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We study the discrete energy spectrum of curved graphene sheets in the presence of a magnetic field. The shifting of the Landau levels is determined for complex and realistic geometries of curved graphene sheets. The energy levels follow a similar square root dependence on the energy quantum number as for rippled and flat graphene sheets. The Landau levels are shifted towards lower energies proportionally to the average deformation and the effect is larger compared to a simple uni-axially rippled geometry. Furthermore, the resistivity of wrinkled graphene sheets is calculated for different average space curvatures and shown to obey a linear relation. The study is carried out with a quantum lattice Boltzmann method, solving the Dirac equation on curved manifolds.

The propagation of a free scalar field ϕ with mass m in a curved background is generally described by the equation (g^μν∇_μ∇_ν + m² + ξR)ϕ = 0. There exist some arguments in the literature that seem to favor the conformal coupling to the detriment of the minimal one. However, the majority of these claims are inconclusive. Here we show that the exact Foldy–Wouthuysen transformation for spin-0 particle coupled to a wide class of static spacetime metrics exists independently of the value of ξ. Nevertheless, if the coupling is of the conformal type, the gravitational Darwin-like term has an uncomplicated structure and it is proportional to the corresponding term in the fermionic case. In addition, an independent computation of this term, which has its origin in the zitterbewegung fluctuation of the boson's position with the mean square <(δr)²> ≈ 1/m², gives a result that coincides with that obtained using the aforementioned exact transformation with ξ = 1/6.

The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor exp[-βV^{eff cl}(x₀)], where V^{eff cl}(x₀) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average . We show how to generalize this concept to paths q^μ(τ) in curved space with metric g_μν(q), and calculate perturbatively the high-temperature expansion of V^{eff cl}(q₀). The requirement of independence under coordinate transformations q^μ(τ)→ q′^μ(τ) introduces subtleties in the definition and treatment of the path average , and covariance is achieved only with the help of a suitable Faddeev–Popov procedure.

The soft breaking of gauge or other symmetries is the typical Quantum Field Theory phenomenon. In many cases one can apply the Stückelberg procedure, which means introducing some additional field (or fields) and restore the gauge symmetry. The original softly broken theory corresponds to a particular choice of the gauge fixing condition. In this paper we use this scheme for performing quantum calculations for fermion-torsion theory, softly broken by the torsion mass in arbitrary curved spacetime.

In this paper, we extend the theory of the $q$-deformed quantum mechanics in one dimension${}^{21}$ into three-dimensional case. We relate the $q$-deformed quantum theory to the quantum theory in a curved space. We discuss the diagonal metric based on $q$-addition in the Cartesian coordinate system and core radius of neutron star. We also discuss the diagonal metric based on $q$-addition in the spherical coordinate system and $q$-deformed Heisenberg atom model.

The effect of the geometry (deviation from the flat space) on the quantum evolution of the momentum and position of a free particle is discussed. It is shown that beginning with a wave-packet of minimum uncertainty (a Gaussian wave), there is a usual increase in the product of the volume uncertainties in the momentum and position space, as seen in the quantum mechanics on a flat spaces. But there is also a contribution from geometry. The leading order of this contribution is calculated.

In this paper, we introduce matrix operator algebra involving a universal curvature constant and using the Dunkl derivative. Consequently, the Dirac equation can be written without spin connections. Iterating the Dirac equation gives the Klein–Gordon equation in its canonical form without first-order Dunkl derivatives. This leads to a new form for the static metric based on the Dunkl operator in curved space.

The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.

We present a mathematical model for physical systems. A large class of functions is built through the functional quantization method and applied to the geometric study of the model. Quantized equations of motion along the Hamiltonian vector field are built up. It is seen that the procedure in higher dimension carries more physical information. The metric tensor appears to induce an electromagnetic field into the system and the dynamical nature of the electromagnetic field in curved space arises naturally. In the end, an explicit formula for the curvature tensor in the quantized space is given.

The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. Maxwell was the first to utilize the quaternions to describe the property of electromagnetic fields. Nowadays, the scholars introduce the complex-octonions to depict the electromagnetic and gravitational fields. And the complex-sedenions can be applied to study the field equations of the four interactions in the classical mechanics and quantum mechanics. Further, it is able to extend the field equations from the flat space into the curved space described with the complex-sedenions, by means of the tangent-frames and tensors. The research states that a few physical quantities will make a contribution to certain spatial parameters of the curved spaces. These spatial parameters may exert an influence on some operators (such as, divergence, gradient, and curl), impacting the field equations in the curved spaces, especially, the field equations of the four quantum-fields in the quantum mechanics. Apparently, the paper and General Relativity both confirm and succeed to the Cartesian academic thought of ‘the space is the extension of substance’.

We show in this paper that the technologically relevant field-like spin–orbit torque (SOT) shows resilience against the geometrical effect of electron backscattering. As a device grows smaller in size, the effect of geometry on physical properties like spin torque, and hence switching current could place a physical limit on the continued shrinkage of such a device — a necessary trend of all memory devices (MRAM). The geometrical effect of curves has been shown to impact quantum transport and topological transition of Dirac and topological systems. In our work, we have ruled out the potential threat of line curves degrading the effectiveness of SOT switching. In other words, SOT switching will be resilient against the influence of curves that line the circumferences of defects in the events of electron backscattering, which commonly happens in the channel of modern electronic devices.