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Majorana-Oppenheimer Approach to Proca Field Equations

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Abstract

A Dirac-like equation for a massive field obeying the classical Proca equations of motion (PMO) is proposed in close analogy with Majorana’s construct for Maxwell electrodynamics. Its underlying algebraic structure is examined and a plausible physical interpretation is discussed. The behavior of the PMO equations in the presence of an external electromagnetic field is also investigated in the low energy limit, via unitary transformations similar to the Foldy-Wouthuysen canonical transformation for a Dirac fermion.

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Notes

  1. We are using Heaviside-Lorentz and natural units through the article.

  2. The spin operator can also be written in a more suggestive form as

    $$\begin{aligned} \vec {S}_{MO}=-i\vec {\alpha }_{MO}\times \vec {\alpha }_{MO}, \end{aligned}$$

    which is completely analogous to the corresponding operator of Dirac’s theory:

    $$\begin{aligned} \vec {S}_{D}=-\frac{i}{2}\vec {\alpha }_{D}\times \vec {\alpha }_{D}; \end{aligned}$$

    here, we label by (MO) the quantities related to the former theory and by (D) the ones related to the latter.

  3. There is an arbitrariness in the first column of every \(\alpha ^{\mu }\) matrices, due to the null element in the first component of both wave and source vectors. Then, we can explore this freedom to chose the unconstrained elements so that \(\alpha ^0\) be symmetric and \(\alpha ^i\) anti-hermitian.

  4. Also, the gauge relation

    $$\begin{aligned} \partial _t\varphi +\vec {\nabla }.\vec {A}=0, \end{aligned}$$

    emerges from (7) independently from the equations of motion in spite of being a natural consequence of the Proca equations.

  5. Here we denote by “\(T\)” the transpose of the matrix.

  6. The hamitonian form of the PMO equation will be discussed in the next section.

  7. Henceforth we shall always consider the PMO equation in the absense of sources (\(\Phi =0\)). Also, the momentum representation for the operators is to be assumed in all the following sections (unless explicit mention to another representation is made).

  8. Naturaly, this tranformation also induces a change

    $$\begin{aligned} \psi ^{'}=\hat{U}\psi \end{aligned}$$

    in the wave vector.

  9. The parameter \(\phi (\vec {p})\) was also chosen in order to provide \(\cos (2\left| \vec {p}\right| \phi )=\frac{m}{\sqrt{m^2+\left| \vec {p}\right| ^2}}\), with the positive sign, so that the particle’s energy be positive.

  10. Note that \(A_{\mu }\) is now an external field and should not be mistaken by the original Proca field from whose components the wave vector (5) is constructed.

  11. Hereafter we will suppress the explicit reference to the time-dependency of the quantities in the mathematical expressions for the sake of brevity.

  12. In order to be more precise, the term \(\big [\hat{M}(t),\big [\hat{M}(t),\big [\hat{M}(t),\big [\hat{M}(t),\hat{H}(t)\big ]\big ]\big ]\big ]\) is of maximal order \(1/m^3\), so that the higher commutators of this kind lay out of our approximation even in maximal order (giving higher order corrections to the hamiltonian). Analogously, the terms involving commutators with \(\dot{\hat{M}}(t)\) contribute only up to \(\big [\hat{M}(t),\big [\hat{M}(t),\dot{\hat{M}}(t)\big ]\big ]\), which is of maximal order \(1/m^3\).

  13. The commutator \(\big [\hat{M}^{'},\big [\hat{M}^{'},\hat{H}^{'}\big ]\big ]\) is of maximal order \(1/m^3\), so that the higher commutators of the same kind are of lower order and don’t contribute to the new hamiltonian in the approximation limit. Also, the commutators involving \(\dot{\hat{M}}^{'}\) do not contribute either since \(\big [\hat{M}^{'},\dot{\hat{M}}^{'}\big ]\) is already of order \(1/m^4\).

  14. The term \(\Big [\hat{M}^{''},\hat{H}^{''}\Big ]\) is of maximal order \(1/m^2\), so the higher commutators exceed the approximation range, as well as all the comutators involving \(\dot{\hat{M}}^{''}\).

  15. The term involving \(\hat{O}^{4}\) is still of order \(1/m^3\); however, it contributes to put in evidence the physical interpretation of some other terms in the final expression of the hamiltonian and, therefore, was intentionally retained.

  16. In these calculations, we return to the coordinate representation for the operators.

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Acknowledgments

G.A.M.A. Fernandes thanks Capes for the financial support.

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Authors and Affiliations

  1. Departamento de Física, Universidade Federal de Santa Catarina, Caixa Postal 476, CEP, Florianópolis, SC, 88040-970, Brazil

    J. L. Tomazelli & G. A. M. A. Fernandes

Authors
  1. J. L. Tomazelli
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  2. G. A. M. A. Fernandes
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Correspondence to G. A. M. A. Fernandes.

