Next Article in Journal / Special Issue
Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations
Previous Article in Journal
Symmetry-Adapted Fourier Series for the Wallpaper Groups

Symmetry 2012, 4(3), 427-440; doi:10.3390/sym4030427

Article
Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection
Andrzej Okniński
Physics Division, Kielce University of Technology, Al. 1000-lecia PP 7, 25-314 Kielce, Poland; Email: fizao@tu.kielce.pl; Tel.: +48-41-3424382; Fax: +48-41-3424306
Received: 18 June 2012; in revised form: 15 July 2012 / Accepted: 26 July 2012 /
Published: 7 August 2012

Abstract

: In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 × 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 × 3 subequations which are thus a supersymmetric link between fermionic and bosonic degrees of freedom.
Keywords:
relativistic wave equations; supersymmetry

1. Introduction

Recently, several supersymmetric systems, concerned mainly with anyons in 2 + 1 dimensions [1,2,3,4,5] as well as with the 3 + 1 dimensional Majorana–Dirac–Staunton theory [6], uniting fermionic and bosonic fields, have been described. Furthermore, bosonic symmetries of the Dirac equation have been found in the massless [7] as well as in the massive case [8]. Our results derived lately fit into this broader picture. We have demonstrated that certain subsolutions of the free Duffin–Kemmer–Petiau (DKP) and the Dirac equations obey the same Dirac equation with some built-in projection operators [9]. We shall refer to this equation as supersymmetric since it has bosonic (spin 0 and 1) as well as fermionic Symmetry 04 00427 i001 degrees of freedom. In the present paper we extend our results to the case of interacting fields.

The paper is organized as follows. In Section 2 relativistic wave equations as well as conventions and definitions used in the paper are described. In particular, several classical and not-so-classical subsolutions of the free Dirac equation are reviewed in Subsection 2.2. The notion of supersymmetry is invoked since some subequations arising in the context of the Dirac equation appear also in the Duffin–Kemmer–Petiau theory of massive bosons. In Section 3 the Dirac equation in longitudinal fields is split into two 3 × 3 subequations which can be written as two Dirac equations with built-in projection operators. In the next Section variables are separated in the subequations to yield 2D Dirac equations in Symmetry 04 00427 i002 subspace and 2D Pauli equations in Symmetry 04 00427 i003 subspace. In Section 5 the Duffin–Kemmer–Petiau equation for spin 0 in crossed fields is split into two 3 × 3 subequations—these equations have the same structure as subequations arising in the Dirac theory. It follows that the free 3 × 3 equations provide a supersymmetric link between the Dirac and DKP theories—this is described in Section 6. In the last Section we discuss our results in a broader context of supersymmetry and Lorentz covariance.

2. Relativistic Wave Equations

In what follows tensor indices are denoted with Greek letters: μ = 0,1,2,3. We shall use the following convention for the Minkowski space-time metric tensor: gμv = diag (1,−1,−1,−1) and we shall always sum over repeated indices. For example, Symmetry 04 00427 i004. Four-momentum operators are defined as Symmetry 04 00427 i005 where natural units have been used: c = 1, Symmetry 04 00427 i006. The interaction will be introduced via minimal coupling,

Symmetry 04 00427 i007

with a four-potential Aμ and a charge q. In what follows we shall work with external fields of special configuration, so-called crossed and longitudinal fields, non-standard but Lorentz covariant, see [10]. We shall also need elements of spinor calculus. Four-vectors Symmetry 04 00427 i008 and spinors Symmetry 04 00427 i009 are related by the formula Symmetry 04 00427 i010:

Symmetry 04 00427 i011

where Symmetry 04 00427 i012 number rows and columns, respectively, Symmetry 04 00427 i013 denotes vector built of the Pauli matrices and σ0 is the 2 × 2 unit matrix. Spinor with lowered indices Symmetry 04 00427 i014 reads:

Symmetry 04 00427 i015

For details of the spinor calculus reader should consult [11,12,13].

