Related papers: The Quantum Darboux Theorem,
It has been established that endowing classical phase space with a Riemannian metric is sufficient for describing quantum mechanics. In this letter we argue that, while sufficient, the above condition is certainly not necessary in passing…
The Darboux transformation operator technique is applied to construct exactly solvable anharmonic singular oscillator potentials and to study their coherent states. Classical system corresponding to a transformed quantum system is…
Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and…
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined…
Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this…
Darboux's theorem guarantees the existence of local canonical coordinates on symplectic manifolds under certain conditions. We demonstrate a general method to construct such Darboux coordinates in the vicinity of a fixed point of a…
We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the…
Supersymmetric or Darboux transformations are used to construct local phase equivalent deep and shallow potentials for $\ell \neq 0$ partial waves. We associate the value of the orbital angular momentum with the asymptotic form of the…
Gauge-invariant observables for quantum gravity are described, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics.…
The Darboux method is commonly used in the coordinate variable to produce new exactly solvable (stationary) potentials in quantum mechanics. In this work we follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to include…
The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the $N$-fold application of the transformation is also established, and…
We consider the evolution of quantum fields on a classical background space-time, formulated in the language of differential geometry. Time evolution along the worldlines of observers is described by parallel transport operators in an…
We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as…
Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs…
We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference…
Geometrical model for quantum objects is suggested. It is shown that equations for free material Dirac field and for Maxwell electromagnetic field can be considered as relations describing propagation of the space topological defects. This…
In the KdV context we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann-Hilbert problem and yields a new explicit formula for perturbation of the…
This paper presents exploratory investigations on the concept of generalized geometrical frequency in electrical systems with an arbitrary number of phases by using Geometric Algebra and Differential Geometry. By using the concept of…
Darboux transformations of the singular harmonic oscillator are considered. Analytical expressions for the propagators are obtained, using the image method applied to formal singular propagators. Two-well and three-well families of…
Two sets of spatially diffeomorphism invariant operators are constructed in the loop representation formulation of quantum gravity. This is done by coupling general relativity to an anti- symmetric tensor gauge field and using that field to…