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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.

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 [

The well-known sixteen-component hypercomplex numbers, sedenions, are obtained from octonions by the Cayley-Dickson extension procedure [

where O_{i} is an octonion and the parameter of duplication e is similar to imaginary unit

is defined as

where

Recently we have developed an alternative approach to constructing the multicomponent values based on our scalar-vector conception realized in associative eight-component octons [

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

where components _{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 _{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

a_{1} | a_{2} | a_{3} | |
---|---|---|---|

a_{1} | −1 | a_{3} | −a_{2} |

a_{2} | −a_{2} | −1 | a_{1} |

a_{3} | a_{2} | −a_{1} | −1 |

e_{1} | e_{2} | e_{3} | |
---|---|---|---|

e_{1} | −1 | e_{3} | −e_{2} |

e_{2} | −e_{2} | −1 | e_{1} |

e_{3} | e_{2} | −e_{1} | −1 |

Note that the unit vectors a_{1}, a_{2}, a_{3} and the space-time units e_{1}, e_{2}, e_{3} generate the anticommutative algebras:

for_{1}, e_{2}, e_{3} commute with a_{1}, a_{2}, a_{3}:

for any n and m. Besides, we assume the associativity of e_{1}, e_{2}, e_{3}, a_{1}, a_{2}, a_{3} multiplication.

Then we can introduce the sixteen-component space-time sedenion

The sedenionic components

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

2) Time scalars

3) Space scalars

4) Space-time scalars

Further we will use the symbol 1 instead units a_{0} and e_{0} 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

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

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

The rotation of sedenion

and by conjugated sedenion

with

The transformed sedenion

Thus, the transformed sedenion

It is clearly seen that rotation does not transform the sedenion-scalar part, but the sedenionic vector

The operations of time inversion_{1}, e_{2}, e_{3} basis and can be presented as

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

The transformed event four-vector

Separating the values with e_{1} and e_{2} we get the well known formulas for time and coordinates transformation [

where x is the coordinate along the

Let us also consider the Lorentz transformation of the full sedenion

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

Thus, the transformed sedenion has the following components:

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

where z_{1} and z_{2} 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_{t}, e_{r}, e_{tr}. For example, performing the operation of time inversion (see (27)) with the quaternion

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

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

Here c is the speed of light, m_{0} is the particle rest mass,

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 [

There is a special class of particles described by the first-order wave equation [

In fact, this equation describes the special quantum field with zero field strengths [

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

The generalized sedenionic wave equation

enables another interpretation. It can be considered as the equation for the force massive field [_{0} is the mass of quantum of field and

Seemingly this equation describes the baryon (strong) field [

describes the lepton (weak) field, where

In the special case, when the mass of quantum m_{0} 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 [

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 [

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

where

where

we obtain following nonhomogeneous equation for the neutrino field:

It follows that in this case only scalar field strength

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

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 [_{n} and e_{n} coincide with the rules for quaternion units introduced by W. R. Hamilton [

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.

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.

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