Realization and breaking of conformal symmetry in quantum field theory: the open questions left by finite-component field representations of the conformal group

The 1969 Mack–Salam classification of finite-component field representations of the conformal group left open a bundle of linked questions about the field-theoretic realization of conformal symmetry: whether infinite-component fields, built from infinite-dimensional little-group representations, can yield physically viable conformal field theories; whether the breaking of conformal symmetry in interactions is always "minimal", governed by the ansatz $\partial^{\nu}\mathscr{K}_{\nu\mu}=2x_{\mu}\partial^{\nu}\mathscr{D}_{\nu}$ with no additional term; whether the induced representations with $\kappa_{\mu}\neq0$, which generate conformally invariant four-fermion interactions but obstruct a conserved hermitian current, admit a consistent symmetry-breaking scheme; and whether realistic interacting models can be constructed in the manifestly covariant six-dimensional formalism.

Background

In four-dimensional Minkowski space the conformal group extends the Poincaré group of translations and Lorentz transformations by dilatations (scale transformations $x\to\lambda x$) and special conformal transformations (the transformations obtained by composing a spatial inversion, a translation and a second inversion, which act as $x^{\mu}\mapsto\frac{x^{\mu}-b^{\mu}x^{2}}{1-2b\cdot x+b^{2}x^{2}}$); it is (a cover of) $SO(4,2)\simeq SU(2,2)$. In the late 1960s, current algebra (the study of the commutation relations of symmetry currents) and the approximate scaling then being observed in deep-inelastic scattering (electron–nucleon scattering at short wavelength, probing the short-distance structure of the strong interaction) made it natural to ask whether this larger symmetry could be realized — and in what pattern it could be broken — in a relativistic quantum field theory (QFT); a theory that realizes the full unbroken symmetry is a conformal field theory (CFT). That is the systematic question Mack and Salam addressed in 1969 for fields with finitely many components.

Their framework starts from the induced-representation construction. The transformation law of a (quasi-)primary field — a field that at each point transforms by a matrix acting on its components, labeled by Lorentz spin and a scaling dimension, with no derivatives of the field entering; this is the ancestor of the modern term "primary field" — is determined, under infinitesimal conformal transformations, by a matrix representation of the Lorentz algebra (generators $\Sigma_{\mu\nu}$), a scaling dimension $\Delta$ (the power-law exponent with which the field rescales under $x\to\lambda x$), and, in the most general induced case, a vector of matrices $\kappa_{\mu}$ acting on the field components. The representation is induced from the stability subgroup of the conformal group that fixes the origin — the "little group" of this construction — whose algebra is generated by the Lorentz transformations $M_{\mu\nu}$, the dilatation $D$, and the special conformal generators, the last four forming a translation-like abelian ideal $T_{4}$ (symbolically, the little-group algebra is $(SO(3,1)\otimes\{D\})\ltimes T_{4}$); the matrices $\kappa_{\mu}$ are exactly the representation of this translation-like ideal on the field components, which is why they are nilpotent in finite-dimensional representations. A field built from an infinite-dimensional representation of this little group carries infinitely many Lorentz spins and is called an infinite-component field; such fields were studied intensively in the 1960s as a way to accommodate the proliferating hadron resonances within a single multiplet. When the symmetry is realized dynamically, its charges are carried by local currents: the dilatation current $\mathscr{D}_{\nu}$, whose spatial integral generates scale transformations, and the special conformal currents $\mathscr{K}_{\nu\mu}$. Mack and Salam derived the general divergence relation

$\partial^{\nu}\mathscr{K}_{\nu\mu}=2x_{\mu}\,\partial^{\nu}\mathscr{D}_{\nu}+V_{\mu},$

where $V_{\mu}$ is an additional local current — the ancestor of what modern literature calls the virial current, the current whose divergence measures the part of the trace of the energy-momentum tensor that cannot be removed by improvement. Their "minimal breaking" ansatz postulates $V_{\mu}=0$: conformal symmetry is broken only through the divergence of the dilatation current. They also developed a manifestly conformally covariant formalism in six dimensions, in which compactified Minkowski space is realized as the projective null cone of $\mathbb{R}^{4,2}$, where the conformal group acts linearly.

The 1970s literature transformed the minimal-breaking question rather than settling it. Callan, Coleman and Jackiw introduced the improved energy-momentum tensor: one may add $(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\Box)$ times a suitable local operator (an "improvement" term) to $T_{\mu\nu}$ without changing the Poincaré charges, and this can remove virial-type terms in scalar theories. Coleman and Jackiw analyzed precisely when dilatation generators fail to generate dilatations; Crewther derived nonperturbative anomaly constraints on the trace; Parisi showed that conformal invariance follows at perturbative renormalization-group (RG) fixed points (coupling values where the beta functions — the derivatives of the couplings with respect to the logarithm of the energy scale — vanish), while Ferrara, Grillo, Parisi and Gatto showed that canonical scaling alone does not force conformal invariance. Combined with the renormalization-group understanding that interacting scale-invariant theories arise as RG fixed points, the question became: does scale invariance at a fixed point always enhance to full conformal invariance?

