On superluminal fermions within the second derivative equation

S. I. Kruglov

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On superluminal fermions within the second derivative equation

S. I. Kruglov

High Energy Physics - Phenomenology · Submitted 5 Jan 2012

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The authors' main claim Thus, the novel tachyonic equation is suggested.

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We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states. For some choice of parameters fermions propagate with the superluminal speed. Thus, the novel tachyonic equation is suggested. The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained. The Lagrangian formulation of the first-order equation is presented and the electric current and energy-momentum tensor are found. The minimal and non-minimal electromagnetic interactions of fermions are considered and Schr\"{o}dinger's form of the equation and the quantum-mechanical Hamiltonian are obtained. The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states.	contribution 0.83	0.84	0.07	speculative 0.51
s1	For some choice of parameters fermions propagate with the superluminal speed.	result 0.98	0.67	0.36	asserted 0.53
s2	Thus, the novel tachyonic equation is suggested.	contribution 0.46	0.77	0.61	hedged possibility 0.90
s3	The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained.	contribution 0.74	0.91	0.07	asserted 1.00
s4	The Lagrangian formulation of the first-order equation is presented and the electric current and energy-momentum tensor are found.	contribution 0.75	0.92	0.05	asserted 1.00
s5	The minimal and non-minimal electromagnetic interactions of fermions are considered and Schr\"{o}dinger's form of the equation and the quantum-mechanical Hamiltonian are obtained.	contribution 0.73	0.91	0.05	asserted 1.00
s6	The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.	contribution 0.50	0.93	0.07	asserted 1.00

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Original abstract

We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states. For some choice of parameters fermions propagate with the superluminal speed. Thus, the novel tachyonic equation is suggested. The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained. The Lagrangian formulation of the first-order equation is presented and the electric current and energy-momentum tensor are found. The minimal and non-minimal electromagnetic interactions of fermions are considered and Schr\"{o}dinger's form of the equation and the quantum-mechanical Hamiltonian are obtained. The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.

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Uses its method or tool 1 paper

Let us introduce, following the method of [ref] (see also [ref] ) the system of first order equations equivalences to Eq.(2) [equation] We define the 16-component wave function [equation] so that $Ψ_0(x)=ψ(x)$, $Ψ_i(x)=(1/m)∂_iψ(x)$.

Modified Dirac equation with Lorentz invariance violation and its solutions for particles in an external magnetic field · Wave equation in the first-order 16-component form · 1.00

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Thus, equation (77) obtained generalizes the expression for the vacuum expectation of chronological pairing of the operators [ref] on the case of $b≠0$.

Quantization of Fermionic Fields with Two Mass States in the First Order Formalism · Field Quantization · 1.00
The first-order relativistic wave equation with minimal and non-minimal interactions reads (see also [ref] for a case $b=0$, $M_0=0$) [equation] where $P_0=ε^0,0⊗I_4$, $P_1=ε^μ,μ⊗I_4$ are the projection operators so that the relations $P_0^2=P_0$, $P_1^2=P_1$, $P_0+P_1=1$ hold.

On the Generalized Dirac Equation for Fermions with Two Mass States · Non-Minimal Electromagnetic Interactions · 0.83

Uses the method or tool of 3 papers

Here, we follow to the work [ref] very closely.

Quantization of Fermionic Fields with Two Mass States in the First Order Formalism · Field Quantization · 0.84
To obtain the Schrödinger form of the equation and quantum-mechanical Hamiltonian we use the method of the work [ref] .

On the Hamiltonian Form of Generalized Dirac Equation for Fermions with Two Mass States · Schrödinger Form of the Equation · 1.00
To obtain the projection matrix extracting solutions to Eq.(24), we use the minimal equation for the matrix $p̌=p_μΓ_μ$ (see Appendix A): [equation] With the help of the method of the work [ref] , one obtains from Eq.(28) the projection matrices [equation] that obey equations [equation] [equation] The operator of the spin projections on the direction of the momentum $p$ is given by [ref] [equation] where $|p| =√(p_1^2 +p_2^2+p_3^2)$, and obeys the matrix equation [equation] From Eq.(33), we find the spin projection operator [ref] [equation] which obeys the equation [equation] One can verify that the operators $p̌$ and $σ_p$ commute $[ p̌,σ_p]=0$ and, therefore, they have the common eigenfunctions.

On the Generalized Dirac Equation for Fermions with Two Mass States · Mass and Spin Projection Operators · 0.86

Compares with 1 paper

At $b=0$, $M_1=0$, one comes to expressions of $j_μ$ and $T_μν$ obtained in [ref] .

On the Generalized Dirac Equation for Fermions with Two Mass States · Electric Current and Energy-Momentum Tensor · 0.97

Cites as support 1 paper

It was shown in [ref] that in HD theory with the degenerated mass states there are not problems with unitarity.

Living with Ghosts · The Novel Tachyonic Equation · 1.00

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