Symplectic and Semiclassical Aspects of the Schl\"afli Identity

The Schl\"afli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schl\"afli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to 3-dimensional spaces of constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin networks, are discussed.Comment: 40 pages, 8 figure

Nonlinear Polarization Transfer and Control of Two Laser Beams Overlapping in a Uniform Nonlinear Medium

A scheme for polarization control using two laser beams in a non-linear optical medium is studied using both co- and counter-propagating beam geometries. In particular, we show that under certain conditions it is possible for two laser beams to exchange their polarization states. A model accounting for a more realistic, 2D propagation geometry is presented. The 2D model produces drastically different results (compared to the 1D propagation geometry), creating difficulties for implementing polarization control in a realistic setting. A proposal for overcoming these difficulties by reducing the non-linear optical medium to a thin slab is presented

Investigation of Beam Instability Under the Effects of Long-Range Transverse Wake Fields in the Berkeley Future Light Source

An ultra-relativistic charged particle bunch moving through a resonator cavity leaves behind a wake field that will affect subsequent bunches (if the bunch is not ultra-relativistic, the wake field will not be exclusively behind it). If the initial bunch enters the cavity off-axis, it will produce a transverse wake field that can then kick later bunches off the axis. Thus, even bunches that were initially traveling on axis could be displaced and, in turn, produce their own transverse wake fields, affecting following bunches. The offsets obtained by bunches could increase along the bunch train, leading to the so-called multi-bunch beam break-up instability [1]. The purpose of our investigation is to see whether such instability will occur in the superconducting, 1.3 GHz, 2.5GeV linac (see Table 1) planned for the Berkeley future light source (BFLS). We assume an initial steady-state situation established for machine operation; i.e. a continuous process where every bunch follows the same trajectory through the linac, with only small deviations from the axis of the rf structures. We will look at a possible instability arising from a bunch having a small deviation from the established trajectory. Such a deviation would produce a wake field that is slightly different from the one produced by the bunches following the established trajectory. This could lead to subsequent bunches deviating further from the established trajectory. We will assume the deviations are small (at first) and so the difference in the wake field caused by a bunch not traveling along the established trajectory is well approximated by a long-range transverse dipole wake. We are concerned only with deviations from the established trajectory; thus, in our models, a transverse position of zero corresponds to the bunch traveling along the established trajectory. Under this assumption, only the additional long-range transverse dipole wake remains in our models

Multisymplectic Geometry with Boundaries

Geometric approaches form the foundation of modern classical mechanics. The prototypical example of a geometric method in mechanics is symplectic geometry applied to the Hamiltonian formulation of a system of particles. Extending this approach to field theories leads to unattractive features, such as an infinite-dimensional phase space and loss of manifest covariance. These deficiencies are particularly glaring in general relativity, where manifest covariance is closely tied to the fundamental symmetries of the theory. Recent progress on covariant Hamiltonian approaches for field theories has led to the development of multisymplectic geometry. Multisymplectic geometry generalizes the symplectic geometry of particle systems to covariant fields, producing a finite-dimensional phase space and retaining manifest covariance. The symplectic 2-form common to symplectic geometry generalizes to the multisymplectic 5-form. The Euler-Lagrange equations for the field can be written in geometric language using the 5-form in a way that is formally identical to the geometric form of Hamilton's equations in particle mechanics. The resulting approach is a powerful geometric tool for understanding classical field theories.In this dissertation, we improve upon the current approach to performing a 3+1 decomposition (also known as space-time split) of multisymplectic geometry. We clarify the relationship between multisymplectic geometry, its 3+1 decomposition, and the traditional symplectic approach to field theory. The key observation is that there exist two intermediate phase spaces between the multisymplectic phase space and the traditional symplectic phase space. We show how a proper understanding of the geometry of these intermediate spaces clarifies aspects of the traditional symplectic formulation. Our improved 3+1 decomposition allows us to easily handle the case when the spatial manifold (in our space-time split) has a boundary. By careful consideration of what happens to the theory at the boundary, we can arrive at appropriate boundary conditions and boundary modifications to various 3+1 quantities. This is the first such decomposition of the multisymplectic phase space with boundaries in the literature.Lastly, we develop a multisymplectic formalism for general relativity. Our approach here is new, and gives great insight into the geometric structure of the theory. In the course of developing multisymplectic general relativity, we introduce local Lorentz transformations as an additional gauge symmetry. We show how reducing by this symmetry after 3+1 decomposition leads to the usual symplectic approach to general relativity

