In order of appearance:

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Saturday, August 18th

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9:35 am - 10:10 am

Phil Morrison, Department of Physics, UT Austin

Magnetic Monopoles: Symmetry Gained is Symmetry Lost

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Dirac's theory and voluminous subsequent work will be briefly reviewed. Despite the manifest symmetry of the  electromagnetic fields, it will be shown that when Dirac's version of Maxwell's equations are coupled to matter models, a fundamental precept of theoretical physics is violated, leading to the introduction of various shenanigans. 

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10:40 am - 11:15 am

Roland Winston, Department of Physics, UC Merced
How Robert Littlejohn Brought Classical Radiometry into the 21st Century

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In the 90's I brought Robert Littlejohn a perplexing problem in classical optics; "brightness" a useful concept in geometrical optics (essentially the distribution of power in phase space), presents challenges when diffraction effects are included. This had previously been encountered in quantum mechanics since position and momentum don't commute. Using the technique of the Wigner Distribution, Robert found a self-consistent description by introducing the instrument function which characterized the measuring apparatus. This was applied to experiments carried out at the Naval Research Laboratory in Washington, DC which provided beautiful confirmation of Robert's formalism. The results are important for infra-red instrumentation and astronomy. Both theory and the experimental verification will be described.

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11:15 am - 11:50 am

Christopher Jarzynski, Department of Chemistry and Biochemistry, University of Maryland

A Classical Route to Quantum Control

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"Shortcuts to adiabaticity" are strategies for guiding a quantum system to evolve quickly from an eigenstate of an initial Hamiltonian to the corresponding eigenstate of a final Hamiltonian -- a result that would ordinarily require slow (adiabatic) driving. While formal solutions to this problem exist, they can be difficult to translate into laboratory settings. I will argue that classical mechanics can help to achieve this goal.

 

For Hamiltonian systems in one degree of freedom, the classical version of this problem is formulated in terms of the classical action, an adiabatic invariant. For a Hamiltonian H(q,p,t), I will show how to construct a "fast-forward" potential energy function V_FF(q,t) that deftly guides all trajectories with a given initial action to end with exactly the same value of the action, even when the time-dependence of H is rapid. The solution is exact and simple, relying only on elementary manipulations of Hamilton’s equations [1]. When this classical solution is applied to the quantum Hamiltonian, the result is a shortcut to adiabaticity that guides a wavefunction to the desired final energy eigenstate with high accuracy.

 

[1] C. Jarzynski, S. Deffner, A. Patra and Y. Subasi, "Fast forward to the classical adiabatic invariant", Phys Rev E 95, 032122 (2017)

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1:30 pm - 2:05 pm

Eugenio Bianchi, Department of Physics, Penn State

Entanglement Semiclassics

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Entanglement is a genuine quantum phenomenon. Remarkably, classical phase-space methods provide a powerful tool for computing the evolution of the entanglement entropy in a large class of systems. In this talk I will present a relation between the Kolmogorov-Sinai rate and the entanglement entropy growth of a subsystem, together with its applications to the physics of the early universe and black hole evaporation.

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2:05 pm - 2:40 pm

Mark Gotay, Assistant Director, Pacific Institute for the Mathematical Sciences

On Second Class Constraints

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After touring the landscape of second-class constraints, we show how a second class Lagrangian system can be converted into a purely first class one in a nontrivial manner.

 

This has numerous applications, in particular allowing one to use the extensive machinery involving first class constraints, gauge groups and so on in order to understand such systems. This makes life simpler in many ways, but has the disadvantage of increasing the computational complexity of such systems.

 

The technique introduced here is reminiscent of how one uses (bosonic) BRST theory in order to understand non-unimodular constrained systems.

