**Steven Duplij**

Steven Duplij (Stepan Douplii) is a theoretical and mathematical physicist from the University of Munster, Germany. He was born in Chernyshevsk–Zabaikalsky, Russia, and studied at Kharkiv University, Ukraine, where he gained his PhD in 1983. While working at Kharkiv, he received the title Doctor of Physical and Mathematical Sciences by Habilitation in 1999.

Dr Duplij is the editor-compiler of 'Concise Encyclopedia of Supersymmetry' (2005, Springer), and is the author of more than a hundred scientific publications and several books. He is listed in the World Directory Of Mathematicians, Marques Who Is Who In America, the Encyclopedia of Modern Ukraine, the Academic Genealogy of Theoretical Physicists and the Mathematics Genealogy Project. His scientific directions include supersymmetry and quantum groups, advanced algebraic structures, gravity and nonlinear electrodynamics, constrained systems and quantum computing.

#### **Michael L. Walker**

Dr Michael Walker completed his Ph.D. at the Australian National University studying chiral symmetry breaking in supersymmetric QED. He then went to study the monopole condensate in the QCD vacuum using the Cho-Dun-Ge decomposition which he adapted to Higgsless symmetry breaking and supersymmetry breaking. His more recent work uses the Clairaut-based formalism to study the implications for the particle interpretation in second quantization, both in QCD and gravity.

His work also includes the application of machine learning to drug design, computer modelling of hormone transport in plants and epidemiological modelling. Currently, he is working on a model of spacetime in which time and relativity emerge spontaneously from four dimensional Euclidean space. His academic affiliation is with the Kirby Institute at the University of New South Wales.

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

**Michael L. Walker 1 and Steven Duplij 2,\***


"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 [1–5].

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 Clairautbased 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.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Citation:** Walker, M.L.; Duplij, S. Editorial: Selected Topics in Gravity, Field Theory and Quantum Mechanics. *Universe* **2022**, *8*, 572. https://doi.org/10.3390/ universe8110572

Received: 24 October 2022 Accepted: 24 October 2022 Published: 30 October 2022

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