2.1. The Dirac Equation

The Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 providing a description of elementary spin Symmetry 04 00427 i016 particles, such as electrons and quarks, consistent with both the principles of quantum mechanics and the theory of special relativity [14,15]. The Dirac Equation is [11,16,17]:

Symmetry 04 00427 i017

where m is the rest mass of the elementary particle. The γ’s are 4 × 4 anticommuting Dirac matrices: Symmetry 04 00427 i018 where I is the 4 × 4 unit matrix. In the spinor representation of the Dirac matrices we have:

Symmetry 04 00427 i019

where σj are the Pauli matrices and σ0 is again the 2 × 2 unit matrix. The wave function is a bispinor, i.e., consists of 2 two-component spinors ξ, η: Symmetry 04 00427 i020 where T denotes transposition of a matrix. Sometimes it is more convenient to use the standard representation:

Symmetry 04 00427 i021

2.2. Subsolutions of the Dirac Equation and Supersymmetry

In the m = 0 case it is possible to obtain two independent equations for spinors ξ, η by application of projection operators Symmetry 04 00427 i022 to Equation (4) since γ5 = −iγ0γ1γ2γ3 anticommutes with γμpμ:

Symmetry 04 00427 i023

In the spinor representation of the Dirac matrices [11] we have γ5 = diag (−1,−1,1, 1) and thus Symmetry 04 00427 i024, Symmetry 04 00427 i025 and separate equations for ξ, η follow:

Symmetry 04 00427 i026
Symmetry 04 00427 i027

Equations (8) and (9) are known as the Weyl equations and are used to describe massless left-handed and right-handed neutrinos. However, since the experimentally established phenomenon of neutrino oscillations requires non-zero neutrino masses, theory of massive neutrinos, which can be based on the Dirac equation, is necessary [18,19,20,21]. Alternatively, a modification of the Dirac or Weyl equation, called the Majorana equation, is thought to apply to neutrinos. According to Majorana theory, neutrino and antineutrino are identical and neutral [22].

Although the Majorana equations can be introduced without any reference to the Dirac theory, they are subsolutions of the Dirac Equation [18]. Indeed, demanding in Equation (4) that Symmetry 04 00427 i028 where C is the charge conjugation operator, Symmetry 04 00427 i029, we obtain in the spinor representation Symmetry 04 00427 i030, Symmetry 04 00427 i031 and the Dirac Equation (4) reduces to two separate Majorana equations for two-component spinors:

Symmetry 04 00427 i032
Symmetry 04 00427 i033

It follows from the condition Symmetry 04 00427 i028 that Majorana particle has zero charge built-in condition. The problem whether neutrinos are described by the Dirac equation or the Majorana equations is still open [18,19,20,21].

Let us note that the Dirac Equation (4) in the spinor representation of the γμ matrices can be also separated in form of second-order Equations:

Symmetry 04 00427 i034
Symmetry 04 00427 i035

Such equations, valid also in the interacting case, were used by Feynman and Gell-Mann to describe weak decays in terms of two-component spinors [23].

More exotic subsolutions of the Dirac equation, related to supersymmetry, are also possible. In the massless case Simulik and Krivsky demonstrated that the following substitution,

Symmetry 04 00427 i036

when introduced into the Dirac Equation (4), converts it for m = 0 and standard representation of the Dirac matrices Equation (6) into the set of Maxwell equations [7]. In the massive case the Dirac Equation (4) can be written as a set of two Equations:

Symmetry 04 00427 i037
Symmetry 04 00427 i038

with P4 = diag (1,1,1,0), P3 = diag (1,1,0,1) and spinor representation of the γμ matrices Equation (5). Equations analogous to (15,16) appear also in the Duffin–Kemmer–Petiau theory of massive bosons [9].

Let us note finally that as shown in [24] the square of the Dirac operator is indeed supersymmetric, and this can be used for a convenient description of fluctuations around a self-dual monopole. Similar behavior has also been observed in the Taub-NUT case, see [25].

