This paper is with the permission of Stepan Moskaliuk similar to what he will put in the confer-ence proceedings of the summer teaching school and workshop for Ukrainian PhD physics stu-dents as given in Bratislava, a...This paper is with the permission of Stepan Moskaliuk similar to what he will put in the confer-ence proceedings of the summer teaching school and workshop for Ukrainian PhD physics stu-dents as given in Bratislava, as of summer 2015. With his permission, this paper will be in part reproduced here for this journal. First of all, we restate a proof of a highly localized special case of a metric tensor uncertainty principle first written up by Unruh. Unruh did not use the Roberson-Walker geometry which we do, and it so happens that the dominant metric tensor we will be examining, is variation in δg<sub>tt</sub>. The metric tensor variations given by δg<sub>rr</sub>, and are negligible, as compared to the variation δg<sub>tt</sub>. Afterwards, what is referred to by Barbour as emergent duration of time is from the Heisenberg Uncertainty principle (HUP) applied to δg<sub>tt </sub>in such a way as to give, in the Planckian space-time regime a nonzero minimum non zero lower ground to a massive graviton, m<sub>graviton</sub>. The lower bound to the massive graviton is influenced by δg<sub>tt </sub>and kinetic energy which is in the Planckian emergent duration of time δt as (E-V) . We find from δg<sub>tt </sub>version of the Heisenberg Uncertainty Principle (HUP), that the quantum value of the Δt·ΔE Heisenberg Uncertainty Principle (HUP) is likely not recoverable due to δg<sub>tt </sub>≠ Ο(1)~g<sub>tt</sub> ≡ 1. i.e. δg<sub>tt</sub>≠ Ο(1) . i.e. is consistent with non-curved space, so Δt · ΔE ≥ no longer holds. This even if we take the stress energy tensor approximation T<sub>ii</sub>= diag (ρ ,-p,-p,-p) where the fluid approximation is used. Our treatment of the inflaton is via Handley et al., where we consider the lower mass limits of the graviton as due to when the inflaton is many times larger than a Potential energy, with a kinetic energy (KE) proportional to ρ<sub>w</sub> ∝ a<sup>-3(1-w)</sup> ~ g*T<sup>4</sup> , with g* initial degrees of freedom, and T initial temperature. Leading t展开更多
We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplaci...We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].展开更多
In 1935 Dirac established the physical wave equations in the de-Sitter spaces but neither energy-momentum operators nor their conservative laws were given. In this article it is proved that in the de-Sitter group ther...In 1935 Dirac established the physical wave equations in the de-Sitter spaces but neither energy-momentum operators nor their conservative laws were given. In this article it is proved that in the de-Sitter group there is a subgroup group isomorphic to the Heisenberg group and the generators of this groups are the energy-momentum operators which obey a conservative law.展开更多
In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 a...In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.展开更多
The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutativ...The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.展开更多
In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrodinger flow for maps from a compact Riemannian manifold into a complete Kahler manifold, or from a Euclidean sp...In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrodinger flow for maps from a compact Riemannian manifold into a complete Kahler manifold, or from a Euclidean space Rm into a compact Kahler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.展开更多
In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg gro...In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.展开更多
This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappin...This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.展开更多
We consider the g-function related to a class of radial functions which gives a characterization of the L^p-norm of a function on the Heisenberg group.
