We reconsider the fundamental commutation relations for non-commutative $\mathbb{R}^{2}$ described in polar coordinates with non-commutativity parameter $θ$. Previous analysis found that the natural transition from Cartesian coordinates to polars led to a representation of $\left[\hat{r}, \hatφ\right]$ as an everywhere diverging series. We compute the Borel resummation of this series, showing that it can subsequently be extended throughout parameter space and hence provide an interpretation of this commutator. Our analysis provides a complete solution for arbitrary $r$ and $θ$ that reproduces the earlier calculations at lowest order. We compare our results to previous literature in the (pseudo-)commuting limit, finding a surprising spatial dependence for the coordinate commutator when $θ\gg r^{2}$. We raise some questions for future study in light of this progress.
We extend the non-commutative coordinates relationship into other than the Minkowski space-time. We clarify the non-commutativity dependency to the geometrical structure. As well as, we find an inverse map between Riemann's normal and global coordinates. Furthermore, we show that behavior of the corresponding coordinates non-commutativity like as a tensor. All results summarized for the Schwarzschild metric. And they summarized for the space-time in the presence of weak gravitational waves.
Some ambiguities have recently been found in the definition of the partial derivative (in the case of presence of both explicit and implicit dependencies of the function subjected to differentiation). We investigate the possible influence of this subject on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates and those of 4-momenta are {\it not} equal to zero. We postulate the non-commutativity of 4-momenta and we derive mass splitting in the Dirac equation. Moreover, two iterated limits may not commute each other, in…
We present a moduli dependent target space effective field theory action for (truncated) heterotic string toroidal compactifications. When moving continuously along moduli space, the stringy gauge symmetry enhancement-breaking effects, which occur at particular points of moduli space, are reproduced. Besides the expected fields, originated in the ten dimensional low energy effective theory, a new vector and scalar fields are included. These fields depend on "double periodic coordinates" as usually introduced in Double Field Theory. Their mode expansion encodes information about string states, …
We consider the double field formulation of the closed bosonic string theory, and calculate the Poisson bracket algebra of the symmetry generators governing both general coordinate and local gauge transformations. Parameters of both of these symmetries depend on a double coordinate, defined as a direct sum of the initial and T-dual coordinate. When no antisymmetric field is present, the $C$-bracket appears as the Lie bracket generalization in a double theory. With the introduction of the Kalb-Ramond field, the $B$-twisted $C$-bracket appears, while with the introduction of the non-commutativit…
The Poisson gauge algebra is a semi-classical limit of complete non-commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity $Θ^{ab}(x)$, especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the cov…
Singular Behavior of the Laplace Operator in Polar Spherical Coordinates and Some of Its Consequences for the Radial Wave Function at the Origin of Coordinates
Bose-Einstein condensation and superfluidity on a fuzzy sphere
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Institute for Theoretical Solid State Physics, IFW Dresden（德累斯顿亥姆霍兹中心IFW凝聚态理论物理研究所）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理学研究所）
It is shown that in the noncommutative spacetime defined by the generalized Moyal product, consistent noncommutativity can be obtained independent of the coordinate system such as Cartesian or polar one. In addition, based on the fact that the generalized Moyal product can be applied to arbitrary spacetime with non-trivial curvature, the effect of noncommutativity in the axisymmetric spacetime with central mass is investigated. The results demonstrate how the noncommutativity of spacetime modify the shape of the stationary limit surface in the noncommutative Kerr spacetime, which implies that …
We point out the necessity of resolving the apparent gauge dependence in the quantum corrections of cosmological observables for Higgs-like inflation models. We highlight the fact that this gauge dependence is due to the use of an asymmetric background current which is specific to a choice of coordinate system in the scalar manifold. Favoring simplicity over complexity, we further propose a practical shortcut to gauge-independent inflationary observables by using effective potential obtained from a polar-like background current choice. We demonstrate this shortcut for several explicit examples…