2.3. The Duffin–Kemmer–Petiau Equations

The DKP equations for spin 0 and 1 are written as:

Symmetry 04 00427 i039

with 5 × 5 and 10 × 10 matrices βμ, respectively, which fulfill the following commutation relations [26,27,28,29]:

Symmetry 04 00427 i040

In the case of 5 × 5 (spin 0) representation of βμ matrices Equation (17) is equivalent to the following set of equations:

Symmetry 04 00427 i041

if we define Ψ in Equation (17) as:

Symmetry 04 00427 i042

Let us note that Equation (19) can be obtained by factorizing second-order derivatives in the Klein–Gordon equation Symmetry 04 00427 i043.

In the case of 10 × 10 (spin 1) representation of matrices βμ Equation (17) reduces to:

Symmetry 04 00427 i044

with Ψ in Equation (17) defined as Symmetry 04 00427 i045:

Symmetry 04 00427 i046

Where Ψλ are real and Ψμν are purely imaginary (in alternative formulation we have Symmetry 04 00427 i047, Symmetry 04 00427 i048, where Ψλ, Ψμν are real). Because of antisymmetry of Ψμν we have pνΨν = 0what implies spin 1 condition. The set of Equation (21) was first written by Proca [30,31] and in a different context by Lanczos, see [32] and references therein. More on the history of the formalism of Duffin, Kemmer and Petiau can be found in [33].

3. Splitting the Dirac Equation in Longitudinal External Fields

The interaction is introduced into the Dirac Equation (4) via minimal coupling Equation (1). We consider a special class of four-potentials obeying the condition:

Symmetry 04 00427 i049

where Symmetry 04 00427 i050 is a commutator. The condition Equation (23) is fulfilled in the Abelian case for

Symmetry 04 00427 i051

This is the case of longitudinal potentials for which several exact solutions of the Dirac equation were found [10].

The Dirac Equation (4) can be written in spinor notation as [11]:

Symmetry 04 00427 i052

where Symmetry 04 00427 i053, Symmetry 04 00427 i054 are given by Equations (2) and (3) (note that Symmetry 04 00427 i055, Symmetry 04 00427 i056, Symmetry 04 00427 i057, Symmetry 04 00427 i058). Obviously, due to relations between components of Symmetry 04 00427 i053 and Symmetry 04 00427 i059 the Equation (25) can be rewritten in terms of components of Symmetry 04 00427 i053 only. Equation (25) corresponds to Equation (4) in the spinor representation of γ matrices and Symmetry 04 00427 i060. We assume here that we deal with four-potentials fulfilling condition Equation (23).

In this Section we shall investigate a possibility of finding subsolutions of the Dirac equation in longitudinal external field, analogous to subsolutions found for the free Dirac equation in ([9]). For m ≠ 0 we can define new quantities:

Symmetry 04 00427 i061
Symmetry 04 00427 i062

where we have:

Symmetry 04 00427 i063
Symmetry 04 00427 i064

In spinor notation Symmetry 04 00427 i065, Symmetry 04 00427 i066, Symmetry 04 00427 i067, Symmetry 04 00427 i068.

The Dirac Equation (25) can be now written with help of Equations (26) and (27) as (we are now using components Symmetry 04 00427 i053 throughout):

Symmetry 04 00427 i069

It follows from Equations (26) and (27) and Equation (23) that the following identities hold:

Symmetry 04 00427 i070
Symmetry 04 00427 i071

Taking into account the identities Equations (31) and (32) we can decouple Equation (30) and write it as a system of the following two Equations:

Symmetry 04 00427 i072
Symmetry 04 00427 i073

System of Equations (33) and (34) is equivalent to the Dirac Equation (25) if the definitions Equations (28) and (29) are invoked.

Due to the identities, Equations (31–34) can be cast into form:

Symmetry 04 00427 i074
Symmetry 04 00427 i075

Let us consider Equation (35). It can be written as:

Symmetry 04 00427 i076

where P4 is the projection operator, P4 = diag (1,1,1,0) in the spinor representation of the Dirac matrices and Symmetry 04 00427 i077. There are also other projection operators which lead to analogous three component equations, P1= diag (0,1,1,1), P2= diag (1,0,1,1), P3= diag (1,1,0,1). Acting from the left on Equation (37) with P4 and (1−P4)we obtain two Equations:

Symmetry 04 00427 i078
Symmetry 04 00427 i079

In the spinor representation of γμ matrices, Equation (38) is equivalent to Equation (33) while Equation (39) is equivalent to the identity Equation (31), respectively. The operator P4 can be written as Symmetry 04 00427 i080 where γ5 = iγ0γ1γ2γ3 (similar formulae can be given for other projection operators P1, P2, P3, see [13] where another convention for γμ matrices was however used). It thus follows that Equation (37) is given representation independent form and is Lorentz covariant (in [9] subsolutions of form Equation (37) were obtained for the free Dirac equation).

