Scattering of Geometric Algebra Wave Functions and Collapse in Measurements

The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.


Introduction
The current research is based on long list of the author works on geometric algebra formalism in quantum mechanics [1] [2]. Previous results allow to implement the process of creating "particles" in "collapse" of wave function. The work aims at receiving explicit formulas identifying geometric objects appearing in interaction of wave functions with other objects.
The research considers wavelike objects that are elements of even subalgebra 3 G + of geometric algebra 3 G [1]. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit i in quantum mechanics [3], and introduces clear definition of wave functions.
The wave likeness follows from the fact that a 3 G + element scalar part and the coefficient of bivector part both can depend on time and three-dimensional position as: Such elements are in the current research wave functions, also called in the case 1 R = g-qubits since they are generalization of commonly known quantum mechanical qubits when formal imaginary unit i is replaced by a geometrically clear unit bivector S I in three dimensions.
The wave functions can be transformed, and scattered, for example, when another 3 G + element multiplies it (from the left.) The result of such time proportional actions is described in the geometric algebra terms by Schrodinger equation [4].
When a wave function acts through the Hopf fibration on a localized 3 G + element, that's executing a measurement, the result can be named as "collapse" of the wave function though it is physical identity of different nature, output of Hopf fibration.

Scattering of a Wave Function by G 3 + Element
In the following, the wave functions will be taken of specific form: The expression in the second square bracket is sum of operator of continuous rotation in plane The variant with 0 E I follows from similar calculation.

Measurements by Wave Functions
Let a 3 G + element The result in the current case is:  The very senseless wave/particle "dualism" cannot be seriously considered since the two are different things. Wave functions, thoroughly considered by P. Dirac as operators, under the name of "states", act on observables, producing "particles" if you prefer, through measurements.

cos cos
Our states as elements of 3 G + are naturally mapped onto unit sphere 3  . Two-state system is then just a couple of points on 3  , say   I  I  I  I  I  I  I  I   I  I  I  I This is geometrically clear and unambiguous explanation of strict connectivity 2 It is universally possible due to the hedgehog theorem. 3 Difference in exponent signs from usual measurement definition is made just for some convenience. It means that the angle has opposite sign or can be thought that the bivector plane was flipped. Journal of Applied Mathematics and Physics of the results of measurements instead of quite absurd "entanglement" in conventional quantum mechanics.
The received formula:     In measurements, through applying wave function to a static 3 G + element, the result is new 3 G + element rotating in not changing plane with constant angular velocity-"particle", that is "collapse" of wave function.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.