Symmetry properties of non-Hermitian PT-symmetric quantum field theories

We describe recent progress in understanding the continuous symmetry properties of non-Hermitian, ${\mathscr{PT}}$-symmetric quantum field theories. Focussing on a simple non-Hermitian theory composed of one complex scalar and one complex pseudoscalar, we revisit the derivation of Noether's theorem to show that the conserved currents of non-Hermitian theories correspond to transformations that do not leave the Lagrangian invariant. We illustrate the impact that this has on the consistent formulation of (Abelian) gauge theories by studying a non-Hermitian extension of scalar quantum electrodynamics. We consider the spontaneous breakdown of both global and local symmetries, and describe how the Goldstone theorem and the Englert-Brout-Higgs mechanism are borne out for non-Hermitian, ${\mathscr{PT}}$-symmetric theories

Emmy Noether: her heritage

Emmy Noether is my role model for the following reasons. She is one of the most important mathematician and physicists of the 20 century; She fought to be a scientist in times where women were not allowed to be one; She was a leader that gathered around her a large school of students and collaborators. In the lecture these reasons will be substantiated with historical facts. In addition, a general assessment of the significance of her ideas and works, from her times until today, will be advanced.Postprint (published version

Noether's theorem and gauge transformations. Application to the bosonic string and CP(2,n-1) model

New results on the theory of constrained systems are applied to characterize the generators of Noethers symmetry transformations. As a byproduct, an algorithm to construct gauge transformations in Hamiltonian formalism is derived. This is illustrated with two relevant examples

Retract (-mathitimathit{i}) rationality and its necessary conditions expressed by unramified presheaves : Noether's problem of a finite group GG as an example (Algebraic Number Theory and Related Topics)

This is another short introduction to the author's study of the rationality problem, which centers the hierarchies of the form: lower rationality = higher ruledness. A particular emphasis is given for Noether's problem of a finite group G, where a technical difficult emerges because the relevant geometric object BG is not approximated by smooth proper varieties. The author's novelty here is a construction of the stable birational subsheaf for any unramified presheaf in the sense of Morel. This gives us a very strong necessary condition for retract (-i) rationality of smooth, not necessary proper, varieties over a perfect field

Geometric Foundations of Gravity and Applications

The thesis is split into three parts: In the first part we describe the Geometric Trinity of Gravity, i.e. the three alternative formulations of gravitational interactions. General Relativity uses the curvature of spacetime to describe gravity. However, there are two other alternative but dynamically equivalent formulations: the Teleparallel theory of gravity, which suggests that gravity is mediated through the torsion of spacetime and the Symmetric Teleparallel gravity that assigns gravity to the non-metricity of spacetime. In addition, we discuss possible modifications in each case. In the second part, we use Lie and Noether symmetries of modified theories of gravity as a geometric criterion to classify them on those that are invariant under point transformations. Furthermore, we calculate the invariants of each symmetry and use them to reduce the dynamics of each system in order to find exact cosmological solutions. However, modified theories should also behave ``correctly'' at astrophysical scales too. That is why, in the last part, we use the notion of the maximum turnaround radius of a structure as a stability criterion to test theories of gravity. Specifically, we derive a general formula for the maximum turnaround radius, which denotes the maximum size that a structure can have, for all theories that respect the Einstein Equivalence Principle. Finally, we apply this formula to the Brans-Dicke and the $f(R)$ theories and discuss the requirements for the stability of large scale structures in their framework