Journal of Mathematical Physics Current Issue
https://pubs.aip.org/jmp
en-usTue, 17 Sep 2024 00:00:00 GMTTue, 17 Sep 2024 22:45:43 GMTSilverchaireditor@pubs.aip.org/jmpwebmaster@pubs.aip.org/jmpHow the non-metricity of the connection arises naturally in the classical theory of gravity
https://pubs.aip.org/aip/jmp/article/65/9/092501/3312881/How-the-non-metricity-of-the-connection-arises
Tue, 17 Sep 2024 00:00:00 GMT<span class="paragraphSection">Spacetime geometry is described by two–<span style="font-style:italic;">a priori</span> independent–geometric structures: the symmetric connection Γ and the metric tensor <span style="font-style:italic;">g</span>. Metricity condition of Γ (i.e. ∇<span style="font-style:italic;">g</span> = 0) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of Γ. Traditionally, instead of the (first order) Palatini principle, we use in this case the (second order) Hilbert principle, assuming metricity condition <span style="font-style:italic;">a priori</span>. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein’s theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions.</span>65909250110.1063/5.0208497https://pubs.aip.org/aip/jmp/article/65/9/092501/3312881/How-the-non-metricity-of-the-connection-arisesVertex coupling interpolation in quantum chain graphs
https://pubs.aip.org/aip/jmp/article/65/9/092102/3312879/Vertex-coupling-interpolation-in-quantum-chain
Tue, 17 Sep 2024 00:00:00 GMT<span class="paragraphSection">We analyze the band spectrum of the periodic quantum graph in the form of a chain of rings connected by line segments with the vertex coupling which violates the time reversal invariance, interpolating between the <span style="font-style:italic;">δ</span> coupling and the one determined by a simple circulant matrix. We find that flat bands are generically absent and that the negative spectrum is nonempty even for interpolation with a non-attractive <span style="font-style:italic;">δ</span> coupling; we also determine the high-energy asymptotic behavior of the bands.</span>65909210210.1063/5.0208361https://pubs.aip.org/aip/jmp/article/65/9/092102/3312879/Vertex-coupling-interpolation-in-quantum-chainIntegrable decompositions for the (2 + 1)-dimensional multi-component Ablowitz–Kaup–Newell–Segur hierarchy and their applications
https://pubs.aip.org/aip/jmp/article/65/9/093501/3312157/Integrable-decompositions-for-the-2-1-dimensional
Wed, 11 Sep 2024 00:00:00 GMT<span class="paragraphSection">This paper investigates integrable decompositions of the (2 + 1)-dimensional multi-component Ablowitz-Kaup-Newell-Segur (AKNS in brief) hierarchy. By utilizing recursive relations and symmetric reductions, it is demonstrated that the (<span style="font-style:italic;">n</span><sub>2</sub> − <span style="font-style:italic;">n</span><sub>1</sub> + 1)-flow of the (2 + 1)-dimensional coupled multi-component AKNS hierarchy can be decomposed into the corresponding <span style="font-style:italic;">n</span><sub>1</sub>-flow and <span style="font-style:italic;">n</span><sub>2</sub>-flow of the coupled multi-component AKNS hierarchy. Specifically, except for two specific scenarios, the (<span style="font-style:italic;">n</span><sub>2</sub> − <span style="font-style:italic;">n</span><sub>1</sub> + 1)-flow of the (2 + 1)-dimensional reduced multi-component AKNS hierarchy can similarly be decomposed into the corresponding <span style="font-style:italic;">n</span><sub>1</sub>-flow and <span style="font-style:italic;">n</span><sub>2</sub>-flow of the reduced multi-component AKNS hierarchy. Through the application of these integrable decompositions and Darboux transformation techniques, multiple solitons for the standard focusing multi-component “breaking soliton” equations, as well as singular, exponential, and rational solitons for the nonlocal defocusing multi-component “breaking soliton” equations, are systematically presented. Furthermore, the elastic interactions and dynamical behaviors among these soliton solutions are thoroughly investigated without loss of generality.</span>65909350110.1063/5.0203907https://pubs.aip.org/aip/jmp/article/65/9/093501/3312157/Integrable-decompositions-for-the-2-1-dimensionalTopological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries
https://pubs.aip.org/aip/jmp/article/65/9/092302/3312156/Topological-recursion-of-the-Weil-Petersson
Wed, 11 Sep 2024 00:00:00 GMT<span class="paragraphSection">The Weil–Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani’s recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be <span style="font-style:italic;">tight</span>. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil–Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in (0, <span style="font-style:italic;">π</span>) in addition to geodesic boundaries. Moreover, the generating function of Weil–Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani’s formula. This work is largely inspired by recent works by Bouttier, Guitter, and Miermont [Ann. Henri Lebesgue <strong>5</strong>, 1035–1110 (2022)] on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani’s recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with Jackiw–Teitelboim (JT) gravity. We show that the multi-boundary correlators of JT gravity with defects are expressible in the tight Weil–Petersson volume generating functions, using a tight generalization of the JT trumpet partition function.</span>65909230210.1063/5.0192711https://pubs.aip.org/aip/jmp/article/65/9/092302/3312156/Topological-recursion-of-the-Weil-PeterssonRescaling transformations and the Grothendieck bound formalism in a single quantum system