In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Be...In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.展开更多
The breakdown of the Heisenberg Uncertainty Principle occurs when energies approach the Planck scale, and the corresponding Schwarzschild radius becomes similar to the Compton wavelength. Both of these quantities are ...The breakdown of the Heisenberg Uncertainty Principle occurs when energies approach the Planck scale, and the corresponding Schwarzschild radius becomes similar to the Compton wavelength. Both of these quantities are approximately equal to the Planck length. In this context, we have introduced a model that utilizes a combination of Schwarzschild’s radius and Compton length to quantify the gravitational length of an object. This model has provided a novel perspective in generalizing the uncertainty principle. Furthermore, it has elucidated the significance of the deforming linear parameter β and its range of variation from unity to its maximum value.展开更多
We investigated the quantum entanglement in spin-1 Heisenberg XY chain for two-spin-qutrit and multi-particle systems. As a measure of the entanglement, the negativity of this state was analyzed as a function of the t...We investigated the quantum entanglement in spin-1 Heisenberg XY chain for two-spin-qutrit and multi-particle systems. As a measure of the entanglement, the negativity of this state was analyzed as a function of the temperature and the magnetic field. We gave some numerical results and discussed them in detail. We found that the negativity increases monotonously with the coupling constants |J1| and |J2|, and it showed a symmetry with respect to the point of J1=0 and J2=0. In addition to the above features, there is evidence that the critical temperature is independent of the length of the chain.展开更多
In Part I of this paper, an inequality satisfied by the vacuum energy density of the universe was derived using an indirect and heuristic procedure. The derivation is based on a proposed thought experiment, according ...In Part I of this paper, an inequality satisfied by the vacuum energy density of the universe was derived using an indirect and heuristic procedure. The derivation is based on a proposed thought experiment, according to which an electron is accelerated to a constant and relativistic speed at a distance L from a perfectly conducting plane. The charge of the electron was represented by a spherical charge distribution located within the Compton wavelength of the electron. Subsequently, the electron is incident on the perfect conductor giving rise to transition radiation. The energy associated with the transition radiation depends on the parameter L. It was shown that an inequality satisfied by the vacuum energy density will emerge when the length L is pushed to cosmological dimensions and the product of the radiated energy, and the time duration of emission is constrained by Heisenberg’s uncertainty principle. In this paper, a similar analysis is conducted with a chain of electrons oscillating sinusoidally and located above a conducting plane. In the thought experiment presented in this paper, the behavior of the energy radiated by the chain of oscillating electrons is studied in the frequency domain as a function of the length L of the chain. It is shown that when the length L is pushed to cosmological dimensions and the energy radiated within a single burst of duration of half a period of oscillation is constrained by the fact that electromagnetic energy consists of photons, an inequality satisfied by the vacuum energy density emerges as a result. The derived inequality is given by where is the vacuum energy density. This result is consistent with the measured value of the vacuum energy density, which is 5.38 × 10<sup>-10</sup> J/m. The result obtained here is in better agreement with experimental data than the one obtained in Part I of this paper with time domain radiation.展开更多
The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets...The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets and explains their ordering in ferromagnetic materials.These solutions are exerted via the new extended FAN sub-equation method.We successfully obtain dark,bright,combined bright-dark,combined dark-singular,periodic,periodic singular,and elliptic wave solutions to this equation which are interesting classes of nonlinear excitation presenting spin dynamics in classical and semi-classical continuum Heisenberg systems.3D figures are illustrated under an appropriate selection of parameters.The applied technique is suitable to be used in gaining the exact solutions of most nonlinear partial/fractional differential equations which appear in complex phenomena.展开更多
We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C. It is found that for impurity entanglement, C appears onl...We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C. It is found that for impurity entanglement, C appears only when J 1 > J for J > 0, and J 1 > 0 for J < 0, and in these two regions C increases with the increase of J 1, so is the critical temperature T c. When J 1 ? | J |, C reaches its maximum value 0.5 and T c reaches the asymptotic value T c = 3.41448J 1. For entanglement between the normal lattices, C appears only when J > 0 and ?2J < J 1 < J, and initially increases with the increase of J 1 and arrives at the maximum value C max = (e4J/T ?3)/(e4J/T + 3) before it decays to zero gradually, so is the critical temperature T c with, however, the maximum value T cmax = 4J/In3.展开更多
