The Equations of Dirac and Maxwell as a Result of Combining Minkowski Space and the Space of Orientations into Seven
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Equations of Dirac and Maxwell as a Result of Combining Minkowski Space and the Space of Orientations into Seven-Dimensional Space-Time R. A. Sventkovsky1* 1
Center for Information Technologies and Systems of Executive Power Bodies, Moscow, 127557 Russia Received January 16, 2020; in final form, January 16, 2020; accepted April 4, 2020
Abstract—The multicomponent wave function of the spin and vector fields is presented as a one-component function depending on the position and orientation of a zero-size moving rotating observer. It is shown that the Dirac and Maxwell equations and the fine structure constant are the result of the connection of two spaces: the Minkowski space and the space of orientations (the observer), and this relationship is not mathematical, but physical in nature. DOI: 10.1134/S0001434620090072 Keywords: seven-dimensional space-time, spinor representation of the Lorentz group.
1. INTRODUCTION In the quantum theory of the angular momentum of a solid, many transformations (addition of vectors, of basis states with spin, rotation of the meter) are defined as properties of functions of the Euler angles ϕ, θ, γ immediately for all points in space without reference to a certain point in space x1 , x2 , x3 . The Euler angles ϕ, θ, γ define the orientation of a rotating observer having unit vectors X(1) , X(2) , X(3) relative to the meter having unit vectors X1 , X2 , X3 . Directions, angles of orientation do not depend on the initial position, as they are always determined through differentiation by coordinates x1 , x2 , x3 . In standard quantum mechanics, there are no explicit descriptions of the bases of spin states, as certain functions of x1 , x2 , x3 from which the well-known transformation properties follow. Note that this should not be confused with equivalent transformation of amplitudes which is postulated but is not proven. There is only a matrix description of the basis states that are denoted as |k = (0, 0 . . . , 1, . . . , 0) which describe an object with number k. This follows from the fact that in Minkowski space there are no two functions of x1 , x2 , x3 that are converted using the spinor group SU (2) for j = 1/2 under spatial rotation x1 , x2 , x3 . In the space of orientations of a solid such objects exist, these are Wigner functions D j∗ for half-integer values j [1], [2]. Therefore, to describe the basis functions of the rotation group SO(3), two angles are enough, and for describing the basis functions of the SU (2) group, three angles are necessary [1], [3]. The well-known group property for spin 1/2 states: the sum of spin 1/2 states is always spin 1/2; 1/2∗ they are implemented by the Wigner functions D 1/2 . For example, ψu = D1/2,1/2 is the state of spin 1/2∗
up (along x3 ), ψd = D−1/2,1/2 is the state of spin down (obtained from ψup by turning about x2 by 180 degrees). Then any sum of these functions ψu , ψd will give a function in the form of ψup in some rotated coordinate system, which is always there [1]–[3]. The property of the group when summing the vector fields E, H, A
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