Embedded Files
Boundary conditions and quantum billiards
The study of quantum systems confined in a bounded domain requires a careful description of the physical properties of its boundary, and thus of the interaction between the system and the boundary, effectively encoded via a proper choice of boundary conditions. In recent years, quantum boundary conditions have increasingly attracted interest in different branches of quantum physics, some examples being the analysis of quantum Hall systems, the study of geometric phases, quantum control theory, topological insulators and QCD, quantum gravity and topology change, as well as the Casimir effect in quantum field theory, to name a few (see e.g. the references contained in [1, 2]).
Quantum billiards are a paradigmatic example of such systems which, despite the apparently innocuous setup, have a rich physical phenomenology as well as many interesting mathematical properties. Therefore, after determining the admissible boundary conditions (e.g. preserving the self-adjointness of the Hamiltonian), it is crucial to characterize their physical properties and their influence on the bulk of the system.
We carried out this program for magnetic quantum billiards in [3], highlighting in particular the relation between boundary conditions and gauge transformations.
Isospectrality and topology change
In a famous paper of 1966, "Can one hear the shape of a drum?", Mark Kac described an interesting inverse spectral problem involving bounded regions of the plane [4]. Many related questions (which may generally fall under the umbrella of "isospectrality") are still open and object of research. Here we briefy mention some topics, involving the isospectrality problem for regions with a fractal boundary, relativistic billiards, quantum graphs, photonic systems, and also the experimental construction of some isospectral regions.
In [5] we studied an analogous problem, determining which boundary conditions, associated with a free quantum particle in a ring with a junction, have the same energy eigenvalues. In other words, we fixed the shape of the "drum" and we determined whether it is possible to distinguish different quantum boundary conditions just by the energy spectrum of the particle.
Although the physical configuration is fixed, the topology of the quantum system is not, being determined by the boundary conditions [6]: while non-local boundary conditions effectively describe a particle in a ring, local conditions model instead a particle confined in a segment.
Moreover, we observed that also the space of boundary conditions carries a non-trivial topological structure, a fact that signals the presence of spectral anholonomies and geometric phases.
In [7] I extended the analysis to the relativistic counterpart of this system described by the Dirac equation.
Quantum Hall systems
Since its discovery, the quantum Hall effect (QHE) has generated huge interest in the scientific community, both from the theoretical and experimental perspective. In [8] we formulate a self-consistent model of the integer QHE, using boundary conditions to investigate finite-size effects associated with the Hall conductivity. Boundary conditions have already been applied in the past for the description of the QHE, with a focus on the phenomenology of the edge states, but their influence on the quantization of the Hall conductivity is still scarcely investigated. For our purposes boundary conditions provide a simple effective framework to describe the QHE, reducing the latter to its key ingredients: a charged particle, a bounded two-dimensional system, and a strong magnetic field.
By assuming the invariance with respect to longitudinal translations, we were able to characterize the general spectral properties of the system. Then we focused, for physical reasons, on the case of Robin boundary conditions. By determining the spectrum corresponding to the selected boundary conditions, we have been able to predict a new kind of states with no classical analogues. The latter have a finite propagation velocity as classical edge states, but their (orbital) chirality is the same of classical bulk states.
Moreover, boundary conditions turned out to add a finer structure to the quantization pattern of the Hall conductivity: the integer plateau are substantially preserved, but a transition region between two consecutive plateaux appears, smoothly controlled by the boundary conditions.
An open problem is whether, in the thermodynamic limit of infinite volume, finite-size effects disappear in the same way regardless of the choice of the boundary conditions.
Dimensional reduction
While the most natural setting of a physical theory in spacetime is a (3+1)-dimensional manifold, there is no obstruction, in principle, to formulating self-consistent theories in a higher or lower-dimensional manifold. In particular, the interest in low-dimensional theories is twofold. On one hand, they are of tantamount importance in the formulation of quantized field theories, on the other hand, the technological developments of the last few years have enabled us to engineer and control truly low-dimensional systems, yielding some fascinating dimension-dependent features. It is thus worth studying whether self-consistent low-dimensional theories can be obtained starting from a more familiar (3+1)-dimensional one (dimensional reduction).
In [9] we investigated the behavior of the Dirac equation, both in the absence and presence of an electromagnetic field, under dimensional reduction from a (3+1)-dimensional to a (2+1)-dimensional Minkowski spacetime, determining how the theory splits in two inequivalent sectors. In particular, some particular solutions admit a sterile electromagnetic sector that is no longer coupled to the Dirac equation.
In [10] we then extended this program to a spacetime with an arbitrary number of space dimensions, clarifying the different behavior of odd and even-dimensional spacetimes.
Current research is devoted to the analysis of other theories (such as Yang-Mills) and of more general spacetimes.
References
[1] M. Asorey, A. Ibort, and G. Marmo, Global Theory of Quantum Boundary Conditions and Topology Change, Int. J. Mod. Phys. A 20, 1001–1025 (2005)
[2] M. Asorey, A. Ibort, and G. Marmo, The topology and geometry of selfadjoint and elliptic boundary conditions for Dirac and Laplace operators, Int. J. Geom. Methods Mod. Phys. 12, 1561007 (2015)
[3] G. Angelone, P. Facchi, and D. Lonigro, Quantum magnetic billiards: boundary conditions and gauge transformations, Ann. Phys. 442, 168914 (2022)
[4] M. Kac, Can one hear the shape of a drum?, Am. Math. Monthly 73, 1 (1966)
[5] G. Angelone, P. Facchi and G. Marmo, Hearing the shape of a quantum boundary condition, Mod. Phys. Lett. A 37, 2250114 (2022)
[6] A. P. Balachandran, G. Bimonte, G. Marmo, and A. Simoni, Topology change and quantum physics, Nucl. Phys. B 446, 299–314 (1995)
[7] G. Angelone, Hearing the boundary conditions of the one-dimensional Dirac operator, arXiv:2311.17561 [quant-ph]
[8] G. Angelone, M. Asorey, P. Facchi, D. Lonigro and Y. Martinez, Boundary conditions for the quantum Hall effect, J. Phys. A: Math. Theor. 56, 025301 (2023)
[9] G. Angelone, E. Ercolessi, P. Facchi, D. Lonigro, R. Maggi, G. Marmo, S. Pascazio and F. V. Pepe, Dimensional reduction of the Dirac theory, J. Phys. A: Math. Theor. 56, 065201 (2023)
[10] D. Lonigro, R. Maggi, G. Angelone, E. Ercolessi, P. Facchi, G. Marmo, S. Pascazio and F. V. Pepe, Dimensional reduction of the Dirac equation in arbitrary spatial dimensions, Eur. Phys. J. Plus 138, 324 (2023)
Page updated
Google Sites
Report abuse