## Abstract

Majorana's arbitrary spin theory is considered in a hyperbolic complex representation. The underlying differential equation is embedded into the gauge field theories of Sachs and Carmeli. In particular, the approach of Sachs can serve as a unified theory of general relativity and electroweak interactions. The method is extended to conformal space with the intention to introduce the strong interaction. It is then possible to use the wave equation, operating on representation functions of the conformal group, to describe the dynamics of matter fields. The resulting gauge groups resemble closely the gauge symmetries of Glashow–Salam–Weinberg and the standard model.

## 1. Introduction

Relativistic quantum physics can be founded on the Dirac equation or a Klein–Gordon equation with spin, as both equations can be formulated in a way that they include the same mathematical content. On the one hand, one has to deal with the simple structure of a first-order differential equation. The price to pay is matrix structures which are more complex than necessary. On the other hand, there is the higher complexity of a second-order differential equation, which is accompanied by a simplification in the applied matrix algebra. Feynman had a preference for the Klein–Gordon fermion equation as the path integral formalism can be handled more easily in this representation [1]. The Klein–Gordon fermion equation, which is equivalent to the squared Dirac equation, has been initially derived by Kramers [2,3] and Lanczos [4]. One could denote Klein–Gordon equations with spin, which are congruent to this representation, as Dirac–Klein–Gordon equations. The hyperbolic complex second-order differential equation introduced in [5] belongs to this category.

The Dirac equation and its quadratic counterpart are used for spin 1/2 particles. Furthermore, it is possible to formulate wave equations of arbitrary spin. This has been done by Majorana [6], Dirac [7], Fierz & Pauli [8,9], Gelfand & Yaglom [10], Nambu [11,12], Fronsdal [13], Gitman & Shelepin [14] and others. Consider in this context also the review articles of Fradkin [15] and Esposito [16]. An important feature of the work of Majorana, Gelfand and Yaglom is the resulting mass spectrum in dependence of the spin of the considered quantum state.

Recently, this mass spectrum could be reproduced with an ordinary d'Alembert operator acting on representation functions of the Poincaré group [17]. The result is a generalized Klein–Gordon equation, where a parameter that represents spin appears beside the mass parameter as part of the fundamental single particle field equation. One could denote this equation as Majorana–Klein–Gordon equation in order to distinguish from the Dirac–Klein–Gordon equation mentioned above. In the Majorana–Klein–Gordon equation, aspects of the underlying relativistic Clifford algebra are moved to the solution space, whereas the differential operator itself is free of spin representations.

The spectrum of the d'Alembert operator, when applied to the representation functions of the Poincaré group, gives rise to three parameters. One of them can be considered as a constant coefficient, which induces a relation between mass and spin in analogy to investigations of Gelfand & Yaglom [10], Barut *et al.* [18] and Varlamov [19]. In fact, one has the choice of which parameters are considered to be physical as the interpretation of the constant is still unclear. Based on this choice, one can reinterpret momentum and mass of the Poincaré group representation functions as given in [17] and see them only as coefficients in a series expansion. What is considered as energy, momentum and mass has to be derived at a later stage, for example in correspondence with the Fourier transformation of the Green function of the considered system. More details can be found in §4.

Sachs developed a unified theory of gravitation and electroweak interactions with a Maxwell-like representation of Einstein's general relativity [20,21]. The method is based on a quaternionic paravector algebra, which is congruent to the hyperbolic complex Pauli algebra applied in [17]. Sachs states that the Majorana equation is the most general irreducible expression of relativistic quantum mechanics in special relativity. It is thus natural to introduce the Majorana–Klein–Gordon equation in the representation of [17] into the Sachs framework of quaternionic physics. The notion of a quaternion has to be understood now in a generalized sense as it covers hyperbolic complex arbitrary spin representations. Consider here §§3–6.

