Quasi-Hopf twist and elliptic Nekrasov factor

We investigate the quasi-Hopf twist of the quantum toroidal algebra of $$ {\mathfrak{gl}}_1 $$ gl 1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach to characterizing Lie polynomials on the generators [Formula: see text] of an algebra with a defining relation in the form of a twisted commutation relation [Formula: see text]. Here, the twisting map [Formula: see text] is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses [Formula: see text]-deformed Heisenberg algebras, rotation algebras, and some types of [Formula: see text]-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

Quantum cosmology with dynamical vacuum in a minimal-length scenario

In this work, we consider effects of the dynamical vacuum in quantum cosmology in presence of a minimum length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation $$[{\hat{X}},{\hat{P}}] := \frac{i\hbar }{ 1 - \beta {\hat{P}}^2 }$$ [ X ^ , P ^ ] : = i ħ 1 - β P ^ 2 . We determine the wave function of the Universe $$ \psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) , which is solution of the modified Wheeler–DeWitt equation in the representation of the quasi-position space, in the limit where the scale factor of the Universe is small. Although $$\psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) is a physically acceptable state it is not a realizable state of the Universe because $$ \psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) has infinite norm, as in the ordinary case with no minimal length.

Gravitational domain wall and stability with some symmetry algebra

In this paper, first, we will try to introduce the gravitational domain wall as a physical system. In the second step, we also introduce the Hun differential equation as a mathematical tools. We factorize the known Heun’s equation as form of operators [Formula: see text], [Formula: see text] and [Formula: see text]. Then we compare the differential equation of gravitational domain wall with corresponding Hun equation. In that case the above-mentioned operators can be obtained for the gravitational system by the comparing process. Finally, we employ such operators and achieve the corresponding symmetry algebra with the usual commutation relation of operators to each other. Here, by having such operators, we investigate the stability of system.

A Remark on the Transformations between Different Representations

The paper respectively studies the transformations of spin component operators and their eigenfunctions from to representation. The studies demonstrate that if using the matrix of obtaining from the equation of transforming the eigenfunctions of and in terms of the transformation arising from Ref. [7] for all of the spin component operators from to representation, a negative symbol will occur in the result, which is not concordant with the commutation relation between and . Moreover, this matrix of transformation can not be applied to the transformation of the eigenfunctions of or . It therefore concludes that with respect to the transformation of representation of an operator and its eigenfunctions, the matrix of transformation should be obtained from the equation of transforming the eigenfunctions of the same operator from the old to the new representation.

Almost Hermitian Identities

We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds.

What is Quantum Uncertainty?

Eve attempts to eavesdrop on Henry, who discovers that her eavesdropping strategy is susceptible to hindrances imposed by the QM uncertainty principle. Eve can detect Henry’s whereabouts to any precision she chooses. However, a precise position measurement induces a large spread of Henry’s velocities and renders his position extremely uncertain, defeating Eve’s attempt to further track him. Such restriction on the maximal accuracy of measuring “complementary” observables was viewed by Bohr as a fundamental limitation on human perception of the world. A deeper explanation of complementarity may assume that the observer’s cognition and the observed object are inseparable, in the spirit of Spinoza, who considered nature as one substance with both mental and physical modalities. By studying these modalities, human knowledge may some day be freed from complementarity constraints. The appendix to this chapter introduces operators and their commutation relation and clarifies the position–momentum uncertainty relation.

Einstein’s E=mc2 Derivable from Heisenberg’s Uncertainty Relations

Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the O ( 2 , 1 ) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group O ( 3 , 2 ) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s E = m c 2 .