On representation-theoretic properties of fermionic fields in de Sitter spacetime and symmetries underlying the conservation of the electromagnetic zilches

This journal-style thesis presents chapters 2-6 in a format suitable for peer-reviewed publication. In chapter 2, we study the quantum spinor field on $N$-dimensional de Sitter spacetime ($dS_{N}$ ) with $(N − 1)$-sphere ($S^{N−1}$) spatial sections. We construct the mode solutions and study their transformation properties under the de Sitter (dS) algebra, spin$(N, 1)$. We reproduce the expression for the massless spinor Wightman two-point function using the mode-sum method. Then, taking advantage of the maximal symmetry of $dS_{N}$ , we construct the massive Wightman two-point function. In chapter 3, we construct the dictionary between the spaces of mode solutions for totally symmetric spin-$s = 3/2, 5/2$ tensor-spinors with any mass parameter on $dS_{N}$ ($N \geq 3$) and Unitary Irreducible Representations (UIRs) of spin$(N, 1)$. Remarkably, we find that the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on $dS_{N}$, are not unitary unless $N = 4$. Chapter 4 provides a technical explanation for the results of chapter 3 by investigating the (non-)existence of positive-definite, dS invariant scalar products for the mode solutions. In chapter 5, we uncover a ‘conformal-like’ $spin$(4, 2) symmetry for strictly massless spin-$s \geq 3/2$ tensor-spinors on $dS_{4}$. We also show that the mode solutions form UIRs of not only the dS algebra but also of spin$(4, 2)$. In chapter 6, we shift focus to the ‘zilches’, a set of little-known conserved quantities for the free electromagnetic (EM) field in four-dimensional Minkowski spacetime. We present, for the first time, the derivation of all zilch conservation laws from ‘zilch symmetries’ of the standard EM action using Noether’s theorem. We also show that the zilch symmetries belong to the enveloping algebra of a "hidden" invariance algebra of free Maxwell’s equations in potential form.