Lagrangian formalism for the new Dirac equation

Antimatter Gravity: Second Quantization and Lagrangian Formalism

Physics ◽

10.3390/physics2030022 ◽

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.

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Chiral Dirac Equation and Its Spacetime and CPT Symmetries

Symmetry ◽

10.3390/sym13091608 ◽

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.

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Dirac equation

Modern Particle Physics ◽

10.1017/cbo9781139525367.022 ◽

2018 ◽

pp. 515-522

Keyword(s):

Dirac Equation

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Bosonic Symmetries, Solutions, and Conservation Laws for the Dirac Equation with Nonzero Mass

Ukrainian Journal of Physics ◽

10.15407/ujpe58.06.0523 ◽

2013 ◽

Vol 58 (6) ◽

pp. 523-533 ◽

Cited By ~ 12

Author(s):

V.M. Simulik ◽

◽

I.Yu. Krivsky ◽

I.L. Lamer ◽

◽

...

Keyword(s):

Dirac Equation ◽

Conservation Laws

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Efficient Treatment of Fixed and Induced Dipolar Interactions

MRS Proceedings ◽

10.1557/proc-653-z7.22 ◽

2000 ◽

Vol 653 ◽

Author(s):

Celeste Sagui ◽

Thoma Darden

Keyword(s):

Molecular Dynamics ◽

Md Simulations ◽

Lagrangian Formalism ◽

Dipolar Interactions ◽

Time Step ◽

Efficient Treatment ◽

Particle Mesh Ewald ◽

Fixed Charges ◽

Self Consistency ◽

Induced Dipoles

AbstractFixed and induced point dipoles have been implemented in the Ewald and Particle-Mesh Ewald (PME) formalisms. During molecular dynamics (MD) the induced dipoles can be propagated along with the atomic positions either by interation to self-consistency at each time step, or by a Car-Parrinello (CP) technique using an extended Lagrangian formalism. The use of PME for electrostatics of fixed charges and induced dipoles together with a CP treatment of dipole propagation in MD simulations leads to a cost overhead of only 33% above that of MD simulations using standard PME with fixed charges, allowing the study of polarizability in largemacromolecular systems.

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SPIN-SPIN INTERACTION IN ONE PARTICLE DIRAC EQUATION

Scientific Herald of Uzhhorod University Series Physics ◽

10.24144/2415-8038.1999.4.66-68 ◽

1999 ◽

Vol 4 (0) ◽

pp. 66-68

Author(s):

І. І. Гайсак ◽

В. С. Морохович

Keyword(s):

Dirac Equation ◽

Spin Interaction

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Introduction to Relativistic Quantum Chemistry

10.1093/oso/9780195140866.001.0001 ◽

2007 ◽

Cited By ~ 30

Author(s):

Kenneth G. Dyall ◽

Knut Faegri

Keyword(s):

Dirac Equation ◽

Quantum Chemistry ◽

Relativistic Theory ◽

Density Functional ◽

Relativistic Quantum ◽

Relativistic Effects ◽

Basis Sets ◽

Spin Orbit ◽

Approximate Methods ◽

Nonrelativistic Theory

This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.

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