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The goal of this paper is to present a spectral theory of advanced photon-counting computerized X-ray tomography, which is a promising microelectronical semiconductor wafer X-ray technique on the verge of becoming clinically feasible and has the potential to dramatically alter the clinical application of computer-aided medical diagnosis in the upcoming decades. In analogy to the deep fact that any real division algebra has dimension 1, 2, 4 or 8, the isogeneous third Galois cohomological, mirror symmetrical approach to high spatial resolution spectral photon-counting computerized tomography establishes that the exceptional Lie group Spin (8 , R) is the only one of the spin group spectrum { Spin (n , R) ∣ n ≥ 1 } to admit a triality automorphism. It is the exceptional phenomenon of triality which is able to produce by the spin representation theory of the exceptional Lie group of octonions O the spectral shaped digital signals of spectral photon-counting computerized X-ray tomography. Similarly to spin echo magnetic resonance tomography, the mathematical approach to spectral high performance computerized tomography is based on the isogeneously invariant, projectively weighted, symplectic coadjoint orbit picture Lie (N) ∗ / CoAd (N) of the real 3-dimensional Heisenberg unipotent Lie group N of type GL (3 , R) , and its solvable diamond toric Lie group extension. [ABSTRACT FROM AUTHOR]

After three generations of quarks-leptons and Higgs boson have been confirmed, we research some possible development directions of particle physics. First, particles of three generations are extended. Second, the hadronic theory must precision quantification such as heavy flavor hadrons and the lifetime formula, etc. Third, various unifications of interactions are discussed. Fourth, some basic principles (Pauli Exclusion Principle, duality, uncertainty principle, etc.) are probably violated under some cases. Fifth, Santillli, Penrose, et al., proposed some new mathematical methods. Sixth, particle astrophysics, the negative matter as unified dark matter and dark energy, and the extensive quantum theory are searched. [ABSTRACT FROM AUTHOR]

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z 3 -skew-symmetric covariant SO (3) -tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group SO (3) ↪ SU (5) . We find two SO (3) -invariants of Z 3 -skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by K (z , z) . We study the invariant properties of K (z , z) and find its stabilizer. Making use of these invariant properties, we define an SO (3) -irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an SO (3) -irreducible geometric structure and find its curvature and torsion. [ABSTRACT FROM AUTHOR]

In this review, the modern achievements in studies of the Pauli exclusion principle (PEP) and the properties of the identical particle systems when PEP is not fulfilled are discussed. The validity of conception of the spin in the framework of density functional theory (DFT) approaches is analyzed. The modern state of the recently discovered Fe-based superconductors is discussed in detail. These materials belong to the paramagnetic semimetal family and become superconductors upon doping. Recently, in 2020, room-temperature superconductivity was realized. However, from the following discussion in the SC community, it was not evident that the results of room-temperature superconductivity have been repeated by other laboratories. Thus, the question "is room temperature really achieved?" is still open. In the concluding remarks, we present the explanation of why the PEP limitations on the symmetry of identical particles system exist in nature, and following from it, some important consequences. [ABSTRACT FROM AUTHOR]

Modern physics lays its foundations on the pillars of Quantum Mechanics (QM), which has been proven successful to describe the microscopic world of atoms and particles, leading to the construction of the Standard Model. Despite the big success, the old open questions at its very heart, such as the measurement problem and the wave function collapse, are still open. Various theories consider scenarios which could encompass a departure from the predictions of the standard QM, such as extra-dimensions or deformations of the Lorentz/Poincaré symmetries. At the Italian National Gran Sasso underground Laboratory LNGS, we search for evidence of new physics proceeding from models beyond standard QM, using radiation detectors. Collapse models addressing the foundations of QM, such as the gravity-related Diósi–Penrose (DP) and Continuous Spontaneous Localization (CSL) models, predict the emission of spontaneous radiation, which allows experimental tests. Using a high-purity Germanium detector, we could exclude the natural parameterless version of the DP model and put strict bounds on the CSL one. In addition, forbidden atomic transitions could prove a possible violation of the Pauli Exclusion Principle (PEP) in open and closed systems. The VIP-2 experiment is currently in operation, aiming at detecting PEP-violating signals in Copper with electrons; the VIP-3 experiment upgrade is foreseen to become operative in the next few years. We discuss the VIP-Lead experiment on closed systems, and the strong bounds it sets on classes of non-commutative quantum gravity theories, such as the θ –Poincaré theory. [ABSTRACT FROM AUTHOR]

