Applied Science and Innovative Research ISSN 2474-4972 (Print) ISSN 2474-4980 (Online) Vol. 6, No. 1, 2022 www.scholink.org/ojs/index.php/asir 46 Original Paper Explaining Double Split Experiment with Geometrical Algebra Formalism Alexander Soiguine1* 1 SOiGUINE Quantum Computing, 31 Aurora, Aliso Viejo, CA, USA * E-mail: alex@soiguine.com Received: February 8, 2022 Accepted: February 15, 2022 Online Published: February 18, 2022 doi:10.22158/asir.v6n1p46 URL: http://doi.org/10.22158/asir.v6n1p46 Abstract The Geometric Algebra formalism opens the door to developing a theory upgrading conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions; unambiguous definition of states, observables, measurements bring into reality clear explanations of conventional weird quantum mechanical features, particularly the results of double split experiments where particles create diffraction patterns inherent to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties in interpretation of quantum mechanics. Keywords geometric algebra, quantum states, observables, measurements 1. Introduction. Working with G-qubits Instead of Qubits Theory of upgrading conventional quantum mechanics has been under development (Soiguine, 1996, 2014, 2015, 2020). The main novel features are: - Replacing complex numbers with elements of even subalgebra of geometric algebra in three dimensions, that’s by elements of the form “scalar plus bivector”. - The objects identifying physical media are of the same structure: explicitly defined plane along with angle of rotation in that plane. - Operators acting on the objects are operators of rotation having the same structure: scalar plus bivector. That is the measurement operation. - Mapping of operator to the result of measurement, that is collapse, creates a “particle”. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 47 Published by SCHOLINK INC. - The operators can be identified by points on the three-sphere 𝕊3 and are connected, due to hedgehog theorem, by Clifford translations. Modifying the operators due to Clifford translation is identified by the generalization of Schrodinger equation containing unit bivectors in three dimensions instead of formal imaginary unit Qubits, identifying states in conventional quantum mechanics, mathematically are elements of two-dimensional complex spaces: .