Let us note finally that Equation (36) can be alternatively written as

Symmetry 04 00427 i081

where Symmetry 04 00427 i082, Symmetry 04 00427 i083, note that Symmetry 04 00427 i084.

4. Separation of Variables in Subequations

It is possible to separate variables in Equations (33) and (34) following procedures described in [10]. Substituting Symmetry 04 00427 i085 and Symmetry 04 00427 i086 from the first two equations into the third in Equation (33) we get:

Symmetry 04 00427 i087

Taking into account definition of Symmetry 04 00427 i053 and property Equation (24) we obtain:

Symmetry 04 00427 i088

where Symmetry 04 00427 i089, Symmetry 04 00427 i090.

To achieve separation of variables we put:

Symmetry 04 00427 i091
Symmetry 04 00427 i092
Symmetry 04 00427 i093

We now substitute Equation (43) into Equation (42) to get:

Symmetry 04 00427 i094
Symmetry 04 00427 i095

where Symmetry 04 00427 i096 is the separation constant and we note that Equations (46a) and (46b) are analogous to Equations (12.15) and (12.19) in [10].

Combining now Equation (46a) with the first of Equation (33) and rescaling, Symmetry 04 00427 i097, we obtain 2D Dirac Equation:

Symmetry 04 00427 i098
Symmetry 04 00427 i099

with effective mass Symmetry 04 00427 i100.

On the other hand, combining Equation (46b) with the second of Equation (33) we get equations:

Symmetry 04 00427 i101
Symmetry 04 00427 i102

which can be written as the Pauli Equation:

Symmetry 04 00427 i103

The same procedure applied to Equation (34) yields the equation for Symmetry 04 00427 i104:

Symmetry 04 00427 i105

Carrying out separation of variables we get 2D Dirac Equation:

Symmetry 04 00427 i106
Symmetry 04 00427 i107

with effective mass Symmetry 04 00427 i108 and Symmetry 04 00427 i109 and equation:

Symmetry 04 00427 i110
Symmetry 04 00427 i111

which is written as the Pauli Equation

Symmetry 04 00427 i112

where the following definitions were used:

Symmetry 04 00427 i113
Symmetry 04 00427 i114
Symmetry 04 00427 i115

5. Splitting the Spin 0 Duffin–Kemmer–Petiau Equations in Crossed Fields

We introduce interaction into DKP Equation (19) via minimal coupling Equation (1). We consider four-potentials obeying the condition:

Symmetry 04 00427 i116

The condition Equation (57) means that Symmetry 04 00427 i117 and is fulfilled by crossed fields [10]:

Symmetry 04 00427 i118

with Symmetry 04 00427 i119.

Equation (19) in the interacting case can be written within spinor formalism (cf. Equations (2) and (3)) as:

Symmetry 04 00427 i120

Indeed, it follows from Equation (59) that Symmetry 04 00427 i121 and Symmetry 04 00427 i122. We have Symmetry 04 00427 i123 and the Klein–Gordon Equation Symmetry 04 00427 i124 follows.

Let us note now that for fields obeying Equation (57), the following spinor identities hold:

Symmetry 04 00427 i125

Due to identities Equation (60) we can split the last of Equation (59) and write Equation (59) as a set of two equations:

Symmetry 04 00427 i126
Symmetry 04 00427 i127

each of which describes particle with mass m (we check this by substituting e.g. Symmetry 04 00427 i128, Symmetry 04 00427 i129 or Symmetry 04 00427 i130, Symmetry 04 00427 i131 into the third equations). Equation (59) and the set of two Equations (61) and (62) are equivalent. We described Equations (61) and (62) in non-interacting case in [34,35]. Equations (61) and (62) and Equations (33) and (34) have the same structure (recall that Symmetry 04 00427 i055, Symmetry 04 00427 i056, Symmetry 04 00427 i057, Symmetry 04 00427 i058). However these equations cannot be written in the form of the Dirac Equations (35) and (36) because identities analogous to Equations (31) and (32) do not hold, i.e., Symmetry 04 00427 i132, Symmetry 04 00427 i133.