The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy △ and the am...The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy △ and the amplitude of the magnetic field B are varied. The thermal entanglement of the nearest neighbour always declines when B increases no matter what the value of the anisotropy is. It is very interesting to note that the entanglement of the next-nearest neighbour can increase to a maximum at a certain magnetic field. Regardless of the boundary condition, the nearestneighbour entanglement always decreases and approaches to a constant value when the size of the system is very large. The constant value of open boundary condition is much larger than that of periodic boundary condition.展开更多
文摘This paper is with the permission of Stepan Moskaliuk similar to what he will put in the confer-ence proceedings of the summer teaching school and workshop for Ukrainian PhD physics stu-dents as given in Bratislava, as of summer 2015. With his permission, this paper will be in part reproduced here for this journal. First of all, we restate a proof of a highly localized special case of a metric tensor uncertainty principle first written up by Unruh. Unruh did not use the Roberson-Walker geometry which we do, and it so happens that the dominant metric tensor we will be examining, is variation in δg<sub>tt</sub>. The metric tensor variations given by δg<sub>rr</sub>, and are negligible, as compared to the variation δg<sub>tt</sub>. Afterwards, what is referred to by Barbour as emergent duration of time is from the Heisenberg Uncertainty principle (HUP) applied to δg<sub>tt </sub>in such a way as to give, in the Planckian space-time regime a nonzero minimum non zero lower ground to a massive graviton, m<sub>graviton</sub>. The lower bound to the massive graviton is influenced by δg<sub>tt </sub>and kinetic energy which is in the Planckian emergent duration of time δt as (E-V) . We find from δg<sub>tt </sub>version of the Heisenberg Uncertainty Principle (HUP), that the quantum value of the Δt·ΔE Heisenberg Uncertainty Principle (HUP) is likely not recoverable due to δg<sub>tt </sub>≠ Ο(1)~g<sub>tt</sub> ≡ 1. i.e. δg<sub>tt</sub>≠ Ο(1) . i.e. is consistent with non-curved space, so Δt · ΔE ≥ no longer holds. This even if we take the stress energy tensor approximation T<sub>ii</sub>= diag (ρ ,-p,-p,-p) where the fluid approximation is used. Our treatment of the inflaton is via Handley et al., where we consider the lower mass limits of the graviton as due to when the inflaton is many times larger than a Potential energy, with a kinetic energy (KE) proportional to ρ<sub>w</sub> ∝ a<sup>-3(1-w)</sup> ~ g*T<sup>4</sup> , with g* initial degrees of freedom, and T initial temperature. Leading t
文摘We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].
基金The project partially supported by National Natural Science Foundation of China under Grant No. 10231050/A010109
文摘In 1935 Dirac established the physical wave equations in the de-Sitter spaces but neither energy-momentum operators nor their conservative laws were given. In this article it is proved that in the de-Sitter group there is a subgroup group isomorphic to the Heisenberg group and the generators of this groups are the energy-momentum operators which obey a conservative law.
基金partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University
文摘In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 90303003 and 10575026） and the Natural Science Foundation of Zhejiang Province, China （Grant No M103042）.
文摘The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.
基金National Key Basic Research Fund (Grant Nos. G1999075109 and G1999075107) the National Science Fund for Distinguished Young Scholars (Grant No. 10025104).
文摘In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrodinger flow for maps from a compact Riemannian manifold into a complete Kahler manifold, or from a Euclidean space Rm into a compact Kahler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.
文摘In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.
文摘This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.
基金Supported by the National Natural Science Foundation of China （No. 10371004） and the Specialized Research Fund for the Doctoral Program Higher Education of China （No. 20030001107）
文摘We consider the g-function related to a class of radial functions which gives a characterization of the L^p-norm of a function on the Heisenberg group.
文摘In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.
文摘The breakdown of the Heisenberg Uncertainty Principle occurs when energies approach the Planck scale, and the corresponding Schwarzschild radius becomes similar to the Compton wavelength. Both of these quantities are approximately equal to the Planck length. In this context, we have introduced a model that utilizes a combination of Schwarzschild’s radius and Compton length to quantify the gravitational length of an object. This model has provided a novel perspective in generalizing the uncertainty principle. Furthermore, it has elucidated the significance of the deforming linear parameter β and its range of variation from unity to its maximum value.
