On the stability of Hamiltonian relative equilibria with non-trivial isotropy

We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu and by Lerman and Singer. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subspace of the Lie algebra of the symmetry group, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more easily computed criterion for stability. The result is also extended to apply to systems whose momentum map is not coadjoint equivariant

Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity

We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master Theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.Comment: 33 pages, no figures, LaTe

Models for Modules

We recall the structure of the indecomposable sl(2) modules in the Bernstein-Gelfand-Gelfand category O. We show that all these modules can arise as quantized phase spaces of physical models. In particular, we demonstrate in a path integral discretization how a redefined action of the sl(2) algebra over the complex numbers can glue finite dimensional and infinite dimensional highest weight representations into indecomposable wholes. Furthermore, we discuss how projective cover representations arise in the tensor product of finite dimensional and Verma modules and give explicit tensor product decomposition rules. The tensor product spaces can be realized in terms of product path integrals. Finally, we discuss relations of our results to brane quantization and cohomological calculations in string theory.Comment: 18 pages, 6 figure

Mathisson's helical motions for a spinning particle --- are they unphysical?

It has been asserted in the literature that Mathisson's helical motions are unphysical, with the argument that their radius can be arbitrarily large. We revisit Mathisson's helical motions of a free spinning particle, and observe that such statement is unfounded. Their radius is finite and confined to the disk of centroids. We argue that the helical motions are perfectly valid and physically equivalent descriptions of the motion of a spinning body, the difference between them being the choice of the representative point of the particle, thus a gauge choice. We discuss the kinematical explanation of these motions, and we dynamically interpret them through the concept of hidden momentum. We also show that, contrary to previous claims, the frequency of the helical motions coincides, even in the relativistic limit, with the zitterbewegung frequency of the Dirac equation for the electron

On transversally elliptic operators and the quantization of manifolds with ff-structure

An $f$-structure on a manifold $M$ is an endomorphism field \phi\in\Gamma(M,\End(TM)) such that $\phi^3+\phi=0$. Any $f$-structure $\phi$ determines an almost CR structure E_{1,0}\subset T_\C M given by the $+i$-eigenbundle of $\phi$. Using a compatible metric $g$ and connection $\nabla$ on $M$, we construct an odd first-order differential operator $D$, acting on sections of $\S=\Lambda E_{0,1}^*$, whose principal symbol is of the type considered in arXiv:0810.0338. In the special case of a CR-integrable almost $\S$-structure, we show that when $\nabla$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator $D$ is given by D = \sqrt{2}(\dbbar+\dbbar^*), where \dbbar is the tangential Cauchy-Riemann operator. We then describe two "quantizations" of manifolds with $f$-structure that reduce to familiar methods in symplectic geometry in the case that $\phi$ is a compatible almost complex structure, and to the contact quantization defined in \cite{F4} when $\phi$ comes from a contact metric structure. The first is an index-theoretic approach involving the operator $D$; for certain group actions $D$ will be transversally elliptic, and using the results in arXiv:0810.0338, we can give a Riemann-Roch type formula for its index. The second approach uses an analogue of the polarized sections of a prequantum line bundle, with a CR structure playing the role of a complex polarization.Comment: 31 page

Reduction and approximation in gyrokinetics

The gyrokinetics formulation of plasmas in strong magnetic fields aims at the elimination of the angle associated with the Larmor rotation of charged particles around the magnetic field lines. In a perturbative treatment or as a time-averaging procedure, gyrokinetics is in general an approximation to the true dynamics. Here we discuss the conditions under which gyrokinetics is either an approximation or an exact operation in the framework of reduction of dynamical systems with symmetryComment: 15 pages late

C^{2} formulation of Euler fluid

The Hamiltonian formalism for the continuous media is constructed using the representation of Euler variables in $\mathcal{C}^{2}\times \infty$ phase space.Comment: 8 page

Anisotropic Hubble expansion of large scale structures

We investigate the dynamics of an homogenous distribution of galaxies moving under the cosmological expansion through Euler-Poisson equations system. The solutions are interpreted with the aim of understanding the cosmic velocity fields in the Local Super Cluster, and in particular the presence of a bulk flow. Among several solutions, we shows a planar kinematics with constant (eternal) and rotational distortion, the velocity field is not potential

SOT-MRAM 300mm integration for low power and ultrafast embedded memories

We demonstrate for the first time full-scale integration of top-pinned perpendicular MTJ on 300 mm wafer using CMOS-compatible processes for spin-orbit torque (SOT)-MRAM architectures. We show that 62 nm devices with a W-based SOT underlayer have very large endurance (> 5x10^10), sub-ns switching time of 210 ps, and operate with power as low as 300 pJ.Comment: presented at VLSI2018 session C8-

Non-abelian Harmonic Oscillators and Chiral Theories

We show that a large class of physical theories which has been under intensive investigation recently, share the same geometric features in their Hamiltonian formulation. These dynamical systems range from harmonic oscillations to WZW-like models and to the KdV dynamics on $Diff_oS^1$. To the same class belong also the Hamiltonian systems on groups of maps. The common feature of these models are the 'chiral' equations of motion allowing for so-called chiral decomposition of the phase space.Comment: 1