Mathematical Physics

As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painlevé II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov-Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit unique solution to the inverse scattering transform of the Zakharov-Shabat system in terms of a Fredholm determinant.

We consider the Klein-Gordon operator on an n -dimensional asymptotically anti-de Sitter spacetime (M,g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ?�M of order up to 2 . Using techniques from b -calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterpart.

A novel method to make Lagrangians Galilean invariant is developed. The method, based on null Lagrangians and their gauge functions, is used to demonstrate the Galilean invariance of the Lagrangian for Newton's law of inertia. It is suggested that this new solution of an old physics problem may have implications and potential applications to all gauge-based theories of physics.

Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and that they allow converting an undriven physical system into a driven one. This is a novel phenomenon in dynamics that resembles the role of gauges in quantum field theories.

We study the coupling of spectral triples with twisted real structures to gauge fields in the framework of noncommutative geometry and, adopting Morita equivalence via modules and bimodules as a guiding principle, give special attention to modifying the inner fluctuations of the Dirac operator. In particular, we analyse the twisted first-order condition as a possible alternative to the approach of arXiv:1304.7583, and elaborate upon the special case of gauge transformations accordingly. Applying the formalism to a toy model, we argue that under certain physically-motivated assumptions the spectral triple based on the left-right symmetric algebra should reduce to that of the Standard Model of fundamental particles and interactions, as in the untwisted case.

We consider the evolution, for a time-dependent Schrödinger equation, of the so called Dirac comb. We show how this evolution allows us to recover explicitly (indeed optically) the values of the quadratic generalized Gauss sums. Moreover we use the phenomenon of superoscillatory sequences to prove that such Gauss sums can be asymptotically recovered from the values of the spectrum of any sufficiently regular function compactly supported on $\R$. The fundamental tool we use is the so called Galilean transform that was introduced and studied in the context on non-linear time dependent Schrödinger equations. Furthermore, we utilize this tool to understand in detail the evolution of an exponential e iωx in the case of a Schrödinger equation with time-independent periodic potential.

In this monograph, two sets of parabolic differential equations are studied, each with nonlinear medium response. The equations are generally referred to as "Newell-Whitehead-Segel equation," which model a wide variety of nonlinear physical, mechanical and biological systems. Nonlinear medium response can be viewed in many perspectives, such as, memory response from the medium, whereby, the medium "remembers" earlier influences; reactive responses, whereby, the medium is actively responsive to input, such as, chemical reactivity, turbulence and many other circumstances; these equations arise often in the biological sciences when modeling population dynamics, whether the population be genomic, such as, alleles, or animal species in the environment; finally, these sets of equations are often employed to model neurological responses from excitable cellular media. The solutions provided are of a very general nature, indeed, whereby, a canonical set of solutions are given for a class of nonlinear parabolic partial differential equations with nonlinear medium response expressed as either a p-times iterative convolution or p-times multiplicative response. The advantage of canonical solution sets are these solutions involve classic representative forms, such as, Green's function or Green's heat kernel and aid researchers in further complication, analysis and understanding of the systemic behavior of these types of nonlinear systems.

A method to construct general null Lagrangians and their exact gauge functions is developed. The functions are used to define classical forces independently from Newtonian dynamics. It is shown that the forces generated by the exact gauge functions allow for the conversion of the first Newton equation into the second one. The presented approach gives new insights into the origin of forces in Classical Mechanics.

A general theory of dynamics is formulated with the aim of its application in emergent quantum mechanics. In such a framework it is argued that the fundamental dynamics of emergent quantum mechanics must be non-reversible.