International Journal of Geometric Methods in Modern Physics最新文献

In this paper, we investigate null Cartan geodesic isophote curves in the Minkowski $3$-space, and give examples where such curves actually exist. By categorizing the types of light vectors, we characterize different types of null Cartan geodesic isophote curves. Moreover, we present the relationship between null Cartan geodesic isophote curves and other special curves.

It is remarkable that Heisenberg’s position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg’s uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time.

In this paper, we present the coisotropic embedding theorem as a tool to provide a solution for the inverse problem of the calculus of variations for a particular class of implicit differential equations, namely the equations of motion of free Electrodynamics.

In this work, we study a constant-roll inflation model embedded in the Barrow entropy scenario. In this regards, we derive the modified of the Friedmann–Robertson–Walker (FRW) universe from the Barrow entropy using the first law of thermodynamics for the apparent horizon of the universe. We consider the inflation dynamics of the early universe under the constant-roll condition where the inflation is driven by a power-law scalar potential field, $V_{(\mathrm{\Phi })}={\mathrm{\Phi }}^n$. We calculated the tensor-to-scalar ratio $r$ and scalar spectral index $n_s$ by applying the constant-roll condition with some other parameters and compared them with the Planck 2020 observable data. To reveal the effect of the Barrow parameter $\mathrm{\Delta }$ on the inflation, we fixed the constant-roll inflation parameter as $\gamma =0.014$ and focused on the exponent of potential in the range $0<n<1$. We observed that as the value of the Barrow parameter approaches zero in the range $0\le \mathrm{\Delta }\le 1$, a long and sufficient inflation occurs, consistent with the observation data. This strengthens the claim that the observable values of the Barrow parameter occur at very small values. In addition, the obtained results were also examined numerically.

In this paper, we classify and describe totally geodesic and parallel hypersurfaces for the entire class of Siklos spacetimes. A large class of minimal hypersurfaces is also described.

Our study employs a connected correlation matrix to quantify quantum entanglement. The matrix encompasses all necessary measures for assessing the degree of entanglement between particles. We begin with a three-qubit state and involve obtaining a mixed state by performing partial tracing over one qubit. Our goal is to exclude the non-connected sector by focusing on the connected correlation. This suggests that the connected correlation is deemed crucial for capturing relevant entanglement degrees. The study classifies mixed states and observes that separable states exhibit the lowest correlation within each class. We demonstrate that the entanglement measure monotonically increases concerning the correlation measure. This implies that connected correlation serves as an effective measure of quantum entanglement. Finally, our proposal suggests that interpreting quantum entanglement from a local perspective is possible. The observable is described as a vector with locality but violates freedom of choice.

The Raychaudhuri equation (RE) for a congruence of curves in a general non-Riemannian geometry is derived. A formal connection is established between the expansion scalar and the cross-sectional volume of the congruence. It is found that the expansion scalar is equal to the fractional rate of change of volume, weighted by a scalar factor that depends on the non-Riemannian features of the geometry. Treating the congruence of curves as a dynamical system, an appropriate Lagrangian is derived such that the corresponding Euler–Lagrange equation is the RE. A Hamiltonian formulation and Poisson brackets are also presented.