This interdisciplinary research paper delves into the intriguing juxtaposition of God's omnipotence versus the Pauli exclusion principle. Combining theological contemplation with the principles of quantum mechanics, the paper navigates the complexities of divine power and the physical constraints observed at the subatomic level. In the quest for common ground between theology and science, we explore historical perspectives on God's omnipotence, the origin and significance of the Pauli exclusion principle, and the interplay between quantum mechanics and classical physics. The paper examines logical paradoxes that arise when contemplating God's omnipotence and explores alternative theological views on divine power. Embracing the significance of interdisciplinary dialogue, we identify shared principles between theology and science, emphasizing epistemic humility, rational inquiry, and the unity of knowledge. The paper concludes with reflections on the ethical considerations and the potential avenues for future research into the intricate interplay between God's omnipotence and the Pauli exclusion principle. This exploration celebrates the beauty of intellectual integration and inspires an ongoing quest for truth, wisdom, and unity of thought in understanding the cosmos and the divine. [ABSTRACT FROM AUTHOR]

We study a new inequality arising from the principle of inclusion and exclusion by mixing the idea from Hlawka's inequality and Tverberg's combinatorial sum. We obtain sharp lower bounds for the sum when the number of variables is small. [ABSTRACT FROM AUTHOR]

The VIP Collaboration is performing high sensitivity tests of the Pauli exclusion principle for electrons in the extremely-low cosmic background environment of the Gran Sasso underground National Laboratory of INFN. In its open-systems configuration, the experiment checks the continuously renewed symmetry state of the conductive target, constantly supplied with electrons through a direct current. Consequently, VIP is operating the sole experiment challenging the spin-statistics connection in compliance with the Messiah-Greenberg superselection rule. The strongest bounds set by the VIP-2 experiment on the Pauli exclusion principle violation probability, by exploiting a copper target, will be reviewed. The future VIP-3 experiment will be presented, the aim of which is to map the Pauli exclusion principle violation probability as a function of the atomic number of the target under test. [ABSTRACT FROM AUTHOR]

The empirical rule that systems of identical particles always obey either Bose or Fermi statistics is customarily imposed on the theory by adding it to the axioms of nonrelativistic quantum mechanics, with the result that other statistical behaviors are excluded a priori. A more general approach is to ask what other many-particle statistics are consistent with the indistinguishability of identical particles. This strategy offers a way to discuss possible violations of the Pauli Exclusion Principle, and it leads to some interesting issues related to preparation of states and a superselection rule arising from invariance under the permutation group. [ABSTRACT FROM AUTHOR]

Monte Carlo method allowing to account for the effect of Pauli Exclusion Principle in the case of spin polarized electron gas is demonstrated. Modeling requires calculation of electron states occupancy accounting for the direction of the spin of the scattered electron. As an example of application, calculations for the case of spin and energy relaxation of initially polarized electrons in monolayer graphene have been performed. Model includes D’yakonov-Perel’ and Elliot-Yafet relaxation mechanisms. It is demonstrated that electron distribution function and energy relaxation follow the spin polarization relaxation and they are mainly governed by spin related scattering processes. [ABSTRACT FROM AUTHOR]

In this review, the modern achievements in studies of the Pauli exclusion principle (PEP) and the properties of the identical particle systems when PEP is not fulfilled are discussed. The validity of conception of the spin in the framework of density functional theory (DFT) approaches is analyzed. The modern state of the recently discovered Fe-based superconductors is discussed in detail. These materials belong to the paramagnetic semimetal family and become superconductors upon doping. Recently, in 2020, room-temperature superconductivity was realized. However, from the following discussion in the SC community, it was not evident that the results of room-temperature superconductivity have been repeated by other laboratories. Thus, the question “is room temperature really achieved?” is still open. In the concluding remarks, we present the explanation of why the PEP limitations on the symmetry of identical particles system exist in nature, and following from it, some important consequences.

