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So far the quantized space is a speculative subject. Theories applied to this subject are alternative theories. Solutions to the open questions of physics are searched also from there.
One example of a quantized space is the memory space of a computer. It consists of bits. The bits are abstract and independent of background.The physical space can also be modeled to be made of abstract cells. The cells themselves are a substance, which will form the space.
In quantized spacetime the directions, the lengths and the time are quantized. The shortest possible length is so called Planck length and the shortest time is Planck time. The three directions are perpendicular to each other. The quantized space is in scale of quantum effects totally different than the observer's space. Because of the differences there appear two pictures of the same space. The first one is the picture of quantized space and the second one is the picture of isotropic observer's space appeared by coarsening.
Geometric examination shows that the quantized cell-structured space needs to be quadratic in comparison with the observer's space. It is also nonlinear and non-unique when observed in the observer's space. Quadraticness means here the transform in set of coordinates x2 => X , y2 => Y and z2 => Z, where (x,y,z)-space represents the observer's space and (X,Y,Z)-space represents the cell-structured space. The shape of octahedron in cell-structured space is transformed into the shape of sphere in the observer's space. This will lead to several phenomena of quantum mechanics like nonlocality and the wave function collapse in measurement. For example the nonlinear quantum correlation in a space, where the directions are quantized, differs from the linear classical correlation as the observations prove.
The quantized space can be perceived only theoretically without any direct observations. All unobserved elementary particles stand in the quantized space until they are observed. Because of the nonlinearity a location (or a point) of the quantized space spreads out when observed in the linear observer's space. Therefore also an unobserved elementary particle needs to spread out like a wave fuction does.
In the wave function collapse the pictures of space will interchange. An elementary particle gets in measurement its location in the linear observer's space after standing earlier in the nonlinear cell-structured space. Thus the measurement leads to changing the picture of space. The observer's space exists only as a consequence of coarsened observations. Without any observations there exists only the nonlinear cell-structured space, which stays unobserved. The ability to form the picture of observer's space by coarsening is called here for the consciousness.
Note that Minkowski space is quadratic too. It is invariant for every observer, which means that it is in certain terms an absolute space (likewise the cell-structured space). The interval between two points in that space is defined as:
where c is the speed of light, and Δt and Δr denote differences of the time and space coordinates, respectively, between the points.
When the observer's space is isotropic, a rigid macroscopic stick needs to turn in space length-preserving. This requires a certain structure of cell-structured space. The 3-dimensional cell-structured space forms a closed surface. Outside the surface exists a cell-structured complex space reaching to a certain distance from the 3D-surface. Flexibility of the complex space leads also to a local phase invariance, when the structure of complex space determines the phase of the wave function in any force field.
- R. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, 4th Ed. Addison-Wesley 1999.
- J. Matousek, J. Nesetril, Invitation to Discrete Mathematics. Oxford University Press 1998.
- Taylor E. F., John A. Wheeler, Spacetime Physics, publisher W. H. Freeman, 1963.