Complementary Local and Nonlocal Measurements of Conjugate Transformation - Unified Logic Representation of Conjugate 0-1 Vectors

- The author has confirmed that a statement listing potential conflicts of interest or lack thereof is included in the text.

Bohr proposed the complementarity principle in 1927 as the foundation of quantum mechanics, since then relevant debates have been critically discussed for many years. Applying a pair of spin particles, Einstein proposed the EPR paradox in 1935. Using nonlocal potential properties, Aharonov and Bohm proposed the AB effect to test complementary measurement results. Under locality conditions, Bell established a Bell inequality under classical logic. Using a pair of particles and double sets of ZMI devices for complementarity measurements, Hardy proposed the Hardy paradox in 1992. During the past 50 years, locality and nonlocality tests on complementarity were hot-topics among the advanced quantum information, computing and measurement directions with various theoretical extensions and solid experimental results.

These complementarity approaches separated local/nonlocal parameters to form different equations without an integrated logic framework to describe these equations including both local and nonlocal features consistently. The main results provide a series of paradoxes that conflict with each other.

This paper uses conjugate transformation. Based on the m+1 kernel form of 0-1 states, n pairs of conjugate partitions were established. Under a given configuration in N bits, a set of 2n 0-1 feature vectors are applied to construct conjugate transformation operators in logic levels with intrinsic measurements to be a set of measurement operators.

The key results of the paper are listed in Theorem 5. Two special functions of vector logic (CNF or DNF expression) and four equivalent expressions of the elementary equation are examples to show local and nonlocal variables in equations consistently. Applying two pairs of conjugate sets <A, B> and their complementary sets <A', B''>, 4 meta measures are established corresponding to <±aA;±bB> quantitative features under measurement operators. The main results of the paper are represented in Lemma (1-4), Theorem (1-5) and Corollary (1-7). From a vector logic viewpoint, conjugate complementary scheme can organize local and nonlocal variables to satisfy the comprehensive properties of modern logic constructions on completeness, non-conflict and consistence in a united logic framework.

Figure 1

Figure 2

Due to technical limitations, full-text HTML conversion of this manuscript could not be completed. However, the manuscript can be downloaded and accessed as a PDF.

Reading your papers “Complementary Local and Nonlocal Measurements of Conjugate Transformation - Unified Logic Representation of Conjugate 0-1 Vectors” and “Measurement Operators in Conjugate Transformation Structure - Conjugate Hierarchy of Multiple Levels on Logic Constructions, Pairs of 0-1 Feature Vectors and Hamiltonian Dynamics”, I see that a major thrust is like mine. The important aspect for me is its relationship to a deeper theory of semantic structure, language and thought, but I also see a relevance to the new approach in quantum mechanics called ‘constructor theory’. Put in my jargon, you are looking at semantic structure based on extending Dirac dualization omega = omega+ + omega- to an extended twistor theory in which you apply a Lorentz rotation to the usual i-complex octonion multiplication table to get the corresponding hyperbolic complex split-complex (h-complex) multiplication table, though your emphasis is in seeing it all as based on a binary vector representation (which might be useful for a spin-coding theory of everything). I note that the full table for multiplying the different flavours of i with i, h with h, and h with i was done and analysed by Charles Arthur Muses, but I am having some trouble finding his original table; that is because he was also a philosopher and mystic so that those kind of weird publications drown out the solid core of his mathematics work. The full table is moderately easy to deduce from the standard octonion, of course, knowing that if you reverse the order of multiplication for different indices you take the negative. hi = -ih. However, the usual imaginary number i interests me less because reasoning in everyday life and medicine is not usually about waves (with important exceptions). In my opinion, the above Lorentz rotation i to h underlies the collapse of the wave function/decoherence, so logic in the everyday world of human experience depends on this rotation of wave mechanics to QM for the everyday world. Note exp(-h 2pi mx^2/Planckh t) gives rise on appropriate Dirac vector normalization of the conjugate variable form not to a periodic function but to a normal (Gaussian) distribution in which the probably of location of particle of mass m at all x on being observed spreads with time t. Dirac went beyond wave mechanics of course because he had mixed i-complex and h-complex wave functions, leading to quantum field theory. Interestingly, discussing QM as the above Lorenz transform of Schrodinger’s purely i-complex wave mechanics, and seeing the above Dirac combined forms, seem consistent (lead to the same results). It is just a matter of when one puts h into the developing description.

Posted 16 Sep, 2020

###### Community comments: 2

Complementary Local and Nonlocal Measurements of Conjugate Transformation - Unified Logic Representation of Conjugate 0-1 Vectors

Posted 16 Sep, 2020

###### Community comments: 2

- The author has confirmed that a statement listing potential conflicts of interest or lack thereof is included in the text.

