Related papers: On complex structures in physics
Complex systems are characterized by specific time-dependent interactions among their many constituents. As a consequence they often manifest rich, non-trivial and unexpected behavior. Examples arise both in the physical and non-physical…
When several inequivalent supercharges form a closed superalgebra in Quantum Mechanics it entails the appearance of hidden symmetries of a Super-Hamiltonian. We examine this problem in one-dimensional QM for the case of periodic potentials…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the…
A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points.…
The development of dark energy models has stimulated interest to cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We review a broad class of phenomena connected with soft cosmological…
A basic problem in quantizing a field in curved space is the decomposition of the classical modes in positive and negative frequency. The decomposition is equivalent to a choice of a complex structure in the space of classical solutions. In…
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…
Integrable quantum mechanical systems with magnetic fields are constructed in two-dimensional Euclidean space. The integral of motion is assumed to be a first or second order Hermitian operator. Contrary to the case of purely scalar…
While general relativity provides a complete geometric theory of gravity, it fails to explain the other three forces of nature, i.e., electromagnetism and weak and strong interactions. We require the quantum field theory (QFT) to explain…
An algebraic formalism for the study of a system of charged particles interacting with an external quantum field is developed. The notion of monoidal categories with duality is used for the description of composite systems and corresponding…
We extend the framework of submanifolds in Riemannian geometry to Riemann-Cartan geometry, which addresses connections with torsion. This procedure naturally introduces a 2-form on submanifolds associated with the nontrivial ambient…
The development of Noncommutative Geometry is creating a reworking and new possibilities in physics. This paper identifies some of the commutation and derivation structures that arise in particle and field interactions and fundamental…
The search for a geometrical understanding of dualities in string theory, in particular T-duality, has led to the development of modern T-duality covariant frameworks such as Double Field Theory, whose mathematical structure can be…
The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following…
In this note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier \cite{BorgesZeMarcio}. As in the theory of functions of a complex variable, it is expected that…
The geometry of antisymmetric fields with nontrivial transitions over a base manifold is described in terms of exact sequences of cohomology groups. This formulation leads naturally to the appearance of nontrivial topological charges…
The two ways of constrained systems quantization are considered from the point of view of their self-consistency at the quantum level. With a transparent example of a particle in the external electromagnetic field we demonstrate that the…
We study T-duality of $(p,q)$-hermitian geometries in backgrounds with non-Abelian gauge fields $A$ in heterotic string theories. We introduce a gauge-dressed complex geometry characterized by a shifted metric $\bar{g} = g + \frac{1}{2}…
There were many attempts to geometrize electromagnetic field and find out new interpretation for quantum mechanics formalism. The distinctive feature of this work is that it combines geometrization of electromagnetic field and…
A new symmetry of $(1,0)$ supersymmetric non-linear $\sigma$-models in two dimensions with Fermi and mass sectors is introduced. It is a generalisation of the so-called special holonomy $W$-symmetry of Howe and Papadopoulos associated with…