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2:40 pm - 3:15 pm

Nicolai Reshetikhin, Department of Mathematics, UC Berkeley

Integrable Systems and the Semiclassical Asymptotics of 6j and q6j symbols

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Racah-Wigner coefficients also known as 6j symbols can be regarded as scalar products of two quantum integrable systems on the same Hilbert space. In this talk it will be explained how the semiclassical asymptotic of such scalar products for can be expressed in terms of underlying symplectic geometry of co-adjoint orbits and how this analysis can be extended to q6j symbols. We will focus only on the multiplicity free case for an arbitrary simple Lie algebra, not necessary restricted to the su2 case, when it becomes the PonzanoRegge asymptotic for 6j symbols and Taylor-Woodward for q6j symbols.

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3:45 pm - 4:20 pm

Josh Burby, Courant Institute of Mathematical Sciences, New York University

Robert’s Weaponization of Darboux’s Theorem

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On the cyclotron timescale, charged particles moving through a strong magnetic field gyrate rapidly around magnetic field lines. On longer timescales, and in the presence of magnetic gradients, the "guiding center" of the cyclotron gyration slowly drifts while preserving an adiabatic invariant known as the magnetic moment. In the early 1960's, Kruskal gave an abstract proof that the motion of a particle's guiding center must be described by a Hamiltonian system. However, before Robert began his thesis work in the late 1970's, the only Hamiltonian guiding center model that could be found required the use of a special field-aligned coordinate system that does not exist for many inhomogeneous magnetic fields of practical relevance. Robert revolutionized the study of guiding center motion by discovering a Hamiltonian theory of guiding center motion that was applicable, and practical to use, in any coordinate system whatsoever. Today this theory is an essential tool in the study magnetic confinement fusion (MCF). In particular, without Robert's theory, the most promising designs for steady-state MCF reactors -- the so-called quasisymmetric stellarators -- would have been impossible to find. I will frame Robert's approach to Hamiltonian guiding center theory as a synergistic combination of classical perturbation methods with Darboux's theorem on the local flexibility of symplectic manifolds. I will then describe an infinite-dimensional extension of Robert's idea. The extension, which I refer to as variational slow manifold reduction, is a tool for modeling the slow motions of field-theoretic models with dynamically-irrelevant fast timescales.

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4:20 pm - 4:55 pm

Robert MacKay, Department of Physics, University of Warwick

Could Many Gamma-Ray Bursts be Optical Illusions?

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Gamma-ray bursts are flashes of gamma-rays lasting from milliseconds to a few minutes, which then soften progressively to X-rays and ultimately to radio waves. They are observed from all directions in space, roughly uniformly. They have been attributed to cataclysmic events. We propose, however, that many of them may be optical illusions, simply the result of our entry into the region illuminated by a continuously emitting object. At such an entry, the emitter appears infinitely blue-shifted and infinitely bright. We demonstrate the phenomenon in de Sitter space, where much can be calculated explicitly, and then extend the idea to more general space-times. Joint work with Colin Rourke.

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4:55 pm - 5:30 pm

Robert Littlejohn, Department of Physics, UC Berkeley

Nearby Orbits in Classical and Quantum Mechanics

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I will talk about impressions from a conference I attended a few weeks ago, mainly about nearby orbits in classical and quantum mechanics, the semiclassical evolution of wave packets, the Herman-Kluk propagator, and related topics.

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Sunday, August 19th

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9:00 am - 9:35 am

James Meiss, Department of Applied Mathematics, University of Colorado at Boulder

Quadfurcation

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Fixed points in Hamiltonian systems or symplectic maps are typically created in saddle-center bifurcations, giving rise to the characteristic fish-shaped phase space portrait for a cubically nonlinear potential energy. However, there is another bifurcation that appears to be an organizing center for the possible behaviors in 4D systems; we call it a quadfurcation, since it can correspond to the creation of four fixed points simultaneously. An excellent model to study this phenomenon is Moser’s normal form for the local dynamics of a 4D symplectic map, which generalizes Hénon's famous 2D quadratic map.

 

Moser's map has at most four fixed points, and they can be viewed as being created in a quadfurcation. Rather surprisingly the resulting structures are usually quite different from the ``cross product of two fish” that one might expect from coupling a pair of 2D maps. Two of the fixed points can be center-center points, leading to the formation of a double-bubble of two-tori that is bounded by manifolds o two center-saddles. We propose that this bifurcation is an organizing center for the dynamics of saddle-center bifurcations in 4D maps, and in particular is a normal form for the formation of accelerator modes in maps such as the Froeschlé family.