𝑥1+𝑖𝑦1 𝑥2+𝑖𝑦2 / , ‖𝑥1 + 𝑖𝑦1‖ 2 + ‖𝑥2 + 𝑖𝑦2‖ 2 = 1, unit value element of 𝐶2. Imaginary unit 𝑖 is used formally, without geometrical identification in three dimensions. In another accepted notations a qubit is: 𝐶2 ∋ ( 𝑧1 𝑧2 ) = 𝑧1 ( 1 0 ) + 𝑧2 ( 0 1 ) = 𝑧1|0⟩ + 𝑧2|1⟩ In the suggested formalism complex numbers 𝑥 + 𝑖𝑦 are replaced with elements of 𝐺3 +, subalgebra of 𝐺3. The 𝐺3 elements of the form 𝑀3 = 𝛼 + 𝐼𝑆𝛽, where 𝐼𝑆 is some unit bivector arbitrary placed in three-dimensional space, comprise so called even subalgebra of algebra 𝐺3. This subalgebra is denoted by 𝐺3 + (Soiguine, 2015, 2020). Elements of 𝐺3 + can be depict as in Figure 1. Figure 1. An Element of 𝑮𝟑 + Unit value elements of 𝐺3 + , 𝛼2 + 𝛽2 = 1 , will be called g-qubits. The wave functions (states) implemented as g-qubits store much more information than qubits, see Figure 2. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 48 Published by SCHOLINK INC. Figure 2. Geomectrically Picted Qubits and G-qubits 2. Lift of Qubits to G-qubits, Fiber Bundles and Probabilities Take right-hand screw oriented basis *𝐵1, 𝐵2, 𝐵3+ of unit value bivectors, with the multiplication rules 𝐵1𝐵2 = −𝐵3 , 𝐵1𝐵3 = 𝐵2 , 𝐵2𝐵3 = −𝐵1 , 𝐼3𝐵1𝐼3𝐵2𝐼3𝐵3 = 𝐼3 (or equivalently 𝐵1𝐵2𝐵3 = 1 ), where 𝐼3 is oriented unit value volume in three dimensions named also pseudoscalar. Quantum mechanical qubit state, |𝜓⟩ = 𝑧1|0⟩ + 𝑧2|1⟩ , is linear combination of two basis states |0⟩ and |1⟩. In the 𝐺3 + terms these two states correspond to the following classes of equivalence in 𝐺3 +, depending on which basis bivector is selected as complex plane: If 𝐵1 is taken as complex plane, then - State |0⟩ has fiber (level set) of the 𝐺3 + elements 𝑠𝑜(𝛼, 𝛽, 𝑆) |0⟩ (0-type 𝐺3 + states): • 𝛼 + 𝛽1𝐵1, 𝛼 2 + 𝛽1 2 = 1 - State |1⟩ has fiber of the 𝐺3 + elements 𝑠𝑜(𝛼, 𝛽, 𝑆) |1⟩ (1-type 𝐺3 + states): • 𝛽3 𝐵3 + 𝛽2 𝐵2 = (𝛽3 + 𝛽2𝐵1)𝐵3, 𝛽3 2 + 𝛽2 2 = 1 If 𝐵2 is taken as complex plane, then - State |0⟩ has fiber (level set) of the 𝐺3 + elements 𝑠𝑜(𝛼, 𝛽, 𝑆) |0⟩ (0-type 𝐺3 + states): • 𝛼 + 𝛽2𝐵2, 𝛼 2 + 𝛽2 2 = 1 - State |1⟩ has fiber of the 𝐺3 + elements 𝑠𝑜(𝛼, 𝛽, 𝑆) |1⟩ (1-type 𝐺3 + states): • 𝛽1 𝐵1 + 𝛽3 𝐵3 = (𝛽1 + 𝛽3𝐵2)𝐵1, 𝛽1 2 + 𝛽3 2 = 1 If 𝐵3 is taken as