Associative Space-Time Sedenions and Their Application in Relativistic Quantum Mechanics and Field Theory

We present an alternative sixteen-component hypercomplex scalar-vector values named “space-time sedenions”, generating associative noncommutative space-time Clifford algebra. The generalization of relativistic quantum mechanics and field theory equations based on sedenionic wave function and space-time operators is discussed.

• Clifford Algebra
• Space-Time Sedenions
• Relativistic Quantum Mechanics
• Sedenionic Klein-Gordon Equation
• Sedenionic Dirac Equation
• Sedenionic Maxwell Equiations

1. Introduction

The multicomponent hypercomplex numbers such as quaternions and octonions are widely used for the reformulation of quantum mechanics and field theory equations. The first generalization of quantum mechanics and electrodynamics was made on the basis of four-component quaternions, which were interpreted as scalar-vector structures [1] -[5] . The next step was taken on the basis of eight-component octonions, which were interpreted as the sum of scalar, pseudoscalar, polar vector and axial vector [6] - [11] . Scalars and axial vectors are not transformed under spatial inversion, while pseudoscalars and polar vectors change their sign under spatial inversion. Therefore, this interpretation takes only the symmetry with respect to the spatial inversion into account. However, a consistent relativistic approach requires taking full time and space symmetries into consideration that leads to the sixteen-component space-time algebras.

The well-known sixteen-component hypercomplex numbers, sedenions, are obtained from octonions by the Cayley-Dickson extension procedure [12] [13] . In this case the sedenion is defined as

where O_i is an octonion and the parameter of duplication e is similar to imaginary unit. The algebra of sedenions has the specific rules of multiplication. The product of two sedenions

,

,

is defined as

where is conjugated octonion. The sedenionic multiplication (2) allows one to introduce a well-defined norm of sedenion. However, such procedure of constructing the higher hypercomplex numbers leads to the fact that the sedenions as well as octonions generate normed but nonassociative algebra [14] - [16] . It complicates the use of the Cayley-Dickson sedenions in the physical applications.

Recently we have developed an alternative approach to constructing the multicomponent values based on our scalar-vector conception realized in associative eight-component octons [17] - [19] and sixteen-component sedeons [20] - [24] . In particular, we have demonstrated the method, which allows one to reformulate the equations of relativistic quantum mechanics and field theory on the basis of sedeonic space-time operators and scalar-vector wave functions. In this paper we present an alternative version of the sixteen-component associative space-time hypercomplex algebra and demonstrate some of its application to the generalization of relativistic quantum mechanics and field theory equations.

2. Sedenionic Space-Time Algebra

It is known, the quaternion is a four-component object

where components (Greek indexes) are numbers (complex in general), is scalar units and values a_m (Latin indexes m = 1, 2, 3) are quaternionic units, which are interpreted as unit vectors. The rules of multiplication and commutation for a_m are presented in Table 1. We introduce also the space-time basis e_t, e_r, e_tr, which is responsible for the space-time inversions. The indexes t and r indicate the transformations (t for time inversion and r for spatial inversion), which change the corresponding values. The value is a scalar unit. For convenience we introduce numerical designations (time scalar unit); (space scalar unit) and (space-time scalar unit). The rules of multiplication and commutation for this basis we choose similar to the rules for quaternionic units (see Table 2).

Table 1

Multiplication rules for unit vectors a m

	a1	a2	a3
a1	−1	a3	−a2
a2	−a2	−1	a1
a3	a2	−a1	−1

Table 2

Multiplication rules for space-time units

	e1	e2	e3
e1	−1	e3	−e2
e2	−e2	−1	e1
e3	e2	−e1	−1

Note that the unit vectors a₁, a₂, a₃ and the space-time units e₁, e₂, e₃ generate the anticommutative algebras:

for, but e₁, e₂, e₃ commute with a₁, a₂, a₃:

for any n and m. Besides, we assume the associativity of e₁, e₂, e₃, a₁, a₂, a₃ multiplication.

Then we can introduce the sixteen-component space-time sedenion in the following form:

The sedenionic components are numbers (complex in general). Introducing designation of scalar and vector values in accordance with the following relations

we can represent the sedenion in the following scalar-vector form:

Thus, the sedenionic algebra encloses four groups of values, which are differed with respect to spatial and time inversion.

1) Absolute scalars and absolute vectors are not transformed under spatial and time inversion.

2) Time scalars and time vectors are changed (in sign) under time inversion and are not transformed under spatial inversion.