The conformal group as a generalization of the Poincaré group has been applied extensively to describe physical systems, see the review article of Kastrup [22]. In the 1960s, there was the hope that the new experimental observations in nuclear and particle physics could be explained with the help of conformal symmetries. One may consider for example publications of Kastrup [23], Hepner [24], Mack & Salam [25], Wess [26,27] or Flato *et al.* [28]. Attempts to describe the baryon spectrum in terms of the conformal group have been published by Nambu [11,12] and Barut & Kleinert [29]. Meson decay rates and proton–proton scattering observables have been calculated in good agreement with experiment in a conformal model by Barut and colleagues [30,31]. Furthermore, the conformal group does not seem to be of relevance only in the context of the strong interaction, but also with respect to cosmology. It has been found that the conformal invariance produces a good correlation to experimental data of the cosmological redshift, see Segal [32,33]. In addition, a conformal extension of Einstein's general relativity has been applied by Mannheim in order to explain velocity curves of rotating galaxies [34].

Thus, there are arguments to apply the conformal compactification to Minkowski space to obtain a conformal hyperbolic complex paravector algebra. It is then possible to extend the theory of Sachs to conformal space with the intention to incorporate the strong interaction. The hyperbolic complex conformal algebra is discussed starting from §7. From the perspective of general relativity, the extension is related to the twistor approach of Penrose [35] and the supergravity theories of Freedman and colleagues [36,37] and Volkov & Soroka [38,39]. Consider in this context also the twistor string theory of Witten [40]. The relation to these approaches points also into the direction of the anti-de Sitter/conformal field theory correspondence of Maldacena [41], Gubser *et al.* [42] and Witten [43].

## 2. The hyperbolic Pauli algebra in Minkowski space

The paravector model of Minkowski space based on complex quaternions has been applied to relativistic physics for example by Sachs [20], Baylis [44] and Carmeli [45]. See Gsponer & Hurni [46] for an extensive list of references in this research area. The algebra of complexified quaternions corresponds to the real Clifford algebra *e*_{μ}=(1,*e*_{i}), which include the three basis elements *e*_{i} of

The product of two basis elements has been separated in equation (2.1) into symmetric and antisymmetric contributions. The symmetric part defines the metric tensor
*n*×*n* matrix, where *n* corresponds to the dimension of the spin representation [17].

The antisymmetric contributions in equation (2.1) define the spin tensor, which is given by the following expression
*n*×*n* matrices. The tensor can be used to define the relativistic spin angular momentum operator

The pseudoscalar is interpreted in geometric algebra as a directed volume element or volume form, see Doran & Lasenby [47]. The pseudoscalar of the Clifford algebra

The hyperbolic unit in the pseudoscalar can be traced back to the hyperbolic complex representation of

Equation (2.1) can be considered from another perspective if one takes into account that a Clifford algebra encodes the group manifold of the orthogonal transformations in the considered space. The basis elements of the Clifford algebra form a non-coordinate basis and equation (2.1) may be reformulated as
*σ*_{μν}.

## 3. Matter fields

Matter fields can be expanded in terms of plane wave representation functions of the Poincaré group. In the terminology of [17], these representations were called quaternion waves. They are labelled by
*n* refers to the number of polarization states, and *υ* is a four vector which has been identified with the momentum in [17], but following the discussion in §1, the vector *υ* is interpreted now as an expansion coefficient. The invariant obtained by the squared vector coefficient *υ*^{2} was omitted in equation (3.1). The representation functions are in fact matrix valued. One may see them as a whole, introduce additional indices referring to the matrix elements or append a spinor for comparison with standard spinor physics. In the latter case, one works with the vector bundle associated with the Poincaré group principle bundle. The corresponding group transformations are then operating in spaces

The notation *τ* is referring to quantum numbers of the charge, parity and time (CPT) transformations. The transformations form the cyclic group *τ*≡{+++} may be chosen as
*τ* can be obtained by changing the sign of the exponent, the position of *x* and *υ*, and the position of the bar symbol.

For the following investigations, it is useful to introduce an alternative representation of the wave function, which includes a spin vector coefficient
*υ* of the representation functions
*υ*. For more details on biparavectors, see Baylis [44]. Equation (3.5) displays furthermore how the spin vector coefficients are related to the relativistic spin angular momentum operator.

