Editorial: Selected Topics in Gravity, Field Theory and Quantum Mechanics

“Selected topics in Gravity, Field Theory and Quantum Mechanics” is for physicists wanting a fresh perspective into quantum gravity. Its content therefore does not include refinements of established approaches but rather brings new methods and approaches to various aspects of the problem. Our expectation that this will lead to further insight is supported by some papers having been cited already [1,2,3,4,5].

The first four contributions bring new, or at least unconventional, mathematical tools to describe the Hamiltonian dynamics of either conformable manifolds or non-trivial background curvature, with consequences for second quantization, spacetime dynamics and the constants of motion. The opening article by the editors [6] uses the Clairaut-based generalisation of the Hamiltonian formalism to study the effects of a non-trivial ground state in a gauged Lorentz symmetry theory on second quantisation. The Clairaut formalism alters the Poisson bracket to rigorously incorporate degrees of freedom which are not dynamic in the usual sense. In a similar vein, Hounnkonnou et al. consider a Poisson algebra whose bracket is based on a conformable differential and construct, among other things, Hamiltonian vector fields and other related objects on conformable Poisson-Schwarzchild and FLRW manifolds [7]. The paper by Znojil [1] addresses the issues of using the Wheeler-de Witt equation to describe the quantum evolution of the cosmos near the big bang singularity. The problem of solutions being “void of a physical meaning” is addressed by replacing the (non-Hermitian) Schroedinger picture with the corresponding Dirac interaction picture. A highly detailed review of quantum current algebra symmetry representations in integrable Hamiltonian systems from both a geometric and analytical perspective is provided by Prykarpatski [8].

The next three papers focus on quantum mechanics. Krivoruchenko [9] presents a logical construction of the linear vector nature of the quantum state, and by extension linear superposition, from the basic principles of quantum statics, number theoretic basis of physics and quantum covariance. The following paper [2] generalises Huygens-Fresnel superposition to massive particles and non-linear field theories using Kirchhoff’s integral theorem. Zooming in from quantum mechanics to quantum gravity [3] shows that the non-Abelian component of the dynamic algebra is essential to general covariance. We have also included detailed analyses of the polyadic and ternary algebraic properties of quantum mechanics. One of the editors (S. Duplij) generalised the algebra of the direct product [4] in quantum mechanics with implications for the particle content of any elementary particle model. Also exploring the generalised algebraic properties of quantum mechanics, Bruce [5] reviews the construction of semiheaps and their operators on a Hilbert space and explores how symmetries in a quantum induce homomorphisms between semiheaps and ternary algebras. The final paper [10] is a review covering topics which intersect with the other papers in this collection.