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Two research papers on the foundation of mathematical physics after the first trustworthy comments by an expert of quantum foundation

** ****Abstract **The three key points include in the two papers:

**The Logic foundation of quantum mechanics has constructed from the 0-1 vector logic, if two papers can pass examination and confirmation****A new constructor theory supports quantum mechanics****Quantum mechanics on the Clifford algebra is rigorous and reliable**

**Highlight****s**** **

This blog briefly describes the background and conclusion of two papers published as preprints.

**Results The Logic Foundation of quantum mechanics QM has constructed from the 0-1 vector logic, if the two papers can pass the examination and confirmation of top experts** (logic, algebra, quantum information, quantum logic, quantum group theory, quantum foundation, mathematical physics, theoretical physics, etc.)

**Constructor Theory**

Multivariable complex vector functions are generated from the 0-1 vector logic via 0-1 vector, 0-1 variables, conjugate states, conjugate feature clusters, Octonion group, conjugate transformation structure CTS, orthogonal projection, extended logic, complementary measurement, and statistical probability operators on quantization.

The multivariable complex vectors support the Clifford algebra to generate the Hilbert space, the von Neumann quantum mechanics, and the discrete Hamiltonian dynamics.

**A new constructor theory provides the hierarchical organization of multiple levels on whole pairs of conjugate feature vectors** **to support quantum mechanics **from microscopic variables, conjugate states, conjugate clusters to macroscopic conjugate classifications.** **

The constructor theory supports quantum mechanics,** **and the results of the two papers show that **quantum mechanics on the Clifford algebra is rigorous and reliable. **

**Removing Paradoxes**

In relation to a series of logic problems and paradoxes faced by quantum schools and interpretations of quantum mechanics, i.e. multiple quantum measurement paradoxes, double slit interference experiment, the EPR paradox, Schrodinger cat, Bell inequality, Hardy paradox, etc., could be resolved gradually during deeper explorations of the 0-1 vector logic …

**Related Background**

The 23rd and 24th preprints, published on the Research Square in September 16th, 2020 are the first two papers (https://www.researchsquare.com/article/rs-76545/v1 https://www.researchsquare.com/article/rs-76524/v1 ) for my activities on the 0-1 logic for four decades to explore the theoretical research on mathematical physics.

*First Trustworthy Comment*

In the first week of the two papers online (September 21-22, 2020), Professor Barry Robson (a top expert on the Dirac operators) gave a trustworthy review with excellent comments for the two papers posted online.

The key parts of his comments were truncated as follows: “ … a deeper theory of semantic structure, language and thought, … to the new approach in quantum mechanics called ‘constructor theory’. … extending Dirac dualization omega = omega+ + omega- to an extended twistor theory … to the usual i-complex octonion multiplication table … split-complex (h-complex) multiplication table, … the full table … was done and analysed by Charles Arthur Muses, …”.

Briefly in his review, the proposed structure - a new discrete transformation - is represented, and suggested as a constructor theory; the CTS core corresponds to Dirac bracket pair; the Octonion group linked to the quantum research of Charles Arthur Muses, and correspondence with i-complex and h-complex of the Dirac wave functions ... (Check full comments at https://www.researchquare.com/article/rs-76524/v1 , https://www.researchsquare.com/article/rs-76545/v1 )

*Extraordinary Moment*

At this extraordinary moment of my life, it is helpful for the sequel readers to understand the research content around the two papers to summarize the occasional surprises and intrinsic difficulties that have surrounded them for four decades, and then briefly to describe the main result of the papers.

*Trouble in Previous Submissions*

The submission paths of two original research papers on mathematical physics have been suffering all the manner. The papers have been through various national and international Journals on mathematics, logic, physics, quantum information, foundation of physics, mathematical physics and other professional academic presses for many times with experiences to carry out submissions and rejections recursively.

In most cases, the manuscript stands one to two weeks after the internal review of the editorial officer, and then the decision was informed that the paper was rejected. Reason for rejection: the paper does not satisfy research directions of the Journal it is recommended to submit it to other professional magazines.

The latest round of rejections on the two papers were in August 2020 to take them left in the editorial office for more than eight months. Reason for rejection: The editorial office could not find a suitable reviewer, because of the global outbreak of new coronavirus pneumonia ...