https://pubs.aip.org/aip/jmp/article/65/9/092101/3312154/Rescaling-transformations-and-the-Grothendieck
Wed, 11 Sep 2024 00:00:00 GMT<span class="paragraphSection">The Grothedieck bound formalism is studied using “rescaling transformations,” in the context of a single quantum system. The rescaling transformations enlarge the set of unitary transformations (which apply to isolated systems), with transformations that change not only the phase but also the absolute value of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum tunneling, damping and amplification, etc). A special case of rescaling transformations are the dequantisation transformations, which map a Hilbert space formalism into a formalism of scalars. The Grothendieck formalism considers a “classical” quadratic form C(θ) which takes values less than 1, and the corresponding “quantum” quadratic form Q(θ) which takes values greater than 1, up to the complex Grothendieck constant <span style="font-style:italic;">k</span><sub><span style="font-style:italic;">G</span></sub>. It is shown that Q(θ) can be expressed as the trace of the product of <span style="font-style:italic;">θ</span> with two rescaling matrices, and C(θ) can be expressed as the trace of the product of <span style="font-style:italic;">θ</span> with two dequantisation matrices. Values of Q(θ) in the “ultra-quantum” region (1, <span style="font-style:italic;">k</span><sub><span style="font-style:italic;">G</span></sub>) are very important, because this region is classically forbidden [C(θ) cannot take values in it]. An example with Q(θ)∈(1,kG) is given, which is related to phenomena where classically isolated by high potentials regions of space, communicate through quantum tunneling. Other examples show that “ultra-quantumness” according to the Grothendieck formalism (Q(θ)∈(1,kG)), is different from quantumness according to other criteria (like quantum interference or the uncertainty principle).</span>65909210110.1063/5.0201690https://pubs.aip.org/aip/jmp/article/65/9/092101/3312154/Rescaling-transformations-and-the-GrothendieckA reduced ideal MHD system for nonlinear magnetic field turbulence in plasmas with approximate flux surfaces
https://pubs.aip.org/aip/jmp/article/65/9/093101/3311985/A-reduced-ideal-MHD-system-for-nonlinear-magnetic
Tue, 10 Sep 2024 00:00:00 GMT<span class="paragraphSection">This paper studies the nonlinear evolution of magnetic field turbulence in proximity of steady ideal Magnetohydrodynamics (MHD) configurations characterized by a small electric current, a small plasma flow, and approximate flux surfaces, a physical setting that is relevant for plasma confinement in stellarators. The aim is to gather insight on magnetic field dynamics, to elucidate accessibility and stability of three-dimensional MHD equilibria, as well as to formulate practical methods to compute them. Starting from the ideal MHD equations, a reduced dynamical system of two coupled nonlinear partial differential equations for the flux function and the angle variable associated with the Clebsch representation of the magnetic field is obtained. It is shown that under suitable boundary and gauge conditions such reduced system preserves magnetic energy, magnetic helicity, and total magnetic flux. The noncanonical Hamiltonian structure of the reduced system is identified, and used to show the nonlinear stability of steady solutions against perturbations involving only one Clebsch potential. The Hamiltonian structure is also applied to construct a dissipative dynamical system through the method of double brackets. This dissipative system enables the computation of MHD equilibria by minimizing energy until a critical point of the Hamiltonian is reached. Finally, an iterative scheme based on the alternate solution of the two steady equations in the reduced system is proposed as a further method to compute MHD equilibria. A theorem is proven which states that the iterative scheme converges to a nontrivial MHD equilbrium as long as solutions exist at each step of the iteration.</span>65909310110.1063/5.0186445https://pubs.aip.org/aip/jmp/article/65/9/093101/3311985/A-reduced-ideal-MHD-system-for-nonlinear-magneticViscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal equation
https://pubs.aip.org/aip/jmp/article/65/9/092701/3311984/Viscosity-solutions-to-a-Cauchy-type-problem-for
Tue, 10 Sep 2024 00:00:00 GMT<span class="paragraphSection">In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of viscosity solutions on the past set (future set) of the Cauchy surface. Furthermore, when the time orientation of viscosity solution is consistent, the uniqueness and stability of viscosity solutions are also obtained.</span>65909270110.1063/5.0178336https://pubs.aip.org/aip/jmp/article/65/9/092701/3311984/Viscosity-solutions-to-a-Cauchy-type-problem-forSolutions for a flame propagation model in porous media based on Hamiltonian and regular perturbation methods
