Fractional formulation of Podolsky Lagrangian density

Lagrangians which depend on higher-order derivatives appear frequently in many areas of physics. In this paper, we reformulate Podolsky's Lagrangian in fractional form using left-right Riemann-Liouville fractional derivatives. The equations of motion are obtained using the fractional Euler Lagrange equation. In addition, the energy stress tensor and the Hamiltonian are obtained in fractional form from the Lagrangian density. The resulting equations are very similar to those found in classical field theory.

Conformal elasticity of mechanism-based metamaterials

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied—conformal—maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

Spherical Solution of Classical Quantum Gravity

In the general relativity theory, using Einstein’s gravity field equation, we discover the spherical solution of the classical quantum gravity. The careful point is that this theory is different from the other quantum theory. This theory is made by the Einstein’s classical field equation.

Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

We define a new homotopy algebraic structure, that we call a braided $$L_\infty $$ L ∞ -algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel’d twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

First steps in classical field theory

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.

Elements of Classical Field Theory

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.