of high scaling-dimension quasi-primary operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. This suggests a scheme, whereby, truncating the basis to operators of dimens…
We consider SU(N) gauge theory in 1+1 dimensions coupled to chiral fermions in the adjoint representation of the gauge group. With all fields in the adjoint representation the gauge group is actually SU(N)/Z_N, which possesses nontrivial topology. In particular, there are N distinct topological sectors and the physical vacuum state has a structure analogous to a θvacuum. We show how this feature is realized in light-front quantization for the case N=2, using discretization as an infrared regulator. In the discretized form of the theory the nontrivial vacuum structure is associated with the zer…
We perform a lattice investigation of QCD with three colors and 2 flavors of Dirac (staggered) fermions in the adjoint representation, defined on a 4d space with one spatial dimension compactified, and study the phase structure of the theory as a function of the size Lc of the compactified dimension. We show that four different phases take place, corresponding to different realizations of center symmetry: two center symmetric phases, for large or small values of Lc, separated by two phases in which center symmetry is broken in two different ways; the dependence of these results on the quark ma…
A study of two-dimensional QCD on a spatial circle with Majorana fermions in the adjoint representation of the gauge groups SU(2) and SU(3) has been performed. The main emphasis is put on the symmetry properties related to the homotopically non-trivial gauge transformations and the discrete axial symmetry of this model. Within a gauge fixed canonical framework, the delicate interplay of topology on the one hand and Jacobians and boundary conditions arising in the course of resolving Gauss's law on the other hand is exhibited. As a result, a consistent description of the residual $Z_N$ gauge sy…
We discuss bosonized two-dimensional QCD with massless fermions in the adjoint and multi-flavor fundamental representations. We evaluate the massive mesonic spectra of several models by using the light-front quantization and diagonalizing the mass operator $M^2=2P^+P^- $. We recover previous results in the case of one flavor adjoint fermions and we find the exact massive spectrum of multi flavor QCD in the limit of large number of flavors.
We discuss a two dimensional SU(N) Yang-Mills theory coupled to massive adjoint fermions for different worlds classified by the integer k=0,1,...N-1. We study the fermion condensate for these unconnected worlds as a function of the parameter k. We show that the condensate as well as the spectrum of the theory do depend on this vacuum parameter k. Technically, the value of the fermion condensate is related to the value of the gluon condensate via the Operator Product Expansion. We use this to find the leading dependence (in the limit of a heavy quark) of the fermion condensate on the nontrivial…
The twisted reduced model of large $N$ QCD with two adjoint Wilson fermions is studied numerically using the Hybrid Monte Carlo method. This is the one-site model, whose large $N$ limit (large volume limit) is expected to be conformal or nearly conformal. The symmetric twist boundary condition with flux $k$ is used. $k$=0 corresponds to periodic boundary conditions. It is shown that the quark mass and $N$ dependencies of the model with non-vanishing $k$ differ significantly from those of the $k$=0 model. A preliminary result for the string tension calculated at $N$=289 is presented. The string…
We evaluate partition functions $Z_I$ in topologically nontrivial (instanton) gauge sectors in the bosonized version of the Schwinger model and in a gauged WZNW model corresponding to $QCD_2$ with adjoint fermions. We show that the bosonized model is equivalent to the fermion model only if a particular form of the WZNW action with gauge-invariant integrand is chosen. For the exact correspondence, it is necessary to integrate over the ways the gauge group $SU(N)/Z_N$ is embedded into the full $O(N^2 - 1)$ group for the bosonized matter field. For even $N$, one should also take into account the …
We consider 1+1-dimensional QCD coupled to Majorana fermions in the adjoint representation of the gauge group $SU(N)$. Pair creation of partons (fermion quanta) is not suppressed in the large-$N$ limit, where the glueball-like bound states become free. In this limit the spectrum is given by a linear \lc\ Schr\" odinger equation, which we study numerically using the discretized \lcq. We find a discrete spectrum of bound states, with the logarithm of the level density growing approximately linearly with the mass. The wave function of a typical excited state is a complicated mixture of components…
Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We pr…
We show that $QCD_2$ with adjoint fermions involves instantons due to nontrivial $π_1[SU(N)/Z_N]~=~Z_N$. At high temperatures, quasiclassical approximation works and the action and the form of effective (with account of quantum corrections) instanton solution can be evaluated. Instanton presents a localized configuration with the size $\propto g^{-1}$. At $N=2$, it involves exactly 2 zero fermion modes and gives rise to fermion condensate $<\barλ^a λ^a>_T$ which falls off $\propto \exp\{-π^{3/2} T/g\}$ at high $T$ but remains finite. At low temperatures, both instanton and bosonization argumen…
We study strings between static quarks in QCD with $n_f$ adjoint fermions, including $N=1$ Super Yang-Mills (SYM), in the calculable regime on $\mathbb R^3\times \mathbb S^1$. We find that they have many qualitatively new features not previously known. The difference from other realizations of abelian confinement is due to the composite nature of magnetic bions, whose Dirac quantum with fundamental quarks is two, and to the unbroken part of the Weyl group. In particular we show that strings are composed of two domain walls, that quarks are not confined on domain walls, that strings can end on …
