Gerard 't Hooft

Regularization and renormalization of gauge fields

A new regularization and renormalization procedure is presented. It is particularly well suited for the treatment of gauge theories. The method works for theories that were known to be renormalizable as well as for Yang-Mills type theories. Overlapping divergencies are disentangled. The procedure respects unitarity, causality and allows shifts of integration variables. In non-anomalous cases also Ward identities are satisfied at all stages. It is transparent when anomalies, such as the Bell-Jackiw-Adler anomaly, may occur.

On the Phase Transition Towards Permanent Quark Confinement

In quantized gauge field theories one can introduce sets of operators that modify the gauge-topological structure of the fields but whose physical effect is essentially local. In 2 + 1 dimensional non-Abelian gauge theories these operators form scalar fields and it is argued that when the local gauge symmetry is not spontaneously broken then these topological fields develop a vacuum expectation value and their mutual symmetry breaks spontaneously. It is shown that quarks are then permanently confined. In 3 + 1 dimensional non-Abelian gauge theories one finds that the topological operators and the gauge field operators form a closed algebra from which it is deduced that this system can be in one of the four different phases: (i) spontaneous breakdown via an explicit or composite Higgs field, (ii) no Higgs field but permanent confinement of gauge quantum numbers, (iii) Higgs effect and still confinement, presumably only if there is an unbroken subgroup, and (iv) an intermediate phase (critical point?) with massless particles. Finally, the algebra can be realized in a simple model where phases (i) and (ii) can be obtained from each other by dual transformation.

Topology of the gauge condition and new confinement phases in non-abelian gauge theories☆

Abstract The gauge-fixing constraint in a gauge field theory is crucial for understanding both short-distance and long-distance behavior of non-abelian gauge field theories. We define what we call “non-propagating” gauge conditions such as the unitary gauge and “approximately non-propagating” or renormalizable gauge conditions, and study their topological properties. By first fixing the non-abelian part of the gauge ambiguity we find that SU( N ) gauge theories can be written in the form of abelian gauge theories with N − 1 fold multiplicity enriched with magnetic monopoles with certain magnetic charge combinations. Their electric chargesare governed by the instanton angle θ. If θ is continuously varied from 0 to 2π and a confinement mode is assumed for some θ, then at least one phase-transition must occur. We speculate on the possibility of new phases: e.g., “oblique confinement,” where θ ⋍ π , and explain some peculiar features of this mode. In principle there may be infinitely many such modes, all separated by phase transition boundaries.

Scalar one-loop integrals

The completely general one-loop scalar one-, two-, three- and four-point functions are studied. Also an integral occurring in connection with soft bremsstrahlung is considered. Formulas in terms of Spence functions are given. An expansion for Spence functions with complex argument is presented. The whole forms a basis for the calculation of one-loop radiative corrections in the general case, including unstable particles and particles with spin.

A property of electric and magnetic flux in non-abelian gauge theories

Abstract Pure non-Abelian gauge models with gauge group SU( N ) are considered in a box with periodic boundary conditions at various temperatures β −1 . Electric and magnetic flux are defined in a gauge-invariant way. The free energy of the system satisfies an exact duality equation, following from Euclidean invariance. The equation relates properties of the electric and the magnetic fields. Conclusions that can be drawn for instance are that for N ⩽ 3 one cannot have both electric and magnetic confinement, and that the infrared structure of the Georgi-Glashow model is self-dual.

On the Quantum Structure of a Black Hole

The assumption is made that black holes should be subject to the same rules of quantum mechanics as ordinary elementary particles or composite systems. Although a complete theory for reconciling this requirement with that of general coordinate transformation invariance is not yet in sight, a number of observations can be made and a general framework is suggested.

Renormalizable Lagrangians for massive Yang-Mills fields

Abstract Renormalizable models are constructed in which local gauge invariance is broken spontaneously. Feynman rules and Ward identities can be found by means of a path integral method, and they can be checked by algebra. In one of these models, which is studied in more detail, local SU(2) is broken in such a way that local U(1) remains as a symmetry. A renormalizable and unitary theory results, with photons, charged massive vector particles, and additional neutral scalar particles. It has three independent parameters. Another model has local SU(2)⊗U(1) as a symmetry and may serve as a renormalizable theory for ϱ-mesons and photons. In such models electromagnetic mass-differences are finite and can be calculated in perturbation theory.

Three-dimensional Einstein gravity: dynamics of flat space

In three spacetime dimensions, the Einstein equations imply that source-free regions are flat. Localized sources can therefore only affect geometry globally rather than locally. Some of these effects, especially those generated by mass and angular momentum are discussed.

Renormalization of massless Yang-Mills fields

The problem of renormalization of gauge fields is studied. It is observed that the use of non-gauge invariant regulator fields is not excluded provided that in the limit of high regulator mass gauge invariance can be restored by means of a finite number of counter-terms in the Lagrangian. Massless Yang-Mills fields can be treated in this manner, and appear to be renormalizable in the usual sense. Consistency of the method is proved for diagrams with non-overlapping divergencies by means of gauge invariant regulators, which however, cannot be interpreted in terms of regulator fields. Assuming consistency the S-matrix is shown to be unitary in any order of the coupling constant. A restriction must be made: no local, parity-changing transformations must be contained in the underlying gauge group. The interactions must conserve parity.

Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking

A properly called “naturalness” is imposed on gauge theories. It is an order-of-magnitude restriction that must hold at all energy scales μ. To construct models with complete naturalness for elementary particles one needs more types of confining gauge theories besides quantum chromodynamics. We propose a search program for models with improved naturalness and concentrate on the possibility that presently elementary fermions can be considered as composite. Chiral symmetry must then be responsible for the masslessness of these fermions. Thus we search for QCD-like models where chiral symmetry is not or only partly broken spontaneously. They are restricted by index relations that often cannot be satisfied by other than unphysical fractional indices. This difficulty made the author’s own search unsuccessful so far. As a by-product we find yet another reason why in ordinary QCD chiral symmetry must be broken spontaneously.