The modern status is as follows. In two dimensions, scale invariance plus unitarity does imply conformal invariance, via Zamolodchikov's c-theorem (a quantity that decreases monotonically along RG flows) and holomorphicity. In four dimensions, Polchinski proved the enhancement perturbatively (using dimensional regularization); Dorigoni and Rychkov clarified why the two-dimensional proof does not extend and what unitarity contributes; Luty, Polchinski and Rattazzi ruled out perturbative scale-without-conformal scenarios via a-theorem arguments (the four-dimensional analogue of the c-theorem) and gave a conditional nonperturbative argument for theories whose stress tensor is "sufficiently nontrivial". Farnsworth, Luty and Prelipina claimed a complete nonperturbative proof in 2013, but that argument was shown incomplete by a follow-up analysis (Dymarsky, Farnsworth, Komargodski, Luty and Prilepina), which closes the related generalized-free-field loophole — the possibility that the trace of the energy-momentum tensor behaves as a generalized free field, an operator whose correlation functions factorize as if it were free though it is not — only under further conditions on dimension-2 scalar operators; Yonekura's independent argument requires additional assumptions of crossing symmetry, momentum-space analyticity and locality criteria. No proof is accepted as complete. Nakayama's review fixes the standard assumptions under which the question is posed: (1) unitarity, (2) Poincaré invariance, (3) a discrete spectrum of scaling dimensions, (4) existence of a scale current, and (5) unbroken scale invariance in the vacuum; under these, no counterexample is known and no complete nonperturbative proof exists.

On the representation side, Mack proved in 1977 that every irreducible unitary ray representation of the covering of $SU(2,2)$ with positive energy is a finite-component field representation, labeled by a scaling dimension and Lorentz spins — exactly the primary fields of modern conformal field theory — so infinite-component fields and non-fully-reducible induced representations with $\kappa_{\mu}\neq0$ are excluded from the unitary positive-energy class. Meanwhile the six-dimensional cone formalism of the 1969 paper survives as the embedding-space formalism of the modern conformal bootstrap (the numerical program that constrains CFT data from consistency conditions on correlation functions), as documented in Rychkov's historical review.

Problem Statement

The problem, as posed by Mack and Salam (1969), is to determine how conformal symmetry can be realized — and in what pattern it is broken — in a physically viable relativistic quantum field theory, through four linked questions:

(i) Admissibility of infinite-component realizations. Can a conformal field theory be built on infinite-component fields — fields transforming under infinite-dimensional representations of the little group (the stability subgroup of the conformal group that fixes the origin, generated by the Lorentz transformations, the dilatation, and the translation-like special conformal generators) — that is physically viable (causal, consistent with spin-statistics and PCT) once the unitarity requirement on the representation is relaxed, given that the finite-component primary-field scheme already exhausts the unitary positive-energy class (Mack 1977)?

(ii) The minimal-breaking hypothesis. Is the breaking of conformal symmetry in interacting quantum field theory always "minimal" — does the special conformal current satisfy

$\partial^{\nu}\mathscr{K}_{\nu\mu}=2x_{\mu}\,\partial^{\nu}\mathscr{D}_{\nu}$

with vanishing extra term $V_{\mu}$, so that breaking enters only through the divergence of the dilatation current? In the modern reformulation this asks: must a four-dimensional unitary Poincaré-invariant quantum field theory with unbroken scale invariance, a conserved scale current, and a discrete spectrum of scaling dimensions be conformally invariant — that is, must its symmetry algebra enhance from the Poincaré algebra plus dilatations to the full conformal algebra, equivalently must its energy-momentum tensor be improvable to a traceless one so that the virial current is removable — nonperturbatively, not merely in perturbation theory?

(iii) Consistency of $\kappa_{\mu}\neq0$ representations. Do the induced representations with $\kappa_{\mu}\neq0$, which yield conformally invariant four-fermion interactions but obstruct a hermitian conserved current, admit a symmetry-breaking scheme that preserves the desired physical properties while retaining useful conformal structure?

(iv) Interacting models in the six-dimensional formalism. Can realistic interacting conformal field theories — including their renormalization and anomaly consistency, and comparison with physical spectra and scattering — be constructed in the manifestly conformally covariant six-dimensional (cone) formalism?

These four questions share one objective — the field-theoretic realization of conformal symmetry.