complex plane, then - State |0⟩ has fiber (level set) of the 𝐺3 + elements 𝑠𝑜(𝛼, 𝛽, 𝑆) |0⟩ (0-type 𝐺3 + states): • 𝛼 + 𝛽3𝐵3, 𝛼 2 + 𝛽3 2 = 1 - State |1⟩ has fiber of the 𝐺3 + elements 𝑠𝑜(𝛼, 𝛽, 𝑆) |1⟩ (1-type 𝐺3 + states): • 𝛽1 𝐵1 + 𝛽2 𝐵2 = (𝛽2 + 𝛽1𝐵3)𝐵2, 𝛽2 2 + 𝛽1 2 = 1 General definition of measurement in the suggested approach includes: - the set of observables to be 𝐺3 +, - the set of states to be elements of 𝐺3 + (g-qubits up to some scalar factor), - measurement of an observable www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 49 Published by SCHOLINK INC. 𝐶 = 𝐶0 + 𝐶1𝐵1 + 𝐶2𝐵2 + 𝐶3𝐵3 by g-qubit (wave function) 𝛼 + 𝐼𝑆𝛽 = 𝛼 + 𝛽1𝐵1 + 𝛽2𝐵2 + 𝛽3𝐵3 is defined as (𝛼 − 𝐼𝑆𝛽)𝐶(𝛼 + 𝐼𝑆𝛽) with the result: 𝐶0 + 𝐶1𝐵1 + 𝐶2𝐵2 + 𝐶3𝐵3 𝛼+𝛽1𝐵1+𝛽2𝐵2+𝛽3𝐵3 → 𝐶0 + (𝐶1,(𝛼 2 + 𝛽1 2) − (𝛽2 2 + 𝛽3 2)- + 2𝐶2(𝛽1𝛽2 − 𝛼𝛽3) + 2𝐶3(𝛼𝛽2 + 𝛽1𝛽3))𝐵1 + (2𝐶1(𝛼𝛽3 + 𝛽1𝛽2) + 𝐶2,(𝛼 2 + 𝛽2 2) − (𝛽1 2 + 𝛽3 2)- + 2𝐶3(𝛽2𝛽3 − 𝛼𝛽1))𝐵2 + (2𝐶1(𝛽1𝛽3 − 𝛼𝛽2) + 2𝐶2(𝛼𝛽1 + 𝛽2𝛽3) + 𝐶3,(𝛼 2 + 𝛽3 2) − (𝛽1 2 + 𝛽2 2)-)𝐵3 Probabilities of observed values are relative measures of the g-qubit fibers for each observable value received by the action of the states on observable. The lift from 𝐶2 to 𝐺3 + needs some *𝐵1, 𝐵2, 𝐵3+ reference frame of unit value bivectors. This frame, as a solid, can be arbitrary rotated in three dimensions. In that sense we have principal fiber bundle 𝐺3 + → 𝐶2 with the standard fiber as group of rotations which is also effectively identified by elements of 𝐺3 +. Suppose we are interested in the probability of the result of measurement in which the observable component 𝐶1𝐵1 does not change. This is relative measure of states √𝛼2 + 𝛽1 2 ( 𝛼 √𝛼2+𝛽1 2 + 𝛽1 √𝛼2+𝛽1 2 𝐵1) in the measurements: √𝛼2 + 𝛽1 2 ( 𝛼 √𝛼2 + 𝛽1 2 − 𝛽1 √𝛼2 + 𝛽1 2 𝐵1)𝐶√𝛼 2 + 𝛽1 2 ( 𝛼 √𝛼2 + 𝛽1 2 + 𝛽1 √𝛼2 + 𝛽1 2 𝐵1) That measure is equal to 𝛼2 + 𝛽1 2, that is equal to 𝑧1 2 in the down mapping from 𝐺3 + to 𝑧1|0⟩ + 𝑧2 |1⟩. Thus, we have clear explanation of common quantum mechanics wisdom on “probability of finding system in state |0⟩”. Similar calculations explain correspondence of 𝛽3 2 + 𝛽2 2 to 𝑧2 2 in 𝑧1|0⟩ + 𝑧2 |1⟩ when the component 𝐶1𝐵1 in measurement just got flipped. Any arbitrary 𝐺3 + state 𝑠𝑜(𝛼, 𝛽, 𝑆) = 𝛼 + 𝛽1𝐵1 + 𝛽2𝐵2 + 𝛽3𝐵3 can be rewritten either as 0-type state or 1-type state: 𝛼 + 𝛽1𝐵1 + 𝛽2𝐵2 + 𝛽3𝐵3 = 𝛼 + 𝐼𝑆(𝛽1,𝛽2,𝛽3)√𝛽1 2 + 𝛽2 2 + 𝛽3 2, where 𝐼𝑆(𝛽1,𝛽2,𝛽3) = 𝛽1𝐵1 +𝛽2𝐵2 +𝛽3𝐵3 √𝛽1 2+𝛽2 2+𝛽3 2 , 0-type, or 𝛼 + 𝛽1𝐵1 + 𝛽2𝐵2 + 𝛽3𝐵3 = (𝛽3 + 𝛽2𝐵1 − 𝛽1𝐵2 − 𝛼𝐵3)𝐵3 = .