文摘We investigated the quantum entanglement in spin-1 Heisenberg XY chain for two-spin-qutrit and multi-particle systems. As a measure of the entanglement, the negativity of this state was analyzed as a function of the temperature and the magnetic field. We gave some numerical results and discussed them in detail. We found that the negativity increases monotonously with the coupling constants |J1| and |J2|, and it showed a symmetry with respect to the point of J1=0 and J2=0. In addition to the above features, there is evidence that the critical temperature is independent of the length of the chain.
文摘In Part I of this paper, an inequality satisfied by the vacuum energy density of the universe was derived using an indirect and heuristic procedure. The derivation is based on a proposed thought experiment, according to which an electron is accelerated to a constant and relativistic speed at a distance L from a perfectly conducting plane. The charge of the electron was represented by a spherical charge distribution located within the Compton wavelength of the electron. Subsequently, the electron is incident on the perfect conductor giving rise to transition radiation. The energy associated with the transition radiation depends on the parameter L. It was shown that an inequality satisfied by the vacuum energy density will emerge when the length L is pushed to cosmological dimensions and the product of the radiated energy, and the time duration of emission is constrained by Heisenberg’s uncertainty principle. In this paper, a similar analysis is conducted with a chain of electrons oscillating sinusoidally and located above a conducting plane. In the thought experiment presented in this paper, the behavior of the energy radiated by the chain of oscillating electrons is studied in the frequency domain as a function of the length L of the chain. It is shown that when the length L is pushed to cosmological dimensions and the energy radiated within a single burst of duration of half a period of oscillation is constrained by the fact that electromagnetic energy consists of photons, an inequality satisfied by the vacuum energy density emerges as a result. The derived inequality is given by where is the vacuum energy density. This result is consistent with the measured value of the vacuum energy density, which is 5.38 × 10<sup>-10</sup> J/m. The result obtained here is in better agreement with experimental data than the one obtained in Part I of this paper with time domain radiation.
基金the Basic Science Research Unit,Scientific Research Deanship at Majmaah University,project number RGP-2019-4。
文摘The principal objective of this article is to construct new and further exact soliton solutions of the(2+1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets and explains their ordering in ferromagnetic materials.These solutions are exerted via the new extended FAN sub-equation method.We successfully obtain dark,bright,combined bright-dark,combined dark-singular,periodic,periodic singular,and elliptic wave solutions to this equation which are interesting classes of nonlinear excitation presenting spin dynamics in classical and semi-classical continuum Heisenberg systems.3D figures are illustrated under an appropriate selection of parameters.The applied technique is suitable to be used in gaining the exact solutions of most nonlinear partial/fractional differential equations which appear in complex phenomena.
基金the Natural Science Research Project of Shaanxi Province(Grant No. 2004A15)
文摘We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C. It is found that for impurity entanglement, C appears only when J 1 > J for J > 0, and J 1 > 0 for J < 0, and in these two regions C increases with the increase of J 1, so is the critical temperature T c. When J 1 ? | J |, C reaches its maximum value 0.5 and T c reaches the asymptotic value T c = 3.41448J 1. For entanglement between the normal lattices, C appears only when J > 0 and ?2J < J 1 < J, and initially increases with the increase of J 1 and arrives at the maximum value C max = (e4J/T ?3)/(e4J/T + 3) before it decays to zero gradually, so is the critical temperature T c with, however, the maximum value T cmax = 4J/In3.
文摘The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy △ and the amplitude of the magnetic field B are varied. The thermal entanglement of the nearest neighbour always declines when B increases no matter what the value of the anisotropy is. It is very interesting to note that the entanglement of the next-nearest neighbour can increase to a maximum at a certain magnetic field. Regardless of the boundary condition, the nearestneighbour entanglement always decreases and approaches to a constant value when the size of the system is very large. The constant value of open boundary condition is much larger than that of periodic boundary condition.
