The eigenmodes for spinor quantum field theory in global de Sitter space-time

The mode solutions of the Dirac equation on $N$-dimensional de Sitter space-time ($dS_{N}$) with $(N-1)$-sphere spatial sections are obtained by analytically continuing the spinor eigenfunctions of the Dirac operator on the $N$-sphere ($S^{N}$). The analogs of flat space-time positive frequency modes are identified and a vacuum is defined. The transformation properties of the mode solutions under the de Sitter group double cover (Spin($N$,1)) are studied. We reproduce the expression for the massless spinor Wightman two-point function in closed form using the mode-sum method. By using this closed-form expression and taking advantage of the maximal symmetry of $dS_{N}$ we find an analytic expression for the spinor parallel propagator. The latter is used to construct the massive Wightman two-point function in closed form.Comment: 33 page

New conformal-like symmetry of strictly massless fermions in four-dimensional de Sitter space

We present new infinitesimal `conformal-like' symmetries for the field equations of strictly massless spin-$s \geq 3/2$ totally symmetric tensor-spinors (i.e. gauge potentials) on 4-dimensional de Sitter spacetime ($dS_{4}$). The corresponding symmetry transformations are generated by the five conformal Killing vectors of $dS_{4}$, but they are not conventional conformal transformations. We show that the algebra generated by the ten de Sitter (dS) symmetries and the five conformal-like symmetries closes on the conformal-like algebra $so(4,2)$ up to gauge transformations of the gauge potentials. Furthermore, we demonstrate that the two sets of physical mode solutions, corresponding to the two helicities $\pm s$ of the strictly massless theories, form a direct sum of Unitary Irreducible Representations (UIRs) of the conformal-like algebra. We also fill a gap in the literature by explaining how these physical modes form a direct sum of Discrete Series UIRs of the dS algebra $so(4,1)$.Comment: 44 pages, no figure

The eigenmodes for spinor quantum field theory in global de Sitter space-time

The mode solutions of the Dirac equation on N-dimensional de Sitter space-time (dS_{N)) with (N−1)-sphere spatial sections are obtained by analytically continuing the spinor eigenfunctions of the Dirac operator on the N-sphere (S^{N}). The analogs of flat space-time positive frequency modes are identified and a vacuum is defined. The transformation properties of the mode solutions under the de Sitter group double cover (Spin(N,1)) are studied. We reproduce the expression for the massless spinor Wightman two-point function in closed form using the mode-sum method. By using this closed-form expression and taking advantage of the maximal symmetry of dS_{N} we find an analytic expression for the spinor parallel propagator. The latter is used to construct the massive Wightman two-point function in closed form

A hidden invariance algebra of Maxwell's equations and the conservation of all Lipkin's zilches from symmetries of the standard electromagnetic action

In 1964, Lipkin discovered a set of conserved quantities for free electromagnetism without a clear physical interpretation, known as the zilches. In 2010, Tang and Cohen realized that one of the zilches, termed as optical chirality, provides a measure of the handedness of light. Although the zilch symmetries of Maxwell's equations in terms of the electromagnetic (EM) tensor are known, the question of how to derive all zilch conservation laws from symmetries of the standard EM action using Noether's theorem has been answered only in the case of optical chirality. In this Letter, we provide the full answer by showing that the zilch symmetry transformations of the 4-potential, $A_{\mu}$, leave invariant the standard EM action. In the rest of the article, we provide new insight concerning the conservation of the zilches and their underlying symmetries. First, we show that the zilch symmetries belong to the enveloping algebra of a ``hidden'' invariance algebra of free Maxwell's equations in potential form. The ``hidden'' algebra closes on $so(6,\mathbb{C})_{\mathbb{R}}$ up to gauge transformations of $A_{\mu}$. The generators of the ``hidden'' algebra consist of familiar conformal symmetry transformations and certain ``hidden'' symmetry transformations of $A_{\mu}$. We discuss the generalization of these ``hidden'' symmetries in the presence of a material 4-current, $J^{\mu}$. The ``hidden'' symmetries are also discussed for the theory of a complex Abelian gauge field. Finally, we show that the zilch symmetries of the standard free EM action can be extended to symmetries of the standard interacting action by considering simultaneous transformations of $A_{\mu}$ and $J^{\mu}$. This allows us to give a new derivation of the continuity equation for optical chirality in the presence of electric charges and currents, while we also derive new continuity equations for the rest of the zilches.Comment: no figures; v7: misprints corrected; v8: minor changes in title and abstract, literature review included in the Introduction, discussion of "hidden'' symmetries in Sec. IV extende

(Non-)unitarity of strictly and partially massless fermions on de Sitter space

Abstract We present the dictionary between the one-particle Hilbert spaces of totally symmetric tensor-spinor fields of spin s = 3/2, 5/2 with any mass parameter on D-dimensional (D ≥ 3) de Sitter space (dS D ) and Unitary Irreducible Representations (UIR’s) of the de Sitter algebra spin(D, 1). Our approach is based on expressing the eigenmodes on global dS D in terms of eigenmodes of the Dirac operator on the (D − 1)-sphere, which provides a natural way to identify the corresponding representations with known UIR’s under the decomposition spin(D, 1) ⊃ spin(D). Remarkably, we find that four- dimensional de Sitter space plays a distinguished role in the case of the gauge-invariant theories. In particular, the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on dS D , are not unitary unless D = 4

A hidden invariance algebra of Maxwell’s equations and the conservation of all Lipkin’s zilches from symmetries of the standard electromagnetic action

In 1964, Lipkin discovered a set of conserved quantities for free electromagnetism without a clear physical interpretation, known as the \textit{zilches}. In 2010, Tang and Cohen realized that one of the zilches, termed as \textit{optical chirality}, provides a measure of the handedness of light, motivating novel investigations into the interactions of light with chiral matter. Although the \textit{zilch symmetries} of Maxwell's equations underlying the conservation of the zilches are known, the question of how to explicitly derive all zilch conservation laws from symmetries of the standard free EM action using Noether's theorem has been answered only in the case of optical chirality. In this Letter, we provide the answer to this question by showing that the {zilch symmetries} leave invariant the standard free EM action. In the rest of the article, we provide new insight concerning the conservation of the zilches and their underlying symmetries. First, we show that the zilch symmetries belong to the enveloping algebra of a \textit{``hidden'' invariance algebra} of free Maxwell's equations in potential form. The ``hidden'' algebra closes on \texorpdfstring{$so(6,\mathbb{C})_{\mathbb{R}}$}{so(6,R)} up to gauge transformations of the four-potential~\texorpdfstring{$A_{\mu}$}{A}. The generators of the ``hidden'' algebra consist of familiar conformal symmetry transformations and certain \textit{``hidden'' symmetry transformations} of~\texorpdfstring{$A_{\mu}$}{A}. We discuss the generalization of these ``hidden'' symmetries of Maxwell's equations in the presence of a material four-current,~\texorpdfstring{$J^{\mu}$}{J}. The ``hidden'' symmetries are also discussed for the theory of a complex Abelian gauge field (this is related to the complex formulation of duality-symmetric electromagnetism). Finally, we show that the zilch symmetries of the standard free EM action can be extended to zilch symmetries of the standard interacting action,~\texorpdfstring{$S'$}{S'}, by considering simultaneous transformations of both~\texorpdfstring{$A_{\mu}$}{A} and~\texorpdfstring{$J^{\mu}$}{J}. This allows us to give a new derivation of the continuity equation for optical chirality in the presence of electric charges and currents, while we also derive new continuity equations for the rest of the zilches