Violations of the Pauli Exclusion Principle (PEP), albeit small, could be motivated by physics beyond the Standard Model, ranging from violation of Lorentz invariance to extra space dimensions. This scenario can be experimentally constrained through dedicated, state-of-the-art X-ray spectroscopy, searching for a forbidden atomic transition from the L shell to the K shell already occupied by two electrons. The VIP-2 Experiment located at the underground Gran Sasso National Laboratories of INFN (Italy) tests PEP violations by introducing new electrons via a direct current in a copper conductor, measuring the X-ray energies through a silicon drift detector. Bayesian and frequentist analyses of approximately six months of data taken with the fully operational setup is presented, setting the strongest limit to date on the PEP violation shown by the VIP collaboration. The upper bound on PEP violation are placed at 90% CL β 2 / 2 ≤ 6.8 × 10 − 42 with the Bayesian approach, and β 2 / 2 ≤ 7.1 × 10 − 42 with the frequentist CLs technique. [ABSTRACT FROM AUTHOR]

Modern physics lays its foundations on the pillars of Quantum Mechanics (QM), which has been proven successful to describe the microscopic world of atoms and particles, leading to the construction of the Standard Model. Despite the big success, the old open questions at its very heart, such as the measurement problem and the wave function collapse, are still open. Various theories consider scenarios which could encompass a departure from the predictions of the standard QM, such as extra-dimensions or deformations of the Lorentz/Poincaré symmetries. At the Italian National Gran Sasso underground Laboratory LNGS, we search for evidence of new physics proceeding from models beyond standard QM, using radiation detectors. Collapse models addressing the foundations of QM, such as the gravity-related Diósi–Penrose (DP) and Continuous Spontaneous Localization (CSL) models, predict the emission of spontaneous radiation, which allows experimental tests. Using a high-purity Germanium detector, we could exclude the natural parameterless version of the DP model and put strict bounds on the CSL one. In addition, forbidden atomic transitions could prove a possible violation of the Pauli Exclusion Principle (PEP) in open and closed systems. The VIP-2 experiment is currently in operation, aiming at detecting PEP-violating signals in Copper with electrons; the VIP-3 experiment upgrade is foreseen to become operative in the next few years. We discuss the VIP-Lead experiment on closed systems, and the strong bounds it sets on classes of non-commutative quantum gravity theories, such as the θ–Poincaré theory.

Exact schemes for the embedding of density functional theory (DFT) and wave function theory (WFT) methods into lower-level DFT or WFT approaches are introduced utilizing orbital localization. First, a simple modification of the projector-based embedding scheme of Manby and co-workers [J. Chem. Phys. 140, 18A507 (2014)] is proposed. We also use localized orbitals to partition the system, but instead of augmenting the Fock operator with a somewhat arbitrary level-shift projector we solve the Huzinaga-equation, which strictly enforces the Pauli exclusion principle. Second, the embedding of WFT methods in local correlation approaches is studied. Since the latter methods split up the system into local domains, very simple embedding theories can be defined if the domains of the active subsystem and the environment are treated at a different level. The considered embedding schemes are benchmarked for reaction energies and compared to quantum mechanics (QM)/molecular mechanics (MM) and vacuum embedding. We conclude that for DFT-in-DFT embedding, the Huzinaga-equation-based scheme is more efficient than the other approaches, but QM/MM or even simple vacuum embedding is still competitive in particular cases. Concerning the embedding of wave function methods, the clear winner is the embedding of WFT into low-level local correlation approaches, and WFT-in-DFT embedding can only be more advantageous if a non-hybrid density functional is employed. [ABSTRACT FROM AUTHOR]

