Narrow Results

We present a proof that the quantum Yang–Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang–Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensures conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles.

A version of the Exact Renormalization Group Equation consistent with gauge symmetry is presented. A discussion of its regularization and renormalization is given. The relation with the Callan–Symanzik equation is clarified.

A general calculational method is applied to investigate symmetry relations among divergent amplitudes in a free fermion model. A very traditional work on this subject is revisited. A systematic study of one, two and three-point functions associated to scalar, pseudoscalar, vector and axial-vector densities is performed. The divergent content of the amplitudes are left in terms of five basic objects (external momentum independent). No specific assumptions about a regulator is adopted in the calculations. All ambiguities and symmetry violating terms are shown to be associated with only three combinations of the basic divergent objects. Our final results can be mapped in the corresponding Dimensional Regularization calculations (in cases where this technique could be applied) or in those of Gertsein and Jackiw which we will show in detail. The results emerging from our general approach allow us to extract, in a natural way, a set of reasonable conditions (e.g. crucial for QED consistency) that could lead us to obtain all Ward Identities satisfied. Consequently, we conclude that the traditional approach used to justify the famous triangular anomalies in perturbative calculations could be questionable. An alternative point of view, dismissed of ambiguities, which lead to a correct description of the associated phenomenology, is pointed out.

The structures and the associated gauge algebra of ABJM theory in ${𝒩}=1$ superspace are reviewed. We derive the Ward identities of the theory in the class of Lorentz-type gauges at quantum level to justify the renormalizability of the model. We compute the Nielsen identities for the two-point functions of the theory with the help of enlarged BRST transformation. The identities are derived in ABJM theory to ensure the gauge independence of the physical poles of the Green’s functions.

The Faddeev–Popov rules for a local and Poincaré-covariant Lagrangian quantization of a gauge theory with gauge group are generalized to the case of an invariance of the respective quantum actions, $S_{(N)}$, with respect to $N$-parametric Abelian SUSY transformations with odd-valued parameters ${\lambda }_p$, $p=1,\dots ,N$ and generators $s_p$: $s_p{s}_q+s_q{s}_p=0$, for $N=3,4$, implying the substitution of an $N$-plet of ghost fields, $C^p$, instead of the parameter, $\xi$, of infinitesimal gauge transformations: $\xi =C^p{\lambda }_p$. The total configuration spaces of fields for a quantum theory of the same classical model coincide in the $N=3$ and $N=4$ symmetric cases. The superspace of $N=3$ SUSY irreducible representation includes, in addition to Yang–Mills fields ${{𝒜}}^{\mu }$, $(3+1)$ ghost odd-valued fields $C^p$, $\widehat{B}$ and $3$ even-valued $B^{pq}$ for $p$, $q=1,2,3$. To construct the quantum action, $S_{(3)}$, by adding to the classical action, $S_0({𝒜})$, of an $N=3$-exact gauge-fixing term (with gauge fermion), a gauge-fixing procedure requires $(1+3+3+1)$ additional fields, ${\bar{\mathrm{\Phi }}}_{(3)}$: antighost $\bar{C}$, $3$ even-valued $B^p$, 3 odd-valued ${\widehat{B}}^{pq}$ and Nakanishi–Lautrup $B$ fields. The action of $N=3$ transformations on new fields as $N=3$-irreducible representation space is realized. These transformations are the $N=3$ BRST symmetry transformations for the vacuum functional, $Z_3(0)=\int d{\mathrm{\Phi }}_{(3)}d{\bar{\mathrm{\Phi }}}_{(3)}exp\{(ı/\hslash )S_{(3)}\}$. The space of all fields $({\mathrm{\Phi }}_{(3)},{\bar{\mathrm{\Phi }}}_{(3)})$ proves to be the space of an irreducible representation of the fields ${\mathrm{\Phi }}_{(4)}$ for $N=4$-parametric SUSY transformations, which contains, in addition to ${{𝒜}}^{\mu }$ the $(4+6+4+1)$ ghost–antighost, $C^r=(C^p,\bar{C})$, even-valued, $B^{rs}=-B^{sr}=(B^{pq},B^{p4}=B^p)$, odd-valued ${\widehat{B}}^r=(\widehat{B},{\widehat{B}}^{pq})$ and $B$ fields. The quantum action is constructed by adding to $S_0({𝒜})$ an $N=4$-exact gauge-fixing term with a gauge boson, $F_{(4)}$. The $N=4$ SUSY transformations are by $N=4$ BRST transformations for the vacuum functional, $Z_4(0)=\int d{\mathrm{\Phi }}_{(4)}exp\{(ı/\hslash )S_{(4)}\}$. The procedures are valid for any admissible gauge. The equivalence with $N=1$ BRST-invariant quantization method is explicitly found. The finite $N=3,4$ BRST transformations are derived and the Jacobians for a change of variables related to them but with field-dependent parameters in the respective path integral are calculated. They imply the presence of a corresponding modified Ward identity related to a new form of the standard Ward identities and describe the problem of a gauge-dependence. An introduction into diagrammatic Feynman techniques for $N=3,4$ BRST invariant quantum actions for Yang–Mills theory is suggested.

A Legendre transform of the recently discovered conformal fixed-point equation is constructed, providing an unintegrated equation encoding full conformal invariance within the framework of the effective average action.

The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one-dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in nonsolvable models has been achieved; it is based on exact Renormalization Group methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below.

This is a joint work with T. Spencer and M. Zirnbauer. We consider a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. We prove [1] that in three or more dimensions the model has a 'diffusive' phase at low temperatures. Localization is expected at high temperatures. The classical tools to study this kind of problem are multiscale analysis (renormalization) together with cluster+Mayer expansions or convexity bounds. Our analysis uses instead estimates on non-uniformly elliptic Green's functions and a family of Ward identities coming from internal supersymmetry.

In this talk I review recent advances on the understanding of the ground state properties of interacting electrons on the honeycomb lattice. In the case of weak short range interactions, renormalization group methods allowed us to give a complete construction of the ground state of the half-filled system and to prove analyticity in the coupling constant of the thermodynamic functions and of the equilibrium correlations. In the case that the electrons interact with a three-dimensional quantum electromagnetic field, the ground state can be constructed order by order in renormalized perturbation theory, with the n-th order admitting n!-bounds. Ward Identities are needed in order to control the flow of the effective charges. Lorentz invariance is dynamically restored, thanks to lattice gauge invariance. This talk is based on joint work with V. Mastropietro and M. Porta.

The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one-dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in nonsolvable models has been achieved; it is based on exact Renormalization Group methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below.