References

1. G. Mack and Abdus Salam, "Finite-component field representations of the conformal group", Annals of Physics 53 (1969) 174-202, DOI 10.1016/0003-4916(69)90278-4, https://doi.org/10.1016/0003-4916(69)90278-4
2. Ernest Abers, I. T. Grodsky and Richard E. Norton, "Diseases of Infinite-Component Field Theories", Physical Review 159 (1967) 1222, DOI 10.1103/PhysRev.159.1222, https://doi.org/10.1103/PhysRev.159.1222
3. R. P. Zaikov and V. Cholakov, "Infinite Component Conformal Fields. Spectral Representation", Bulgarian Journal of Physics 3 (1976) 113-120, no DOI, https://inspirehep.net/literature/100877
4. Curtis G. Callan, "Broken Scale Invariance in Scalar Field Theory", Physical Review D 2 (1970) 1541, DOI 10.1103/PhysRevD.2.1541, https://doi.org/10.1103/PhysRevD.2.1541
5. Curtis G. Callan, Sidney Coleman and Roman Jackiw, "A new improved energy-momentum tensor", Annals of Physics 59 (1970) 42, DOI 10.1016/0003-4916(70)90394-5, https://doi.org/10.1016/0003-4916(70)90394-5
6. Sidney Coleman and Roman Jackiw, "Why dilatation generators do not generate dilatations", Annals of Physics 67 (1971) 552, DOI 10.1016/0003-4916(71)90153-9, https://doi.org/10.1016/0003-4916(71)90153-9
7. R. J. Crewther, "Nonperturbative Evaluation of the Anomalies in Low-Energy Theorems", Physical Review Letters 28 (1972) 1421, DOI 10.1103/PhysRevLett.28.1421, https://doi.org/10.1103/PhysRevLett.28.1421
8. G. Parisi, "Conformal invariance in perturbation theory", Physics Letters B 39 (1972) 643, DOI 10.1016/0370-2693(72)90020-2, https://doi.org/10.1016/0370-2693(72)90020-2
9. S. Ferrara, A. F. Grillo, G. Parisi and R. Gatto, "Canonical scaling and conformal invariance", Physics Letters B 38 (1972) 333, DOI 10.1016/0370-2693(72)90259-6, https://doi.org/10.1016/0370-2693(72)90259-6
10. G. Mack, "All unitary ray representations of the conformal group SU(2,2) with positive energy", Communications in Mathematical Physics 55 (1977) 1, DOI 10.1007/BF01613145, https://doi.org/10.1007/BF01613145
11. A. B. Zamolodchikov, "Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory", JETP Letters 43 (1986) 730, no DOI, https://inspirehep.net/literature/240292
12. Joseph Polchinski, "Scale and conformal invariance in quantum field theory", Nuclear Physics B 303 (1988) 226, DOI 10.1016/0550-3213(88)90179-4, https://doi.org/10.1016/0550-3213(88)90179-4
13. Daniele Dorigoni and Slava Rychkov, "Scale Invariance + Unitarity => Conformal Invariance?", arXiv:0910.1087 (2009), https://arxiv.org/abs/0910.1087
14. Markus A. Luty, Joseph Polchinski and Riccardo Rattazzi, "The a-theorem and the Asymptotics of 4D Quantum Field Theory", arXiv:1204.5221 (2012), https://arxiv.org/abs/1204.5221
15. Kara Farnsworth, Markus A. Luty and Valentina Prelipina, "Scale Invariance plus Unitarity Implies Conformal Invariance in Four Dimensions", arXiv:1309.4095 (2013), https://arxiv.org/abs/1309.4095
16. Anatoly Dymarsky, Kara Farnsworth, Zohar Komargodski, Markus A. Luty and Valentina Prilepina, "Scale invariance, conformality, and generalized free fields", Journal of High Energy Physics 02 (2016) 099, DOI 10.1007/JHEP02(2016)099, arXiv:1402.6322, https://arxiv.org/abs/1402.6322
17. Kazuya Yonekura, "Unitarity, Locality, and Scale versus Conformal Invariance in Four Dimensions", arXiv:1403.4939 (2014), https://arxiv.org/abs/1403.4939
18. Yu Nakayama, "Scale invariance vs conformal invariance", Physics Reports 569 (2015) 1-93, DOI 10.1016/j.physrep.2014.12.003, arXiv:1302.0884, https://arxiv.org/abs/1302.0884
19. Miguel S. Costa, Joao Penedones, David Poland and Slava Rychkov, "Spinning Conformal Correlators", arXiv:1107.3554 (2011), https://arxiv.org/abs/1107.3554
20. Slava Rychkov, "Conformal bootstrap: from Polyakov to our times", arXiv:2509.02779 (2025), https://arxiv.org/abs/2509.02779