The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_N$ admits topologically nontrivial gauge field configurations associated with nontrivial $π_1[SU(N)/Z_N] = Z_N$. The topological sectors are labelled by an integer $k=0,\ldots, N-1$. However, in contrast to $QED_2$ and $QCD_4$, this topology is not associated with an integral invariant like the magnetic flux or Pontryagin index. These instantons may admit fermion zero modes, but there is always an equal number of left-handed and right-handed modes, so that the Atiyah-Singer theorem, which determines in other cases the number of the …
't Hooft anomaly matching is powerful for constraining the low energy phases of gauge theories. In 3d one common anomaly is the parity anomaly in a $T$-symmetric theory where one cannot gauge the global symmetry group without breaking the time-reversal symmetry. We find that a $T$-symmetric $\text{SU}(N)$ gauge theory with either fermionic or bosonic matter in the fundamental representation of the gauge group has a parity anomaly between the flavor group and $T$-symmetry provided that there is also a massless Majorana fermion in the adjoint representation of the gauge group. We then analyze th…
We consider a modification of QCD in which conventional fundamental quarks are replaced by Weyl fermions in the adjoint representation of the color SU(N). In the case of two flavors the low-energy chiral Lagrangian is that of the Skyrme-Faddeev model. The latter supports topologically stable solitons with mass scaling as N^2. Topological stability is due to the existence of a nontrivial Hopf invariant in the Skyrme-Faddeev model. Our task is to identify, at the level of the fundamental theory, adjoint QCD, an underlying reason responsible for the stability of the corresponding hadrons. We argu…
We investigate the relationship between asymptotic symmetries of QCD and vacuum transitions induced during scattering. Starting with the Fock vacuum in the far past, the infinity of conservation laws associated to (non-Abelian) asymptotic symmetries in QCD can be used to determine the vacuum in the far future. We show that the corresponding asymptotic Hilbert space in the future is generated by a class of dressed states in which each finite energy particle is dressed by a cloud of interacting soft gluons. We identify the precise structure of the dressing using conservation laws and show that t…
Large-N QCD with heavy adjoint fermions emulates pure Yang-Mills theory at long distances. We study this theory on a four- and three-torus, and analytically argue the existence of a large-small volume equivalence. For any finite mass, center symmetry unbroken phase exists at sufficiently small volume and this phase can be used to study the large-volume limit through the Eguchi-Kawai equivalence. A finite temperature version of volume independence implies that thermodynamics on R^3 x S^1 can be studied via a unitary matrix quantum mechanics on S^1, by varying the temperature. To confirm this no…
We address the issue of topological angles in the context of two dimensional SU(N) Yang-Mills theory coupled to massive fermions in the adjoint representation. Classification of the resulting multiplicity of vacua is carried outin terms of asymptotic fundamental Wilson loops, or equivalently, charges at the boundary of the world. We explicitly demonstrate that the multiplicity of vacuum states is equal to N for SU(N) gauge group. Different worlds of the theory are classified by the integer number k=0,1,...N-1 (superselection rules) which plays an analogous role to the $θ$ parameter in QCD. Via…
We show that adjoint QCD features very strong Bose-Fermi cancellations in the large $N$ limit, despite the fact that it is manifestly non-supersymmetric. The difference between the bosonic and fermionic densities of states in large $N$ adjoint QCD turns out to have a `two-dimensional' scaling $\sim \exp{(\sqrt{\ell E})}$ for large energies $E$ in finite spatial volume, where $\ell$ is a length scale associated with the curvature of the spatial manifold. In particular, all Hagedorn growth cancels, and so does the growth $\exp{(V^{1/4} E^{3/4})}$ expected in a standard local 4d theory in spatial…
We study four dimensional $SU(2)$ Yang-Mills theory with two massless adjoint Weyl fermions. When compactified on a spatial circle of size $L$ much smaller than the strong-coupling scale, this theory can be solved by weak-coupling nonperturbative semiclassical methods. We study the possible realizations of symmetries in the $\mathbb R^4$ limit and find that all continuous and discrete $0$-form and $1$-form 't Hooft anomaly matching conditions are saturated by a symmetry realization and massless spectrum identical to that found in the small-$L$ limit, with only a single massless flavor-doublet …
We study the large-N volume reduction of QCD with adjoint quarks regularized on the lattice. Specifically, we use Wilson fermions, and while our d-dimensional lattice has (d-1) infinite dimensions, the remaining direction is reduced to a point. We perform a weak-coupling one-loop calculation of the free energy as a function of the holonomy in the reduced direction, and map the regimes in the bare lattice parameter space where the holonomy averages to zero and a Z_N-center symmetric configuration is the ground state. For d=4 and N_f=1/2,1 and 2 Dirac flavors we see that the center symmetry is i…
We consider adjoint scalar matter coupled to QCD(1+1) in light-cone quantization on a finite `interval' with periodic boundary conditions. We work with the gauge group SU(2) which is modified to ${\rm{SU(2)/Z_2}}$ by the non-trivial topology. The model is interesting for various nonperturbative approaches because it is the sector of zero transverse momentum gluons of pure glue QCD(2+1), where the scalar field is the remnant of the transverse gluon component. We use the Hamiltonian formalism in the gauge $\partial_- A^+ = 0$. What survives is the dynamical zero mode of $A^+$, which in other the…
SU(2) gauge theory coupled to massless fermions in the adjoint representation is quantized in light-cone gauge by imposing the equal-time canonical algebra. The theory is defined on a space-time cylinder with "twisted" boundary conditions, periodic for one colour component (the diagonal 3- component) and antiperiodic for the other two. The focus of the study is on the non-trivial vacuum structure and the fermion condensate. It is shown that the indefinite-metric quantization of free gauge bosons is not compatible with the residual gauge symmetry of the interacting theory. A suitable quantizati…