https://pubs.aip.org/aip/jmp/article/65/9/091501/3311983/Solutions-for-a-flame-propagation-model-in-porous
Tue, 10 Sep 2024 00:00:00 GMT<span class="paragraphSection">This article extends the exploration of solutions to the issue of flame propagation driven by pressure and temperature in porous media that we introduced in earlier papers. We continue to consider a <span style="font-style:italic;">p</span>-Laplacian type operator as a mathematical formalism to model slow and fast diffusion effects, that can be given in the non-homogeneous propagation of flames. In addition, we introduce a forced convection to model any possible induced flow in the porous media. We depart from previously known models to further substantiate our driving equations. From a mathematical standpoint, our goal is to deepen in the understanding of the general behavior of solutions via analyzing their regularity, boundedness, and uniqueness. We explore stationary solutions through a Hamiltonian approach and employ a regular perturbation method. Subsequently, nonstationary solutions are derived using a singular exponential scaling and, once more, a regular perturbation approach.</span>65909150110.1063/5.0149573https://pubs.aip.org/aip/jmp/article/65/9/091501/3311983/Solutions-for-a-flame-propagation-model-in-porousStückelberg-modified massive Abelian 3-form theory: Constraint analysis, conserved charges and BRST algebra
https://pubs.aip.org/aip/jmp/article/65/9/092301/3311762/Stuckelberg-modified-massive-Abelian-3-form-theory
Mon, 09 Sep 2024 00:00:00 GMT<span class="paragraphSection">For the Stückelberg-modified massive Abelian 3-form theory in any arbitrary <span style="font-style:italic;">D</span>-dimension of spacetime, we show that its classical gauge symmetry transformations are generated by the first-class constraints. We establish that the Noether conserved charge (corresponding to the local gauge symmetry transformations) is same as the standard form of the generator for the underlying local gauge symmetry transformations (expressed in terms of the first-class constraints). We promote these <span style="font-style:italic;">classical</span> local, continuous and infinitesimal gauge symmetry transformations to their <span style="font-style:italic;">quantum</span> counterparts Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations which are respected by the coupled (but equivalent) Lagrangian densities. We derive the conserved (anti-)BRST charges by exploiting the theoretical potential of Noether’s theorem. However, these charges turn out to be non-nilpotent. Some of the highlights of our present investigation are (i) the derivation of the off-shell nilpotent versions of the (anti-)BRST charges from the standard non-nilpotent Noether conserved (anti-)BRST charges, (ii) the appearance of the operator forms of the first-class constraints at the <span style="font-style:italic;">quantum</span> level through the physicality criteria with respect to the nilpotent versions of the (anti-)BRST charges, and (iii) the deduction of the Curci–Ferrari-type restrictions from the straightforward equality of the coupled (anti-)BRST invariant Lagrangian densities as well as from the requirement of the absolute anticommutativity of the off-shell nilpotent versions of the conserved (anti-)BRST charges.</span>65909230110.1063/5.0205593https://pubs.aip.org/aip/jmp/article/65/9/092301/3311762/Stuckelberg-modified-massive-Abelian-3-form-theoryA Gaussian integral that counts regular graphs
https://pubs.aip.org/aip/jmp/article/65/9/093301/3311237/A-Gaussian-integral-that-counts-regular-graphs
Thu, 05 Sep 2024 00:00:00 GMT<span class="paragraphSection">In a recent article [Kawamoto, J. Phys. Complexity <strong>4</strong>, 035005 (2023)], Kawamoto evoked statistical physics methods for the problem of counting graphs with a prescribed degree sequence. This treatment involved truncating a particular Taylor expansion at the first two terms, which resulted in the Bender-Canfield estimate for the graph counts. This is surprisingly successful since the Bender-Canfield formula is asymptotically accurate for large graphs, while the series truncation does not <span style="font-style:italic;">a priori</span> suggest a similar level of accuracy. We upgrade this treatment in three directions. First, we derive an exact formula for counting <span style="font-style:italic;">d</span>-regular graphs in terms of a <span style="font-style:italic;">d</span>-dimensional Gaussian integral. Second, we show how to convert this formula into an integral representation for the generating function of <span style="font-style:italic;">d</span>-regular graph counts. Third, we perform explicit saddle point analysis for large graph sizes and identify the saddle point configurations responsible for graph count estimates. In these saddle point configurations, only two of the integration variables condense to significant values, while the remaining ones approach zero for large graphs. This provides an underlying picture that justifies Kawamoto’s earlier findings.</span>65909330110.1063/5.0208715https://pubs.aip.org/aip/jmp/article/65/9/093301/3311237/A-Gaussian-integral-that-counts-regular-graphs