A semiclassical approach to spectra of quantized torus Hamiltonians

• Applicant: Dr. Sebastian Egger
• Subject Area: Mathematics
• Term: from 2014 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 251320072

The project deals with open problems in quantum chaos and provides methods for an application of semiclassical techniques to lattice quantum chromodynamics. In quantum chaos, there is a scientific consensus and supported by extensive numerical calculations that the statistical properties of the spectrum of a quantized Hamiltonian are determined by specific symmetries of the corresponding classical Hamiltonian. A sufficiently mathematically rigorous foundation of this property has, however, not yet been achieved. Torus maps turned out as very useful model systems to test this connection, but also provided surprising counter-examples. I plan to generalize the methods of torus maps to Hamiltonians possessing a toroidal phase space. In particular, I intend to investigate and test the predictions of quantum chaos for the spectral correlations of torus Hamiltonians by a comprehensive semiclassical analysis. As a crucial part of the project I intend to derive a Gutzwiller-type trace formula for Hamiltonians possessing a torus as phase space and to apply the trace formula to an investigation of spectral densities and spectral correlations. By this analysis I hope to gain a better understanding of the connections between the classical dynamics in the toroidal phase space generated by classical Hamiltonians and the spectra of the corresponding quantized Hamiltonians. I plan to develop methods to combine torus quantum mechanics with lattice quantum chromodynamics. Lattice quantum chromodynamics provides methods to determine the physical masses of mesons and hadrons numerically. The idea of my approach is to express the lattice Dirac operators, which are important objects in lattice quantum chromodynamics, by quantized torus Hamiltonians. Combining the results I intend a semiclassical analysis of spectral determinants of lattice Dirac operators. The semiclassical analysis of the spectral determinants possesses the potential to simplify exhausting numerical calculations in lattice quantum chromodynamics.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Host: Professor Dr. Jens Bolte