Minkowski continuum

[AntonioLao]

This is a four-dimensional continuum in the sense that the product of the time parameter, the speed constant of light, and the imaginary unity is equivalent to any of the three spatial coordinates of three-dimensional space such that dx+dy+dz-(icdt) becomes a relativistic invariance commonly known as the proper interval in special theory of relativity and where dx, dy, dz, and dt are the corresponding differentials. These signify they are very, very small infinitesimal physical quantities very close to the value of zero. For them to be physically meaningful there are associated unit vectors: h, i, j, and k that must be orthogonal. Although in 3-space any combination of three unit vectors can be easily become orthogonal in an optimum configuration, orthogonality in 4-space requires extension into the complex domain of imaginary numbers such that there exist 6 combinatory permutations of 2-space orthogonality between pairs of unit vectors: A(i,j), B(i,k), C(i,h), D(j,k), E(j,h), and F(k,h). Out of these six distinct orthogonal 2-spaces only combinations of three can be mutually orthogonal and there are exactly 20 orthogonal combinations: ABC, ABD, ABE, ABF, ACD, ACE, ACF, ADE, ADF, AEF, BCD, BCE, BCF, BDE, BDF, CDE, CDF, CEF, and DEF. Nonetheless, in forming the six sides of an arbitrary cube eight distinct combinations out of these 20 are necessary, no more and no less.

Consequently, to formulate a quantum theory of the Minkowski continuum requires at the most eight distinct combinations such that if four are chosen as ADE, AFB, AEB, and ADF plus the consideration that ADE has by itself again six possible equal permutations (ADE, AED, DAE, DEA, EAD, EDA) then their corresponding complements can only be CDE, CFB, CEB, and CDF. Unfortunately, the number of orthogonal combinations for 20 fundamental combinations taken eight at a time is equal to 125970 possible orthogonal combinations. This number could have a connection with the various patterns of an 8-cube, or 27-cube, or 64-cube, or 125-cube structure of Rubik’s cubes. On the other hand, the number of elements (vertices, edges, faces, and volume etc.) of any dimensional solid increase in direct proportion with the higher dimensions. For example,

(sorry MS Excel table cannot be inserted here)

Surprisingly, all dimensional solids have the same number of elements in one dimension where and when they are contracted into a line segment with two vertices and one edge. The directional properties of this singular edge can then be circumscribed within the surface of a Möbius topology such that the direction of this edge changed into the opposite simply by going around one cycle. Conversely, this change of direction is an indication that one cycle has been completed without the use of a frame of reference as required in other scientific theories, e.g., as in special and general theory of relativity. This justifies the fact that the left-handed and the right-handed system of coordinates are distinctly independent of each other and cannot be ignored or taken for granted in a complete physical theory. It also justifies the independent existence of polar and axial vectors in the physical world for use in the construction of a quantum theory of the space-time continuum, especially a quantum theory of the Minkowski continuum. Moreover, although energy is conventionally defined as the scalar dot product of a vector force and a vector distance, the square of energy can be defined as the scalar dot product of two vector cross products of vector force and vector distance such that the results give both energy and square of energy as absolute scalar physical quantities.

[AntonioLao]

Here is the missing table for the previous post now attached as a PDF file