We analyze the recently proposed classical/quantum mechanical interpretation of Zernike system and establish its equivalence to the Higgs oscillator on sphere or pseudosphere (Lobachevsky plane). We show that the non-reality of the classical Zernike Hamiltonian is an insignificant artifact of imaginary gauge and can be eliminated with a canonical transformation. The quantum counterpart of this canonical transformation is a similarity transformation mapping the system to the quantum Higgs oscillator with integration measure depending on [Formula: see text] parameters. When [Formula: see text], it results in the Hermitian Hamiltonian describing a free particle on (pseudo)sphere, while deviation from this point leads to a pseudo-Hermitian system.
We conjecture the connection between sl and so members of universal, in Vogel's sense, multiplets. The key element is the notion of the vertical componentwise sum circle plus v of Young diagrams. Representations in the decomposition of the power of the adjoint representation of sl(N) algebra can be parameterized by a pair of Young diagrams lambda and tau of equal area. We conjecture that within each universal multiplet there exists an associated pair of sl and so representations such that the Young diagram of the so representation is given by lambda circle plus v tau. This correspondence allows us to derive the universal form of the Casimir eigenvalues on that multiplet. We verify this conjecture for universal decompositions of tensor powers of the adjoint representations up to the fourth power, and for several series of universal representations. Based on this evidence, we propose a set of universal Casimirs for the fifth tensor power of the adjoint representation.We also conjecture that the vertical sum operation is connected with the (dual version of the) folding map of Dynkin diagrams. This connection may explain the intrinsic symmetry of universal formulae under automorphisms of Dynkin diagrams.
Context. Microquasars have emerged as promising candidates to explain the cosmic-ray flux at petaelectronvolt energies. LHAASO observations revealed V4641 Sgr as the most extreme example so far. Its gamma-ray spectrum extends up to 800 TeV, which requires particles with multi-PeV energy. The TeV emission is highly extended, which challenges expectations given the reported low-inclination angle of the V4641 Sgr jets. Aims. We spatially and spectrally resolved the gamma-ray emission from V4641 Sgr and investigated the particle acceleration in the system. Methods. Using ≈100 h of H.E.S.S. data, we performed a spectro-morphological study of the gamma-ray emission around V4641 Sgr. We employed HI and dedicated CO observations of the region to infer the target material for cosmic-ray interactions. Results. We detected multi-TeV emission around V4641 Sgr with a high significance. The emission region is elongated, and its major and minor axes are 0.34°±0.01syst±0.04stat and 0.06°±0.01syst±0.01stat, respectively. We found a power-law spectrum with an index ≈1.8, and together with results from other gamma-ray instruments, this reveals a spectral energy distribution (SED) that peaks at energies of ≈100 TeV for the first time. We found indications (3σ) of a two-component morphology, with indistinguishable spectral properties. The position of V4641 Sgr is inconsistent with the best-fit position of the single-component model and with the dip between the two components. We found no significant evidence of an energy-dependent morphology. No dense gas was found at any distance towards V4641 Sgr, which places an upper limit of ngas ≲ 0.2 cm−3 within the gamma-ray emission region. Conclusions. The peak of the SED at ≈100 TeV identifies V4641 Sgr as a candidate cosmic-ray accelerator beyond the so-called knee. The absence of dense target gas places stringent energetic constraints on hadronic interpretations, however. The H.E.S.S. measurement requires an unusually hard (≈1.5) spectral index for the protons. A leptonic scenario faces fewer obstacles if the particle transport is fast enough to avoid losses and to reproduce the observed energy-independent morphology. The absence of bright X-ray emission across the gamma-ray emission region requires a magnetic field strength ≲3 μG, however. Our findings favour a leptonic origin of the gamma-ray emission. This conclusion does not exclude hadron acceleration in the V4641 Sgr system.
We consider eigenvalues of the Casimir operator on the naturally defined stable sequences of representations of su(N) algebra and prove that eigenvalues are linear over N iff λ_1+2λ_2+...+kλ_k=λ_N-1+2λ_N-2+...+kλ_N-k, where λ_i are Dynkin labels, and λ_i=0 for k<i<N-k, with fixed k. These representations are exactly those which appear in the decomposition of ad(su(N))^⊗ k, therefore this linearity admits the presentation of eigenvalues in the universal, in Vogel's sense, form, and supports the hypothesis of universal decomposition of ad^⊗ k into Casimir eigenspaces.
We consider the most general covariant gravity action up to terms that are quadratic in curvature. These can be endowed with generic form factors, which are functions of the d’Alembert operator. If they are chosen in a specific way as an exponent of an entire function, the theory becomes ghost-free and renormalizable at the price of non-locality. Furthermore, according to power-counting arguments, if these functions grow sufficiently fast along the real axis, divergences may only appear at the first order in loop expansion. Using the heat kernel technique, we compute the one-loop logarithmic divergences in the ultraviolet limit and determine the conditions under which they vanish completely, apart from the Gauss-Bonnet term and a surface term, both of which can be neglected on a four-dimensional manifold without a boundary. We identify form factors both within the Tomboulis class and beyond it that lead to vanishing logarithmic divergences. The general expression for the one-loop beta functions of the dimensionless couplings in quadratic gravity with asymptotically monomial form factors is given.
