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What quantum "state" really is?
Abstract
To get out of logical deadlock in interpreting gedanken experiments like "Schrodinger cat", actual meaning of a "wave function", or a "state", in the case of complex two-dimensional Hilbert space, is shown to be an element of even subalgebra of geometric algebra [1], [2] over three-dimensional Euclidian space. Exodus 23:2 1
... Unambiguous definition of states and observables, does not matter are we in "classical" or "quantum" frame, should follow the general paradigm, [3], [4], [5], [6]: ...
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [3], [4], [7], [6], [8]. ...
... Let ( ) = −ℎ 1 ( ) 3 1 − ℎ 2 ( ) 3 2 − ℎ 3 ( ) 3 3 Then | ( )| = √ℎ 1 2 ( ) + ℎ 2 2 ( ) + ℎ 3 2 ( ), bivector part of (6.1) is 3 ), and the scalar part of the wave function (6.1) is cos(| ( )| ). ...
... Unambiguous definition of states and observables, does not matter are we in "classical" or "quantum" frame, should follow the general paradigm [1], [2], [3]: ...
... All parameters , 1 , 2 , 3 , 1 , 2 , 3 , are (real 3 ) scalars. ...
... 3 form of a Hamiltonianis in one-to-one map with its matrix form in the Pauli matrix basis, see[1] That means that the Schrodinger equation governs evolution, under the Hamiltonian Clifford translations, of states which, in turn, can act on observables. ...
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.
... 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]. ...
... What is different in the current approach to the light propagation in a beam guide is the fact that formally used imaginary unit is replaced with a unit bivector in three dimensional space not necessary orthogonal to the z direction, default beam guide axis. 7 The electric fields should naturally be considered as states, up to the magnitude factor, that's the 3 G operators acting on observables. ...
... Assume we deal with a detectable polarization in the xy plane: circular, elliptic or linear one, which means that the electric field vector end point moves along the corresponding trajectory. The following result takes place: 7 The interest to the transverse light beam spin models is growing intensively, see, for example [15] 8 Similar definitions of polarization are used in some different contents, see, for example [14]. 8 where is angle of the ellipse ...
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.
... "The world is all that occurs, when that is experienced, what happens" [25,2016] "The co∼eventum mechanics is a science of co∼events that occurs, when that is experienced, what happens" [2018] "A man is the ability of living matter to co∼be, to group co∼events and to make a believabilistic choice of them" [2018] For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've nally done it in my co∼eventum mechanics 19 ! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. ...
... The later Taittiriya Upanishad, probably the oldest extant example of the whole range of levels, deals with ve levels which are largely the phenomenological aspects of the human body is related to the emergence of such a change during the Common Era. and psyche. A later text, the Mandukya Upanishad [19], offers a more metaphysical content of four levels, while the seven-fold system of the Tantrics (the kundalini) is clearly linked to the structure of the human nervous system and the interior experiences related to its whole range, from basic biological functions to the most expanded levels of consciousness (MacPhail, 2013, 173-176). Aurobindo Ghosh offers a vefold contemporary version that partakes of Western psychology (MacPhail, 2013, 31-38). ...
∙ Financial and actuarial mathematics
∙ Mathematics in the humanities, socio-economic and natural sciences
∙ Probability theory and statistics
∙ Multivariate statistical analysis
∙ Eventology of multivariate statistics
∙ Co∼eventum mechanics
∙ Theory of experience and chance
∙ Eventology of safety
∙ Eventology of risk and decision-making under risk and uncertainty
∙ Philosophy of probability and event
∙ Eventological economics and psychology
∙ Eventological problems of arti1cial intelligence
∙ Eventological aspects of quantum mechanics information theory
∙ Decision-making under risk and uncertainty
∙ Risk measurement and risk models
∙ Theory of fuzzy events and generalized theory of uncertainty
∙ System analysis and events management
... The living space of objects in the suggested theory is geometric (often called Clifford) algebra in three dimensions, 3 . ...
... In the above result the sense of the orientation and the direction of were assumed to agree with 3 =̂. Opposite orientation, − 3 =̂, that's and compose left hand screw and ̂= 3 , will give solution = ( 0 + 3 Maxwell equation (4.1) is a linear one. Then any linear combination of + and − saving the structure of (4.2) will also be a solution. ...
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.
... What Dirac had effectively done [5] was to remove the distinction between an element of the operator algebra and the wave function without losing any information about the content of what is carried by the wave function. This is exactly what is shown below to be an accurate implementation of the Definitions 2.1 and 2.2 in the case of a qubit as the state in terms of geometric algebra, when action of a state on observable is non-commutative 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 generalize the Dirac's idea and to implement states as the 3 G valued operators (see also [8]). ...
... The item O I is unit value bivector defining the bivector part of the observable, orientation in three dimension space. If its expansion in the basis is [2], [7]. 1 The lift can equivalently be written as ...
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.
... Unambiguous definition of states and observables, does not matter are we in "classical" or "quantum" frame, should follow the general paradigm [3], [4], [5]: ...
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.
... The electromagnetic field F is created by some given distribution of charges and currents, also written as geometric algebra multivector: ρ 3 and multiplication is the geometrical algebra one, to the F. The result is: ...
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.
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++.
Comprehensive and physically consistent model of a tossed coin is presented
in terms of geometric algebra. The model clearly shows that there is nothing
elementary particle specific in the half-spin quantum mechanical formalism. It
also demonstrates what really is behind this formalism, feasibly reveals the
probabilistic meaning of wave function and shows that arithmetic of packed
objects, namely wave functions and Pauli matrices, reduces the amount of
available information.
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.
Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.
- J V Neumann
J. v. Neumann, Mathematical Foundations of Quantum Mechanics, Princeton: Priceton University
Press, 1955.
- D Hestenes
D. Hestenes, New Foundations of Classical Mechanics, Dordrecht/Boston/London: Kluwer Academic
Publishers, 1999.