Dirac Equation Redux by Direct Quantization of the 4-Momentum Vector

Dirac equation (DE) is a cornerstone of quantum physics. We prove that direct quantization of the 4-momentum vector p with modulus 𝑚𝑐 (𝑚 is rest mass) yields a coordinate-free and manifestly covariant equation. In coordinate representation, this is equivalent to DE with spacetime frame vectors xμ replacing Dirac’s γμ -matrices. Remember that standard DE is not manifestly covariant. The two sets {xμ}, {γμ} obey to the same Clifford algebra. Adding an independent Hermitian vector x5 to the spacetime basis {xμ} allows to accommodate the momentum operator in a real vector space with a complex structure generated alone by vectors and multivectors. The real vector space arising from the action of the Clifford product onto the quintet {xμ , x5 } has dimension 32, the same as the equivalent real dimension for the space of Dirac matrices. x5 proves defining for the combined CPT symmetry, axial vs. polar vectors, left and right handed rotors & spinors, etc.; therefore, we name it reflector and {xμ , x5 } – a basis for spacetime-reflection (STR). The pentavector 𝐼 ≡ x05123 in STR substitutes the imaginary unit i. We develop the formalism by deriving all the essential results from the novel STR DE: conserved probability currents, symmetries, nonrelativistic approximation and spin 1/2 magnetic angular momentum. It will become clear that key symmetries follow more directly and with clearer geometric interpretation in STR than in the standard approach. In simple terms, we demonstrate how Dirac matrices are a redundant representation of spacetime-reflection directors.

Dirac Redux - Dirac’s Equation in Physical Spacetime without Internal Degrees of Freedom

The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices is superfluous. One can write down a coordinate-free manifestly covariant equation by direct quantization of the energy-momentum 4-vector P with modulus m: P(psi) = m(psi) (no slash!), the spinor (psi) taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DE properties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors x0, x1, x2, x3 formally substitute Dirac’s gamma-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5, which in 3D transforms a parity-odd vector x into a parity-even vector x5x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4 x 4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for the generators of polar vectors and boosts and the other for the generators of axial vectors and rotors, comprising Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. By proving that Dirac’s matrices are redundant, because all the information from them is contained in spacetime-reflection, it becomes relevant to reexamine those areas of modern physics that take Dirac matrices and their generalizations as fundamental.

Tachyonic Dirac Equation Revisited

In this paper, we revisit the two theoretical approaches for the formulation of the tachyonic Dirac equation. The first approach works within the theory of restricted relativity, starting from a Lorentz invariant Lagrangian consistent with a spacelike four-momentum. The second approach uses the theory of relativity extended to superluminal motions and works directly on the ordinary Dirac equation through superluminal Lorentz transformations. The equations resulting from the two approaches show mostly different, if not opposite, properties. In particular, the first equation violates the invariance under the action of the parity and charge conjugation operations. Although it is a good mathematical tool to describe the dynamics of a space-like particle, it also shows that the mean particle velocity is subluminal. In contrast, the second equation is invariant under the action of parity and charge conjugation symmetries, but the particle it describes is consistent with the classical dynamics of a tachyon. This study shows that it is not possible with the currently available theories to formulate a covariant equation that coherently describes the neutrino in the framework of the physics of tachyons, and depending on the experiment, one equation rather than the other should be used.

Consequent Quantum Mechanics by Applying 8-Dimensional Spinors in the Dirac Equation

Aims: A consequent quantum mechanics was developed by rendering operators also for the charge and rest mass. In this formalism the Dirac equation was extended by applying 8-dimensional spinors for the decomposition of square root in the covariant equation of special relativity. Results: The charge and mass operators defined by 8–dimensional spinors commute with the Hamiltonian of electron and positron in electromagnetic field, but they do not commute for neutrino and quarks. Conclusions: For neutrino the expectation values of the rest mass and charge are zero allowing these particles moving with the speed of light. The momentum of neutrino commutes with the Hamiltonian thus it has a well-defined value for the three types of neutrinos explaining why the neutrinos can oscillate. For quarks neither the rest mass nor the charge operators commute with the Hamiltonian, thus the fractional charge and renormalized mass can be considered as expectation values in the hadron states. Since any charge measurements should give eigenvalues of its operator, no fractional charge can be detected excluding possibility of observing free quarks.

Quantum Approximation for Josephson's Tunneling in Thx DUO2 Nano Material for 535 Tesla at Muon Cyclotron

The convergence quantum states of free covariant equation in Einsteins space with quantum condition is studied using the ABR (Abrikosov-Balseiro-Russell) formulation in convergence approximation for Josephson tunneling is important role for determine of neutrino particle existing, especially after Cerenkovs effect for 517 tesla super magnetic at Large Hadron Collider (LHC) Cyclotron in CERN, Lyon, France based on ThxDUO2nanomaterial. This approaching will be solved the problem for determine the value of interstellar Electrical Conductivity (EC) on DUO2chain reaction, then the post condition of muon has been known exactly. In this research shown the value of EC is 4.32 μeV at 378 tesla magnetic field for 2.1 x 104ci/mm fast thermal neutron floating in 45.7 megawatts adjusted power of CERNs Cyclotron. The resulted by special Electron-Scanning-Nuclear-Absorbtion (ESNA) shown any possibilities of Josephsons tunneling must be boundary by muon particles without neutrino particle existing for 350 456 tesla magnetic field on UO2more enrichment nuclear fuel at CERN, whereas this research has purpose for provide the mathematical formulation to boundary of muons moving at nuclear research reactor to a high degree of accuracy and with Catch-Nuc, one of nuclear beam equipment has a few important value of experimental effort.

Does the manifestly covariant equation ∂α Aα = 0 imply that Aα is a four-vector?

Does the form invariance of an equation ∂αAα = 0 imply that Aα is a four-vector? We present a simple example to answer this question, with a view to clarifying the misconception that still exists.PACS Nos.: 03.30.+p, 03.50.De, 03.50.–z