Prehistory – Starting from ’t Hooft’s (1971) we have a short look at Taylor’s and Slavnov’s works (1971–72) and at the lectures given by Rouet and Stora in Lausanne (1973) which determine the transition from pre-history to history.

History – We give a brief account of the main analyses and results of the BRS collaboration concerning the renormalized gauge theories, in particular the method of the regularization-independent, algebraic renormalization, the algebraic proof of S-matrix unitarity and that of gauge choice independence of the renormalized physics. We conclude this report with a suggestion to the crucial question: what could remain of BRS invariance beyond perturbation theory.

This is a short review of work done in common with Jacques Bros, Michel Gaudin, Ugo Moschella, and Vincent Pasquier. Among results are explicit Källén–Lehmann representations for products of two free-field two-point functions in the de Sitter and the anti-de Sitter spaces and applications to particle decay.

We discuss the transport of a tracer particle through the Bose–Einstein condensate of a Bose gas. The particle interacts with the atoms in the Bose gas through two-body interactions. In the limiting regime where the particle is very heavy and the Bose gas is very dense, but very weakly interacting (‘mean-field limit’), the dynamics of this system corresponds to classical Hamiltonian dynamics. We show that, in this limit, the particle is decelerated by emission of gapless modes into the condensate (Cerenkov radiation). For an ideal gas, the particle eventually comes to rest. In an interacting Bose gas, the particle is decelerated until its speed equals the propagation speed of the Goldstone modes of the condensate. This is a model of ‘Hamiltonian friction’. It is also of interest in connection with the phenomenon of ‘decoherence’ in quantum mechanics. This note is based on work we have carried out in collaboration with D Egli, I M Sigal and A Soffer.

Pauli–Villars regularization of Yang–Mills theories and of supergravity theories is outlined, with an emphasis on BRST invariance. Applications to phenomenology and the anomaly structure of supergravity are discussed.

One challenge in particle physics is that not all the momenta relevant to many processes are observable. Some particles are nearly invisible (neutrinos and hypothetical neutralinos), others escape undetected down the beam pipes of colliders. One faces the task of extracting the maximum information (e.g. on the mass of the unobserved particles) from a set of more unknowns than constraining energy–momentum conservation equations. We study the simplest realistic case of current interest: single-𝑊 production at a hadron collider, followed by its leptonic decay. We derive and discuss the statistically-optimal ‘singularity variable’ relevant to the measurement of the 𝑊 mass. In spite of its simplicity, this process is fairly non-trivial and constitutes a good ‘training’ example for the scrutiny of phenomena involving invisible objects. Our graphical analysis of the phase space is akin to that of a Dalitz plot, extended to such processes.

We report some results on the relation between extremal black holes in locally supersymmetric theories of gravity and groups of type $E_7$, appearing as generalized electric-magnetic duality symmetries in such theories. Some basics on the covariant approach to the stratification of the relevant symplectic representation are reviewed, along with a connection between special Kähler geometry and a ‘generalization’ of groups of type $E_7$.

The algebraic structure of chiral anomalies is made globally valid on non-trivial bundles by the introduction of a fixed background connection. Some of the techniques used in the study of the anomaly are improved or generalized, including a systematic way of generating towers of ‘descent equations’.

In the large-$N_c$ limit of QCD, two-point functions of local operators become harmonic sums. I review some properties which follow from this fact and which are relevant for phenomenological applications. This has led us to consider a class of analytic number theory functions as toy models of large-$N_c$ QCD which also is discussed.

We discuss the notion of Poincaré duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincaré duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.