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Complex Conjugation - Relative to What?
Abstract
Some initial, technically simple but fundamentally important statements concerning the very notion of a complex number are formulated in terms of Clifford (Geometric) algebra generated by vectors in some geometrically and physically sensitive dimensions. It is shown that it makes no sense to speak about complex numbers without identifying a corresponding two-dimensional plane.
... 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 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. ...
... 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. ...
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.
... Thus, we have usual system of Maxwell equations: 9 For any vector we write = ...
... requiring that it satisfies (3.1) Element 0 in (3.2) is a constant element of geometric algebra 3 and is unit value bivector of a plane in three dimensions, generalization of the imaginary unit [9], [1]. The exponent in (3.2) is unit value element of = + 3 ℎ with some initial conditions: ...
The article contains an application of a theory aiming to change the underlying mathematical structure of conventional quantum mechanics which is a no-work-around obstacle to create quantum computers. The theory modifications, along with geometrically feasible generalization of formal imaginary unit to unit value areas of explicitly defined planes in three dimensions, include implementation of idea that commonly used notions "state", "observable", "measurement" require a clear unambiguous redefinition. New definition helps to establish effective formalism which in combination with geometric algebra generalizations brings into reality a kind of physical fields, which are states in terms of the suggested theory, spreading through the whole three-dimensional space and full range of scalar values of the time parameter. The fields can be modified instantly in all points of space and past and future time values, thus eliminating the concepts of cause and effect, and one-directional time.
... equivalent to one equation ( + ∇) = . 5 For any vector we write ̂= | | ⁄ Without charges and currents the equation becomes: ...
... and require that it satisfies (5.1.1) Element 0 in (5.2.1) is a constant element of geometric algebra 3 and is unit value bivector of a plane in three dimensions, generalizing imaginary unit [5]. The exponent in (5.2.1) is unit value element of 3 + : ...
5. Maxwell equation solutions in , their Clifford translations and measurements 5.1. The form of the Maxwell system for electromagnetic field in geometric algebra terms Many formal aspects of the states in terms of the suggested theory have been considered in previous chapters. Now we'll consider physically feasible 3 + states that can be critical for practical implementation of the quantum computing and quantum cryptography schemes, the two most important practical applications. Let's show how the system of the electromagnetic Maxwell equations is formulated as one equation in geometric algebra terms. Take geometric algebra element of the form: = + 3 ℎ (vector plus bivector.) The electromagnetic field is created by some given distribution of charges and currents, that also can be written as geometric algebra multivector: ≡ −. Apply operator + ∇, where ∇=̂+̂+̂ 5 and multiplication is the geometrical algebra one, to the. The result is: (+ ∇) = ∇ • ⏟ + + 3 (∇ ∧ ℎ) ⏟ + ∇ ∧ + 3 ℎ ⏟ + 3 (∇ ⋅ ℎ) ⏟ Comparing component-wise (+ ∇) and multivector we get: { • ≡ () = + 3 (∧ ℎ) ≡ − (ℎ) = − ∧ + 3 ℎ ≡ 3 () + 3 ℎ = 0 3 (⋅ ℎ) ≡ 3 (ℎ) = 0 Thus, we have usual system of Maxwell equations: { () = − (ℎ) = − ℎ + () = 0 (ℎ) = 0 equivalent to one equation (+ ∇) =. 5 For any vector we write ̂= | | ⁄
... As was shown, see [3], [7], [8], a qubit state can be lifted to g-qubit, element of 3 G -even geometric subalgebra of the geometric algebra 3 G in three dimensions. The lift particularly uses the generalization of a formal imaginary plane to explicitly defined planes in three dimensions [2], [3]. The g-qubit state is strictly interpreted as operator acting on observables, also elements of geometric algebra, in the process of measurement. ...
... The elements of the fiber are g-qubits 4 defined as the lift: 2 Due to critical reasons explained later a state should actually be a couple of a g-qubit and integer number that will eliminate ambiguity in the g-qubit angle value. 3 The reference frame ...
