Paul Dirac

Summary

Paul Adrien Maurice Dirac (dɪˈræk; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is considered to be one of the founders of quantum mechanics and quantum electrodynamics. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University and the University of Miami, and a 1933 Nobel Prize in Physics recipient. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory". He also made significant contributions to the reconciliation of general relativity with quantum mechanics. Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul

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Symmetry and topology are fundamental properties of nature. Mathematics provides us with a general framework to understand these concepts. On one side, symmetry describes the invariance properties of an object for specific transformations. On the other side, topology classifies objects under continuous deformations. Two objects with different topologies cannot be deformed one into each other without creating or annihilating a singularity, sometimes referred to as 'node'. These concepts gradually found applications in physics, namely in the description of the electronic properties of solids, which is the focus of this thesis. Symmetry and topology protect special nodes in the band structure of a crystal, where several states are degenerate in energy. Close to a node, the electron wave function obeys the Dirac or Weyl Hamiltonians, which were formally introduced to describe fermions in high-energy particle physics. These exotic fermions exhibit unique optical and transport properties, fully manifesting their quantum nature. Symmetry guides us in the search for which classes of materials may host topological nodes. Recently the attention of the scientific community has been attracted by crystals with non-symmorphic symmetries (screw axes and glide planes), as promising candidates for topological phases. This thesis focuses on two families of non-symmorphic crystals, whose topological properties have been investigated by combining conventional angle-resolved photoelectron spectroscopy (ARPES) with state-of-the-art spin-resolved (spARPES) and time-resolved ARPES (trARPES). The experimental results are supported by calculations carried out in collaboration with theory groups. ZrSiTe and ZrSiSe belong to the same class of non-trivial semimetals. They host Dirac electrons with large mobility. The results of my spARPES experiments clarify that spin-orbit interaction (SOI) not only removes the topological nodes in ZrSiTe, but also induces a 'hidden' spin polarization of its bulk electronic states, otherwise forbidden by the inversion symmetry of the lattice. Moreover, trARPES data provide evidence that the electron-electron interaction is only partially screened in the metallic state of ZrSiSe. As a consequence, the band velocity is enhanced, at odds with general expectations. More importantly, I show that this band renormalization can be controlled at the ultrafast time scale (namely fs scale, with 1 fs=10^{-15} s) via intense optical excitation, paving the way for engineering the band structure of Dirac semimetals. Tellurium is a chiral semiconductor, with a small and direct band gap. The low-symmetry of its lattice and the simple chemical composition make it the ideal case to study the interplay between symmetry and topology. With ARPES I determine several Weyl nodes in its electronic structure. By means of spARPES, I demonstrate that in their surrounding the spin exhibits a hedgehog configuration. This observation is new and it highlights the connection between spin-dependent and topology-related properties in Te. Finally, I illustrate promising preliminary results based on trARPES that explore the appealing possibility of optically controlling the topology of the electronic structure of Te upon excitation of coherent phonons.

EPFL2020

Dirac Group(oid)s and Their Homogeneous Spaces

Madeleine Jotz

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie bialgebra. This result is extended in Liu et al. (1998) to a classification of the Poisson homogeneous spaces of a Poisson groupoid (G⇉P, πG) via a special class of Dirac structures in the Courant algebroid defined by the corresponding Lie bialgebroid. It is hence natural to ask what corresponds via these classifications, or extensions of them, to arbitrary Dirac structures in the Drinfel'd double or the Courant algebroid associated to a Poisson group(oid). We show in this thesis that there is a bigger class of Dirac structures that fits in this correspondence. They correspond naturally to Dirac homogeneous spaces of the Poisson group(oid). Dirac Lie groups and Dirac groupoids have been defined by Ortiz (2009) as a generalization of Poisson Lie groups, Poisson groupoids and presymplectic groupoids. We prove in an alternative manner the existence of a natural Lie bialgebra associated to a Dirac Lie group, and we find good candidates for the infinitesimal data of a Dirac groupoid; a square of morphisms of Lie algebroids associated to the multiplicative Dirac structure. These objects generalize simultaneously the Lie bialgebroid of Poisson groupoids and the IM-2-forms associated to presymplectic groupoids (Bursztyn et al. (2004)). We show also the existence of a Courant algebroid associated to these algebroids. This Courant algebroid structure is induced in a natural way by the ambient standard Courant algebroid TG ×G TG and is isomorphic to the Courant algebroid AG ×P AG in the Poisson case. We show that Dirac homogeneous spaces of Dirac Lie groups (respectively Dirac groupoids) correspond to certain classes of Dirac structures in the Drinfel'd double of the Lie bialgebra (respectively in the Courant algebroid). These results generalize the classification theorems in Drinfel'd (1993) and Liu et al. (1998). Along the way, we study involutive multiplicative foliations on Lie groupoids and show when and how there is a natural groupoid structure defined on the leaf spaces in the set of objects and the set of units of the Lie groupoid. This is a fact that is always true in the Lie group case, where a multiplicative foliation is automatically the left and right invariant image of an ideal in the Lie algebra, i.e., the leaves are the cosets of a normal subgroup of the Lie group.

EPFL2011

Dirac vs. Majorana HNLs (and their oscillations) at SHiP

SHiP is a proposed high-intensity beam dump experiment set to operate at the CERN SPS. It is expected to have an unprecedented sensitivity to a variety of models containing feebly interacting particles, such as Heavy Neutral Leptons (HNLs). Two HNLs or more could successfully explain the observed neutrino masses through the seesaw mechanism. If, in addition, they are quasi-degenerate, they could be responsible for the baryon asymmetry of the Universe. Depending on their mass splitting, HNLs can have very different phenomenologies: they can behave as Majorana fermions - with lepton number violating (LNV) signatures, such as same-sign dilepton decays - or as Dirac fermions with only lepton number conserving (LNC) signatures. In this work, we quantitatively demonstrate that LNV processes can be distinguished from LNC ones at SHiP, using only the angular distribution of the HNL decay products. Accounting for spin correlations in the simulation and using boosted decision trees for discrimination, we show that SHiP will be able to distinguish Majorana-like and Dirac-like HNLs in a significant fraction of the currently unconstrained parameter space. If the mass splitting is of order 10(-6) eV, SHiP could even be capable of resolving HNL oscillations, thus providing a direct measurement of the mass splitting. This analysis highlights the potential of SHiP to not only search for feebly interacting particles, but also perform model selection.

2020

Dirac Group(oid)s and Their Homogeneous Spaces

Madeleine Jotz

EPFL2011

EPFL2020