Substituting first two equations into the third one in Equation (61), we get the Klein–Gordon equation Symmetry 04 00427 i124, which can be solved via separation of variables for the case of crossed fields, see Chapter 3 in [10] (the same can be done in Equation (62)).

6. A Supersymmetric Link between Dirac and DKP Theories

We have shown that subsolutions of the Dirac equation as well as of the DKP equations for spin 0 obey analogous pairs of 3 × 3 Equations (33–62), respectively.

More exactly, Equations (33) and (34) can be written as:

Symmetry 04 00427 i134
Symmetry 04 00427 i135

with

Symmetry 04 00427 i136
Symmetry 04 00427 i137

and πμ = pμ − qAμ, Aμ obeying condition of longitudinality Equation (23).

On the other hand, Equations (61) and (62) can be written in analogous form:

Symmetry 04 00427 i138
Symmetry 04 00427 i139

with the same matrices ρμ, Symmetry 04 00427 i140, cf. Equations (65) and (66), and πμ = pμ − qAμ, Aμ obeying condition Equation (57)—fulfilled by crossed fields.

It thus follows that the 3 × 3 free equations described in [34,35]:

Symmetry 04 00427 i141
Symmetry 04 00427 i142

provide a link between solutions of the Dirac and DKP equations. Namely, Equations (69) and (70) in the interacting case, pμπμ = pμ − qAμ, lead to subsolutions of the Dirac Equations (63) and (64) in the case of longitudinal fields Equation (23), while for crossed fields Equation (57) yield DKP subsolutions Equations (67) and (68).

7. Discussion

We have shown that the Dirac equation in longitudinal external fields is equivalent to a pair of 3 × 3 subequations (33) and (34) which can be further written as Dirac equations with built-in projection operators, Equations (37) and (40). Furthermore, we have demonstrated that the Duffin–Kemmer–Petiau equations for spin 0 in crossed fields can be split into two 3 × 3 subequations (61) and (62) (subequations of the DKP equations for spin 1 were discussed in [36]). It was also shown that all the subequations can be obtained via minimal coupling from the same 3 × 3 subequations (69) and (70), which are thus a supersymmetric link between fermionic and bosonic degrees of freedom. It can be expected that for a combination of crossed and longitudinal potentials these subequations should describe interaction of fermionic and bosonic degrees of freedom. We shall investigate this problem in our future work.

Finally, we shall address problem of Lorentz covariance of the subequations. Let us have a closer look at a single subequation of spin 0 DKP equation, say Equation (67). Although both equations, Equation (67) and (68), are covariant as a whole, this subequation alone is not Lorentz covariant. Moreover, it cannot be written as manifestly covariant Dirac equation, cf. the end of Section 5. There is however another possibility of introducing full covariance. Let us consider left and right eigenvectors of the operator ρμπμ:

Symmetry 04 00427 i143
Symmetry 04 00427 i144

where symbols Symmetry 04 00427 i145, Symmetry 04 00427 i146 mean action of Symmetry 04 00427 i147 to the right or to the left, respectively (left solutions are actually used in the Dirac theory, where they are denoted as Symmetry 04 00427 i148, they are however related to the right solutions by the formula Symmetry 04 00427 i149 (symbol † denotes Hermitian conjugation) [11]).

It turns out that Equation (71), with Symmetry 04 00427 i150 and Symmetry 04 00427 i151, are equivalent to Equations (61) and (62) respectively Symmetry 04 00427 i152 and involve components of the whole spinor Symmetry 04 00427 i153 since Symmetry 04 00427 i154. The same analysis applies to Equation (68), i.e., Symmetry 04 00427 i155, Symmetry 04 00427 i156 and Symmetry 04 00427 i157, Symmetry 04 00427 i158 (note that Symmetry 04 00427 i159 and Symmetry 04 00427 i160, as well as Symmetry 04 00427 i161 and Symmetry 04 00427 i162 are algebraically related).