𝛽3 + 𝐼𝑆(𝛽2,−𝛽1,−𝛼 )√𝛼 2 + 𝛽1 2 + 𝛽2 2/𝐵3, www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 50 Published by SCHOLINK INC. where 𝐼𝑆(𝛽2,−𝛽1,−𝛼 ) = 𝛽2𝐵1−𝛽1𝐵2−𝛼𝐵3 √𝛼2+𝛽1 2+𝛽2 2 , 1-type. All that means that any 𝐺3 + state 𝛼 + 𝛽1𝐵1 + 𝛽2𝐵2 + 𝛽3𝐵3 measuring arbitrary observable 𝐶1𝐵1 + 𝐶2𝐵2 + 𝐶3𝐵3 does not change the observable projection onto plane of 𝐼𝑆(𝛽1,𝛽2,𝛽3) = 𝛽1𝐵1 +𝛽2𝐵2 +𝛽3𝐵3 √𝛽1 2+𝛽2 2+𝛽3 2 and just flips the observable projection onto plane 𝐼𝑆(𝛽2,−𝛽1,−𝛼 ) = 𝛽2𝐵1−𝛽1𝐵2−𝛼𝐵3 √𝛼2+𝛽1 2+𝛽2 2 . 3. Double Slit Experiment Taking the set of g-qubits and projection of them onto 𝐶2: 𝜋: 𝐺3 + → 𝐶2, we get fiber bundle. The projection depends on which basis bivector plane is selected as corresponding to formal imaginary unit plane. If we take, for example 𝐵3, the projection is: 𝜋: 𝑠𝑜(𝛼, 𝛽, 𝑆) = 𝛼 + 𝛽1𝐵1 + 𝛽2𝐵2 + 𝛽3𝐵3 → ( 𝛼 + 𝑖𝛽3 𝛽2 + 𝑖𝛽1 ) Then for any 𝑧 = .𝑥1+𝑖𝑦1 𝑥2+𝑖𝑦2 / ∈ 𝐶2 the fiber in 𝐺3 + consists of all elements 𝐹𝑧 = 𝑥1 + 𝑦2𝐵1 + 𝑥2𝐵2 + 𝑦1𝐵3 with an arbitrary triple of orthonormal bivectors *𝐵1, 𝐵2, 𝐵3+ satisfying multiplication rules. That particularly means that the standard fiber is group of rotations of basis bivectors in the standard fiber 𝐹𝑧. Thus, the fiber bundle is principal fiber bundle. Let one first slit is only open, and the fiber, wave function, is some 𝐹1 = 𝑥1 1 + 𝑦2 1𝐵1 + 𝑥2 1𝐵2 + 𝑦1 1𝐵3. For the only open second slit the fiber is different: 𝐹2 = 𝑥1 2 + 𝑦2 2𝐵1 + 𝑥2 2𝐵2 + 𝑦1 2𝐵3. When both slits are open the corresponding fiber is defined by connection, parallel transport anywhere between fibers 𝐹1 and 𝐹2. Let we have a smooth curve 𝛾(𝑡, 𝑃1, 𝑃2), 0 ≤ 𝑡 ≤ 1, connecting points 𝑃1 = (𝑥1 1, 𝑦2 1, 𝑥2 1, 𝑦1 1) and 𝑃2 = (𝑥1 2, 𝑦2 2, 𝑥2 2, 𝑦1 2), on three-dimensional sphere 𝕊3 such that 𝛾(0, 𝑃1, 𝑃2) = 𝑃1 and 𝛾(1, 𝑃1, 𝑃2) = 𝑃2. The easiest way to define parallel transport is 𝛾(𝑡, 𝑃1, 𝑃2) = (1 − 𝑡)𝑃1 + 𝑡𝑃2. For convenience purposes let us write 𝐹1 and 𝐹2 as exponents: 𝐹1 = 𝑥1 1 + 𝑦2 1𝐵1 + 𝑥2 1𝐵2 + 𝑦1 1𝐵3 = 𝑥1 1 +√(𝑦2 1)2 + (𝑥2 1)2 + (𝑦1 1)2 ( 𝑦2 1 √(𝑦2 1) 2 +(𝑥2 1) 2 +(𝑦1 1) 2 𝐵1 + 𝑥2 1 √(𝑦2 1) 2 +(𝑥2 1) 2 +(𝑦1 1) 2 𝐵2 + 𝑦1 1 √(𝑦2 1) 2 +(𝑥2 