scholarly journals Chiral Dirac Equation and Its Spacetime and CPT Symmetries

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1608
Author(s):  
Timothy B. Watson ◽  
Zdzislaw E. Musielak
Keyword(s):  
Dirac Equation ◽  
Chiral Symmetry ◽  
Irreducible Representations ◽  
Lagrangian Formalism ◽  
Projection Operators ◽  
Fundamental Nature ◽  
Minimal Assumption ◽  
Linearly Independent ◽  
Starting Point ◽  
Novel Method

The Dirac equation with chiral symmetry is derived using the irreducible representations of the Poincaré group, the Lagrangian formalism, and a novel method of projection operators that takes as its starting point the minimal assumption of four linearly independent physical states. We thereby demonstrate the fundamental nature of this form of the Dirac equation. The resulting equation is then examined within the context of spacetime and CPT symmetries with a discussion of the implications for the general formulation of physical theories.

scholarly journals Chiral symmetry in Dirac equation and its effects on neutrino masses and dark matter

2020 ◽  
Vol 35 (30) ◽  
pp. 2050189
Author(s):  
T. B. Watson ◽  
Z. E. Musielak
Keyword(s):  
Dark Matter ◽  
Dirac Equation ◽  
Chiral Symmetry ◽  
Massive Particle ◽  
Irreducible Representations ◽  
Correct Identification ◽  
Neutrino Masses ◽  
Poincaré Group ◽  
Chiral Angle ◽  
Poincare Group

Chiral symmetry is included into the Dirac equation using the irreducible representations of the Poincaré group. The symmetry introduces the chiral angle that specifies the chiral basis. It is shown that the correct identification of these basis allows explaining small masses of neutrinos and predicting a new candidate for Dark Matter massive particle.

scholarly journals Efficient rules for all conformal blocks

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Jean-François Fortin ◽  
Wen-Jie Ma ◽  
Valentina Prilepina ◽  
Witold Skiba
Keyword(s):  
Operator Product Expansion ◽  
General Procedure ◽  
Conformal Block ◽  
Irreducible Representations ◽  
Operator Product ◽  
Projection Operators ◽  
Space Operator ◽  
Conformal Blocks ◽  
General Rules

Abstract We formulate a set of general rules for computing d-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism [1]. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars S and some specific irreducible representations R, namely 〈SSSS〉, 〈SSSR〉, 〈SRSR〉 and 〈SSRR〉 (where, when allowed, R is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.

Lagrangian formalism for the new Dirac equation

2013 ◽  
Vol 7 ◽  
pp. 141-150
Author(s):  
A. Ousmane Manga ◽  
N. V. Samsonenko ◽  
A. Moussa
Keyword(s):  
Dirac Equation ◽  
Lagrangian Formalism

Entanglement classification in the noninteracting Fermi gas

2017 ◽  
Vol 15 (07) ◽  
pp. 1750055 ◽  
Author(s):  
M. A. Jafarizadeh ◽  
F. Eghbalifam ◽  
S. Nami ◽  
M. Yahyavi
Keyword(s):  
Density Matrix ◽  
Fermi Gas ◽  
Reduced Density Matrix ◽  
Irreducible Representations ◽  
Entanglement Measure ◽  
Projection Operators ◽  
Entanglement Witness ◽  
Gas Density ◽  
Weyl Duality ◽  
Reduced Density

In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group [Formula: see text]. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even [Formula: see text] qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of [Formula: see text]. The restricted density matrix to the subspaces of the irreducible representations: [Formula: see text], [Formula: see text] and [Formula: see text] are denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix [Formula: see text] and [Formula: see text], so these classes do not exist in the total Fermi gas density matrix.

Dirac equation in presence of the Hartmann and Higgs oscillator superintegrable potentials with the spin and pseudospin symmetries

2016 ◽  
Vol 31 (35) ◽  
pp. 1650190 ◽  
Author(s):  
V. Mohammadi ◽  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Energy Spectra ◽  
Irreducible Representations ◽  
Relativistic Energy ◽  
Quadratic Algebras ◽  
Vector Potentials ◽  
Dirac Hamiltonian ◽  
Symmetry Algebras

The spin and pseudospin symmetries in the Dirac Hamiltonian are investigated in the presence of the Hartmann and the Higgs oscillator superintegrable potentials. The Pauli-Dirac representation is used in the Dirac equation with scalar and vector potentials of equal magnitude. Then, the Dirac equation is reduced to a Schrödinger-like equation. The symmetry algebras of the Schrödinger-like equation corresponding to the superintegrable potentials are represented. Also, the associated irreducible representations are shown by means of the quadratic algebras. Finally, the relativistic energy spectra of the Hartmann and the Higgs oscillator superintegrable potentials are calculated.

scholarly journals Antimatter Gravity: Second Quantization and Lagrangian Formalism

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 397-411
Author(s):  
Ulrich D. Jentschura
Keyword(s):  
Dirac Equation ◽  
Flat Space ◽  
Space Time ◽  
Charge Conjugation ◽  
Lagrangian Formalism ◽  
Curved Space ◽  
The Earth ◽  
Antimatter Gravity ◽  
Quantized Field ◽  
Charge Conjugation Parity

