Noncommutative sedeons and their application in field theory

We present sixteen-component values"sedeons", generating associative noncommutative space-time algebra. The generalized second-order and first-order equations of relativistic quantum mechanics based on sedeonic wave function and sedeonic space-time operators are proposed. We also discuss the description of fields with massive quantum on the basis of second-order and first-order equations for sedeonic potentials.


Introduction
The application of multicomponent hypercomplex numbers and multivectors in classical and quantum physics has a long history. In particular, the simplest generalization of electrodynamics and quantum mechanics was developed on the basis of quaternions [1]- [6]. The structure of quaternions with four components (scalar and vector) corresponds to the relativistic four-vector approach that allows one to reformulate field equations in terms of quaternionic algebra. However, the essential imperfection of the quaternionic algebra is that the quaternions do not include pseudoscalar and pseudovector components. The consideration of full space symmetry with respect to spatial inversion leads to the eight-component structures enclosing scalar, pseudoscalar, vector and pseudovector. There is a lot of works on application of different eight-component values such as biquaternions and octonions in classical electrodynamics and relativistic quantum mechanics [7]- [20]. However, a consistent relativistic approach implies equally the space and time symmetries that requires the consideration of the extended sixteen-component space-time algebras.
There are a few approaches in the development of theory on the basis of sixteen-component structures. One of them is the application of hypercomplex numbers sedenions, which are obtained from octonions by Cayley-Dickson extension procedure [21]- [25]. But as in the case of octonions the essential imperfection of sedenions is their nonassociativity. Another approach is based on the application of hypercomplex multivectors generating associative space-time Clifford algebras. The basic idea of such multivectors is an introduction of additional noncommutative time unit vector, which is orthogonal to the space unit vectors [26,27]. However, the application of such multivectors in quantum mechanics is considered in general as one of abstract algebraic scheme enabling the reformulation of Klein-Gordon and Dirac equations for the multicomponent wave functions but does not touch the physical entity of these equations.
Recently we have developed an alternative approach based on our scalar-vector concept [28]- [30] realized in sixteen-component sedeons. In present paper we demonstrate the application of the sedeons to the reformulation of relativistic quantum mechanics and massive field equations.

Sedeonic space-time algebra
The sedeonic algebra encloses four groups of values, which are differed with respect to spatial and time inversion. Here indexes t and r indicate the transformations (t for time inversion and r for spatial inversion), which change the corresponding values. All introduced values can be integrated into one space-time sedeonṼ, which is defined by the following expression: Let us introduce scalar-vector basis a 0 , a 1 , a 2 , a 3 , where the value a 0 ≡ 1 is absolute scalar unit and the values a 1 , a 2 , a 3 are absolute unit vectors generating the right Cartesian basis. We introduce also four space-time scalar units e 0 , e 1 , e 2 , e 3 , where value e 0 ≡ 1 is a absolute scalar unit; e 1 ≡ e t is a time scalar unit; e 2 ≡ e r is a space scalar unit; e 3 ≡ e tr is a space-time scalar unit. Using space-time scalar units e j (j = 0, 1, 2, 3) and scalar-vector basis a k (k = 0, 1, 2, 3) we can introduce unified sedeonic components V jk in accordance with the following relations: Then the sedeon (1) can be written in the following expanded form: The sedeonic components V jk are numbers (complex in general). Further we will use symbol 1 instead of units a 0 and e 0 for simplicity. The multiplication and commutation rules for sedeonic absolute unit vectors a 1 , a 2 , a 3 and space-time units e 1 , e 2 , e 3 are presented in tables 1 and 2 respectively.
In the tables and further the value i is the imaginary unit (i 2 = −1). Note that sedeonic units e 1 , e 2 , e 3 and unit vectors a 1 , a 2 , a 3 generate the anticommutative algebras: a n a m = −a m a n , e n e m = −e m e n , for n and m = 1, 2, 3 (n = m), but e 1 , e 2 , e 3 commute with a 1 , a 2 , a 3 : a n e m = e m a n ,  Table 2: for any n and m. Thus the sedeonṼ is the complicated space-time object consisting of absolute scalar, time scalar, space scalar, space-time scalar, absolute vector, time vector, space vector and space-time vector.
Introducing the designations of scalar-vector values we can write the sedeon (3) in the compact form On the other hand, introducing the designations of space-time sedeon-scalars we can write the sedeon (3) in another form or introducing the sedeon-vector it can be represented in following compact form: Further we will indicate the sedeon-scalars and the sedeon-vectors with the bold capital letters. Let us consider the sedeonic multiplication in detail. The sedeonic product of two sedeonsÃ andB can be presented in the following form: Here we denote the sedeonic scalar multiplication of two sedeon-vectors (internal product) by symbol "·" and round brackets and sedeonic vector multiplication (external product) by symbol "×" and square brackets In (13) and (14) the multiplication of sedeonic components is performed in accordance with (8) and table 2. Note that in sedeonic algebra the expression for the vector product has some difference from analogous expression in Gibbs vector algebra. Let us consider three absolute vectors A, B and C. Then the formula for the vector triple product in sedeonic algebra has the following form: Thus, the sedeonic productF has the following components: 2 Sedeonic spatial rotation and space-time conjugation The rotation of the sedeonṼ on the angle θ around the absolute unit vector n is realized by uncompleted sedeonŨ = cos(θ/2) + i n sin(θ/2) (18) and by complex conjugated sedeonŨ * = cos(θ/2) − i n sin(θ/2), which satisfy the relationŨ * Ũ =ŨŨ * = 1.
The transformed sedeonṼ ′ is defined as the sedeonic product Thus the transformed sedeonṼ ′ can be written in the following expanded form: It is clearly seen that rotation does not transform the sedeon-scalar part, but sedeonic vector V is rotated on the angle θ around n. The operations of time conjugation ( R t ), space conjugation ( R r ) and space-time conjugation ( R tr ) are connected with transformations in e 1 , e 2 , e 3 basis and can be presented as

