Search Results (16)

We present a detailed study on the geometrization of the Proca field in the so-called Weyl Gauge-Invariant Theory, shedding new light on the physical interpretation of the Weyl field. We first describe the field equations of the theory. We then obtain a solution for the weak field using a spherically symmetric and static approximate metric. Our analysis revealed that the Weyl field, in the weak field approximation, exhibits a behavior identical to the Yukawa potential, similar to the Proca field. Furthermore, the obtained metric solution is equivalent to the Einstein–Proca case, demonstrating that the description of the Weyl field in the Weyl Gauge-Invariant Theory is consistent with Proca theory in the context of General Relativity. Finally, we conclude that the Weyl field can be formally interpreted as a Proca field of geometrical nature. Full article

In Xinjiang, soybean yield potential is constrained by varietal limitations and abiotic stresses. Although plant growth regulators (PGRs) can enhance crop yield, their specific impacts on soybean production, endogenous hormone regulation, and nutrient dynamics in this region have not been well characterized. This study evaluated the effects of different PGR treatments on yield formation, hormone levels, and nutrient contents through a field experiment conducted in Ili, Xinjiang, from 2023 to 2025. Foliar applications of naphthaleneacetic acid (NAA, 300 g ha⁻¹), prohexadione-calcium (Pro-Ca, 450 g ha⁻¹), and iron chlorine e6 (ICE6, 45 g ha⁻¹) were applied twice—at the fourth trifoliolate and full-pod stages—with an untreated control (CK) for comparison. Compared with CK, PGR treatments increased biomass allocation to reproductive organs by 6.2%, nutrient accumulation of N, P, and K by 12.3%, 25.5%, and 6.5%, respectively, pod number by 6.92 pods per plant, seed number by 4.86 seeds per plant, and 100-seed weight by 0.47 g, resulting in 6.6–12.0% higher grain yield. Seed PGR residues were 0.009 mg kg⁻¹. PGR application enhanced reproductive organ conversion capacity, nutrient uptake efficiency, and regulated endogenous hormone levels, clarifying internal yield-formation mechanisms and offering valuable reference for soybean research, particularly in similar latitudes. Full article

The soybean yield per unit area in Xinjiang has reached a high level, with the crop maturing quickly because of the higher temperatures and levels of mechanization. However, environmental factors cause flowers and pods to shed easily, limiting yield potential. Efficient plant growth regulators (PGRs) used to increase crop yields have gained popularity, but their effectiveness in reducing flower and pod shedding, considering factors such as environment, crop variety, and time of spraying, remains unclear. This study investigated whether spraying several PGRs could reduce soybean flower and pod shedding. Field experiments were conducted from 2022 to 2024 in Ili, Xinjiang, China, using α-naphthaleneacetic acid (NAA), prohexadione-calcium (Pro-Ca), and iron chlorine e6 (ICE6) with foliar applications of 300, 450, and 45 g ha⁻¹ at the four-node stage (V4) and full pod stage (R4). All PGR treatments reduced flower and pod shedding over the years and resulted in an increase in the average flower and pod numbers compared to normal-growth-treated (CK) soybeans. The effective slowing of flower and pod shedding during the critical pod formation stage (R4) ensured a stable yield potential. The flower-to-pod conversion rate was higher after spraying plants with PGRs than for the CK group, and pod retention was higher at the beginning of maturity (R7). Our results demonstrated that spraying PGRs (NAA, Pro-Ca, and ICE6) effectively reduced soybean flower and pod shedding, optimized pod distribution, and increased soybean yield potential. The study findings provide a useful reference for global soybean growers to optimize planting methods. Full article