**Draft Paper**

The initial idea and key development of the first paper dates back to 26 years ago in the period of my PhD study at Monash University in Australia 1990s, where the conjugate transformation structure CTS is a chapter of the doctoral thesis [1]. Returning to this topic again was 20 years ago, and I prepared an English monograph on variant construction 2018 for Springer Nature (https://link.springer.com/book/10.1007/978-981-13-2282-2 ), and occasionally discovered a draft paper from historical document storages.

*Refined Papers*

After carefully translating the paper into Chinese and proofreading the key formula, I realized that various nontrivial problems cannot be ignored in various formulas of the draft document. Fortunately, associated with the continuous development and advanced research (https://www.researchgate.net/profile/Jeffrey_Zheng)

on vector measurement tools and descriptions more than 20 years, and combining the 0-1 vector logic on variant construction [2] established 10 years ago, the significant quantitative formulas have been developed. After translation, modification and optimization, the Chinese paper has been the core content of Chapter 4 of Volume 1 in the variant construction monograph (three volumes) [3] officially published by Chinese Science Press in November 2020. Key contents were expanded and translated as English papers with refined cases, and structured optimization, then making a series of submission activities to relevant professional journals ...

*Existing Results*

Repeated entanglements of submission activities had a go to publish such elementary results publicly. It was grown from the 2010-2013 period that a series of visual distribution results on statistical probability were obtained on simulation algorithms in asynchronous, parallel, same time and separate time, and other control conditions to take the fine simulation of two-way interference on visual interaction, simulation models, and methods. The advanced results of this series have been published in Acta Photonica Sinica, Laser and Optoelectronics Progress, International Journal of Computation and Modelling, and Journal of Modern Optics etc.

Associated with the publication of these professional Journal papers, related funding applications were submitted to the Natural Science Foundation of China for four years. Through these applications could not get the sufficient supports from the majority of review experts, however, a series of academic questions and scientific queries were collected via reviewing feedbacks of both Journal submissions and funding applications from multiple reviewers of professional experts on computational simulation, visual models and methods, quantum foundation, quantum computing, quantum information, double-slit photon interference, quantum optics, optoelectronics and other fields.

*Scientific Questions*

For example, how to confirm the correctness of these simulation results from the perspective of quantum theory? How does the controllable statistical probability distribution correspond to the result of quantum optics, optoelectronics, or double-slit interference experiment? So many clustering distributions correspond to state exhaustion projections on statistical probability procedures, from where has Schrodinger equation, Dirac equation, or Heisenberg equation been involved? ...

*Aim to Descriptions*

Since the specific topics of the elementary levels were focused in the first paper, the series of academic questions and scientific queries cannot be involved. There are no primary obstacles to answer the series of questions and strict descriptions on application levels to apply the advanced researches on mathematical physics, if the core problems can be concretely agreed from the foundational levels of elementary researches.

To briefly describe this new construction, the structure describes their input and output modes among core components with possible correspondences on their classical components. It is the most importance for the most readers to obtain higher levels of structural viewpoints to search targets on a pair of bird eyes from sky to grasp the main components and mapping functionalities in the complex transformation.

**The Main Content of the First Paper**

In the first paper, the constructor theory from logic, complex number to dynamics is composed of three layers: logic (vector logic), algebra (statistical probability), and transformation (discrete dynamics).

*Logical layer - 0-1 vector logic*

The vector logical layer consists of four spaces: m+1 state space, 2n feature space, N 0-1 vector space, and 2^{2n} vector function space (conjugate transformation structure).

m+1 state space

Starting from a fixed-length m+1, m ≥ 0 as a state of m+1 bits, there are 2^{m+1} states. For any 0-1 state, a feature bit is nominated at a fixed position to partition its value of 0 or 1 divided into two state collections with the same number of states, 2^{m+1} states of two sets are conjugated to each other in pairs. i.e., 2^{m} states make up a state collection, while 2^{m} states make up a conjugate state collection.

2n feature space

n groups 1≤ n ≤ 2^{m} of feature classes are made of n partitions and each partition is a feature class on the 2^{m} state collection, and corresponding n groups of conjugate feature classes are grouped on the 2^{m} conjugate state collection. There is a total number of 2n groups of feature classes on the 2^{m+1} states.

N 0-1 vector space

For any length N 0-1 vector, N ≥ 2^{m+1}, its head to tail is linked together as a ring. Under a shift operation on the vector, each m+1 bits consequently linked is a state, two closed neighbor states have m bits overlapped, and the whole bit vector can be represented as a N state vector; 2n feature classes can be used to project the state vector to generate 2n distinguished 0-1 vectors with N lengths, i.e., 2n feature vectors.

2^{2n} conjugate vector function space

Based on 2n feature vectors, 2^{2n} groups of vector functions are generated by four extended vector logical operators (∩, ∪, ￢, ~) : (and, or, not, conjugate). The 2^{2n} vector functions form a vector logic function space. Nothing likes classical vector logic functions, if there are 2n 0-1 vectors as vector variables, the number of vector logic functions is _{2}2^{2n} super exponential power. Since vector logic operations on feature vectors represent the complete orthogonal characteristics, most of the vector functions in classical vector algebra can be significantly reduced as a few results of simple orthogonal projections in the conjugate vector function space. All the results of the operations are contained in any combination of vector logic operators with the 2n feature vectors for the input, and this original vector logical transformation structure is the conjugate transformation structure CTS.

Feature Operator

Under a representation on pairs of feature vectors, the logical vector operators in each CTS can be represented as a form <A, B>. From the vector logic viewpoint, the CTS construction layouts the logical foundation for ensuing Dirac brackets in classical quantum dynamics.

Eight Forms

Based on the CTS, under the action of the five extended vector logical operators (∩, ∪, ￢, ~, ’):(and, or, not, conjugate, complementary), select two feature vector sets A,B for the original representation A,B and complementary representation A’,B’, and any feature operator <A, B> has eight distinguishable forms as follows:

{<A, B>, <A, B’>, <A’, B>, <A’, B’>, <B, A>, <B’, A>, <B, A’>, <B’, A’>}

*Algebra Layer - Measurement Statistics, Complex, Clifford Algebra, Hilbert Space, von Neumann Quantum Formalization*

From the quantitative measurement viewpoint, any feature vectors A(A’), B(B’), under the conditions of quantitative measurement, the corresponding statistical measures are a_{A}(a_{A’}), b_{B}(b_{B’}). From the quantitative representing, two complementary measures meet the opposite of each other as equivalents: a_{A} = - a_{A’}, b_{B} = - b_{B’}. In the algebraic layer, any feature operator <A, B> has eight distinguishable measuring operators, expressed as: {(a_{A},b_{B}), (a_{A},b_{B’}), (a_{A’},b_{B}), (a_{A’},b_{B’}), (b_{B},a_{A}), (b_{B’},a_{A}), (b_{B},a_{A’}), (b_{B’},a_{A’}) }.

Complex Form

Using the correspondence, the paper has proofed that such a measuring operator is equivalent to a complex elementary vector:

(a_{A},b_{B}) = a_{A} + i b_{B};

where i = √-1 is the imaginary number.

Under the condition of statistical measurement, a list of equations shows complex vector functions to be a natural correspondence from a feature operator. Based on this vector logic foundation, multiple complex vector functions correspond to the Clifford algebra under the vector algebra. A distinguishable represent of eight operators of measures is an octonion group for transforms, each element corresponds a meta generator for the transforming group.

Under the corresponding support, it is natural for multivariate complex function to use the C*-algebra to produce the Hilbert space, and then to create the quantum formalization [4] of von Neumann.

*Transform Layer - Dirac Operator, Dynamic Transformation, Discrete Dynamics*

For the representation, the last part of the first paper uses the complicated measuring operator <alpha, beta> as a complex bracket to represent discrete transformation tables created by the two typical operators: < beta, alpha> conjugate operator ~, and <alpha, - beta> complex conjugate operator * on discrete differential derivations. It is convenient to be compared with the traditional discrete differentials to represent a list of conjugate and complex conjugate transformations and the classic dynamic differential formulas. There are 1-1 correspondences between the discrete differential formulas and the classic differential formulas illustrated in the series of representations. From the typical comparison, these special cases show the analytical characteristics of discrete dynamics accompanied by two transformation operators associated with complicated complex time variations.

The transformation corresponding to the measuring operator <alpha, beta> is exactly the same as the Dirac bracket with the same functions and constraints in the combinatorial operations of the composite descriptions on general feature vectors.

*Conclusion for the First Paper*

In short, the first paper introduces a vector function space in the 0-1 vector logic, where two conjugate sets of 2n feature vectors N-length generate an orthogonal vector function space in the CTS. Finally, a statistical measurement converts a feature operator to be a complex function, and two typical transformations of complex operators, conjugate and complex conjugate, show a series of the discrete dynamic formula associated with complicated complex time variations.