1) 2 +(𝑦1 1) 2 𝐵3) = 𝑒 𝐼𝑆1𝜑1, where 𝜑1 = 𝑐𝑜𝑠 −1 𝑥1 1, 𝐼𝑆1 = 𝑦2 1 √(𝑦2 1) 2 +(𝑥2 1) 2 +(𝑦1 1) 2 𝐵1 + 𝑥2 1 √(𝑦2 1) 2 +(𝑥2 1) 2 +(𝑦1 1) 2 𝐵2 + 𝑦1 1 √(𝑦2 1) 2 +(𝑥2 1) 2 +(𝑦1 1) 2 𝐵3. Angle 𝜑1 is not uniquely defined since it can be any of cos−1 𝑥1 1 ± 2𝜋𝑘1 , 𝑘1 = 0,1,2, … , where cos−1 𝑥1 1 is, by definition, taken from interval ,0, 𝜋-. The angle cos−1 𝑥1 1 will be denoted as 𝜑1(0). www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 51 Published by SCHOLINK INC. 𝐹2 = 𝑥1 2 + 𝑦2 2𝐵1 + 𝑥2 2𝐵2 + 𝑦1 2𝐵3 = 𝑥1 2 +√(𝑦2 2)2 + (𝑥2 2)2 + (𝑦1 2)2 ( 𝑦2 2 √(𝑦2 2) 2 +(𝑥2 2) 2 +(𝑦1 2) 2 𝐵1 + 𝑥2 2 √(𝑦2 2) 2 +(𝑥2 2) 2 +(𝑦1 2) 2 𝐵2 + 𝑦1 2 √(𝑦2 2) 2 +(𝑥2 2) 2 +(𝑦1 2) 2 𝐵3) = 𝑒 𝐼𝑆2𝜑2, where 𝜑2 = 𝑐𝑜𝑠 −1 𝑥1 2 , 𝐼𝑆2 = 𝑦2 2 √(𝑦2 2) 2 +(𝑥2 2) 2 +(𝑦1 2) 2 𝐵1 + 𝑥2 2 √(𝑦2 2) 2 +(𝑥2 2) 2 +(𝑦1 2) 2 𝐵2 + 𝑦1 2 √(𝑦2 2) 2 +(𝑥2 2) 2 +(𝑦1 2) 2 𝐵3. As above, 𝜑2 = cos −1 𝑥1 2 ± 2𝜋 𝑘2, 𝑘2 = 0,1,2, … The angle cos−1 𝑥1 2 will be denoted as 𝜑2(0). Measurement of an observable 𝐶 = 𝐶0 + 𝐶1𝐵1 + 𝐶2𝐵2 + 𝐶3𝐵3 = |𝐶| ( 𝐶0 |𝐶| + 𝐶1 |𝐶| 𝐵1 + 𝐶2 |𝐶| 𝐵2 + 𝐶3 |𝐶| 𝐵3) = |𝐶| ( 𝐶0 |𝐶| +√1 − 𝐶0 2 |𝐶|2 ( 𝐶1 |𝐶|√1− 𝐶0 2 |𝐶|2 𝐵1 + 𝐶2 |𝐶|√1− 𝐶0 2 |𝐶|2 𝐵2 + 𝐶3 |𝐶|√1− 𝐶0 2 |𝐶|2 𝐵3 ) ) = |𝐶|𝑒𝐼𝑆𝜑, where |𝐶| = √𝐶0 2 + 𝐶1 2 + 𝐶2 2 + 𝐶3 2, 𝜑 = cos−1 𝐶0 |𝐶| , 𝐼𝑆 = 𝐶1 |𝐶|√1− 𝐶0 2 |𝐶|2 𝐵1 + 𝐶2 |𝐶|√1− 𝐶0 2 |𝐶|2 𝐵2 + 𝐶3 |𝐶|√1− 𝐶0 2 |𝐶|2 𝐵3, by the wave function 𝑒𝐼𝑆1𝜑1 is: 𝑀1 = 𝑒 −𝐼𝑆1𝜑1 |𝐶|𝑒𝐼𝑆𝜑 𝑒𝐼𝑆1𝜑1 Measurement by 𝑒𝐼𝑆2𝜑2 is: 𝑀2 = 𝑒 −𝐼𝑆2𝜑2 |𝐶|𝑒𝐼𝑆𝜑𝑒𝐼𝑆2𝜑2 Measurement by any intermediate parallel transport wave function (1 − 𝑡)𝑒𝐼𝑆1𝜑1 + 𝑡𝑒𝐼𝑆2𝜑2 then reads: (1 − 𝑡)2𝑒−𝐼𝑆1𝜑1|𝐶|𝑒𝐼𝑆𝜑 𝑒𝐼𝑆1𝜑1 + 𝑡2𝑒−𝐼𝑆2𝜑2|𝐶|𝑒𝐼𝑆𝜑𝑒𝐼𝑆2𝜑2 + |𝐶|𝑡(1 − 𝑡)(𝑒−𝐼𝑆1𝜑1𝑒𝐼𝑆𝜑𝑒𝐼𝑆2𝜑2 + 𝑒−𝐼𝑆2𝜑2𝑒𝐼𝑆𝜑𝑒𝐼𝑆1𝜑1) = (1 − 𝑡)2𝑒−𝐼𝑆1𝜑1|𝐶|𝑒𝐼𝑆𝜑𝑒𝐼𝑆1𝜑1 + 𝑡2𝑒−𝐼𝑆2𝜑2|𝐶|𝑒𝐼𝑆𝜑𝑒𝐼𝑆2𝜑2 + 𝑡(1 − 𝑡)(𝑒−𝐼𝑆1𝜑1𝑒𝐼𝑆2𝜑2𝑒−𝐼𝑆2𝜑2|𝐶|𝑒𝐼𝑆𝜑𝑒𝐼𝑆2𝜑2 + 𝑒−𝐼𝑆2𝜑2𝑒𝐼𝑆1𝜑1𝑒−𝐼𝑆1𝜑1|𝐶|𝑒𝐼𝑆𝜑𝑒𝐼𝑆1𝜑1) = (1 − 𝑡)2𝑀1 + 𝑡 2𝑀2 + 𝑡(1 − 𝑡)(𝑒 −𝐼𝑆2𝜑2𝑒𝐼𝑆1𝜑1𝑀1 + 𝑒 −𝐼𝑆1𝜑1𝑒𝐼𝑆2𝜑2𝑀2) Let us make natural for double split experiment assumption 𝑆1 = 𝑆2 = 𝑆0 (that is the two wave functions, measuring states, are of 0-type with identical bivector planes.) Then we get the measurement result by the intermediate parallel transport wave function: (1 − 𝑡)2𝑀1 + 𝑡 2𝑀2 + 𝑡(1 − 𝑡)(𝑒 −𝐼𝑆2𝜑2𝑒𝐼𝑆1𝜑1𝑀1 + 𝑒 −𝐼𝑆1𝜑1𝑒𝐼𝑆2𝜑2𝑀2) = (1 − 𝑡)2𝑀1 + 𝑡 2𝑀2 + 𝑡(1 − 𝑡)(𝑒 𝐼𝑆0(𝜑1−𝜑2)𝑀1 + 𝑒 𝐼𝑆0(𝜑2−𝜑1)𝑀2) It is easily seen that the result of measurement is 𝑀1 when 𝑡 = 0 and 𝑀2 when 𝑡 = 1. Consider the following simplified scenario. