The electron field and the Dirac bracket

A recent quantization rule of Fermi systems starts from the new symmetric brackets of classical mechanics. As a consequence, Fermi and Bose quantization can be put on an equal footing, instead of the standardad hoc procedure. We prove that the rule gives the right anticommutation relations when applied to the case of the relativistic electron. We show that this is a crucial test of the rule.For completeness, Dirac's Hamiltonian mechanics and the plus and minus Dirac bracket formalisms are developed for the electron's field.     

The quantization of Fermi systems and the Dirac bracket

This work is devoted (a) to discussing some problems related to the quantization rule for constrained classical models of Fermi type, and (b) to working with some detail a specific model which is a classical analogue of the quantum Fermi systems. The quantization of this model is shown to depend on the addition of a total time derivative to the corresponding Lagrangian.     

On the Dirac oscillator

In the present work we obtain a new representation for the Dirac oscillator based on the Clifford algebra C?₇. The symmetry breaking and the energy eigenvalues for our model of the Dirac oscillator are studied in the non-relativistic limit.     

On the Dirac equation

A connection is established between solutions of the Dirac equation (for an electron in a constant magnetic field) and those of the wave equation for a particle with zero mass.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 96–100, December, 1973.     

Dirac Neutrinos in the Seesaw Mechanism: Violation of the Number of Dirac Neutrinos

Physics of Atomic Nuclei - The seesaw mechanism, which explains the smallness of the neutrino masses by the involvement of high Majorana masses, leads to particles of the same Majorana nature and...     

Integral invariants of the Buttin bracket

We study the densities (most general objects which may be integrated over supersurfaces in superspace), invariant with respect to supercanonical transformations which do not change the Buttin bracket. The only such nontrivial object is, in a definite sense, the odd semidensity explicitly constructed here.     

Dirac combs

We consider tempered distributions given by linear combinations of delta functions placed at different points and whose Fourier transform is also a sum of the delta functions. We show that they can be characterized as finite superpositions of periodic structures.     

Recursive properties of Dirac and metriplectic Dirac brackets with applications

In this article, we prove that Dirac brackets for Hamiltonian and non-Hamiltonian constrained systems can be derived recursively. We then study the applicability of that formulation in analysis of some interesting physical models. Particular attention is paid to the feasibility of implementation code for Dirac brackets in Computer Algebra System.     

Fluid-dynamical representations of the Dirac equation

A relative kinetic mass operator is defined bym =c ⁻²·(E −eΦ), and it is shown that bt using it in a symmetric form one can correlate the (charge) velocity operatorα in the Dirac theory exactly with the general quantum mechanical momentum —ih∇. Then the net force, defined as the rate of change of the relative momentum with time, is exactly equal to the Lorentz force. The contribution due to the time variation of mass equals the negative of space variation of the scalar potential, the Newtonian force, whereas the time variation of the charge current absorbs the entire vector potential dependence. The analogous Euler equations can be written either in terms of the charge current or in terms of the mass current. For a many particle system one needs the usual net single particle parameters and the consideration of both the direct and exchange contributions of the two particle interaction. These Euler equations yield two different conditions of the stationary state. It is shown that the charge-current condition is necessary but not sufficient, whereas the mass-current condition retains the appropriate scalar potential dependence. These two conditions are compared for the spherically symmetric case. The charge density, charge current and relative mass current are tabulated for atomic spinors. Differences between the quantum and classical forces for the H ₂ ⁺ molecular ion exhibit the inadequacy of ordinary atomic spinor basis in forming molecular spinors.     

Algebraic properties of the Dirac equation

The first-order symmetry operators of the Dirac equation are classified according to their tensor properties under transformations of the homogeneous Lorentz group; a minimal system of generators for the ring of symmetry operators of the free Dirac equation is obtained, and the physical meaning of the spin operators is considered; fields are found which admit symmetry operators of first order.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 84–89, February, 1972.The author is grateful to V. N. Shapovalov for discussions and valuable suggestions.     

Invariance properties of the Dirac monopole

The quantum mechanical motion of a spinless electron in the external field of a magnetic monopole of magnetic charge μ is investigated. It is shown that Dirac's quantum condition $\text{2μe(}\text{h}\text{?}\text{c)}{}^{\mathrm{?1}}\text{= n}$ for the string being unobservable ensures rotation invariance and correct space reflection properties for any integer value of n. The rotation and space reflection operators are found and their group theoretical properties are discussed. When n is odd, half-integer j representations of the SU(2) group emerge without the introduction of spin. A method for constructing conserved quantities in the case when the potential is not explicitly invariant under the symmetry operation is also presented and applied to the discussion of the angular momentum of the electron-monopole system.     

The Geometry of the Dirac Equation

We discuss the problem of the derivation and the interpretation of metric tensors and generalized equations of motion for test particles from quasilinear spinor equations.     

Bosonic symmetries of the Dirac equation

We have found on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass the new physically meaningful features of this equation. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found, among which (together with the 32-dimensional pure matrix algebra of invariance) the new spin s=(1,0) multiplet Poincaré symmetry is proved. In order to carry out the corresponding proofs a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.     

Tensorial representation of the Dirac equation

We discuss tensor representations of the Dirac equation using a geometric approach. We find that the mass zero Dirac equations can be represented by Maxwell equations having a source which obeys the empty space wave equation. We also obtain a relation for the source in terms ofE andH. In the case of mass not equal to zero a difficulty is encountered in removing the constant spinors¯ _Aand¯ _A.We find that the arbitrary constant spinors can be eliminated in a spinor theory based on the Klein-Gordon equation.