The momentum operator is introduced as the derivative with respect to the space–time coordinates multiplied by the pseudoscalar of the algebra

The momentum operator as defined above can be used to perform a translation of the plane wave

Sachs developed a theory of elementary matter based on a complex quaternion algebra, which is congruent to the algebra introduced in §2. With the principle of least action, Sachs derives Maxwell-like field equations of general relativity from a quaternionic Einstein–Hilbert Lagrangian [54]. The basis elements of the quaternion frame of Sachs are affected by coordinate transformations in the following way

Matter fields are identified with sections of the fibre bundle that is associated with the quaternion frame bundle of Sachs. The sections can be expanded in terms of the plane wave representation functions of the Poincaré group. The transition functions are given again by equation (3.10) and they are applied to the sections in the sense of equation (3.8) by multiplication from the left.

## 4. Spin spectrum of the d'Alembert operator

The Klein–Gordon equation encodes the free propagation of a given density distribution as it corresponds to the inverse of the propagator or Green function. The Klein–Gordon equation is formed by a Laplacian acting on a group manifold minus the eigenvalue of the Laplacian with respect to the considered group representation functions. Note that the notion of a Laplacian is applied here in a generalized sense. Thus, for the Laplacian operating on the Poincaré group *E*_{4} and referring to the propagation by virtue of a four-dimensional translation *T*_{4}, one may write

The notation for the Laplacian has been extended to distinguish from a potentially possible propagation related to the second Casimir operator of the Poincaré group, the Pauli–Lubanski vector
*w*, which is defined with the help of the orbital angular momentum operators as

The Laplacian defined by equation (4.1) is equivalent to the ordinary d'Alembert operator, except for a different sign. With the help of equation (3.7), the Laplacian can be expressed in terms of its eigenvalues with respect to the plane wave representation functions of the Poincaré group
*s* with the formula *n*=2*s*+1. Thus, one could use in equation (3.1) also the spin to characterize the representation functions of the Poincaré group. The result of equation (4.6) can be used to formulate the following equation for fields of arbitrary spin

A Green or propagator function can be introduced as the inverse of the Majorana–Klein–Gordon equation with respect to the four-dimensional delta function. The Fourier transformation of the propagator results in the following expression
*E*=*p*_{0}. Thus, the energy spectrum can be determined from the poles of the Green function as
*p*. The eigenvalue of the Laplacian has been parametrized in terms of the physical rest mass
*υ* as they are in fact derived from the Poincaré group representation functions. The physical momentum can then be reintroduced in correspondence with equation (4.10)
*n*=2 applied to electrons and protons, the scattering amplitude derived in [17] has to be scaled down by a factor of 2^{4}. After this transformation, the Klein–Gordon scattering amplitude matches exactly with the corresponding result of the Dirac theory.

## 5. General relativity

Sachs has investigated a spinor representation of general relativity based on complex quaternions [20]. Starting point is the Einstein–Hilbert Lagrangian with a quaternionic curvature scalar
*e*_{α}(*x*) has been suppressed for the sake of simplicity. It should be noted that this expression has been introduced also by Carmeli, who uses the term spinor affine connection [45]. The gauge transformation of the potential is defined by Sachs as

The spin affine connection can be inserted as the gauge potential into the Laplacian of equation (4.1) to introduce interactions into the Majorana–Klein–Gordon equation

Sachs and Carmeli introduce a field strength, which is defined with respect to the gauge potential as
*Γ*^{a}_{bμ} to distinguish the indices referring to the basis elements from the index, which refers to the covariant derivative. This distinction will become relevant if the gauge potential will be introduced into the Laplacian of equation (9.1), where the basis elements of the conformal solution space refer to a different dimensionality compared with the applied differential operator.

Sachs and Carmeli are able to bring the contributions resulting from the variation of the Einstein–Hilbert action into a Maxwell-like gauge field representation. Further, Maxwell-like gauge representations of general relativity have been derived by Rodrigues Jr & Capelas de Oliveira [57] and Mielke [58], see also Carmeli [59]. To have an idea of this correspondence, one may insert the gauge potential of equation (5.3) into the field strength tensor of equation (5.7). In relationship with gauge theories of general relativity, one may consult also the collection of research articles edited by Blagojevic & Hehl [60] and compare with the representations mentioned above. For theories with spin and torsion, one may compare especially with Kibble [61], Sciama [62] and Hehl *et al.* [63].

