We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincare' group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fibre degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between E(2) (the two-fold cover of the 2-dimensional Euclidean group) and its conjugate.
We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localization.
We consider the action of the $q$-deformed Poincaré superalgebra on the massless non-relativistic R-matrix in ordinary (undeformed) integrable $AdS_2 \times S^2 \times T^6$ type IIB superstring theory. The boost generator acts non-trivially on the R-matrix, confirming the existence of a non-relativistic rapidity $γ$ with respect to which the R-matrix must be of difference form. We conjecture that from a massless AdS/CFT integrable relativistic R-matrix one can obtain the parental massless non-relativistic R-matrix simply by replacing the relativistic rapidity with $γ$. We check our conjecture …
An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral propertiessuch as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle…
In this work, we derive from first principles the relativistic wave equation of massless particles of arbitrary helicity. We start from unitary projective irreducible representations of the restricted Poincaré group. We define a weaker notion of localization and find, in particular, a position operator for any massless particle of arbitrary helicity. Therefore, having the position representation for these particles, we obtain the wave equations in this representation. We compare our results with previous findings in the literature.
We consider solutions of the massless scalar wave equation $\Box_gψ=0$, without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It follows from previous analyses in the Kerr exterior that for solutions $ψ$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon $\mathcal H^+$. Using the derived decay rate, we show that $ψ$ is in fact uniformly bounded, $|ψ|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which $ψ$ in…
