*Editorial* **Special Issue: "Symmetries in Quantum Mechanics"**

**G. Jordan Maclay 1,2**


This Special Issue "Symmetries in Quantum Mechanics" describes research using two of the most fundamental probes we have in nature. The 11 papers discuss phenomena in atoms, galaxies, and people, which is a testimonial to the breadth of the influence of symmetry and quantum mechanics. The papers can be broadly divided into four categories: 1. Fundamentals of quantum systems, 2. Algebraic methods in quantum mechanics, 3. Teleportation and scattering, and 4. Cosmology.

We offer some brief comments on these papers:

**Fundamentals of Quantum Systems**: In their paper "Symmetry, Transactions, and the Mechanism of Wave Function Collapse", Professors Cramer and Mead address persistent fundamental questions in quantum mechanics using the transactional theory of quantum mechanics [1]. This theory interprets the psi and psi\* wavefunctions as retarded and advanced waves moving in opposite time directions that meet to form a quantum "handshake". Applying this formalism, they carefully model the transfer of energy from an excited H atom to a nearby H atom in its ground state as a process that evolves continuously in time, without having to invoke any adhoc assumptions such as a collapse of the wavefunction. This is a welcome milestone in understanding fundamental processes in quantum mechanics.

Professor Jussi Lindgren and collaborator Jukka Liukkonen give a derivation of the Uncertainty Principle that is based on the effect of the stochastic fluctuations of spacetime, described in terms of stochastic optimal control theory [2]. This innovative approach, described in their paper, "The Heisenberg Uncertainty Principle as an Endogenous Property of Stochastic Optimal Control Systems in Quantum Mechanics", frees the description of the uncertainly principle from the properties of the measuring apparatus, and makes it a natural consequence of the approach to equilibrium for this stochastic system. The results are generally consistent with Bohmian mechanics, but provide a simpler mechanism, and suggest an objective, realistic interpretation of quantum mechanics.

In a unique contribution titled "Phishing for (Quantum-Like) Phools-Theory and Experimental Evidence", Prof. Ariane Lambert-Mogiliansky and her doctoral student Adrian Calmettes have applied quantum-like decision theory to explore the well-documented influence of distraction from irrelevant information on decision making, and conducted an experiment with 1253 respondents to verify their model [3]. They develop their theory using a quantum cognition model in which the quantum state represents the decision makers representation of the world, classically the beliefs. Uncertainty is seen as a quantum effect, and different influences, informative or distractions, are akin to complementary variables is quantum mechanics. This paper highlights the breadth of the influence of quantum theory in diverse fields.

Prof. Garrett Moddel and his students have built an entirely new type of optical micro-device with the striking property of having a measurable current present with no applied voltage that is described in their paper "Optical-Cavity-Induced Current" [4]. The metal-insulator-metal tunneling device has an optical cavity on only one side so the MIM device is exposed to quantum vacuum fluctuations with a highly asymmetric mode distribution on opposite sides, which may induce asymmetric hot electron currents,

**Citation:** Maclay, G.J. Special Issue: "Symmetries in Quantum Mechanics". *Symmetry* **2021**, *13*, 1620. https:// doi.org/10.3390/sym13091620

Received: 27 August 2021 Accepted: 31 August 2021 Published: 3 September 2021

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resulting in a net current. Very extensive tests were conducted over a period of several years to eliminate a broad variety of artifacts and verify the extraordinary experimental results. Over the years, I have seen many attempts to obtain useful energy by manipulations of vacuum fluctuations, but this is the first device I have seen that I think may represent a real breakthrough.

**Algebraic Methods in Quantum Mechanics:** Post-doctoral researcher Marisol Bermudez-Montana and collaborators develop in their paper "Algebraic DVR Approaches Applied to Describe the Stark Effect" two algebraic approaches using a discrete variable representations (DVR) to compute matrix representations of three dimensional Hamiltonians of the hydrogen atom and the harmonic oscillator in a simple form [5]. The elegant methods are applied to compute the Stark effect.

In the paper "Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts", Prof. Jordan Maclay has given a clear and comprehensive presentation of the group theory for the H atom in which all generators are manifestly Hermitean and expressed in terms of the canonical momenta and positions and the normal inner product is used [6]. Emphasis is on the physics of the hydrogen atom. He has used a complete set of basis states of the inverse of the coupling constant, which have the same quantum numbers as the usual bound states, allowing a uniform treatment of the bound and scattering states. With a new approach to calculating radiative shifts using SO(4,2) symmetries, he has derived a novel generating function for the non-relativistic Lamb shifts for different states.

**Teleportation and Scattering:** Professors Carlos Cardoso-Isidoro and Francisco Delgado describe in "Symmetries in Teleportation Assisted by N-Channels under Indefinite Causal Order and Post-Measurement" a method to enhance the information transfer in quantum teleportation using a superposition of channels of indefinite causal order [7]. A weak measurement can provide further improvement.

In their paper "Permutation Symmetry in Coherent Electron Scattering by Disordered Media," Professors Elena Orlenke and Fedor Orlenko compute the scattering cross section for inelastic scattering from atoms in disordered matter, taking into account the indistinguishability of electrons in the atomic core and in the incident beam [8]. As a consequence, different results are predicted for scattering from He triplet and singlet states.

**Cosmology:** The paper "An Invariant Characterization of the Levi-Civita Spacetimes" by Cooper Watson, a doctoral student of Prof. Gerald Cleaver at Baylor University, and his collaborators, reviews and analyzes different space–time solutions of Einstein's equations of general relativity published by Tullio Levi-Civita from 1917–19 [9]. These little known solutions have, until now, been generally unavailable to English speaking researchers. These results have been cast in modern language and related to other GR results. Today these solutions may be of use in understanding aspects of gravitational waves.

Prof. Abhik Sanyal and collaborators discuss inflation in terms of a Scalar-Tensor theory of gravity in "Inflation with the Scalar-Tensor Theory of Gravity" [10]. The parameters of scalar-tensor theories, which are generalizations of the Brans–Dicke theory, are determined for the particular case of standard non-minimal coupling by exploiting a symmetry associated with a conserved current. All the parameters are determined and are shown to be consistent with the Planck survey data from 2018.

Prof. Pinto gives a careful review in "Gravitational Dispersion Forces and Gravity Quantization" [11] of the history of electrodynamic dispersion forces, detailing the classical and semi-classical explanations and the later quantum developments, and contrasts this development to the purely quantum discussions of gravitational dispersion forces, which, however, can also be considered within a semi-classical framework. He sheds light on the common assumption that the gravitational field is necessarily a quantized field, and suggests that data do not support the conclusion that the gravitational field must be quantized.

We thank all the authors for their important and welcome contributions to "Symmetries in Quantum Mechanics," we thank all the reviewers and editors for their invaluable

and timely help insuring the quality of the papers, and we thank the excellent and responsive staff at MDPI, especially Mia Guo, for making this issue a success.

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

**Conflicts of Interest:** The author declares no conflict of interest.

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