2026-04-07 - 2026-04-10
Hier geht es zur AnmeldungAaron Held
Ecole Normale Superieure (Paris / Rue d'Ulm)
As you may remember from your lectures on classical mechanics, quantum mechanics, electrodynamics, or general relativity, our current descriptions of nature are all governed by second-order differential equations. The underlying reason is that higher time derivatives are associated with dangerous instabilities and thus commonly dismissed as pathological. This dismissal is rooted in a two-step inference: First, a famous theorem by Ostrogradski links (non-degenerate) higher-derivative motion with opposite-energy degrees of freedom – so-called ghosts. Second, the resulting unbounded total energy is assumed to cause an unavoidable instability. While the first inference is a mathematical fact, the second is not – and indeed, insightful counterexamples have by now been established.
The first lecture reviews the Ostrogradski theorem and detail common expectations prior to the above developments. The subsequent three lectures on (i) classical mechanics, (ii) quantum mechanics, and (iii) classical field theory will cover the construction of rigorous counterexamples as well as physical explanations of how common expectations fail. We will also discuss relevant applications, in particular in the context of effective field theory and nonlinear gravitational phenomena such as black-hole binaries.
The course will include pedagogical introductions to mathematical topics required to draw the above conclusions. In particular, this includes the theory of classical and quantum integrable systems as well as the mathematical analysis of partial differential equations. The introduction of these tools will hopefully also be interesting beyond the specific application to ghosts.