Sedeonic Lorentz transformations
The relativistic event four-vector can be represented in the follow sedeonic form: where c is the velocity of light, t is the absolute scalar of time and r is the absolute radius-vector. The square of this value is the Lorentz invariant The Lorentz transformation of event four-vector is realized by sedeons where tanh 2ϑ = v/c; v is velocity of motion along the absolute unit vector m. Note that The transformed event four-vectorS ′ is written as Separating the values with e 1 and e 2 we get the well-known expressions for the time and coordinates transformations [31] : where x is the coordinate along the m vector. Let us also consider the Lorentz transformation of the full sedeonṼ. The transformed sedeoñ V ′ can be written as sedeonic productṼ In expanded form:Ṽ Rewriting the expression (32) with scalar (13) and vector (14) products we get Thus, the transformed sedeon have the following components:

Sedeonic generalization of Klein-Gordon equation
The wave function of relativistic particle satisfies an equation, which is obtained from the Einstein relation between energy and momentum by means of changing classical energy E and momentum p on corresponding quantum-mechanical operators:Ê where c is the speed of light, m 0 is the mass of particle,h is the Planck constant. The absolute vector of gradient has the following form: In sedeonic algebra the Einstein relation (35) can be written as Let us consider the wave function in the form of space-time sedeoñ Then the generalized sedeonic wave equation for sedeonic wave fucntion is written in the following form In this equation the basis elements e t , e r , e tr and a 1 , a 2 , a 3 play the role of the space-time operators, which transform the sedeonic wave functionṼ by means of component permutation. In fact the equation (40) is the system of 16 scalar equations for each component of wave function. Redefining the operators we can rewrite the equation (40) in compact form: Formally, the sedeonic equation (42) can be represented in the form of the system of Maxwelllike first-order equations. Let us consider the sequential action of operators. After the action of the first operator in the left part of equation (42) we obtain Introducing the scalar and vector values the relation (45) is presented as Then the wave equation (42) can be rewritten in the following form: Applying the operator i∂ t − ∇ r − im tr to both parts of equation (47) and separating sedeonscalar and sedeon-vector parts we get the wave equations for the values E 0 and E: On the other hand, performing sedeonic multiplication in expression (47) and separating sedeonscalar and sedeon-vector parts we obtain the Maxwell-like system of first-order equations: As one can see the values E 0 and E can be interpreted as the quantum field intensities. These fields are defined on the whole space and carry information about the kinematic properties of the particle.
The equation (40) can be generalized for a particle in an external electromagnetic field. Let us consider the charged particle with electrical charge e. In this case we have to change operators in (40) by where ϕ is scalar potential and A is vector potential of electromagnetic field. Then we obtain the following wave equation This equation describes the charged particle with spin 1/2 in an external electromagnetic field [32].

Sedeonic generalization of Dirac equation
The sedeonic algebra enables the reformulation of the first-order Dirac equation [33] as a wave equation for the sedeonic wave function: In this equation the basis elements e t , e r , e tr and a 1 , a 2 , a 3 play the role of the space-time operators, which transform the sedeonic wave functionṼ by means of component permutation. In fact, equation (53) describes the special quantum field with zero field intensities E 0 and E (see expression (46)).
The equations (53) can be generalized for a particle in an external electromagnetic field. In this case we have This equation describes the particle with spin 1/2 in an external electromagnetic field [32].
6 Sedeonic second-order equation for massive field 6

