Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup

A class of quadratic deformations of Lie superalgebras

We study certain Z_2-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.Comment: 26pp, LaTeX. Original title: "Finite dimensional quadratic Lie superalgebras"; abstract re-worded; text clarified; 3 references added; rearrangement of minor appendices into text; new subsection 4.

Covariant spinor representation of iosp(d,2/2)iosp(d,2/2) and quantization of the spinning relativistic particle

A covariant spinor representation of $iosp(d,2/2)$ is constructed for the quantization of the spinning relativistic particle. It is found that, with appropriately defined wavefunctions, this representation can be identified with the state space arising from the canonical extended BFV-BRST quantization of the spinning particle with admissible gauge fixing conditions after a contraction procedure. For this model, the cohomological determination of physical states can thus be obtained purely from the representation theory of the $iosp(d,2/2)$ algebra.Comment: Updated version with references included and covariant form of equation 1. 23 pages, no figure

Radio Galaxy populations and the multi-tracer technique: pushing the limits on primordial non-Gaussianity

We explore the use of different radio galaxy populations as tracers of different mass halos and therefore, with different bias properties, to constrain primordial non-Gaussianity of the local type. We perform a Fisher matrix analysis based on the predicted auto and cross angular power spectra of these populations, using simulated redshift distributions as a function of detection flux and the evolution of the bias for the different galaxy types (Star forming galaxies, Starburst galaxies, Radio-Quiet Quasars, FRI and FRII AGN galaxies). We show that such a multi-tracer analysis greatly improves the information on non-Gaussianity by drastically reducing the cosmic variance contribution to the overall error budget. By using this method applied to future surveys, we predict a constraint of sigma_fnl=3.6 on the local non-Gaussian parameter for a galaxy detection flux limit of 10 \muJy and sigma_fnl=2.2 for 1 \muJy. We show that this significantly improves on the constraints obtained when using the whole undifferentiated populations (sigma_fnl=48 for 10 \muJy and sigma_fnl=12 for 1 \muJy). We conclude that continuum radio surveys alone have the potential to constrain primordial non-Gaussianity to an accuracy at least a factor of two better than the present constraints obtained with Planck data on the CMB bispectrum, opening a window to obtain sigma_fnl~1 with the Square Kilometer Array.Comment: 9 pages, 5 figures, submitted to MNRA