We shall now discuss problem of Lorentz covariance of subequations of the Dirac equation, Equations (63) and (64). Let first note that Equations (69) and (70), as well as Equations (63) and (64), can be written in covariant form as the Dirac equation with one zero component as Equations (15,16,37,40), respectively. However, solutions of Equations (63) and (64) do not involve the whole spinor Symmetry 04 00427 i163. We might consider left eigensolutions of the operator Symmetry 04 00427 i164 again but this does not change the picture—Equations (63) and (64) involve components Symmetry 04 00427 i165, Symmetry 04 00427 i166, Symmetry 04 00427 i167, Symmetry 04 00427 i168 only as well as the whole spinor Symmetry 04 00427 i169. It follows that in Equations (63) and (64) we deal with Lorentz symmetry breaking—a hypothetical phenomenon considered in some extensions of the Standard Model [37,38,39].

References

  1. Jackiw, R.; Nair, V.P. Relativistic wave equation for anyons. Phys. Rev. D 1991, 43, 1933–1942. [Google Scholar] [CrossRef]
  2. Plyushchay, M. Fractional spin: Majorana-Dirac field. Phys. Lett. B 1991, 273, 250–254. [Google Scholar] [CrossRef]
  3. Horváthy, P.; Plyushchay, M.; Valenzuela, M. Bosons, fermions and anyons in the plane, and supersymmetry. Ann. Phys. 2010, 325, 1931–1975. [Google Scholar] [CrossRef]
  4. Horváthy, P.A.; Plyushchay, M.S.; Valenzuela, M. Supersymmetry between Jackiw–Nair and Dirac–Majorana anyons. Phys. Rev. D 2010, 81. [Google Scholar] [CrossRef]
  5. Horváthy, P.; Plyushchay, M.; Valenzuela, M. Supersymmetry of the planar Dirac–Deser–Jackiw–Templeton system and of its nonrelativistic limit. J. Math. Phys. 2010, 51. [Google Scholar] [CrossRef]
  6. Horváthy, P.; Plyushchay, M.; Valenzuela, M. Bosonized supersymmetry from the Majorana–Dirac–Staunton theory and massive higher-spin fields. Phys. Rev. D 2008, 77. [Google Scholar] [CrossRef]
  7. Simulik, V.; Krivsky, I. Bosonic symmetries of the massless Dirac equation. Adv. Appl. Clifford Algebr. 1998, 8, 69–82. [Google Scholar] [CrossRef]
  8. Simulik, V.; Krivsky, I. Bosonic symmetries of the Dirac equation. Phys. Lett. A 2011, 375, 2479–2483. [Google Scholar] [CrossRef]
  9. Okninski, A. Supersymmetric content of the Dirac and Duffin–Kemmer–Petiau equations. Int. J. Theor. Phys. 2011, 50, 729–736. [Google Scholar] [CrossRef]
  10. Bagrov, V.; Gitman, D. Exact Solutions of Relativistic Wave Equations; Springer: Berlin, Germany, 1990; Volume 39. [Google Scholar]
  11. Berestetskii, V.; Lifshitz, E.; Pitaevskii, V. Relativistic Quanturn Theory, Part 1; McGraw-Hill Science: New York, NY, USA, 1971. [Google Scholar]
  12. Misner, C.; Thorne, K.; Wheeler, J. Gravitation; WH Freeman Co.: New York, NY, USA, 1973. [Google Scholar]
  13. Corson, E. Theory of Tensors, Spinors, and Wave Equations; Blackie: London, UK, 1953. [Google Scholar]
  14. Dirac, P. The quantum theory of the electron. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Character 1928, 117, 610–624. [Google Scholar] [CrossRef]
  15. Dirac, P. The quantum theory of the electron. Part II. Proc. R. Soc. Lond. Ser. A 1928, 118, 351–361. [Google Scholar] [CrossRef]
  16. Bjorken, J.; Drell, S. Relativistic Quantum Mechanics; McGraw-Hill: New York, NY, USA, 1964; Volume 3. [Google Scholar]
