The Physics Perspective

I originally started out as a physicist, so a lot of my research is guided by me wanting to understand a particular physical phenomenon. That typically involves a lot of tools from very different mathematical disciplines, what matters is that you solve a problem and not that you stay within the confines of one particular mathematical field.

On the other hand, mathematics makes it easier to find commonalities between very different physical theories and it enforces a conceptual rigor I find insightful.

Derivation of Effective Dynamics

One of my main interests is understanding how to derive simpler, approximate models from more fundamental equations. Understanding mathematical and physical mechanisms of why certain approximations work and what assumptions are necessary improves the understanding of the mechanisms behind physical phenomena. Very often one assumes a special choice or class of state(s) or that the system possesses two scales.

For example, I have derived non- and semirelativistic quantum dynamics for a spin-1/2 particle from the Dirac equation with the exact same technique. The only difference is the choice of momentum scale.

A lot of my works concern perturbed periodic systems, e. g. a crystalline solid subjected to an external electromagnetic field or a photonic crystal that is slowly modulated. For those I have derived effective tight-binding models. These are much simpler to analyze as they discretize space and allow the particle to hop from one unit cell to another. Alternatively, one can derive effective semiclassical equations of motion, where properties of the full operator enter into the hamiltonian equations of motion. In photonic crystals, this leads to ray optics equations. In all of these cases the main technical tool is (magnetic) pseudodifferential theory, also known as (magnetic) Weyl quantization in the physics community.

Yet another approach in this direction is linear response theory to obtain macroscopic conductivity properties. We apply an electric field E to drive a current and add control parameters such as temperature and an externally applied magnetic field B. With the analytic-algebraic linear response framework I have developed with Giuseppe De Nittis, we have been able to justify a “Taylor expansion” in |E|.

Semiclassical Dynamics for Dissipative and Open Systems

Recently, my interest has turned to deriving semiclassical dynamics for dissipative and open systems. While the form of the quantum equations has been well-understood for about 50 years, there is no uniformly accepted mathematical framework for dissipative classical dynamics (akin to hamiltonian mechanics, which builds on symplectic geometry).

As a starting point Gihyun Lee and I have introduced the notion of magnetic pseudodifferential super operator, operators that act on other operators, and we plan on using the semiclassical limit as a way to decide which is the right mathematical structure for the classical equations.

Quantum-Wave Analogies

Conceptual and mathematical rigor makes it easier for me to find analogies and similarities between different fields. For example, the interpretation of the Schrödinger equation from quantum mechanics and Maxwell's equations for classical electromagnetism are very different. Electromagnetic fields are directly accessible to measurement whereas the quantum wave function — as a matter of principle — is not. A second is that classical fields are real-valued, which leads to an unbreakable constraint in the equations. To establish quantum-wave analogies it is necessary to work with complex Hilbert spaces, which necessitates to remove superfluous degrees of freedom.

Mathematically, though, a (perturbed) periodic Schrödinger operator describing a crystalline solid and a (perturbed) periodic Maxwell operators share many commonalities. This opens the door to realizing many phenomena first proposed for quantum systems with electromagnetic and other classical waves. Periodic media for classical waves can be fabricated to have certain properties, which are much more difficult to realize in condensed matter physics. However, even if the mathematical mechanism is the same, it is important to take the correct physical interpretation into account in order to arrive at predictions for outcomes of experiments.

Topological Phenomena

One such quantum-wave analogy I have work on since 2012 are topological phenomena in classical waves. This has first been proposed by Raghu and Haldane in 2005 (published in 2009), then experimentally verified by Joannopoulos and Soljacic in 2009. The idea behind topological phenomena is that discrete symmetries enforce certain phase relationships, which need to be met. These tend to be very robust under perturbations, a feature that makes them very useful in applications.

One of the goals of my work was to prove the bulk-boundary correspondence for photonic crystals. Partial results include a bulk classification and an understanding of the effective dynamics.