Conformal Blocks for the 4-Point Function in Conformal Quantum Mechanics

Extending previous work on 2 -- and 3 -- point functions, we study the 4 -- point function and its conformal block structure in conformal quantum mechanics CFT$_1$, which realizes the SO(2,1) symmetry group. Conformal covariance is preserved even though the operators with which we work need not be primary and the states are not conformally invariant. We find that only one conformal block contributes to the four-point function. We describe some further properties of the states that we use and we construct dynamical evolution generated by the compact generator of SO(2.1).Comment: Formula for 4-point function is corrected and generalize

Tutorial on Scale and Conformal Symmetries in Diverse Dimensions

We review the relation between scale and conformal symmetries in various models and dimensions. We present a dimensional reduction from relativistic to non-relativistic conformal dynamics.Comment: Addendum added that includes author S.-H. Ho; Sodano Fest, Perugia, Italy, January 201

Persistence of zero modes in a gauged Dirac model for bilayer graphene

A recently constructed model for low lying excitations in bilayer graphene exhibits mid-gap, zero energy modes in its Dirac-like spectrum, when a scalar order parameter takes a vortex profile. We show that these modes persist when the dynamics is extended by a gauge field interaction, which also renders finite the vortex energy. The effect of the gauge field on the zero energy wave function is to shift the phase of the (damped) oscillatory component of the wave function in the absence of the gauge field.Comment: 7 pages, no figure

About the oscillatory possibilities of the dynamical systems

This paper attempts to make feasible the evolutionary emergence of novelty in a supposedly deterministic world which behavior is associated with those of the mathematical dynamical systems. The work was motivated by the observation of complex oscillatory behaviors in a family of physical devices, for which there is no known explanation in the mainstream of nonlinear dynamics. The paper begins by describing a nonlinear mechanism of oscillatory mode mixing explaining such behaviors and establishes a generic dynamical scenario with extraordinary oscillatory possibilities, including expansive growing scalability. The relation of the scenario to the oscillatory behaviors of turbulent fluids and living brains is discussed. Finally, by considering the scenario as a dynamic substrate underlying generic aspects of both the functioning and the genesis of complexity in a supposedly deterministic world, a theoretical framework covering the evolutionary development of structural transformations in the time evolution of that world is built up.Comment: 40 pages, 12 figures, to appear in Physica

State Space for Planar Majorana Zero Modes

Zero modes arising from a planar Majorana equation in the presence of $N$ vortices require an $\mathcal{N}$-dimensional state-space, where $\mathcal{N} = 2^{N/2}$ for $N$ even and $\mathcal{N} = 2^{(N + 1)/2}$ for $N$ odd. The mode operators form a restricted $\mathcal{N}$-dimensional Clifford algebra.Comment: 6 pages, no figure

Fake Conformal Symmetry in Conformal Cosmological Models

We examine the local conformal invariance (Weyl invariance) in tensor-scalar theories used in recently proposed conformal cosmological models. We show that the Noether currents associated with Weyl invariance in these theories vanish. We assert that the corresponding Weyl symmetry does not have any dynamical role.Comment: References updated, errors corrected, published in Phys. Rev. D. 4 pages, RevTeX, no figure

Reducing the Chern-Simons term by a symmetry

Reducing a 3-dimensional Chern-Simons term by a symmetry yields other topologically interesting structures. Specifically, reducing by radial symmetry results in a 1-dimensional quantum mechanical model, which has recently been used in an analysis of finite-temperature Chern-Simons theory. The radially symmetric expression may be inserted into 3-dimensional monopole or (2+1)-dimensional instanton equations, where it eliminates the monopole or instanton solutions.Comment: Email correspondence to [email protected] ; 7 pages, REVTe