3) Space scalars and space vectors are changed under spatial inversion and are not transformed under time inversion.

4) Space-time scalars and space-time vectors are changed under spatial and time inversion.

Further we will use the symbol 1 instead units a₀ and e₀ for simplicity. Introducing the designations of scalar- vector values

we can write the sedenion (6) in the following compact form:

On the other hand, introducing designations of space-time sedenion-scalars

we can write the sedenion (6) as

or introducing the sedenion-vector

we can rewrite the sedenion in following compact form:

Further we will indicate sedenion-scalars and sedenion-vectors with the bold capital letters.

Let us consider the sedenionic multiplication in detail. The sedenionic product of two sedenions and can be represented in the following form

Here we denoted the sedenionic scalar multiplication of two sedenion-vectors (internal product) by symbol “∙” and round brackets

and sedenionic vector multiplication (external product) by symbol “×” and square brackets,

In (16) and (17) the multiplication of sedenionic components is performed in accordance with (11) and Table 2. Thus the sedenionic product

has the following components:

Note that in the sedenionic algebra the square of vector is defined as

and the square of modulus of vector is

3. Spatial Rotation and Space-Time Inversion

The rotation of sedenion on the angle around the absolute unit vector is realized by sedenion

and by conjugated sedenion:

with

The transformed sedenion is defined as sedenionic product

Thus, the transformed sedenion can be written as

It is clearly seen that rotation does not transform the sedenion-scalar part, but the sedenionic vector is rotated on the angle around.

The operations of time inversion, space inversion and space-time inversion are connected with transformations in e₁, e₂, e₃ basis and can be presented as

4. Sedenionic Lorentz Transformations

The relativistic event four-vector can be represented in the follow sedenionic form:

The square of this valueis the Lorentzinvariant

The Lorentz transformation of event four-vector is realized by sedenions

where, is velocity of motion along the absolute unit vector. Note that

The transformed event four-vector is written as

Separating the values with e₁ and e₂ we get the well known formulas for time and coordinates transformation [25] :

where x is the coordinate along the vector.

Let us also consider the Lorentz transformation of the full sedenion. The transformed sedenion can be written as sedenionic product

Rewriting the expression (35) with scalar (16) and vector (17) products we get

Thus, the transformed sedenion has the following components:

5. Subalgebras of Space-Time Complex Numbers, Quaternions and Octonions

The sedenionic basis introduced above enables constructing different types of low-dimensional hypercomplex numbers. For example, one can introduce space-time complex numbers

where z₁ and z₂ real numbers. These values are transformed under space and time conjugation and Lorentz transformations. Moreover, we can consider the space-time quaternions, which differ in their properties with respect to the operations of the spatial and time inversion and Lorentz transformations:

The absolute quaternion (39) is the sum of the absolute scalar and absolute vector. It remains constant under the transformations of space and time inversion (27). Time quaternion, space quaternion and space-time quaternion are transformed under inversions in accordance with the commutation rules for the basis elements e_t, e_r, e_tr. For example, performing the operation of time inversion (see (27)) with the quaternion we obtain the conjugated quaternion

In addition, the sedenionic basis allows one to construct various types of space-time eight-component octonions:

6. Generalized Sedenionic Equations of Relativistic Quantum Mechanics

The wave function of free quantum particle should satisfy an equation, which is obtained from the Einstein relation for energy and momentum

by means of changing classical energy E and momentum on corresponding quantum-mechanical operators:

Here c is the speed of light, m₀ is the particle rest mass, is the Planck constant. In sedenion algebra the Einstein relation (47) can be written as

Let us consider the wave function in the form of space-time sedenion

Then the generalized sedenionic wave equation for free particle can be written in the following symmetric form:

Note that for electrically charged particle in an external electromagnetic field we have the following sedenionic wave equation:

This equation describes the particles with spin 1/2 in an external electromagnetic field [18] [21] .

There is a special class of particles described by the first-order wave equation [26] . For these particles the sedenionic Dirac-like wave equation has the following form:

In fact, this equation describes the special quantum field with zero field strengths [19] . Analogously the electrically charged particle interacting with external electromagnetic field is described by the following sedenionic first-order wave equation:

This equation also describes particles with spin 1/2 in an external electromagnetic field [19] .