Appendix

Appendix

Some commutators and terms relevant for the FW procedure in Sect. 3.2 are listed below:

$$\begin{aligned}&\Big [\hat{M},\hat{H}\Big ]=-\frac{i}{m}\alpha ^0\hat{O}^2-\frac{i}{2m}\alpha ^0\Big [\hat{O},\hat{\epsilon }\Big ]+i\hat{O};\end{aligned}$$
(46)
$$\begin{aligned}&\Big [\hat{M},\Big [\hat{M},\hat{H}\Big ]\Big ]=\frac{1}{m^2}\hat{O}^3+\frac{1}{4m^2}\Big [\hat{O},\Big [\hat{O},\hat{\epsilon }\Big ]\Big ]+\frac{1}{m}\alpha ^0\hat{O}^2;\end{aligned}$$
(47)
$$\begin{aligned}&\Big [\hat{M},\Big [\hat{M},\Big [\hat{M},\hat{H}\Big ]\Big ]\Big ]=-\frac{i}{m^3}\alpha ^0\hat{O}^4-\frac{i}{8m^3}\alpha ^0\Big [\hat{O},\Big [\hat{O},\Big [\hat{O},\hat{\epsilon }\Big ]\Big ]\Big ]+\frac{i}{m^2}\hat{O}^3;\end{aligned}$$
(48)
$$\begin{aligned}&\Big [\hat{M},\Big [\hat{M},\Big [\hat{M},\Big [\hat{M},m\alpha ^0\Big ]\Big ]\Big ]\Big ]=\frac{1}{m^3}\alpha ^0\hat{O}^4;\end{aligned}$$
(49)
$$\begin{aligned}&\Big [\hat{M},\dot{\hat{M}}\Big ]=\frac{1}{4m^2}\Big [\hat{O},\dot{\hat{O}}\Big ];\end{aligned}$$
(50)
$$\begin{aligned}&\Big [\hat{M},\Big [\hat{M},\dot{\hat{M}}\Big ]\Big ]=-\frac{i}{8m^3}\alpha ^0\Big [\hat{O},\Big [\hat{O},\dot{\hat{O}}\Big ]\Big ];\end{aligned}$$
(51)
$$\begin{aligned}&\Big [\hat{M}^{'},\hat{H}^{'}\Big ]=-\frac{i}{m}\alpha ^0\hat{O}^{'2}-\frac{i}{2m}\alpha ^0\Big [\hat{O}^{'},\hat{\epsilon }^{'}\Big ]+i\hat{O}^{'};\end{aligned}$$
(52)
$$\begin{aligned}&\Big [\hat{M}^{'},\Big [\hat{M}^{'},\hat{H}^{'}\Big ]\Big ]=\frac{1}{m^2}\hat{O}^{'3}+\frac{1}{4m^2}\Big [\hat{O}^{'},\Big [\hat{O}^{'},\hat{\epsilon }^{'}\Big ]\Big ]+\frac{1}{m}\alpha ^0\hat{O}^{'2};\end{aligned}$$
(53)
$$\begin{aligned}&\Big [\hat{M}^{''},\hat{H}^{''}\Big ]=-\frac{i}{2m}\alpha ^0\Big [\hat{O}^{''},\hat{\epsilon }^{''}\Big ]+i\hat{O}^{''};\end{aligned}$$
(54)
$$\begin{aligned}&\hat{O}^2=\Big (\hat{\vec {p}}-e\vec {A}\Big )^2-ie\Theta \vec {\alpha }.\vec {B};\end{aligned}$$
(55)
$$\begin{aligned}&\hat{O}^4=\Big (\hat{\vec {p}}-e\vec {A}\Big )^4-ie\Big (\hat{\vec {p}}-e\vec {A}\Big )^2\Theta \vec {\alpha }.\vec {B}-ie\Theta \vec {\alpha }.\vec {B}\Big (\hat{\vec {p}}-e\vec {A}\Big )^2-e^2\Big (\Theta \vec {\alpha }.\vec {B}\Big )^2;\end{aligned}$$
(56)
$$\begin{aligned}&\Big [\hat{O},\dot{\hat{O}}\Big ]=\Big [\hat{O},\alpha ^0\vec {\alpha }.\hat{\dot{\vec {p}}}\Big ]-e\Big [\hat{O},\alpha ^0\vec {\alpha }.\partial _t\vec {A}\Big ];\end{aligned}$$
(57)
$$\begin{aligned}&\Big [\hat{O},\Big [\hat{O},\hat{\epsilon }\Big ]\Big ]=e\Big (\vec {\nabla }.\vec {E}\Big )-e\Theta \vec {\alpha }.\Big (\vec {\nabla }\times \vec {E}\Big )+2ie\Theta \vec {\alpha }.\left( \vec {E}\times \Big (\hat{\vec {p}}-e\vec {A}\Big )\right) +\nonumber \\&\quad \qquad \qquad \qquad \quad +ie\Big [\hat{O},\alpha ^0\vec {\alpha }.\partial _t\vec {A}\Big ]. \end{aligned}$$
(58)

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Tomazelli, J.L., Fernandes, G.A.M.A. Majorana-Oppenheimer Approach to Proca Field Equations. Found Phys 44, 973–989 (2014). https://doi.org/10.1007/s10701-014-9824-y

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