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Complex Conjugation — Relative to What?
Abstract
Some initial, technically simple but fundamentally important statements concerning the very origin of the notion of a complex number are formulated in terms of the Clifford (Geometric) algebra generated by vectors in some geometrically and physically sensitive dimensions. A new insight into the sense of geometrical product is given. It is shown that it makes no sense to speak about complex numbers without identifying a corresponding two-dimensional plane. This is particularly important if the given physical situation is set in higher dimensions. Because of great importance of these questions in education and because of increasing use of graphical computer programs in mathematical education and research, some components of a computer program implementing the Geometric Algebra approach are outlined in terms of classes of the object-oriented computer language C++.
... In the suggested approach a qubit state will be lifted to gqubit, 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 states are interpreted strictly as operators acting on observables, also elements of geometric algebra, in the process of measurement. ...
... The even sub-algebra 3 G , in the fiber bundle terms, can be taken as total space for base space 2 C and any 2 C qubit ...
... It is plausible to retrieve how the Hamiltonian action on states in conventional quantum mechanics is generalized in the current context. Any conventional quantum mechanics (CQM) 2 C state lift to 3 G can be written as exponent: ...
Keeping in mind that existing problems of conventional quantum mechanics could happen due of a wrong mathematical structure, I suggest an alternative basic structure. The critical part of it is modifying commonly used terms "state", "observable", "measurement" and giving them a clear unambiguous definition. This concrete definition, along with the use of a variable complex plane, is quite natural in geometric algebra terms and helps establish a feasible language for quantum computing. The suggested approach is then used in a fiber optics quantum information transferring/processing scenario.
... | ( 0 )| ≡ ( 0 ) is a generalized imaginary unit, in geometric algebra terms (see [9]). Thus, we get: Let's make measurement of an observable that, for some simplicity, has only bivector part, ( , , ) = 1 1 + 2 2 + 3 3 ≡ ( 1 , 2 , 3 ). ...
... Element 0 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 [9], [3]. The geometric algebra product F is: ...
... Interesting thing is that the component of measurement lying in plane is only defined by the applied Clifford translation parameters and does not change with time 9 . It only depends on the ⃗ value and Clifford translation parameter . ...
The article contains outline 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. Part of that is 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, effect and one-directional time.
... Wave functions act in that context on static 3 G + elements through measurements, creating "particles", see [2]. ...
The superiority of hypothetical quantum computers is not due to faster cal-culations but due to different schemes of calculations running on specialhardware. The core of quantum computing follows the way a state of a quan-tum system is defined when basic things interact with each other. In conven-tional approach it is implemented through tensor product of qubits. In thegeometric algebra formalism simultaneous availability of all the results fornon-measured observables is based on the definition of states as points onthree-dimensional sphere.
(PDF) Parallelizable Calculation of Observables Values on Analog Quantum Computer. Available from: https://www.researchgate.net/publication/382341953_Parallelizable_Calculation_of_Observables_Values_on_Analog_Quantum_Computer#fullTextFileContent [accessed Jul 19 2024].
... is a constant element of geometric algebra 3 G and S I is unit value bivector of a plane S in three dimensions, generalization of the imaginary unit [7]. The exponent in (2.2) is the unit value element of 3 G + : ...
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [3], [4], [7], [6], [8]. ...
Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum theory (Mathematics Subject Classification, item 81). Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example, primitively considering atoms as a kind of planetary system, very familiar from macroscopic experience but recklessly used in a physically very different situation. In the current work the three-sphere becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The states as 3 points evolve, governed by updated Schrodinger equation, and act as operators on observables in measurements. ALEXANDER SOIGUINE 30
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [3], [4], [7], [6], [8]. ...
... The path to the new theory starts with generalization of complex numbers by explicit introduction of a variable "complex" plane in three dimensions that immediately eliminates the questions like "Why do we need imaginary unit in quantum mechanics?" [1] State, wave function, will be a unit value element of even subalgebra of three-dimensional geometric algebra. Such elements will execute twisting of observables. ...
Quantum computing rests upon two theoretical pillars: entanglement and superposition. But some physicists say that this is a very shaky foundation and quantum computing success will have to be based on a different theoretical foundation. The g-qubit theory supports this point of view. Current article is the second one of the two and about the entanglement. It gives different, more physically feasible, not mysterious, explanation of what the entanglement is. The suggested formalism demonstrates that the core of future quantum computing should not be in entanglement which only formally follows in conventional quantum mechanics from representation of the many particle states as tensor products of individual states. The core of quantum computing scheme should be in manipulation and transferring of wave functions on as operators acting on observables and formulated in terms of geometrical algebra. In this way quantum computer will be a kind of analog computer keeping and processing information by sets of objects possessing infinite number of degrees of freedom, contrary to the two value bits or two-dimensional Hilbert space elements, qubits.
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [1], [2], [4], [5]. ...
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of some weird quantum mechanical features, particularly, the results of double-slit experiments where particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar system. Abstract-The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of some weird quantum mechanical features, particularly, the results of double-slit experiments where particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar system.
... Usage of even subalgebra 3 G + of geometric algebra 3 G [1] [2] [3] stems from generalization of complex numbers [3] [4]. The sprefield wave functions (states) received as special 3 G + solutions of Maxwell equations [5] [6]. ...
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions , followed by unambiguous definition of states, observables, measurements , bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere 3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The 3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators .
The Dirac theory has a hidden geometric structure. This talk traces the concep-tual steps taken to uncover that structure and points out significant implications for the interpre-tation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpreta-tion for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer con-nections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation.
This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a geometric product of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.