The application of the CPT (charge-conjugation, parity, and time reversal) theorem to an apple falling on Earth leads to the description of an anti-apple falling on anti–Earth (not on Earth). On the microscopic level, the Dirac equation in curved space-time simultaneously describes spin-1/2 particles and their antiparticles coupled to the same curved space-time metric (e.g., the metric describing the gravitational field of the Earth). On the macroscopic level, the electromagnetically and gravitationally coupled Dirac equation therefore describes apples and anti-apples, falling on Earth, simultaneously. A particle-to-antiparticle transformation of the gravitationally coupled Dirac equation therefore yields information on the behavior of “anti-apples on Earth”. However, the problem is exacerbated by the fact that the operation of charge conjugation is much more complicated in curved, as opposed to flat, space-time. Our treatment is based on second-quantized field operators and uses the Lagrangian formalism. As an additional helpful result, prerequisite to our calculations, we establish the general form of the Dirac adjoint in curved space-time. On the basis of a theorem, we refute the existence of tiny, but potentially important, particle-antiparticle symmetry breaking terms in which possible existence has been investigated in the literature. Consequences for antimatter gravity experiments are discussed.

scholarly journals POVM construction: A simple recipe with applications to symmetric states

2017 ◽  
Vol 15 (06) ◽  
pp. 1750042
Author(s):  
Swarnamala Sirsi ◽  
Karthik Bharath ◽  
S. P. Shilpashree ◽  
H. S. Smitha Rao
Keyword(s):  
Orthonormal Basis ◽  
Dimensional Space ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Irreducible Representations ◽  
Projection Operators ◽  
Simple Method ◽  
Positive Operator Valued Measures ◽  
Higher Dimensional ◽  
Symmetric States

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

scholarly journals Higher derivative fermionic field equation in the first-orderformalism

2007 ◽  
Vol 85 (8) ◽  
pp. 887-897 ◽  
Author(s):  
S I Kruglov
Keyword(s):  
Dirac Equation ◽  
Spin Projection ◽  
Electromagnetic Current ◽  
Pure Spin ◽  
Projection Operators ◽  
Third Order ◽  
First Order ◽  
Generalized Dirac Equation ◽  
Electromagnetic Interactions ◽  
Higher Derivative

The generalized Dirac equation of the third order, describing particles with spin 1/2 and three mass states, is analyzed. We obtain the first-order generalized Dirac equation in the 24-dimensional matrix form. The mass and spin projection operators are found that extract solutions of the wave equation corresponding to pure spin states of particles. The density of the electromagnetic current is obtained, and minimal and nonminimal(anomalous) electromagnetic interactions of fermions are considered by introducing three phenomenological parameters. The Hamiltonian form of the first-order equation is obtained.PACS Nos.: 03.65.Pm, 11.10.Ef; 12.10.Kt

The many electron problem

1972 ◽  
Vol 72 (1) ◽  
pp. 123-134 ◽  
Author(s):  
J. A. De Wet
Keyword(s):  
Angular Momentum ◽  
Shell Structure ◽  
Dirac Particle ◽  
Irreducible Representations ◽  
Projection Operators ◽  
Atomic Shell ◽  
Atomic Shell Structure ◽  
Experimental Justification ◽  
The Many

1. Introduction. In a paper on the many nucleon problem (l), which we shall henceforth call I, the determination of the irreducible representations of the four-dimensional unitary group were found from a decomposition of its infinitesimal ring U04 The method of decomposition made use of the four primitive four-component idempotents (projection operators) of the Dirac ring each of which, as was recognized long ago by Eddington (2), can be identified with a possible charge-spin state of a Dirac particle. Some experimental justification for the representations was also provided, and it is the purpose of this paper to apply the same tools to the many electron problem. In particular, matrices will be derived for the spin multiplets of a system of r electrons, and it will be shown how the model can account for the atomic shell structure and orbital angular momentum.

scholarly journals Improving Centrifugal Compressor Performance by Optimizing the Design of Impellers Using Genetic Algorithm and Computational Fluid Dynamics Methods

2019 ◽  
Vol 11 (19) ◽  
pp. 5409 ◽  
Author(s):  
Mohammad Omidi ◽  
Shu-Jie Liu ◽  
Soheil Mohtaram ◽  
Hui-Tian Lu ◽  
Hong-Chao Zhang
Keyword(s):  
High Energy ◽  
Subject Area ◽  
Hybrid Optimization ◽  
Blade Angle ◽  
Starting Point ◽  
Trailing Edges ◽  
Novel Method ◽  
Compressor Performance ◽  
Hybrid Optimization Model ◽  
High Energy Consumption

It has always been important to study the development and improvement of the design of turbomachines, owing to the numerous uses of turbomachines and their high energy consumption. Accordingly, optimizing turbomachine performance is crucial for sustainable development. The design of impellers significantly affects the performance of centrifugal compressors. Numerous models and design methods proposed for this subject area, however, old and based on the 1D scheme. The present article developed a hybrid optimization model based on genetic algorithms (GA) and a 3D simulation of compressors to examine the certain parameters such as blade angle at leading and trailing edges and the starting point of splitter blades. New impeller design is proposed to optimize the base compressor. The contribution of this paper includes the automatic creation of generations for achieving the optimal design and designing splitter blades using a novel method. The present study concludes with presenting a new, more efficient, and stable design.

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