  17. Thaller, B. The Dirac Equation; Springer-Verlag: New York, NY, USA, 1992. [Google Scholar]
  18. Zralek, M. On the possibilities of distinguishing dirac from majorana neutrinos. Acta Phys. Pol. B 1997, 28, 2225–2257. [Google Scholar]
  19. Perkins, D. Introduction to High Energy Physics; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
  20. Fukugita, M.; Yanagida, T. Physics of Neutrinos and Applications to Astrophysics; Springer Verlag: Berlin, Germany, 2003. [Google Scholar]
  21. Szafron, R.; Zralek, M. Can we distinguish dirac and majorana neutrinos produced in muon decay? Acta Phys. Pol. B 2009, 40, 3041–3047. [Google Scholar]
  22. Majorana, E. Teoria simmetrica dell’elettrone e del positrone. Il Nuovo Cimento (1924–1942) 1937, 14, 171–184. [Google Scholar] [CrossRef]
  23. Feynman, R.; Gell-Mann, M. Theory of the Fermi interaction. Phys. Rev. 1958, 109, 193–198. [Google Scholar] [CrossRef]
  24. Horvathy, P.; Feher, L.; O’Raifeartaigh, L. Applications of chiral supersymmetry for spin fields in selfdual backgrounds. Int. J. Mod. Phys. 1989, A4, 5277–5285. [Google Scholar]
  25. Comtet, A.; Horvathy, P. The Dirac equation in Taub-NUT space. Phys. Lett. B 1995, 349, 49–56. [Google Scholar] [CrossRef]
  26. Duffin, R. On the characteristic matrices of covariant systems. Phys. Rev. 1938, 54. [Google Scholar] [CrossRef]
  27. Kemmer, N. The particle aspect of meson theory. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 1939, 173, 91–116. [Google Scholar] [CrossRef]
  28. Kemmer, N. The Algebra of Meson Matrices. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1943; 39, pp. 189–196. [Google Scholar]
  29. Petiau, G. Contribution a la théorie des équations d’ondes corpusculaires. Acad. R. Belg. Classe Sci. Mem. 1936, 16, 1–136. [Google Scholar]
  30. Proca, A. Wave theory of positive and negative electrons. J. Phys. Radium 1936, 7, 347–353. [Google Scholar] [CrossRef]
  31. Proca, A. Sur les equations fondamentales des particules elementaires. Comp. Ren. Acad. Sci. Paris 1936, 202, 1366–1368. [Google Scholar]
  32. Lanczos, C. Die erhaltungssätze in der feldmäßigen darstellung der diracschen theorie. Zeitschrift Phys. Hadron. Nuclei 1929, 57, 484–493. [Google Scholar]
  33. Bogush, A.; Kisel, V.; Tokarevskaya, N.; Red’kov, V. Duffin–Kemmer–Petiau formalism reexamined: Non-relativistic approximation for spin 0 and spin 1 particles in a Riemannian space-time. Ann. Fond. Louis Broglie 2007, 32, 355–381. [Google Scholar]
  34. Okninski, A. Effective quark equations. Acta Phys. Pol. B 1981, 12, 87–94. [Google Scholar]
  35. Okninski, A. Dynamic theory of quark and meson fields. Phys. Rev. D 1982, 25, 3402–3407. [Google Scholar] [CrossRef]
  36. Okninski, A. Splitting the Kemmer–Duffin–Petiau equations. Available online: http://arxiv.org/pdf/math-ph/0309013v1.pdf (accessed on 30 July 2012).
  37. Colladay, D.; Kostelecký, V.A. CPT violation and the standard model. Phys. Rev. D 1997, 55, 6760–6774. [Google Scholar]
  38. Visser, M. Lorentz symmetry breaking as a quantum field theory regulator. Phys. Rev. D 2009, 80. [Google Scholar] [CrossRef]
  39. Kostelecký, V.A.; Russell, N. Data tables for Lorentz and CPT violation. Rev. Mod. Phys. 2011, 83, 11–31. [Google Scholar] [CrossRef]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert