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The Barut second-order equation, dynamical invariants and interactions

The second-order equation in the (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group has been proposed by A. Barut in the beginning of the 70s, ref. [1]. It permits to explain the mass splitting of leptons (e, μ, τ). Recently, the interest has grown to this model (see, for instance, the papers by S. Kruglov [2] and J. P. Vigier et al. [3]). We continue the research deriving the equation from the first principles, finding the dynamical invariants for this model, investigating the influence of the potential interactions. The Barut main equation is


The Barut main equation is
• It represents a theory with the conserved current that is linear in 15 generators of the 4-dimensional representation of the O(4, 2) group, N ab = i 2 γ a γ b , γ a = {γ µ , γ 5 , i}. • Instead of 4 solutions it has 8 solutions with the correct relativistic relation E = ± p 2 + m 2 i . In fact, it describes states of different masses (the second one is m µ = m e (1 + 3 2α ), α is the fine structure constant), provided that a certain physical condition is imposed on the α 2 parameter (the anomalous magentic moment should be equal to 4α/3).
If we present the 4-spinor as Ψ(p) = column(φ R (p)) φ L (p)) then Ryder states [5] that φ R (0) = φ L (0). Similar argument has been given by Faustov [6]: "the matrix B exists such that Bu λ (0) = u λ (0), B 2 = I for any (2J + 1)-component function within the Lorentz invariant theories" 1 The most general form of the relation in the (1/2, 0) ⊕ (0, 1/2) representation has been given by Dvoeglazov [7,4a]: with 1 The latter statement is more general than the Ryder one, because it admits B = 0 e +iα e −iα 0 , so that Θ J is the Wigner operator for spin J = 1/2, ϕ is the azimuthal angle p → 0 of the spherical coordinate system. Next, we use the Lorentz transformations: Applying the boosts and the relations between spinors in the rest frame, one can obtain: In the Dirac form we have: where In the QFT form we must introduce the creation/annihilation If one applies the unitary transformation to the Majorana representation [8] then γ-matrices become to be pure imaginary, and the equations are pure real.
where Ψ = Ψ 1 +iΨ 2 . It appears as if the real and imaginary parts have different masses. Finally, The physical origin was claimed by Barut to be in the magnetic self-interaction of the electron (the factor n 4 appears due to the Bohr-Sommerfeld rule for the charge moving in circular orbits in the field of a fixed magnetic dipole µ). One can start from (7), but , as opposed to the above-mentioned, one can write the coordinate-space equation in the form: with The operatorΨ may be linear-dependent on the states included in the Ψ. let us apply the most simple form Ψ 1 = −iγ 5 Ψ 4 , Ψ 2 = +iγ 5 Ψ 3 . Then, one can recover the 3rd order Barut-like equation [4b]: Thus, we have three mass states. Let us reveal the connections with other models. For instance, in refs. [3,9] the following equation has been studied: We can note: • The Barut equation is a sum of the Dirac equation and the Feynman-Gell-Mann equation.
• Recently, it was suggested to associate an analogue of Eq. (19) with the dark matter [10], provided that Ψ is composed of the self/anti-self charge conjugate spinors, and it has the dimension [energy] 1 in c =h = 1. The interaction Lagrangian is L H ∼ gΨΨφ 2 . • The term ∼Ψσ µν ΨF µν will affect the photon propagation, and non-local terms will appear in higher orders. • However, it was shown in [3b,c] that a) the Mott cross-section formula (which represents the Coulomb scattering up to the order ∼ e 2 ) is still valid; b) the hydrogen spectrum is not much disturbed; if the electromagnetic field is weak the corrections are small. • The solutions are the eigenstates of γ 5 operator.
The most general conserved current of the Barut-like theories is Let us try the Lagrangian: Then, the equation follows: and its Dirac-conjugate: The derivatives acts to the left in the second equation. Thus, we have the Dirac equation when α 1 = i 2 , α 2 = 0, and the Barut equation when α 2 = 1 m 2α/3 1+4α/3 . In the Euclidean metrics the dynamical invariants are ] .
N Ψ,Ψ µν are the Lorentz group generators. Then, the energy-momentum tensor is Hence, the HamiltonianĤ = −iP 4 The 4-current is Hence, the charge operatorQ = −i J 4 d 3 x iŝ Finally, the spin tensor is In the quantum case the corresponding field operators are written: The 4-spinor normalization isū The commutation relations are with all other to be equal to zero. The dimensions of the Ψ,Ψ are as usual, [energy] 3/2 . Hence, the second-quantized Hamiltonian is written (Remember that α 1 ∼ i 2 , the commutation relations may give another i, so the contribution of the first term to eigenvalues will be real. But if α 2 is real, the contribution of the second term may be imaginary). The charge iŝ However, due to [Λ R,L , σ · p] − = 0 the last term with α 3 does not contribute. The conclusions are: • We obtained the Barut-like equations of the 2nd order and 3rd order in derivatives. The Majorana representation has been used. • We obtained dynamical invariants for the free Barut field on the classical and quantum level. • We found relations with other models (such as the Feynman-Gell-Mann equation).
• As a result of analysis of dynamical invariants, we can state that at the free level the term ∼ α 3 ∂ µΨ σ µν ∂ ν Ψ in the Lagrangian does not contribute. • However, the interaction terms ∼ α 3Ψ σ µν ∂ ν ΨA µ will contribute when we construct the Feynman diagrams and the S-matrix. In the curved space (the 4-momentum Lobachevsky space) the influence of such terms has been investigated in the Skachkov works [12]. Briefly, the contribution will be such as if the 4-potential were interact with some "renormalized" spin. Perhaps, this explains, why did Barut use the classical anomalous magnetic moment g ∼ 4α/3 instead of α 2π . The author acknowledges discussions with participants of recent conferences.