Bohr proposed the complementarity principle in 1927 as the foundation of quantum mechanics, since then relevant debates have been critically discussed for many years. Applying a pair of spin particles, Einstein proposed the EPR paradox in 1935. Using nonlocal potential properties, Aharonov and Bohm proposed the AB effect to test complementary measurement results. Under locality conditions, Bell established a Bell inequality under classical logic. Using a pair of particles and double sets of ZMI devices for complementarity measurements, Hardy proposed the Hardy paradox in 1992. During the past 50 years, locality and nonlocality tests on complementarity were hot-topics among the advanced quantum information, computing and measurement directions with various theoretical extensions and solid experimental results.

These complementarity approaches separated local/nonlocal parameters to form different equations without an integrated logic framework to describe these equations including both local and nonlocal features consistently. The main results provide a series of paradoxes that conflict with each other.

This paper uses conjugate transformation. Based on the m+1 kernel form of 0-1 states, n pairs of conjugate partitions were established. Under a given configuration in N bits, a set of 2n 0-1 feature vectors are applied to construct conjugate transformation operators in logic levels with intrinsic measurements to be a set of measurement operators.

The key results of the paper are listed in Theorem 5. Two special functions of vector logic (CNF or DNF expression) and four equivalent expressions of the elementary equation are examples to show local and nonlocal variables in equations consistently. Applying two pairs of conjugate sets <A, B> and their complementary sets <A', B''>, 4 meta measures are established corresponding to <±aA;±bB> quantitative features under measurement operators. The main results of the paper are represented in Lemma (1-4), Theorem (1-5) and Corollary (1-7). From a vector logic viewpoint, conjugate complementary scheme can organize local and nonlocal variables to satisfy the comprehensive properties of modern logic constructions on completeness, non-conflict and consistence in a united logic framework.

Figure 1

Figure 2

Due to technical limitations, full-text HTML conversion of this manuscript could not be completed. However, the manuscript can be downloaded and accessed as a PDF.

Reading your papers “Complementary Local and Nonlocal Measurements of Conjugate Transformation - Unified Logic Representation of Conjugate 0-1 Vectors” and “Measurement Operators in Conjugate Transformation Structure - Conjugate Hierarchy of Multiple Levels on Logic Constructions, Pairs of 0-1 Feature Vectors and Hamiltonian Dynamics”, I see that a major thrust is like mine. The important aspect for me is its relationship to a deeper theory of semantic structure, language and thought, but I also see a relevance to the new approach in quantum mechanics called ‘constructor theory’. Put in my jargon, you are looking at semantic structure based on extending Dirac dualization omega = omega+ + omega- to an extended twistor theory in which you apply a Lorentz rotation to the usual i-complex octonion multiplication table to get the corresponding hyperbolic complex split-complex (h-complex) multiplication table, though your emphasis is in seeing it all as based on a binary vector representation (which might be useful for a spin-coding theory of everything). I note that the full table for multiplying the different flavours of i with i, h with h, and h with i was done and analysed by Charles Arthur Muses, but I am having some trouble finding his original table; that is because he was also a philosopher and mystic so that those kind of weird publications drown out the solid core of his mathematics work. The full table is moderately easy to deduce from the standard octonion, of course, knowing that if you reverse the order of multiplication for different indices you take the negative. hi = -ih. However, the usual imaginary number i interests me less because reasoning in everyday life and medicine is not usually about waves (with important exceptions). In my opinion, the above Lorentz rotation i to h underlies the collapse of the wave function/decoherence, so logic in the everyday world of human experience depends on this rotation of wave mechanics to QM for the everyday world. Note exp(-h 2pi mx^2/Planckh t) gives rise on appropriate Dirac vector normalization of the conjugate variable form not to a periodic function but to a normal (Gaussian) distribution in which the probably of location of particle of mass m at all x on being observed spreads with time t. Dirac went beyond wave mechanics of course because he had mixed i-complex and h-complex wave functions, leading to quantum field theory. Interestingly, discussing QM as the above Lorenz transform of Schrodinger’s purely i-complex wave mechanics, and seeing the above Dirac combined forms, seem consistent (lead to the same results). It is just a matter of when one puts h into the developing description.

It is great to get your opinion. The basic forms of the conjugate transformation structures are exactly similar to the Dirac brackets plus relevant operations. I am very happy to know that the core of group transformations on the i-complex and h-complex forms was done and analyzed by Charles Arthur Muses. The natural operators are beautiful and expected to see further linkages between the Dirac constructions and the conjugate transformation structures. Thanks for showing so many interesting facts ..

## Jeffrey Zheng

## replied on 23 September, 2020

It is great to get your opinion. The basic forms of the conjugate transformation structures are exactly similar to the Dirac brackets plus relevant operations. I am very happy to know that the core of group transformations on the i-complex and h-complex forms was done and analyzed by Charles Arthur Muses. The natural operators are beautiful and expected to see further linkages between the Dirac constructions and the conjugate transformation structures. Thanks for showing so many interesting facts ..