Triangle Seminars

Abstract:
Inspired by an old idea of von Neumann, we seek a pair of commuting operators X,P which are, in a specific sense, 'close' to the canonical non-commuting position and momentum operators, x,p. Difficulties with von Neumann's original idea (involving an alleged orthogonalization of the coherent states) are discussed. Here these difficulties are avoided by restricting attention to operators acting on density matrices which are reasonably decohered (i.e., spread out in phase space). Such operators could be of use in discussions of emergent classicality from quantum mechanics. Moreover, they may be
used to give a discussion of the relationship between exact and approximate decoherence in the decoherent histories approach to quantum theory.

Abstract:
We analyze the eigenvalue spectrum of the staggered Dirac matrix

in two-color QCD at nonzero chemical potential when the

eigenvalues become complex. The quasi-zero modes and their role for

chiral symmetry breaking and the deconfinement transition are examined.

The bulk of the spectrum and its relation to quantum chaos is considered.

A comparison with predictions from random matrix theory is presented.



We further provide first evidence that matrix models describe the low lying

complex Dirac eigenvalues in a theory with dynamical fermions at nonzero

chemical potential. Lattice data for two-color QCD with staggered fermions

are compared to detailed analytical results from matrix models in the corresponding

symmetry class, the complex chiral symplectic ensemble. They confirm the predicted

dependence on chemical potential, quark mass and volume.

Abstract:
The escape out of open quantum systems can be characterised by quasibound states, which are solutions of the wave equation subject to outgoing boundary conditions. The energy eigenvalue of a quasibound state is complex, and the imaginary part is associated to the decay rate of the state. Quasibound states can be observed, e.g., as the lasing modes of optical microresonators. Random-matrix theory gives a wealth of information on quasibound states in disordered media, such as random dielectrics. Interesting systems are, however, ballistic (clean), and scattering only takes place at the (often complicated) confinements. I discuss the similarities and differences between quasibound states in disordered and ballistic systems. A semiclassical analysis reveals that ballistic systems feature a set of quasibound states which decay very quickly (faster even than the classical time of flight). The remaining long-lived quasibound states obey random-matrix statistics, just as in disordered systems, but renormalized in compliance with a recently proposed fractal Weyl law. I illustrate these results numerically for a model system, the open kicked rotator.