Multisymplectic Geometry with Boundaries

Geometric approaches form the foundation of modern classical mechanics. The prototypical example of a geometric method in mechanics is symplectic geometry applied to the Hamiltonian formulation of a system of particles. Extending this approach to field theories leads to unattractive features, such as an infinite-dimensional phase space and loss of manifest covariance. These deficiencies are particularly glaring in general relativity, where manifest covariance is closely tied to the fundamental symmetries of the theory. Recent progress on covariant Hamiltonian approaches for field theories has led to the development of multisymplectic geometry. Multisymplectic geometry generalizes the symplectic geometry of particle systems to covariant fields, producing a finite-dimensional phase space and retaining manifest covariance. The symplectic 2-form common to symplectic geometry generalizes to the multisymplectic 5-form. The Euler-Lagrange equations for the field can be written in geometric language using the 5-form in a way that is formally identical to the geometric form of Hamilton's equations in particle mechanics. The resulting approach is a powerful geometric tool for understanding classical field theories.In this dissertation, we improve upon the current approach to performing a 3+1 decomposition (also known as space-time split) of multisymplectic geometry. We clarify the relationship between multisymplectic geometry, its 3+1 decomposition, and the traditional symplectic approach to field theory. The key observation is that there exist two intermediate phase spaces between the multisymplectic phase space and the traditional symplectic phase space. We show how a proper understanding of the geometry of these intermediate spaces clarifies aspects of the traditional symplectic formulation. Our improved 3+1 decomposition allows us to easily handle the case when the spatial manifold (in our space-time split) has a boundary. By careful consideration of what happens to the theory at the boundary, we can arrive at appropriate boundary conditions and boundary modifications to various 3+1 quantities. This is the first such decomposition of the multisymplectic phase space with boundaries in the literature.Lastly, we develop a multisymplectic formalism for general relativity. Our approach here is new, and gives great insight into the geometric structure of the theory. In the course of developing multisymplectic general relativity, we introduce local Lorentz transformations as an additional gauge symmetry. We show how reducing by this symmetry after 3+1 decomposition leads to the usual symplectic approach to general relativity

Nonlinear polarization transfer and control of two laser beams overlapping in a uniform nonlinear medium.

A scheme for polarization control using two laser beams in a non-linear optical medium is studied using both co- and counter-propagating beam geometries. In particular, we show that under certain conditions it is possible for two laser beams to exchange their polarization states. A model accounting for a more realistic, 2D propagation geometry is presented. The 2D model produces drastically different results (compared to the 1D propagation geometry), creating difficulties for implementing polarization control in a realistic setting. A proposal for overcoming these difficulties by reducing the non-linear optical medium to a thin slab is presented

Photochemically-induced acousto-optics in gases

Acousto-optics consists of launching acoustic waves in a medium (usually a crystal) in order to modulate its refractive index and create a tunable optical grating. In this article, we present the theoretical basis of a new scheme to generate acousto-optics in a gas, where the acoustic waves are initiated by the localized absorption (and thus gas heating) of spatially-modulated UV light, as was demonstrated in Y. Michine and H. Yoneda, Commun. Phys. 3, 24 (2020). We identify the chemical reactions initiated by the absorption of UV light via the photodissociation of ozone molecules present in the gas, and calculate the resulting temperature increase in the gas as a function of space and time. Solving the Euler fluid equations shows that the modulated, isochoric heating initiates a mixed acoustic/entropy wave in the gas, whose high-amplitude density (and thus refractive index) modulation can be used to manipulate a high-power laser. We calculate that diffraction efficiencies near 100 percent can be obtained using only a few millimeters of gas containing a few percent ozone fraction at room temperature, with UV fluences of less than 100 mJ/cm2, consistent with the experimental measurements by Michine and Yoneda. Gases have optics damage thresholds two to three times beyond those of solids; these optical elements should therefore be able to manipulate kJ-class lasers. Our analysis suggest possible ways to optimize the diffraction efficiency by changing the buffer gas composition