 

*This is joint work with Arnd Bäcker.

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9:35 am - 10:10 am

Richard Montgomery, Department of Mathematics, UC Santa Cruz

Stumbling around Shape Space

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I will present three vignettes:

 

1. Robert in the 80s.

 

2. A brutally simple tokamak. How to trap charged particles using a magnetic well with infinitely steep sides. I review recent work of Gabriel Martins, Yves Colin-de-Verdiere and Truc.

 

3. Dynamical consequences of shape space geometry. I draw a picture or two summarizing the dynamics of the planar three body problem. I then focus on understanding ``Marchal's lemma''—the failure of collision paths to minimize action—in terms of a conical metric structure buried in the Jacobi-Maupertuis metric.

 

Caveat: I may delete one of these vignettes in the interest of time.

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10:40 am - 11:15 am

Hal Haggard, Physics Program, Bard College

Curved Polyhedra, Group-Valued Momenta, and the Cosmological Constant

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We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. A concrete realization of this is provided by the relation with the spin-network states of loop quantum gravity. Many prominent approaches to quantum gravity struggle when it comes to incorporating a positive cosmological constant in the models. Using this framework and a quantization of complex SL(2,ℂ) Chern-Simons theory we include a cosmological constant, of either sign, into a model of quantum gravity. 

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11:15 am - 11:50 am

Kevin A. Michell, Department of Physics, UC Merced

Computing Maslov Indices from Tangle Topologies

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Maslov indices are integers defined for periodic orbits of Hamiltonian systems. In semiclassical theories of quantum mechanics, Maslov indices determine a global phase shift upon traversing the periodic orbit. This phase shift is essential for semiclassical quantization conditions. We focus on the Maslov indices of unstable hyperbolic orbits in two degree-of-freedom Hamiltonian systems. In the event that a global surface-of-section exists for this system, it can be reduced to an area-preserving map of a two-dimensional phase space. The topological dynamics of the chaotic sea for such systems can typically be described by a network of heteroclinic/homoclinic tangles---the one-dimensional stable and unstable manifolds anchored to a select set of periodic orbits of the system. We show how the Maslov indices of specific periodic orbits can be directly computed from the topology of the tangle. Furthermore, in the case where symbolic dynamics has been extract from the tangle using the homotopic lobe dynamics technique, we should how the Maslov index of a periodic orbit can be computed directly from the symbolic itinerary, or label, of the periodic orbit. Example computations are drawn from atomic physics---the electron dynamics for a hydrogen atom in parallel electric and magnetic fields.

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1:30 pm - 2:05 pm

William Miller, Department of Chemistry, UC Berkeley

Classical Molecular Dynamics Simulations of Electronically Non-Adiabatic Processes

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A recently described symmetrical quasi-classical (SQC) windowing methodology for classical trajectory simulations has been applied to the Meyer-Miller (MM) model for the electronic degrees of freedom in electronically non-adiabatic dynamics. The approach treats nuclear and electronic degrees of freedom (DOF) equivalently (i.e., by classical mechanics, thereby retaining the simplicity of standard molecular dynamics), providing ``quantization" of the electronic states through the symmetrical quasi-classical (SQC) windowing model. The approach is seen to be capable of treating extreme regimes of strong and weak coupling between the electronic states, as well as accurately describing coherence effects in the electronic DOF (including the de-coherence of such effects caused by coupling to the nuclear DOF). It is able to provide the full electronic density matrix from the one ensemble of trajectories, and the SQC windowing methodology correctly describes detailed balance (unlike the traditional Ehrenfest approach). Calculations can be (equivalently) carried out in the adiabatic or a diabatic representation of the electronic states, and most recently it has been shown that a modification of the canonical equations of motion in the adiabatic representation eliminates (without approximation) the need for second-derivative coupling terms.