We show the existence of the |$\alpha+\alpha^*$| cluster structure at the highly excited energy around |$E_x=20$| MeV in |$^{8}$| Be for the first time in the coupled-channel calculations. An extended double-folding model derived using a realistic precise cluster wave function with a well-developed |$N+3N$| cluster structure for the first excited state of |$^4$| He was employed. The calculation reproduces the experimental phase shifts in |$\alpha + \alpha$| scattering up to |$E_{\rm c.m.}=21$| MeV well. The result shows that the well-developed core-excited |$\alpha+\alpha^*$| structure appears as resonances for |$L=0$| and |$2$| near the |$\alpha+\alpha^*$| threshold which correspond to the experimental states at |$E_x=20.20$| MeV and |$E_x=22.24$| MeV in |$^8$| Be. [ABSTRACT FROM AUTHOR]

It is important that any dynamics method approaches the correct population distribution at long times. In this paper, we derive a one-body reduced density matrix dynamics for electrons in energetic contact with a bath. We obtain a remarkable equation of motion which shows that in order to reach equilibrium properly, rates of electron transitions depend on the density matrix. Even though the bath drives the electrons towards a Boltzmann distribution, hole blocking factors in our equation of motion cause the electronic populations to relax to a Fermi-Dirac distribution. These factors are an old concept, but we show how they can be derived with a combination of time-dependent perturbation theory and the extended normal ordering of Mukherjee and Kutzelnigg for a general electronic state. The resulting non-equilibrium kinetic equations generalize the usual Redfield theory to many-electron systems, while ensuring that the orbital occupations remain between zero and one. In numerical applications of our equations, we show that relaxation rates of molecules are not constant because of the blocking effect. Other applications to model atomic chains are also presented which highlight the importance of treating both dephasing and relaxation. Finally, we show how the bath localizes the electron density matrix. [ABSTRACT FROM AUTHOR]

Violations of the Pauli Exclusion Principle (PEP), albeit small, could be motivated by physics beyond the Standard Model, ranging from violation of Lorentz invariance to extra space dimensions. This scenario can be experimentally constrained through dedicated, state-of-the-art X-ray spectroscopy, searching for a forbidden atomic transition from the L shell to the K shell already occupied by two electrons. The VIP-2 Experiment located at the underground Gran Sasso National Laboratories of INFN (Italy) tests PEP violations by introducing new electrons via a direct current in a copper conductor, measuring the X-ray energies through a silicon drift detector. Bayesian and frequentist analyses of approximately six months of data taken with the fully operational setup is presented, setting the strongest limit to date on the PEP violation shown by the VIP collaboration. The upper bound on PEP violation are placed at 90% CL β2/2≤6.8×10−42 with the Bayesian approach, and β2/2≤7.1×10−42 with the frequentist CLs technique.

The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrödinger equation may have any permutation symmetry. It is demonstrated that the proof in some textbooks on quantum mechanics that only symmetric and antisymmetric states can exist is wrong. However, the scenarios, in which arbitrary permutation symmetry (degenerate permutation states) is permitted, lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature of particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate should not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in the future some unknown elementary particles can be discovered that are not fermions or bosons. [ABSTRACT FROM AUTHOR]

Topological phases and the associated multiple edge states are studied for parity and time-reversal (⁠|$\mathcal{PT}$|⁠)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining |$\mathcal{PT}$| symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the |$\mathbb{Z}$| topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to |$\mathbb{Z}_2$| from |$\mathbb{Z}$|⁠. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution. [ABSTRACT FROM AUTHOR]

The underlying structure and mechanisms of the fundamental hydrogen atom are not clear in current models; they are still found to be incomplete (Linnyk et al. [8]). For example, in heavy-ion collisions, “The photons provide a critical test for the theoretical model and in conclusion (Linnyk et al. [8]): the standard dynamical models – constructed to reproduce the ’hadronic world’ – fail to explain the photon experimental data.“ This ‘comments study’ builds upon a recent article in the Results in Physics journal entitled: ‘Mechanisms explaining Coulomb’s electric force & Lorentz’s magnetic force from a classical perspective’ (Correnti, 2018 [1]). For example, we use the electric force mechanism and the attending electron structure derived in the linked article (Correnti, 2018 [1]) to develop the atomic hydrogen structure. The H2 molecule and hydrogen gas structures then become accessible in this study and (Correnti, 2012 [2]).