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 52 Published by SCHOLINK INC. Assume we are only interested in the projections of 𝑀1 and 𝑀2 onto the plane of their rotations, 𝑆0, 𝑀1(𝑆0) and 𝑀2(𝑆0). Then from the general formula 𝑒𝐼𝑆2𝜑2𝑒𝐼𝑆1𝜑1 = cosφ1 cosφ2 − (𝑠1 ⋅ 𝑠2) sinφ1 sinφ2 + 𝐼3𝑠2 cosφ1 sinφ2 + 𝐼3𝑠1 cosφ2 sinφ1 − 𝐼3(𝑠2 × 𝑠1) sinφ1 sinφ2 we get that up to some factors 𝑒𝐼𝑆0(𝜑1−𝜑2)𝑀1(𝑆0) is 𝑀1(𝑆0) rotated in 𝑆0 by angle 𝜑1 − 𝜑2 and 𝑒𝐼𝑆0(𝜑2−𝜑1)𝑀2(𝑆0) is 𝑀2(𝑆0) rotated in 𝑆0 by angle 𝜑2 − 𝜑1. Without loss of generality suppose that the angles 𝜑1(0) and 𝜑2(0)are equal by values but opposite in sign: 𝜑1(0) = −𝜑0, 𝜑2(0) = 𝜑0, 𝜑1(0) − 𝜑2(0) = −2𝜑0 𝜑2(0) − 𝜑1(0) = 2𝜑0 Then it follows that in Clifford translations the projection 𝑀1(𝑆0) rotates in 𝑆0 additionally by −2(𝜑0 ± 𝜋(𝑘1 − 𝑘2)) , 𝑘1 = 0,1,2, … , 𝑘2 = 0,1,2, … , and projection 𝑀2(𝑆0) rotates in 𝑆0 additionally by 2(𝜑0 ± 𝜋(𝑘1 − 𝑘2)), 𝑘1 = 0,1,2, … , 𝑘2 = 0,1,2, … . Thus, we get infinite number of copies of 𝑀1(𝑆0) and 𝑀2(𝑆0) with varying values depending every time on uniformly distributed, by assumption, value of 𝑡, 0 ≤ 𝑡 ≤ 1, and separated by ±𝜋 along the big circle of intersection of plane 𝑆0 with the sphere 𝕊3. 4. Conclusions It was demonstrated that the geometric algebra formalism along with generalization of complex numbers and subsequent lift of the two-dimensional Hilbert space valued qubits to geometrically feasible elements of even subalgebra of geometric algebra in three dimensions allows, particularly, to resolve the double split experiment results with diffraction patterns inherent to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties in interpretation of conventional quantum mechanics. References Soiguine, A. (2014). What quantum “state” really is? Retrieved from http://arxiv.org/abs/1406.3751 Soiguine, A. (2015). Geometric Algebra, Qubits, Geometric Evolution, and All That. Retrieved from http://arxiv.org/abs/1502.02169 Soiguine, A. (2020). The Geometric Algebra Lift of Qubits and Beyond. s.l.:LAMBERT Academic Publishing. Soiguine, A. M. (1996). Complex Conjugation - Relative to What? In Clifford Algebras with Numeric and Symbolic Computations (pp. 284-294). Boston: Birkhauser.