## 6. Gravitation and electroweak interactions

There are a number of proposals in the context of general relativity for unified theories of gravitational and electromagnetic forces. One may have a look at publications of Ferraris & Kijowski [64,65], Hammond [66], Evans [67], Poplawski [68,69] or Giglio & Rodrigues Jr [70]. The gauge theory of Sachs can be compared with these approaches based on the following discussion.

As already mentioned, the gauge transformation of the matter fields is implemented with the transition functions introduced in equations (3.10) and (3.11)
*Ω*^{μν} is kept constant, and the basis vectors of the Clifford algebra are coordinate-dependent *Ω*^{μν}(*x*), which gives rise to the electromagnetic part of the unified theory.

The interpretation of the spin contributions as electromagnetism has been discarded by Rodrigues Jr and Capelas de Oliveira, because of Sachs' wrong interpretation of symmetric and antisymmetric tensor contributions within the quaternionic representation of general relativity [72]. Spinor representations of general relativity are considered to be pure gravitational also by other authors. Carmeli interprets his representation in this sense and adds additional internal symmetries to include electromagnetic or Yang–Mills fields [45]. However, the gauge transformation of equation (6.1) has not been considered in these two references. Blagojevic & Hehl [60] have made a general statement on fallacies about torsion with respect to unified theories of gravitation and electromagnetism. Sachs does not identify electromagnetism with torsion, but with the matrix structures inside metric and spin tensor, see again §2 for more details on these matrix structures.

Electromagnetism in its common understanding results from a one parameter gauge group. It should be noted however that Sachs considers in fact an electroweak theory and uses the notion of electromagnetism in a unified sense. The method is applied for example to the weak decay of neutrons [73]. The electroweak interactions are described within the Glashow–Salam–Weinberg theory as a four parameter gauge group. The corresponding gauge symmetry appears also within the algebra of Sachs as the spatial basis elements *e*_{i} satisfy the *e*_{0}=1 gives rise to a

The product of basis elements *Pin*(1,3). Thus, the electroweak interaction can be understood in this representation as being related to the following group structure

Note that there is an alternative identification of gravitational and non-gravitational contributions in the gauge group. Gravitation could appear already in flat space–time as a hyperbolic complex Maxwell theory [76]. Thus, gravitation would be a substructure in the *G*_{EW} in equation (6.3) will not be adjusted with respect to this interpretation.

## 7. Conformal compactification of Minkowski space

The methodology discussed in the previous sections is situated in the non-compact Minkowski space. One possibility to enclose the unlimited space–time geometry is to add infinity by virtue of the conformal compactification, see Penrose & Rindler [77]. In the context of Clifford algebras, this procedure is described for example by Porteous [78,79]. The Minkowski space is extended to six dimensions, where the first four dimensions are identical *u*^{μ}=*x*^{μ}. Furthermore, two additional coordinates are introduced
*u*^{4} is assigned to the additional negative sign coordinate with respect to this metric.

Conformal structures and twistors in the paravector model of space–time have been investigated before by Elstrodt *et al.* [80], Maks [81] and da Rocha & Vaz [82]. It is thus natural to reintroduce this formalism under consideration of the hyperbolic complex number system. The basis elements for a paravector in conformal space *e*_{μ}=(1,*e*_{i}) are then defined in terms of the Dirac algebra, which corresponds to the Clifford algebra *u*=*u*^{μ}*e*_{μ}.

Most of the formulae in the first sections remain valid owing to this formal analogy. They can be also applied in conformal space under consideration of the corresponding dimension, metric and basis elements. The antisymmetric contributions *σ*_{μν} are defined again by equation (2.3), but now the spin tensor has a 6×6 matrix structure, and each of the matrix elements is a 4×4 matrix in its fundamental representation. The generators of the spin angular momentum are given again by equation (2.4), but now calculated with the basis elements of the conformal algebra

The pseudoscalar of the conformal algebra *ı*_{E} and the pseudoscalar of conformal space by *ı*_{P}
_{E} or ı_{P} pseudoscalars is leading to a positive sign and thus repulsion.

