arXiv subjects

Guowu Meng

Publications and source records attributed to Guowu Meng.

At least 19 recordsLinked to original sources

arXiv2026

A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures

We exhibit an explicit integer polynomial in five variables, of total degree $14$ and with constant Hessian determinant $128$, whose gradient is not injective. Consequently its formal Legendre transform is not a polynomial, and the Hessian conjecture $\HC_5$ is false. The counterexample is obtained from the six-variable doubling of Alp\"oge's 2026 Jacobian counterexample by a one-variable \emph{Schur descent}---a partial Legendre transform in a single variable. Combined with de~Bondt's theorem that $\HC_n$ holds for $n\le3$, with the elementary doubling and stabilization bridges relating the Jacobian conjectures $\JC_n$ to the Hessian conjectures $\HC_n$, and with Alp\"oge's refutation of $\JC_3$, this decides the Hessian conjecture in every dimension except $n=4$: $\HC_n$ is true for $n\le3$, false for $n\ge5$, and open only at $n=4$. Exactly two statements of the two families remain unsettled, $\JC_2$ and $\HC_4$, linked by $\HC_4 \Rightarrow \JC_2$. Along the way we record, as a warm-up, an explicit six-variable counterexample to $\HC_6$ with constant Hessian determinant $-4$ and non-injective gradient. This note adds the five-variable counterexample to, and updates the status recorded in, the first author's earlier educational preprint \cite{MengRG2026}.

math.AC↗

arXiv2026

Symmetry Regularization of 1D Generalized Coulomb Problems

For the 1D generalized Coulomb problems--a family that includes the quantizations of the 1D generalized Kepler problems of Ma-Meng-Xiao \cite{MMX2025}--we construct two explicit unitary intertwiners $\hat\iota_{\pm}$, the quantum analogs of the classical symmetry regularization maps $\iota_{\pm}$ of \cite{MMX2025}, that unitarily identify each $H_{\kappa}$-energy-definite portion of the Hilbert space $L^{2}(\mathbb{R}_{>0},\mathrm{d}q)$ with a unitary lowest-weight representation of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$.

math-ph↗

arXiv2016

The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems

A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $\mathfrak {so}^*(4n)$ on the Poisson manifold $T^*\mathbb H_*^n/\mathrm{Sp}(1)$. (Here $\mathbb H_*^n:=\mathbb H^n\backslash \{0\}$ and the Hamiltonian action of $\mathrm{Sp}(1)$ on $T^*\mathbb H_*^n$ is induced from the natural right action of $\mathrm{Sp}(1)$ on $\mathbb H_*^n$. )

math-ph↗

arXiv2015

The classical dynamic symmetry for the $\mathrm{U}(1)$-Kepler problems

For the Jordan algebra of hermitian matrices of order $n\ge 2$, we let $X$ be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map $π_X$: $T^*X\to X$ with the canonical map $X\to \mathbb{C}P^{n-1}$ (i.e., the map that sends a hermitian matrix to its column space), pulls back the Kähler form of the Fubini-Study metric on $\mathbb{C}P^{n-1}$ to a real closed differential two-form $ω_K$ on $T^*X$. Let $ω_X$ be the canonical symplectic form on $T^*X$ and $μ$ be a real number. A standard fact says that $ω_μ:=ω_X+2μ\,ω_K$ turns $T^*X$ into a symplectic manifold, hence a Poisson manifold with Poisson bracket $\{\, ,\,\}_μ$. In this article we exhibit a Poisson realization of the simple real Lie algebra $\mathfrak {su}(n, n)$ on the Poisson manifold $(T^*X, \{\, ,\,\}_μ)$, i.e., a Lie algebra homomorphism from $\mathfrak {su}(n, n)$ to $\left(C^\infty(T^*X, \mathbb R), \{\, ,\,\}_μ\right)$. Consequently one obtains the Laplace-Runge-Lenz vector for the classical $\mathrm{U}(1)$-Kepler problem with level $n$ and magnetic charge $μ$. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the $\mathrm{U}(1)$-Kepler problems with level $2$, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin [ J. Math. Phys. $\bf 12$ (1971), 841-843] on the classical dynamic symmetry for the MICZ-Kepler problems.