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2:05 pm - 2:40 pm

Rafael de la Llave, School of Mathematics, Georgia Institute of Technology

Melnikov Theory for Normally Hyperbolic Manifolds and Arnold Diffusion

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We consider systems that have a normally hyperbolic manifold (in particular a peridodic or quasi-periodic orbit) and a perturbation depending arbitrarily on time. We show how to compute the effects of the perturbation in a geometrically natural way.

 

Furthermore, we show how this first order effects, under some mild hypothesis can be chained to produce effects of order 1.

 

This is joint work with M. Gidea and T. M.-Seara.

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2:40 pm - 3:15 pm

Melvin Leok, Department of Mathematics, UC San Diego

Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation Spaces

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Variational integrators are geometric structure-preserving numerical methods that preserve the symplectic structure, satisfy a discrete Noether's theorem, and exhibit exhibit excellent long-time energy stability properties. An exact discrete Lagrangian arises from Jacobi's solution of the Hamilton-Jacobi equation, and it generates the exact flow of a Lagrangian system. By approximating the exact discrete Lagrangian using an appropriate choice of interpolation space and quadrature rule, we obtain a systematic approach for constructing variational integrators. The convergence rates of such variational integrators are related to the best approximation properties of the interpolation space.

 

Many gauge field theories can be formulated variationally using a multisymplectic Lagrangian formulation, and we will present a characterization of the exact generating functionals that generate the multisymplectic relation. By discretizing these using group-equivariant spacetime finite element spaces, we obtain methods that exhibit a discrete multimomentum conservation law. We will then briefly describe an approach for constructing group-equivariant interpolation spaces that take values in the space of Lorentzian metrics that can be efficiently computed using a generalized polar decomposition. The goal is to eventually apply this to the construction of variational discretizations of general relativity, which is a second-order gauge field theory whose configuration manifold is the space of Lorentzian metrics.

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3:45 pm - 4:20 pm

Matthias Reinsch, Department of Physics, UC Berkeley

Semiclassical physics, neuroimaging, brain–computer interfaces and potentially large piles of money

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Audience members will be encouraged to jump on this bandwagon before someone else gets all the money.  There are fascinating problems involved in the construction of low cost, real-time brain imaging technology using arrays of antennas similar to cell phone antennas.  Picosecond pulses are used.  The dielectric properties (real and imaginary) of the various types of tissue in the brain are well known, and the problem involves three-dimensional vector electrodynamics, including polarization effects.  The stage is set for the application of some of our favorite topics such as WKB theory of vector wave equations.  Applications such as deploying software onto existing cell phones to use the on-board antennas to do functional brain imaging and monitor people's thoughts are too speculative and will not be addressed (but if this does happen, you heard it here first).

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4:20 pm - 4:55 pm

Alain Brizard, Department of Physics, St. Michael’s College

In Search of the Guiding Center

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A retrospective of the nearly 40-year period since the publication of Littlejohn’s 1979 paper on “A guiding center Hamiltonian: A new approach” is presented. The progress toward a variational formulation of the guiding-center Vlasov-Maxwell equations will be reviewed, with special emphasis on the guiding-center polarization and magnetization in a nonuniform magnetized plasma.

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4:55 pm - 5:30 pm

Vincenzo Aquilanti, Department of Physics and Geology, Università degli Studi di Perugia

Jacobi’s ladder: Quantum, Semiclassical and Classical Mechanics

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A brief account is given of approaches elaborated in collaboration between myself (and my group in Perugia) and Robert (and his group in Berkeley) over the last 20 years, initially on the three, four, …, n-body problem from a hyperspherical coordinate perspective.

 

The extensive use of the quantum theory of angular momentum became for us the starting point of extensive investigations, including semiclassical approaches and the general theory of spin network, in relationships with orthonormal complete polynomial sets and their asymptotic properties of interest in various areas of applied quantum mechanics.

 

We enjoyed collaborating in particular with Roger Anderson (Santa Cruz) and Annalisa Marzuoli (Pavia).

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