## 8. Conformal plane waves

Plane wave representation functions of the conformal group can be introduced in analogy to the representation functions of the Poincaré group. The conformal group space is strictly speaking extended by translations to provide a formal analogy to the Poincaré group wave functions. Nevertheless, the notion of conformal group will be used for the sake of simplicity in this context. The wave functions now have the form
*u*=*u*^{μ}*e*_{μ}, expressed with the coordinates and basis elements introduced in §7, can also be written as
*u*≡*u*(*x*) is a matrix valued function of the Minkowski paravector *x*. The non-diagonal elements are abbreviated in terms of the function
*υ*, which labels the Poincaré group representation functions, can be represented in analogy to equation (8.2)

The conformal vector given by equation (8.2) has been represented in the context of Möbius transformations in a similar form by Vahlen [85], Ahlfors [86], Fillmore & Springer [87] and Cnops [88,89]. In the terminology of Kisil [90], which is thought to be generalized according to Minkowski space, the conformal vector is classified as a *h*-zero radius cycle within the geometries of the Erlangen program of Klein. The *h* stands for a hyperbola. The notion of a cycle combines the geometrical objects of circles, parabolas and hyperbolas as conformal invariant conic sections. These three cases are distinguished by their corresponding hypercomplex units, which square to −1, 0 and 1. The terminology of a cycle traces here back to Yaglom [91]. It has been extended and adjusted by Kisil [90]. The zero radius hyperbola is interpreted geometrically as a light cone with centre at *x*, see for example Kisil in [92]. Thus, the mathematical structures resulting from the conformal compactification are elements of a cycle space. The cycles can be transformed by Möbius transformations and they are displayed in a point space, which corresponds to the four-dimensional Minkowski space.

## 9. The wave equation

It is now possible to calculate the spectrum of the Laplacian with respect to the wave functions introduced in §8
*C*_{4} has been introduced to indicate that the Laplacian is operating on the representation functions of the conformal group. The argument in the exponential of the group representation functions is determined by cycles, as discussed in §8. The cycles are displayed in the four-dimensional Minkowski point space. Consequently, the momentum operator remains a four-dimensional vector derivative acting in the Minkowski point space. The Green function, which is derived from the Laplacian, refers to the propagation of a density distribution by virtue of a four-dimensional translation. This is indicated by the notation *T*_{4}. Compared with equation (4.1), the momentum operator is defined with the volume element of the conformal space as introduced in equation (7.5).

The Laplacian is acting on the conformal plane waves of equation (8.1). As an alternative to the calculation which leads to equation (4.6), one can operate directly with the four-dimensional Minkowski space derivative on the conformal plane waves, in detail

The conformal plane waves thus satisfy the wave equation for all higher spin representations and masses, as they appear within the Poincaré subgroup. To see the above solutions in a wider context, one may compare here with the conformal zero rest mass fields of arbitrary spin, which have been investigated by Penrose [93,94] and Penrose & MacCallum [95]. With respect to the differential operator, it should be noted that the idea to trace quantum physics back to the wave equation has been suggested before by Sallhofer [96] in the context of the Schrödinger theory.

In §10, it will be argued that the extension to conformal symmetries induce the strong interaction. The electroweak interactions have been positioned in §6 within a reduced solution space, where the corresponding differential equation is invariant only with respect to the Poincaré transformations. Spontaneous and anomalous symmetry breaking may reduce the symmetry of the overall mathematical structure and makes mass and spin visible. Equation (9.3) is free of these parameters and thus provides a more general view. One may therefore select the wave equation for the development of a relativistic field theory. The wave equation, as the inverse of the propagator, can be understood as the mathematical representation of Newton's first law of motion applied to fields. The propagation rule of Newton is the conceptual source of the method.

