10,958 research outputs found
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 and quantization of the spinning relativistic particle
A covariant spinor representation of 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 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
- …