math.DG↗

arXiv2015

On the trajectories of O(1)-Kepler Problems

The trajectories of the $\mathrm{O}(1)$-Kepler problem at level $n\ge 2$ are completely determined. It is found in particular that a non-colliding trajectory is an ellipse, a parabola or a branch of hyperbola according as the total energy is negative, zero or positive. Moreover, it is shown that the group $\mathrm{GL}(n, \mathbb R)/\mathrm{O}(1)$ acts transitively on both the set of oriented elliptic trajectories and the set of oriented parabolic trajectories. The method employed here is similar to the one used by Levi-Civita in the study of planar Kepler problem in 1920.

math-ph↗

arXiv2014

The Universal Kepler Problem

For each simple euclidean Jordan algebra $V$, we introduce the analogue of hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. Being referred to as the universal hamiltonian, universal angular momentum and universal Laplace-Runge-Lenz vector respectively, they are elements in (essentially) the TKK (Tits-Kantor-Koecher) algebra of $V$ and satisfy commutation relations similar to the ones for the hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. We also give some examples of Poisson realization of the TKK algebra, along with the resulting classical generalized Kepler problems. For the simplest simple euclidean Jordan algebra (i.e., $\mathbb R$), we give examples of operator realization for the TKK algebra, along with the resulting quantum generalized Kepler problems.

math-ph↗

arXiv2014

Tulczyjew's Approach for Particles in Gauge Fields

Around mid-1970s W. M. Tulczyjew discovered an approach which brings the two formalisms under a common geometric roof: the dynamics of a particle with configuration space $X$ is determined by a Lagrangian submanifold $D$ of $TT^*X$ (the total tangent space of $T^*X$), and the description of $D$ by its Hamiltonian $H$: $T^*X\to \mathbb R$ (resp. its Lagrangian $L$: $TX\to\mathbb R$) yields the Hamilton (resp. Euler-Lagrange) equation. It is reported here that Tulczyjew's approach also works for the dynamics of (charged) particles in gauge fields, in which the role of the total cotangent space $T^*X$ is played by Sternberg phase spaces. In particular, it is shown that, for a particle in a gauge field, the equation of motion can be locally presented as the Euler-Lagrange equation for a Lagrangian which is the sum of the ordinary Lagrangian $L(q, \dot q)$, the Lorentz term, and an extra new term which vanishes whenever the gauge group is abelian. A charge quantization condition is also derived, generalizing Dirac's charge quantization condition from $\mathrm{U}(1)$ gauge group to any compact connected gauge group.

math-ph↗

arXiv2013

The Poisson Realization of so(2, 2k+2) on Magnetic Leaves

Let ${\mathbb R}^{2k+1}_*={\mathbb R}^{2k+1}\setminus\{\vec 0\}$ ($k\ge 1$) and $π$: ${\mathbb R}^{2k+1}_*\to \mathrm{S}^{2k}$ be the map sending $\vec r\in {\mathbb R}^{2k+1}_*$ to ${\vec r\over |\vec r|}\in \mathrm{S}^{2k}$. Denote by $P\to {\mathbb R}^{2k+1}_*$ the pullback by $π$ of the canonical principal $\mathrm{SO}(2k)$-bundle $\mathrm{SO}(2k+1)\to \mathrm{S}^{2k} $. Let $E_\sharp\to {\mathbb R}^{2k+1}_*$ be the associated co-adjoint bundle and $E^\sharp\to T^*{\mathbb R}^{2k+1}_*$ be the pullback bundle under projection map $T^*{\mathbb R}^{2k+1}_*\to {\mathbb R}^{2k+1}_*$. The canonical connection on $\mathrm{SO}(2k+1)\to \mathrm{S}^{2k} $ turns $E^\sharp$ into a Poisson manifold. The main result here is that the real Lie algebra $\mathfrak{so}(2, 2k+2)$ can be realized as a Lie subalgebra of the Poisson algebra $(C^\infty(\mathcal O^\sharp), \{, \})$, where $\mathcal O^\sharp$ is a symplectic leave of $E^\sharp$ of special kind. Consequently, in view of the earlier result of the author, an extension of the classical MICZ Kepler problems to dimension $2k+1$ is obtained. The hamiltonian, the angular momentum, the Lenz vector and the equation of motion for this extension are all explicitly worked out.