.1 Homogeneous equation
The Einstein relation between energy and momentum (35) allows another field interpretation. In this case E, p and m 0 can be interpreted as energy, momentum and mass of a quantum of field. Then the equation is the wave equation for the field potentialW and relation can be considered as the dispersion relation for the free wave of massive field. Let us introduce new operators Then we can rewrite the equation (55) in compact form: Let us choose the potential in the following form: where the components a, b, c, d, A, B, C and D are the functions of spatial coordinates and time.
Introducing the scalar and vector fields strengths according to the following definitions: we get and the wave equation (58) takes the form Performing the action of operator in the left part of the equation (62), and separating the terms with different space-time properties, we obtain the system of equations for the field's strengths, similar to the system of Maxwell's equations in electrodynamics: The proposed equations for massive field possess a specific gauge invariance. It is easy to see that fields strengths (60) and equations (63) are not changed under the following substitutions for potentials: where ε a , ε b , ε c , ε d , are arbitrary scalar functions, which satisfy homogeneous Klein-Gordon equation. These gauge conditions are different from those taken in electrodynamics [34]. Multiplying each of the equations (63) to the corresponding field strength and adding these equations to each other, we obtain: Let us introduce the following notations: Then the equation (65) can be written as: This expression is an analog of the Poynting theorem for massive field. The value w plays the role of the field energy density and P is a vector of energy flux density. The minus sign in expressions (66) and (67) are chosen with respect to the attractive character of charge interaction (see further Section 6.2.).

Nonhomogeneous equation
Let us consider the sedeonic nonhomogeneous equation for massive field whereJ is the source of massive field. By analogy with electrodynamics we consider the source in the following form [29] where ρ B is a volume density of charge and j B is density of current. In this case we can describe the field by sedeonic potentialW written in the following form where b( r, t) is a scalar part and C( r, t) is a vector part of field potential. In this case we have only the following nonzero field's strengths and the equation (69) can be rewritten as Then we obtain the following equations for the field strengths: On the other hand, applying the operator (ie 1 ∂ − e 2 ∇ − ie 3 m) to the equation (73) we obtain the following wave equations for the field strengths: Assuming the charge conservation we can choose the field strength e equal to zero. This is equivalent to the following gauge condition (see (72)): ∂b + ( ∇ · C) = 0 (77) similar to the Lorentz gauge. Let us consider the simplest case of stationary field of point scalar source. In the stationary case j B = 0 and field potential can be chosen in a scalar form Then we have only two nonzero field components g = −mb, and the following field equations: Let us calculate the field produced by a scalar stationary point sourcẽ where q B is the point charge and δ( r) is delta function. Then stationary wave equation can be written in spherical coordinates as The partial solution of the equation (82), which decays at r → ∞, is Thus in this case the stationary field has scalar and vector components where r 0 is a unit radial vector. Two point charges interact due to the overlap of their fields. Taking into account that the field in this case is the sum of the two fields g = g 1 + g 2 and H = H 1 + H 2 the energy of interaction is equal (see (66)) where the integral is over all space. This expression can be derived analytically: where R is the distance between the point charges. This expression coinsides with a well-known law of interaction between two baryons, which is described by Yukawa potential [35], therefore q B can be interpreted as a baryon charge.
7 Sedeonic first-order equation for massive field

Homogeneous equation
Let us consider a special massive field that is described by sedeonic first-order equation. In sedeonic algebra the homogeneous first-order Dirac-like equation corresponding to the equation (55) is written as Choosing potential in the form (59) we find that sedeonic equation (88) is equivalent to the following system ∂a + ( ∇ · D) + mc = 0, ∂b + ( ∇ · C) + md = 0, In fact, these equations describe the special field with zero field strengths (see for comparison the expressions (60)). Let us consider the plane wave solution of equation (88) in detail. In this case the potential can be written asW where ω is a frequency and k is an absolute wave vector; the amplitude of the wave U does not depend on the coordinates and time. In this case, the dependence of frequency on the wave vector has two branches: Let us consider the amplitude of the wave function in the form of (59): where a, b, c, d, A, B, C and D are arbitrary constants. Then the solution can be written as Substituting this expression in the original equation (53) we get: For convenience we introduce the following notation: then equation (94) can be rewritten as For fixed k let us represent the vector constants in (92) in the form where the vectors A , B , C and D are parallel to the vector k while the vectors A ⊥ , B ⊥ , C ⊥ and D ⊥ are perpendicular to k. Then performing the multiplication in (96), we obtain the following system of algebraic equations: where the values A , B , C and D are the projections of the vectors A , B , C and D on the vector k. Let us solve this system of equations. From (108) and (109) we find Using (91) one can easily check that for arbitrary vector constants A ⊥ and B ⊥ equations (106) and (107) are fulfilled. As a next step from equations (98)-(101) we obtain: One can check that these solution fulfill the equations (102)-(105). Thus the sedeonŨ has the formŨ Note that this expression can be rewritten in the following form: Substituted this amplitude into (96) one can see that this equation is satisfied for any parameters a, b, c, d, A ⊥ , B ⊥ because the expression in round brackets is sedeonic zero divisor. Indeed it is simple to check that In general, the plane wave solution for the equation (88) can be written in the following sedeonic form: whereM is an arbitrary sedeon with constant components. In this case after performing multiplication in (119) we obtain that the components of the resulting sedeon are defined only by 8 independent combinations of the sedeonM components. Note that the internal structure of this wave is changed under space and time inversion. In massless case the dispersion relation is and plane wave solution can be written as Let us analyze the structure of the plane wave (121) in detail. We suppose that wave vector is directed along z axis. Then the first-order equation (88) can be rewritten in the following equivalent form: whereW ′ = ie tW . Using (120) and (121) we can write solution of (122) in the following form: Note that the wave functionW ′ + describes the positive branch of dispersion law (120) that corresponds, for example, to the "antiparticle", whileW ′ − describes the negative branch that corresponds to the "particle" state. Besides, as it is seen the wave functions (123) and (124) Indeed it is simple to check thatŜ where eigenvalue S z = ±1/2. It is seen that plane waves (123) and (124) correspond to the different eigenvalues S z . ThusW ′ + describes "antiparticle" state with spirality S z = +1/2, whilẽ W ′ − describes "particle" state with spirality S z = −1/2. However in the case of massive field the plane wave (119) has more complicated space-time structure.

