Research Article | Open Access

# Generalized Solutions of the Dirac Equation, Bosons, and Beta Decay

**Academic Editor:**Smarajit Triambak

#### Abstract

We study the Hagen-Hurley equations describing spin 1 particles. We split these equations, in the interacting case, into two Dirac equations with nonstandard solutions. It is argued that these solutions describe decay of a virtual boson in beta decay.

#### 1. Introduction

Recently, we have shown that in the free case covariant solutions of the and Duffin-Kemmer-Petiau (DKP) equations are generalized solutions of the Dirac equation [1]. These wavefunctions are nonstandard since they involve higher-order spinors. We have demonstrated recently that in the case the generalized solutions describe decay of a pion [2]. The aim of this work is to interpret spin solutions, possibly in the context of weakly decaying particles.

There are several relativistic equations describing spin particles; see [3, 4] for the reviews. The most common approach to study properties of spin bosons is based on the DKP equations (the DKP particles are bosons [5]). Several classes of potentials were used in DKP equations to investigate interactions of spin particles [6–15]. However, we will apply the Hagen-Hurley equations [16–18] in spinor form [1, 19, 20]. Our motivation stems from the observation that these equations violate parity and thus should describe weakly interacting particles (see [21] for weak interactions and violation of parity).

In the next section we transform the Hagen-Hurley equations, in the interacting case, into two Dirac equations with nonstandard solutions involving higher-order spinors, extending our earlier results described in [1]. These generalized solutions bear some analogy to generalized solutions of the Dirac equation argued to describe a lepton and three quarks [22]. In Section 3 we describe transition from nonstandard solutions of two Dirac equations to the Dirac equation for a lepton and the Weyl equation for a neutrino. In the last section we show that the transition is consistent with decay of a virtual boson in beta decay. In what follows we are using definitions and conventions of [23].

#### 2. Generalized Solutions of the Dirac Equation in the Interacting Case

We have shown recently that, in the noninteracting case, solutions of the and DKP equations are generalized solutions of the Dirac equation [1]. In our derivation we have split the DKP equations for into two Hagen-Hurley equations [16–18]. Let us note here that in the case of interaction with external fields such splitting is not possible since the identities (27) of [24], enabling the splitting, are not valid in the interacting case. Therefore, we will base our theory on the formulation; see (18) and (19) in [1] and Subsection 6(ii) in [19]. These equations violate parity [19], where , . Since parity is violated in weak interactions [21], these equations seem to describe weakly interacting particles.

We write one of these equations (Eq. (19) of [1]), in the interacting case, in the form where (1b) is the spin constraint [19]. In (1a) we have , , where are the Pauli matrices and is the unit matrix. Equations (1a) and (1b), which were first proposed by Dirac [20], can be written in the Hagen-Hurley form:with matrices given in [3], with , where and stands for transposition of a matrix.

Equations (1a) in explicit form read where condition (1b), , is not imposed. We thus get two Dirac equations or, alternatively, a single Dirac equation with generalized solution : generalizing (24) of [1].

#### 3. Decay of Spin 1 Bosons

We note that solutions of two Dirac equations (3a) and (3b) are nonstandard since they involve higher-order spinors rather than spinors , . To interpret (3a) and (3b) we put where is the Weyl spinor, describing massless neutrinos, while and are the Dirac spinors. Our analysis below will not be exact since neutrinos are massive [21]. On the other hand, their masses are very small so this approximation should not lead to significant errors.

Note that now and, accordingly, the spin is not determined: more exactly, the spin is in the space. It means that we consider not real but virtual (off-shell) bosons [25]. This substitution is in the spirit of the method of fusion of de Broglie [26, 27] (similar ansatz was used in the case [2]). After the substitution of (5a) and (5b) into (3a) and (3b) we obtain two equations:where and, after substituting solution of the Weyl equation,, , we get a single Dirac equation for spinors and :where , since components and cancel out.

Equations (7) and (8) describe two spin particles, whose spins can couple to or : that is, .

#### 4. Conclusions

Results obtained in Sections 2 and 3 cast new light on the Hagen-Hurley equations as well as on weak decays of spin 1 bosons. We have shown that transition from (1a) and (1b), describing a spin particle, to (7) and (8), via substitution of (5a) and (5b), which means that now , corresponds to decay of this particle into a Weyl antineutrino, cf. (7), and a Dirac lepton, cf. (8). Indeed, it should be a weak decay since (1a) and (1b) violate parity. The spin of this particle becomes undetermined in the process of decay; more exactly it belongs to the space: this suggests that this is a virtual particle. Therefore, the products, a lepton and a Weyl antineutrino (this suggests maximal parity violation [25]), should have total spin 0 or 1 and there should be a third particle to secure spin conservation.

The above description fits a (three-body) beta decay with formation of a virtual boson, decaying into a lepton and antineutrino. This is most conveniently explained in the case of a mixed beta decay [28]:where products of the boson decay (see [29]) are shown in square brackets and denotes spin : this seems to correspond well to the proposed transition from (1a) and (1b) to (7) and (8). Since spin of the products of decay of the virtual boson belongs to the space, their spin can be or . Moreover, in the case of the Gamow-Teller transition there must be a spin-flip in the decaying nucleon. Let us add here the fact that in reaction (9) some neutrons (82%) decay according to the Gamow-Teller mechanism while some (%) undergo the Fermi transition [28]. This mixed mechanism is explained by decoupled spins of the just born products: indeed, the condition for the spinor , due to substitution (5a), does not hold and spin of the products is in the space. More about properties of bosons can be found in [21].

It is now obvious that another set of equations, involving spinor rather than (see (18) of [1]), describes a decay with intermediate boson. Let us note finally that kinematics of the neutrino appears in the Dirac equation for the electron with arbitrary neutrino four-momentum, suggesting a continuous distribution of neutrino energy.

#### Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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#### Copyright

Copyright © 2016 Andrzej Okniński. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.