The next step is to impose the principle of gauge invariance. This results in the introduction of a gauge potential, which modifies the wave equation according to equation (5.5)
*e*_{μ} of the conformal solution space can be used in the definitions of §5. The gauge potential implements the invariance of the solutions with respect to gauge transformations, which are induced by the global symmetries of the underlying differential equation. Thus, the principal bundle is naturally defined. It couples the differential equation, the propagation rule for densities defined in the base manifold of the bundle, with the bundle structure group that refers to the symmetries of the differential equation.

## 10. Strong interactions and skyrmions

Since the 1960s, various models have been proposed that consider conformal symmetries to understand the structure of the strong interaction, see the discussion in §1. In this sense, one may extend the gauge theory of Sachs to conformal space and search for indications which can be interpreted as an extension of the theory of electroweak interactions to a theory which also includes the strong interaction. The gauge transformation in the conformal solution space is formally defined as in equation (6.1)
*Ω* is given formally by equation (3.11), but now the basis elements of the conformal Clifford algebra *U*(4)×*U*(4) corresponds to an extension of the Pati–Salam model [97]. The Pati–Salam model is a so-called preon model, which postulates the existence of substructures of quarks. The *U*(4)×*U*(4) symmetry appears explicitly in the unified gauge theory of conformal gravity, Maxwell and Yang-Mills fields of Castro [98,99]. It forms furthermore a slight extension of the model of Fayyazuddin [100]. The *U*(4) symmetry is considered in the gauge invariant spinor theory of Dürr [101].

Alternatively, one can assign linear combinations of the 32 elements of the Clifford algebra

Matrix algebras of multiple *SU*(2) and *U*(1) groups can be related to the 32-dimensional conformal Clifford algebra

One may think also of an extension of the *SU*(2)×*SU*(2) chiral Skyrme model [108], whose solitons can be interpreted as the baryons of QCD, see Witten [109,110] and Adkins *et al.* [111]. For example, Pomarol & Wulzer [112] extended to a *U*(2)×*U*(2) gauge symmetry to describe baryons as Skyrme-like solitons. Ma *et al.* [113] investigated a *U*(2) extension of the original Skyrme model, where the extension is understood as a hidden local symmetry. More background on skyrmions and solitons can be found in Alkofer & Reinhardt [114], Manton & Sutcliffe [115], Brown & Rho [116], Weigel [117] and Dunajski [118]. The investigations which focus on effective meson models are founded on the large number of colours limit of 't Hooft [119]. One may ask now whether the description can be extended beyond pure strong interactions in accordance with the above gauge groups. This question is of importance if also leptons are interpreted as solitons, see for example Weiner [120]. In this context, one may consider the geometric models of matter of Atiyah *et al.* [121], who describe electrons, protons, neutrinos and neutrons with a method that was inspired by Skyrme's baryon theory.

The correspondence between the gauge groups mentioned in this section has to be understood in the following sense. The generators of the subgroups can be represented in terms of generators of the higher dimensional groups. For example, there are three possibilities to express the generators of *U*(2)=*SU*(2)×*U*(1) in terms of the *SU*(3) Gell-Mann matrices. One of them is up to factors given by
*U*(4) generators. With the help of these representations, the full group space can be spanned in accordance with equation (10.2) starting from equations (10.3), (10.4) or (10.5).

The discussion indicates that the gauge field structures resulting from the conformal compactification have the potential to describe the strong interaction. The partition of the Pin group into subgroups is not unique. Accordingly, there are different theoretical models that explain manifestations of the strong interaction depending on the considered energy region.

## 11. Summary

The Majorana–Klein–Gordon equation is a hyperbolic complex second-order differential equation describing the dynamics of matter fields with arbitrary spin. The equation is embedded into the gauge field theories of Sachs and Carmeli, where Sachs considers a unified representation of general relativity and electroweak interactions.

The method can be generalized to conformal space with the intention to incorporate the strong interaction. The conformal compactification of a Minkowski vector can be interpreted as a light cone, which is still situated in Minkowski space. It is then possible to trace relativistic physics back to the wave equation acting on representation functions of the conformal group.

- Received January 10, 2014.
- Accepted May 21, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.