Nonhomogeneous equation
Let us consider the nonhomogeneous equation corresponding to the equation (88) HereĨ is the field source. Choosing the potentialW in the form (59), we obtain the following equation for the field strengths: This equation means that the strengths of this field are nonzero only in the region of field source. Let us consider the sedeonic source in the following form: where ρ L is a volume density of charge and j L is volume density of current. In this case the equation (128) is rewritten as Applying the operator (ie 1 ∂ − e 2 ∇ − ie 3 m) to the equation (130) and separating the values with different space-time properties we obtain the following equations for the field strengths: Assuming charge conservation we have the following gauge condition: which is similar to the Lorentz gauge, but for the field strengths. Let us consider a stationary field generated by a scalar point source where q L is the point charge. Then the intensity of the scalar field is This field is non-zero only in the region of source. It indicates that two point charges interact only if they are at the same point of space. The interaction energy for two point charges q L1 and q L2 is equal where R is the vector of distance between point charges. Such a law of interaction is typical for leptons involved in a weak interaction. So the q L can be interpreted as a lepton charge. Moreover one can suppose the interaction between q B and q L charges due to the overlap of scalar fields g B and g L . In the case of point q B and q L the fields are determined by the expressions (84) and (135), so that the interaction energy is equal to: As a result, we get: where R is the distance between q B and q L charges.

Concluding remarks
Thus, in this paper we have presented the sixteen-component sedeons generating associative noncommutative space-time algebra. This algebra can be considered as the scalar-vector variant of complexified Clifford algebra with specific commutation and multiplication rules. The sedeonic basis elements a 1 , a 1 , a 3 are responsible for the spatial rotation, while the elements e t , e r and e tr are responsible for the space-time inversions. Mathematically, these two bases are equivalent, and the different physical properties attributed to them are an important physical essence of our sedeonic hypothesis. In contrast to the Gibbs-Heaviside vector algebra the multiplication rules for vector basis in sedeonic algebra contain the imaginary unit (see Table 1). It enables the realization of scalar-vector algebra whith Clifford product [32]. Apparently, such possibility of vector basis multiplication was pointed first by A. Macfarlane [36]. Later the similar multiplication rules for matrix basis were applied by W.Pauli [37] and P.A.M.Dirac [38] in their spinor equations.
The important point is that the sedeonic basis elements simultaneously play a role of the operators and space-time basis of the wave function. From a physical point of view, this allows us to reformulate the Klein-Gordon equation of relativistic quantum mechanics as the wave equation for special scalar-vector field that carries information about the kinematic properties of quantum particles. This sedeonic Klein-Gordon equation can be reformulated as Maxwell-like equations for the field intensities. At the same time the sedeonic first-order Dirac wave equation can be interpreted as the equation describing special field with zero field intensities.
On the other hand the sedeonic Klein-Gordon equation allows another interpretation as the wave equation for the potentials of a force massive field. In this case the Einstein relation between energy and momentum can be interpreted as the relation for the energy, momentum and mass of a quantum of force field. The sources of this field are corresponding charges q B and currents j B . At the same time the sedeonic first-order wave equation describes the special force field with zero field strengths. The sources of this field are corresponding charges q L and currents j L . We defined the concept of energy and energy flux for the force massive field and derived an expression that describes the energy conservation for a massive field, similar to the Poynting theorem in electrodynamics. Based on this concept, we have considered the interaction of point charges due to the overlap of scalar and vector fields.