7. Generalized Sedenionic Equations for Massive Field

The generalized sedenionic wave equation

enables another interpretation. It can be considered as the equation for the force massive field [27] . In this case the parameter m₀ is the mass of quantum of field and is field potential. Considering the phenomenological source of field we can propose the following nonhomogeneous wave equation for the field potential:

Seemingly this equation describes the baryon (strong) field [23] [24] and is baryon current. On the other hand, corresponding nonhomogeneous first-order equation

describes the lepton (weak) field, where is a lepton current [23] [24] .

8. Generalized Sedenionic Equations for Massless Field

In the special case, when the mass of quantum m₀ is equal to zero, the Equation (56) coincides with the equation for electromagnetic field. Indeed, choosing the potential as

and the source of field as

we obtain the following wave equation:

After the action of the first operator in the left-hand side of Equation (60) we obtain

In sedenionic algebra the electric and magnetic fields are defined as

Besides we can define the scalar field

Assuming electric charge conservation the scalar field f can be chosen equal to zero, that coincides with Lorentz gauge condition [22] . In Lorentz gauge we can rewrite the expression (61) as

Then the wave Equation (60) can be represented in the following form:

Performing sedenionic multiplication in the left-hand side of Equation (65) we get

Separating space-time values we obtain the system of equations in the following form:

The system (67) coincides with the Maxwell equations.

Among the solutions of the homogeneous sedeonic wave equation of electromagnetic field (60) there is a special class that satisfies the sedeonic first-order equation of the following form [22] :

This equation describes the free neutrino field. On the other hand, let us consider the nonhomogeneous equation of neutrino field

where is phenomenological source. We choose the scalar source in the form

where is the density of neutrino charge. Choosing the potential in the form (58):

we obtain following nonhomogeneous equation for the neutrino field:

It follows that in this case only scalar field strength (see (63)) is nonzero:

The density of neutrino charge for point source is equal

where q_v is point neutrino charge. Then the interaction energy of two point neutrino charges can be represented as follows:

Substituting (73) and (74), we obtain

where is the vector of distance between first and second charges.

9. Discussion

The algebra of sedenions proposed in this article is the anticommutative associative space-time Clifford algebra. The sedenionic basis elements a_n are responsible for the spatial rotation, while the elements e_n 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 sedenionic hypothesis.

In contrast to the previously discussed sedeonic algebra [20] -[23] , which uses the multiplication rules of basic elements and proposed by A. Macfarlane [28] , the multiplication rules for sedenionic basis elements a_n and e_n coincide with the rules for quaternion units introduced by W. R. Hamilton [29] . There is a close connection between these two basses. The transition from the sedeonic basis to sedenionic basis is performed by following replacement:

,

.

There is one disadvantage of sedenions connected with the fact that the square of the vector is a negative value. However, on the other side the sedenionic rules of cross-multiplying do not contain the imaginary unit and this leads to the some simplifications in the calculations. But of course, the physical results do not depend on the choice of algebra, so these two algebras are equivalent.

10. Conclusion

Thus, in this paper we presented the sixteen-component hypercomplex values sedenions, generating associative noncommutative space-time algebra. We considered the generalization of the relativistic quantum mechanics and theory of massive and massless fields based on hypercomplex scalar-vector wave functions and sedenionic space-time operators.

Acknowledgements

The authors are very thankful to G. V. Mironova for kind assistance and moral support.

References

1. Adler, S.L. (1995) Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York.

2. Imaeda, K. (1976) A New Formulation of Classical Electrodynamics. Nuovocimento, 32, 138-162. http://dx.doi.org/10.1007/BF02726749

3. Majernik, V. (1999) Quaternionic Formulation of the Classical Fields. Advances in Applied Clifford Algebras, 9, 119-130. http://dx.doi.org/10.1007/BF03041944

4. Davies, A.J. (1990) Quaternionic Dirac Equation. Physical Review D, 41, 2628-2630. http://dx.doi.org/10.1103/PhysRevD.41.2628

5. Schwartz, C. (2006) Relativistic Quaternionic Wave Equation. Journal of Mathematical Physics, 47, Article ID: 122301. http://dx.doi.org/10.1063/1.2397555

6. Liu, Y.-F. (2002) Triality, Biquaternion and Vector Representation of the Dirac Equation. Advances in Applied Clifford Algebras, 12, 109-124. http://dx.doi.org/10.1007/BF03161242

7. Gogberashvili, M. (2006) Octonionic Electrodynamics. Journal of Physics A: Mathematics in General, 39, 7099-7104. http://dx.doi.org/10.1088/0305-4470/39/22/020