We study the Stueckelberg field in de Sitter space, which is a massive vector field with the gauge fixing (GF) term ${\displaystyle \frac{1}{2\zeta }}{({A^{\mu }}_{;\mu })}^2$. We obtain the vacuum stress tensor, which consists of the transverse, longitudinal, temporal, and GF parts, and each contains various UV divergences. By the minimal subtraction rule, we regularize each part of the stress tensor to its pertinent adiabatic order. The transverse stress tensor is regularized to the 0th adiabatic order, while the longitudinal, temporal, and GF stress tensors are regularized to the 2nd adiabatic order. The resulting total regularized vacuum stress tensor is convergent and maximally symmetric, has a positive energy density, and respects the covariant conservation, and thus, it can be identified as the cosmological constant that drives the de Sitter inflation. Under the Lorenz condition ${A^{\mu }}_{;\mu }=0$, the regularized Stueckelberg stress tensor reduces to the regularized Proca stress tensor that contains only the transverse and longitudinal modes. In the massless limit, the regularized Stueckelberg stress tensor becomes zero, and is the same as that of the Maxwell field with the GF term, and no trace anomaly exists. If the order of adiabatic regularization were lower than our prescription, some divergences would remain. If the order were higher, say, under the conventional 4th-order regularization, more terms than necessary would be subtracted off, leading to an unphysical negative energy density and the trace anomaly simultaneously. Full article

We consider the Weyl invariant theory of gravity as an alternative approach to the problem of the origin of dark matter. According to this theory, the geometric Weyl 1-form effectively behaves as a Proca field. In this work, our starting point is to consider the existence of a gas of Weyl–Proca particles in a Bose–Einstein condensate and investigate its behavior in a cosmological context. The results obtained show that, for appropriate values of the free parameter of the model, the Weyl field behaves approximately as a dust fluid in the matter-dominated era as expected for a dark matter candidate. Full article

We study an extended Proca model with one scalar field and one massive vector field in one space dimension and one time dimension. We construct the soliton solution and subsequently compute the vacuum polarization energy (VPE), which is the leading quantum correction to the classical energy of the soliton. For this calculation, we adopt the spectral methods approach, which heavily relies on the analytic properties of the Jost function. This function is extracted from the interaction of the quantum fluctuations with a background potential generated by the soliton. Particularly, we explore eventual non-analytical components that may be induced by mass gaps and the unconventional normalization for the longitudinal component of the vector field fluctuations. By numerical simulation, we verify that these obstacles do not actually arise and that the real and imaginary momentum formulations of the VPE yield equal results. The Born approximation to the The Jost function is crucial when implementing standard renormalization conditions. In this context, we solve problems arising from the Born approximation being imaginary for real momenta associated with energies in the mass gap. Full article

We revisit Weyl’s unified field theory, which arose in 1918, shortly after general relativity was discovered. As is well known, in order to extend the program of the geometrization of physics started by Einstein to include the electromagnetic field, H. Weyl developed a new geometry which constitutes a kind of generalization of Riemannian geometry. In this paper, our aim is to discuss Weyl’s proposal anew and examine its consistency and completeness as a physical theory. We propose new directions and possible conceptual changes in the original work. Among these, we investigate with some detail the propagation of gravitational waves, and the new features arising in this recent modified gravity theory, in which the presence of a massive vector field appears somewhat unexpectedly. We also speculate whether the results could be examined in the context of primordial gravitational waves. Full article

This article reviews recent progress in computational quantum gravity caused by the framework that efficiently computes Feynman’s rules. The framework is implemented in the FeynGrav package, which extends the functionality of the widely used FeynCalc package. FeynGrav provides all the tools to study quantum gravitational effects within the standard model. We review the framework, provide the theoretical background for the efficient computation of Feynman rules, and present the proof of its completeness. We review the derivation of Feynman rules for general relativity, Horndeski gravity, Dirac fermions, Proca field, electromagnetic field, and $SU\left(N\right)$ Yang–Mills model. We conclude with a discussion of the current state of the FeynGrav package and discuss its further development. Full article

Dark matter could accumulate around neutron stars in sufficient amounts to affect their global properties. In this work, we study the effect of a specific model for dark matter—a massive and self-interacting vector (spin-1) field—on neutron stars. We describe the combined systems of neutron stars and vector dark matter using Einstein–Proca theory coupled to a nuclear matter term and find scaling relations between the field and metric components in the equations of motion. We construct equilibrium solutions of the combined systems, compute their masses and radii, and also analyze their stability and higher modes. The combined systems admit dark matter (DM) core and cloud solutions. Core solutions compactify the neutron star component and tend to decrease the total mass of the combined system. Cloud solutions have the inverse effect. Electromagnetic observations of certain cloud-like configurations would appear to violate the Buchdahl limit. This could make Buchdahl-limit-violating objects smoking gun signals for dark matter in neutron stars. The self-interaction strength is found to significantly affect both mass and radius. We also compare fermion Proca stars to objects where the dark matter is modeled using a complex scalar field. We find that fermion Proca stars tend to be more massive and geometrically larger than their scalar field counterparts for equal boson masses and self-interaction strengths. Both systems can produce degenerate masses and radii for different amounts of DM and DM particle masses. Full article