Motivated by the recent interest in underground experiments phenomenology (see Refs. 1–3), we review the main aspects of one specific noncommutative space–time model, based on the Groenewold–Moyal plane algebra, the 𝜃 -Poincaré space–time. In the 𝜃 -Poincaré scenario, the Lorentz co-algebra is deformed introducing a noncommutativity of space–time coordinates. In such a theory, a new quantum field theory in noncommutative space–time can be reformulated. Tackling on several conceptual misunderstanding and technical mistakes in the literature, we will focus on several issues such: (i) the construction of fields theories in 𝜃 -Poincaré; (ii) the unitarity of the S-matrix; (iii) the violation of locality, (iv) the violation of the spin-statistic theorem and the Pauli principle; (v) the observables for underground experiments. [ABSTRACT FROM AUTHOR]

So far the universality of Pauli exclusion principle (PEP) was queried some times. First, we look out some hypothesizes of violation of PEP (VIP), and proposed violation at high energy and some tests. Next, we discuss another possibility on VIP at ultra-low energy, which is similar to superconductivity, superfluids and Bose-Einstein condensation (BEC), etc. Further, we propose a possible mechanism of VIP: Cooper pairs extend to general fermion pairs, so they transform to bosons with VIP. From this we may predict some characters of this like-boson, as in nuclei and atoms, etc. Moreover, we research some possible VIP in mathematics and physics. Based on the extensive quantum theory, its PEP may be violated. [ABSTRACT FROM AUTHOR]

A mathematically strict analysis of the consequences of the requirement of wave function antisymmetry is presented. It is shown that the probability density for symmetric electron configurations is zero, i.e., these electron configurations in atoms are inaccessible owing to the action of fundamental forces termed by the author "the forces of symmetry". As a consequence, a fundamentally new, in computational aspect, quantum chemistry can be developed. A previously reported hypothesis that forbids two electrons with the same spins to have equal potentials is confirmed. [ABSTRACT FROM AUTHOR]

The correlation of electron counts from two detectors illuminated by a coherent electron beam is analyzed by associating the path of the electron beam through the lenses with the direct Coulomb interaction between two individual electrons. This is shown to lead to a full statistical description of the electron counts. The dominant contribution to the correlation is found to be due to the trajectory displacement caused by the repulsive Coulomb interaction between the first anode and the cathode tip, and the correlation of electron counts is found to depend on the amount of defocusing on the shift of the virtual source for two electrons within the correlation time. The Coulomb scatterings, which altered the direction of two neighbor electrons, occur during the acceleration, leading to a significant decrease in the electron density. The Coulomb potential with no screening will then cause large-angle scattering of nearest-neighbor electrons within the correlation time. These results are consistent with those obtained by a previous experiment. [ABSTRACT FROM AUTHOR]

The Pauli exclusion principle (PEP) can be considered from two aspects. First, it asserts that particles that have half-integer spin (fermions) are described by antisymmetric wave functions, and particles that have integer spin (bosons) are described by symmetric wave functions. It is called spin-statistics connection (SSC). The physical reasons why SSC exists are still unknown. On the other hand, PEP is not reduced to SSC and can be consider from another aspect, according to it, the permutation symmetry of the total wave function can be only of two types: symmetric or antisymmetric. They both belong to one-dimensional representations of the permutation group, while other types of permutation symmetry are forbidden. However, the solution of the Schrödinger equation may have any permutation symmetry. We analyze this second aspect of PEP and demonstrate that proofs of PEP in some wide-spread textbooks on quantum mechanics, basing on the indistinguishability principle, are incorrect. The indistinguishability principle is insensitive to the permutation symmetry of wave function. So, it cannot be used as a criterion for the PEP verification. However, as follows from our analysis of possible scenarios, the permission of states with permutation symmetry more general than symmetric and antisymmetric leads to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature particles only in symmetric and antisymmetric permutation states is not accidental, since all symmetry options for the total wave function, except the antisymmetric and symmetric, cannot be realized. From this an important conclusion follows, we may not expect that in future some unknown elementary particles that are not fermions or bosons can be discovered.