Following the B. Hiley belief [1] that unresolved problems of conventional quantum mechanics could be the result of a wrong mathematical structure, an alternative basic structure is suggested. Critical part of the structure is modification of the sense of commonly used terms " state " , " observable " , " measurement " giving them a clear unambiguous definition. This concrete definition, along with using of variable complex plane [2], is quite natural in geometric algebra terms [3]. It helps to establish a feasible language for the area of quantum computing. The suggested approach is used then in the fiber optics quantum information transferring/processing scenario.
... polarized electromagnetic waves are the only type of waves following from the solution of Maxwell equations in free space done in geometric algebra terms.Indeed, let's take the electromagnetic field in the form: in (4.2) is a constant element of geometric algebra 3 and is unit value bivector of a plane in three dimensions, generalization of the imaginary unit[5],[1]. The exponent in (4.2) is unit value element of 3 + [1]: = cos + sin , = − • Solution of (4.1) should be sum of a vector (electric field ) and bivector (magnetic field 3 ℎ): 4 For any vector we write ̂= | do we write ( − • ) or ( − • ) . ...
The Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in that terms, bring into reality a kind of physical fields, states in the suggested theory, spreading through the whole three-dimensional space and values of the time parameter. The fields can be modified instantly in all points of space and time values, thus eliminating the concept of cause and effect and perceiving of one-directional time.
... where 1 and 2 are complex numbers in their usual meaning. When complex numbers are generalized to explicit geometrical three dimensional objects, elements of 3 + , even subalgebra of geometric algebra 3 , [4], [5], then a qubit, for some chosen triple of basis bivectors in three dimensions { 1 , 2 , 3 }, gets lift to 3 + , defined up to arbitrary permutation of elements { 1 , 2 , 3 }, for example: It follows that qubit state 1 |0⟩ = ( 1 1 + 1 2 )|0⟩ has the lift 1 1 + 1 2 1 and qubit state 2 |1⟩ = ( 2 1 + 2 2 )|1⟩ has the lift 2 2 2 + 2 1 3 = ( 2 1 + 2 2 1 ) 3 since basis bivectors satisfy multiplication rules (in the righthand screw orientation of 3 ): ...
... where 1 and 2 are complex numbers in their usual meaning. When complex numbers are generalized to explicit geometrical three dimensional objects, elements of 3 + , even subalgebra of geometric algebra 3 , [4], [5], then a qubit, for some chosen triple of basis bivectors in three dimensions { 1 , 2 , 3 }, gets lift to 3 + , defined up to arbitrary permutation of elements { 1 , 2 , 3 }, for example: It follows that qubit state 1 |0⟩ = ( 1 1 + 1 2 )|0⟩ has the lift 1 1 + 1 2 1 and qubit state 2 |1⟩ = ( 2 1 + 2 2 )|1⟩ has the lift 2 2 2 + 2 1 3 = ( 2 1 + 2 2 1 ) 3 since basis bivectors satisfy multiplication rules (in the righthand screw orientation of 3 ): ...
Conventional quantum mechanical qubits can be lifted to geometric algebra states valued operators that act on observables. That operators may be implemented via the two types of Maxwell equations' solution polarizations. Solution of Maxwell equation in geometric algebra formalism gives g-qubits which are exact lifts of conventional qubits. Therefore, it unambiguously reveals actual meaning of complex parameters of qubits of the commonly accepted Hilbert space quantum mechanics and, particularly, directly demonstrates the option of instant nonlocality of states.
... are generally elements of 3 G , though mainly we will consider elements of 3 G . Another critical thing is explicit generalization of formal "imaginary unit" to a unit value bivector from 3 G specified by a process under consideration [6] [7]. All that allows to 3 generalize the Dirac's idea and to implement states as the 3 G valued operators (see also [8]). ...
Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum phenomena.
... are generally elements of 3 G , though mainly we will consider elements of 3 G . Another critical thing is explicit generalization of formal "imaginary unit" to a unit value bivector from 3 G specified by a process under consideration [6] [7]. All that allows to 3 generalize the Dirac's idea and to implement states as the 3 G valued operators (see also [8]). ...
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.
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