math-ph↗

arXiv2013

5th Force and Quark Mixing

In a recent article, this author proposed a program for physics beyond the Standard Model, solely based on modifying the twin pillars of fundamental physics by replacing Lorentz structure with Euclidean Jordan algebra while keeping quantum theory. This program predicts not only quarks and leptons but also a short-range 5th fundamental force accompanying gravity. This 5th force predicts quark mixing and the related CP violation, which in fact was a phenomena observed in labs about fifty years ago. Thus, there are two conflicting theories as of now, the one based on the 5th force which {\em predicts} this phenomena and the established Cabibbo-Kobayashi-Maskawa (CKM) theory which was invented to {\em explain} this phenomena. In this article a test of these two theories against the recent experimental data is presented. It is found in this test that the CKM theory fares poorly, whereas the one based on the 5th force withstands the test well, in both accuracy and precision. For example, for the CKM matrix entry $V_{cd}$, we have $$|V_{cb}^{\tiny \mbox{experiment}}|=0.0409 \pm 0.0011, \quad |V_{cb}^{\tiny \mbox{CKM}}|= 2.37\pm 1.82,\quad |V_{cb}^{\tiny \mbox{5th force}}|=0.0408 \pm 0.0028. $$

physics.gen-ph↗

arXiv2013

On the Orbits of the Magnetized Kepler Problems in Dimension 2k+1

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the "external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.

math-ph↗

arXiv2012

What are the Fundamental Matter Particles?

Quantum theory and Lorentz structure are the twin pillars of fundamental physics today. With quantum theory kept and Lorentz structure replaced by Euclidean Jordan algebra --- a more fundamental structure, one naturally arrives at the notion of abstract fundamental matter particles. These abstract particles fall into distinct abstract universes according to their symmetry groups. If it is assumed that the charged particle count for such an abstract universe is 32, then this abstract universe must be conformally-symmetric and 11-dimensional Lorentzian when it is extremely hot; furthermore, it matches our real world universe in quite a few aspects, from the charged particle content to the existence of dark matter. Based on this match, a few predictions can be made: 1) the electric-weak force symmetry must be broken if the macroscopic spatial dimension is 2 or higher, 2) there are infinitely many generations of quarks and leptons, and 3) there exists a 5th fundamental force. This 5th force predicts quark mixing and the related CP violation because it transforms quarks among its various generations and violates the CP symmetry. A quantitative check concerning the Cabibbo-Kobayashi-Maskawa matrix entries shows a good agreement between experiments and the 5th force based computations.

physics.gen-ph↗

arXiv2011

Lorentz Group and Oriented MICZ-Kepler Orbits

The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz vector $\mathbf A$, with the parameter space consisting of the pairs of 3D vectors $(\mathbf A, \mathbf L)$ with ${\mathbf L}\cdot {\mathbf L} > (\mathbf L\cdot \mathbf A)^2$. The recent 4D perspective of the Kepler problem yields a new parametrization, with the parameter space consisting of the pairs of Minkowski vectors $(a,l)$ with $l\cdot l =-1$, $a\cdot l =0$, $a_0>0$. This new parametrization of orbits implies that the MICZ-Kepler orbits of different magnetic charges are related to each other by symmetries: \emph{${\mathrm {SO}}^+(1,3)\times {\mathbb R}_+$ acts transitively on both the set of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits}. This action extends to ${\mathrm {O}}^+(1,3)\times {\mathbb R}_+$, the \emph{structure group} for the rank-two Euclidean Jordan algebra whose underlying Lorentz space is the Minkowski space.

math-ph↗

arXiv2011

Generalized Kepler Problems I: Without Magnetic Charges

For each simple euclidean Jordan algebra $V$ of rank $ρ$ and degree $δ$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter $ν$, is obtained. Here, $ν\in{\mathcal W}(V):=\{k {δ\over 2}\mid k=1, ..., (ρ-1)\}\cup((ρ-1){δ\over 2}, \infty)$ and was introduced by N. Wallach to parametrize the set of nontrivial scalar-type unitary lowest weight representations of the conformal group of $V$. For the quantum dynamic problem labelled by $ν$, the bound state spectra is $-{1/2\over (I+ν{ρ\over 2})^2}$, I=0, 1, ... and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by $ν$. A few results in the literature about these representations become more explicit and more refined. The Lagrangian for a classical Kepler-type dynamic problem introduced here is still of the simple form: ${1\over 2} ||\dot x||^2+{1\over r}$. Here, $\dot x$ is the velocity of a unit-mass particle moving on the space consisting of $V$'s semi-positive elements of a fixed rank, and $r$ is the inner product of $x$ with the identity element of $V$.