Scalar Tensor Vector Gravity (STVG) or MOdified Gravity (MOG) is a metric theory of gravity with dynamical scalar fields and a massive vector field introduced in addition to the metric tensor. In the weak field approximation, MOG modifies the Newtonian acceleration with a Yukawa-like repulsive term due to a Maxwell–Proca type Lagrangian. This associates matter with a fifth force and a modified equation of motion. MOG has been successful in explaining galaxy rotation curves, cosmological observations and all other solar system observations without the need for dark matter. In this article, we discuss the key concepts of MOG theory. Then, we discuss existing observational bounds on MOG weak field parameters. In particular, we will present our original results obtained from the X-COP sample of galaxy clusters. Full article

In this work, we present the general differential geometry of a background in which the space–time has both torsion and curvature with internal symmetries being described by gauge fields, and that is equipped to couple spinorial matter fields having spin and energy as well as gauge currents: torsion will turn out to be equivalent to an axial-vector massive Proca field and, because the spinor can be decomposed in its two chiral projections, torsion can be thought as the mediator that keeps spinors in stable configurations; we will justify this claim by studying some limiting situations. We will then proceed with a second chapter, where the material presented in the first chapter will be applied to specific systems in order to solve problems that seems to affect theories without torsion: hence the problem of gravitational singularity formation and positivity of the energy are the most important, and they will also lead the way for a discussion about the Pauli exclusion principle and the concept of macroscopic approximation. In a third and final chapter, we are going to investigate, in the light of torsion dynamics, some of the open problems in the standard models of particles and cosmology which would not be easily solvable otherwise. Full article

We prove that vector fields described by the generalized Proca class of theories do not admit consistent coupling with a gravitational sector defined by a scalar–tensor theory of the degenerate type. Under the assumption that there exists a frame in which the Proca field interacts with gravity only through the metric tensor, our analysis shows that at least one of the constraints associated with the degeneracy of the scalar–tensor sector is inevitably lost whenever the vector theory includes coupling with the Christoffel connection. Full article

In this work, we study cylindrically symmetric solutions within SU(3) non-Abelian Proca theory coupled to a Higgs scalar field. The solutions describe tubes containing either the flux of a color electric field or the energy flux and momentum. It is shown that the existence of such tubes depends crucially on the presence of the Higgs field (there are no such solutions without this field). We examine the dependence of the integral characteristics (linear energy and momentum densities) on the values of the electromagnetic potentials at the center of the tube, as well as on the values of the coupling constant of the Higgs scalar field. The solutions obtained are topologically trivial and demonstrate the dual Meissner effect: the electric field is pushed out by the Higgs scalar field. Full article

We present a comparative analysis of the self-gravitating solitons that arise in the Einstein–Klein–Gordon, Einstein–Dirac, and Einstein–Proca models, for the particular case of static, spherically symmetric spacetimes. Differently from the previous study by Herdeiro, Pombo and Radu in 2017, the matter fields possess suitable self-interacting terms in the Lagrangians, which allow for the existence of Q-ball-type solutions for these models in the flat spacetime limit. In spite of this important difference, our analysis shows that the high degree of universality that was observed by Herdeiro, Pombo and Radu remains, and various spin-independent common patterns are observed. Full article

Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary but fixed spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. Here, the supercharges transform between energy eigenstates of positive and negative energy. For such supersymmetric Hamiltonians, an exact Foldy–Wouthuysen transformation exists which brings it into a block-diagonal form separating the positive and negative energy subspaces. The relativistic dynamics of a charged particle in a magnetic field are considered for the case of a scalar (spin-zero) boson obeying the Klein–Gordon equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation. In the latter case, supersymmetry implies for the Landé g-factor $g=2$. Full article