8. Gamba A. ,et al. (1998)Maxwell’s Equations in Octonion Form NuovoCimento A 111, 293-302.

9. Tolan, T., ?zdas, K. and Tanisli, M. (2006) Reformulation of Electromagnetism with Octonions. Nuovo Cimento B, 121, 43-55. http://dx.doi.org/10.1393/ncb/i2005-10189-9

10. Gogberashvili, M. (2006) Octonionic Version of Dirac Equations. International Journal of Modern Physics A, 21, 3513-3523. http://dx.doi.org/10.1142/S0217751X06028436

11. De Leo, S. and Abdel-Khalek, K. (1996) Octonionic Dirac Equation. Progress in Theoretical Physics, 96, 833-846. http://dx.doi.org/10.1143/PTP.96.833

12. Dickson, L.E. (1919) On Quaternions and Their Generalization and the History of the Eight Square Theorem. Annals of Mathematics, 20, 155-171. http://dx.doi.org/10.2307/1967865

13. Imaeda, K. and Imaeda, M. (2000) Sedenions: Algebra and Analysis. Applied Mathematics and Computations, 115, 77-88. http://dx.doi.org/10.1016/S0096-3003(99)00140-X

14. Carmody, K. (1988) Circular and Hyperbolic Quaternions, Octonions, and Sedenions. Applied Mathematics and Computations, 28, 47-72. http://dx.doi.org/10.1016/0096-3003(88)90133-6

15. Carmody, K. (1997) Circular and Hyperbolic Quaternions, Octonions, and Sedenions—Further Results. Applied Mathematics and Computations, 84, 27-47. http://dx.doi.org/10.1016/S0096-3003(96)00051-3

16. K?plinger, J. (2006) Dirac Equation on Hyperbolic Octonions. Applied Mathematics and Computations, 182, 443-446. http://dx.doi.org/10.1016/j.amc.2006.04.005

17. Mironov, V.L. and Mironov, S.V. (2009) Octonic Representation of Electromagnetic Field Equations. Journal of Mathematical Physics, 50, Article ID: 012901. http://dx.doi.org/10.1063/1.3041499

18. Mironov, V.L. and Mironov, S.V. (2009) Octonic Second-Order Equations of Relativistic Quantum Mechanics. Journal of Mathematical Physics, 50, Article ID: 012302. http://dx.doi.org/10.1142/S0217751X09045480

19. Mironov, V.L. and Mironov, S.V. (2009) Octonic First-Order Equations of Relativistic Quantum Mechanics. International Journal of Modern Physics A, 24, 4157-4167. http://dx.doi.org/10.1142/S0217751X09045480

20. Mironov, V.L. and Mironov, S.V. (2011) Noncommutative Sedeons and Their Application in Field Theory. e-Print. http://arxiv.org/abs/1111.4035

21. Mironov, V.L. and Mironov, S.V. (2013) Reformulation of Relativistic Quantum Mechanics Equations with Non-Commutative Sedeons. Applied Mathematics, 4, 53-60. http://dx.doi.org/10.4236/am.2013.410A3007

22. Mironov, V.L. and Mironov, S.V. (2014) Sedeonic Equations of Gravitoelectromagnetism. Journal of Modern Physics, 5, 917-927. http://dx.doi.org/10.4236/jmp.2014.510095

23. Mironov, S.V. and Mironov, V.L. (2014) Sedeonic Equations of Massive Fields. International Journal of Theoretical Physics, e-Print. http://vixra.org/abs/1311.0005

24. Mironov, S.V. and Mironov, V.L. (2014) Space-Time Sedeons and Their Application in Relativistic Quantum Mechanics and Field Theory. Institute for Physics of Microstructures RAS, Nizhny Novgorod. http://vixra.org/abs/1407.0068

25. Landau, L.D. and Lifshits, E.M. (1975) Classical Theory of Fields. 4th Edition, Pergamon Press, New York.

26. Dirac, P.A.M. (1958) The Principles of Quantum Mechanics. Clarendon Press, Oxford.

27. Cafaro, C. and Ali, S.A. (2006) The Spacetime Algebra Approach to Massive Classical Electrodynamics with Magnetic Monopoles. Advances in Applied Clifford Algebras, 17, 23-36. http://dx.doi.org/10.1007/s00006-006-0014-7

28. Macfarlane, A. (1900) Hyperbolic Quaternions. Proceedings of the Royal Society at Edinburgh, 1899-1900 Session, 169-181.

29. Hamilton, W.R. (1853) Lectures on Quaternions. Royal Irish Academy, Dublin.