math-ph↗

arXiv2011

Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems

For a {\em simple Euclidean Jordan algebra}, let $\mathfrak{co}$ be its conformal algebra, $\mathscr P$ be the manifold consisting of its semi-positive rank-one elements, $C^\infty(\mathscr P)$ be the space of complex-valued smooth functions on $\mathscr P$. An explicit action of $\mathfrak{co}$ on $C^\infty(\mathscr P)$, referred to as the {\em hidden action} of $\mathfrak{co}$ on $\mathscr P$, is exhibited. This hidden action turns out to be mathematically responsible for the existence of the Kepler problem and its recently-discovered vast generalizations, referred to as $J$-Kepler problems. The $J$-Kepler problems are then reconstructed and re-examined in terms of the unified language of Euclidean Jordan algebras. As a result, for a simple Euclidean Jordan algebra, the minimal representation of its conformal group can be realized either as the Hilbert space of bound states for its $J$-Kepler problem or as $L^2({\mathscr P}, {1\over r}\mathrm{vol})$, where $\mathrm{vol}$ is the volume form on $\mathscr P$ and $r$ is the inner product of $x\in \mathscr P$ with the identity element of the Jordan algebra.

math-ph↗

arXiv2010

The O(1)-Kepler Problems

Let $n\ge 2$ be an integer. To each irreducible representation $σ$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|σ|)),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. (Here $|σ|=0$ or 1 depending on whether $σ$ is trivial or not.) Furthermore, it is shown that the correspondence $σ\leftrightarrow \mathscr H(σ)$ is the theta-correspondence for dual pair $(\mathrm{O}(1), \mathrm{Sp}_{2n}(\mathbb R))\subseteq \mathrm{Sp}_{2n}(\mathbb R)$.

math-ph↗

arXiv2010

The Sp(1)-Kepler Problems

Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five. The dynamical symmetry group of this system is $\widetilde {\mathrm O}^*(4n)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\barσ)),$$ which occurs at the right-most nontrivial reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. Here $\barσ$ is the highest weight of $σ$. Furthermore, it is shown that the correspondence $σ\leftrightarrow \mathscr H(σ)$ is the theta-correspondence for dual pair $(\mathrm{Sp}(1), \mathrm{O}^*(4n))\subseteq\mathrm{Sp}_{8n}(\mathbb R)$.

math-ph↗

arXiv2008

The U(1)-Kepler Problems

Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mr U(1)$, a $\mr U(1)$-Kepler problem in dimension $(2n-1)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a MICZ-Kepler problem. The dynamical symmetry group of this system is $\widetilde {\mr U}(n, n)$, and the Hilbert space of bound states ${\ms H}(σ)$ is the unitary highest weight representation of $\widetilde {\mr U}(n, n)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_n, 1/2+\bar σ, \underbrace{1/2, ..., 1/2}_{n-1})$$ when $\bar σ\ge 0$ or $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -1/2+\bar σ, \underbrace{1/2, ..., 1/2}_n)$$ when $\bar σ\le 0$. (Here $\barσ$ is the infinitesimal character of $σ$.) Furthermore, it is shown that the correspondence between $σ^*$ (the dual of $σ$) and $\ms H(σ)$ is the theta-correspondence for dual pair $(\mr{U}(1), {\mr U}(n,n))$ in $\mr{Sp}(4n, \bb R)$.

math-ph↗

arXiv2008

A Generalization of the Kepler Problem

We construct and analyze a generalization of the Kepler problem. These generalized Kepler problems are parameterized by a triple $(D, κ, μ)$ where the dimension $D\ge 3$ is an integer, the curvature $κ$ is a real number, the magnetic charge $μ$ is a half integer if $D$ is odd and is 0 or 1/2 if $D$ is even. The key to construct these generalized Kepler problems is the observation that the Young powers of the fundamental spinors on a punctured space with cylindrical metric are the right analogues of the Dirac monopoles.

math-ph↗