**The Main Content of the Second Paper**

Compared with the first paper, the goal of the second paper is much simpler.

*Measuring Problem*

Since the Einstein and Bohr controversy presented the EPR paradox [5] in 1935, the debate over quantum measurements involving local and global variables has continued. Associated with the discovery of the AB effect [6] by Aharonov and Bohm in 1959 on complementary properties, Bell proposed the Bell inequality [7] in 1964 using the local conditions of the measurement, and a series of optoelectronic measurement experiments were performed since 1971, all results showed the cycle distribution of quantum measurement theory, which was significantly different from the square boundary determined by the Bell inequality on local variables, however, the boundary of another square region based on the global variables was much larger than the cycle distribution of the quantum measurement results [8].

*Hardy Paradox*

Due to the conditions of local and global complementarities, Hardy summed up this type of optoelectronic measuring problems as the dilemma of local and global not. In 1992, the Hardy paradox [9] proposed to highlight this trouble measuring condition in front of the public.

*Two Cases for the CTS*

Using the CTS on 0-1 vector extension logic operation, this paper discusses the two simplest sets of vector logic, corresponding to F_{1}(<A, B>X) - DNF and F_{0}(<A, B>X) - CNF formal logical expressions respectively. Using the 0-1 feature vector set, it is shown that under any pair of feature selection, the expression can be made use of neither all local variables, nor all global variables, the appropriate measuring equation always contains both local and global variables in the mixed expression.

*Geometric Distribution*

From the geometric viewpoint, this result coincides with the conditions of the cycle distribution unique to quantum measurement, i.e. neither all local variables, not all global variables. From the logical viewpoint, the CTS forms a complete support to express quantum interaction on the complex complementary variables superior than other existing quantum logic systems [10].

**Conclusion of the Blog**

Based on the 0-1 vector logic from 0-1 variables, states, conjugate classes, feature vectors, etc., to the conjugate transformation structure CTS, the measuring system converts a feature vector into a multivariable complex vector function with imaginary variables. Since the multivariable complex vector supports the Clifford algebra, the Hilbert space and von Neumann quantum mechanics can be created.

The two papers show that quantum mechanics is rigorous and reliable on the Clifford algebra. Various measuring paradoxes, such as EPR, the Schrodinger cat, Bell inequality and Hardy paradox, could be gradually resolved following the 0-1 vector logic.

In other words, the 0-1 vector logic lays the logical foundation of quantum mechanics. However, for the most critical steps, the two papers need to be recognized by high-levels of top experts on logic, mathematics, geometry, quantum information, quantum mechanics, theoretical physics, and quantum foundation etc.

From the perspective of elementary academic researches through multi-disciplinary, multi angle demonstrations, accurate and exact verifications, the new system structure must be rigorous, consistent, complete, satisfied full scientific condition of advanced logical reasoning to experience the most advanced tests on mathematical physics.

Looking forward for the two papers to attracting attention from international professional experts and a series of institutions and organizations for elementary researches of the frontier exploration on both advanced theories and applications.

*Expectation ...*

In the current conditions, how could we use the CTS solving certain specific problems? and could the CTS be utilized to explore quantum mechanics, quantum field theory, quantum information, quantum computing, quantum measurement, quantum interaction, quantum cryptography, quantum gravity and other advanced theoretical researches and explorations, and the latest cutting-edge technology? There are clear big gaps on specific models, systematic examples, and effective results …

The two papers are merely the starting point for everyone to apply the original vector logic from the 0-1 vector logic to discrete Hamiltonian dynamics, and the analysis of local and global complementary paradoxes.

The advanced theoretical theory and applied research works need to be follow-up urgently to coordinate various professional experts work together on multiple levels of basic theory and frontier application researches.

Further explorations and researches can make the ice on quantum foundation for a century break a gap, which has been troubled by various logical paradoxes since the begin of the quantum theory for hundreds of years. Using the 0-1 vector logic, the CTS and relevant methods, it promotes the research and exploration of advanced quantum foundation, and opens up a new path for the global application of various cutting-edge high-tech applications in near future …

**References**

[1] Z. J. Zheng, *Conjugate Transformation of Regular Plane Lattices for Binary Images*, **PhD Thesis**, Dep. Computer Science, Monash University 1994.

[2] Jeffrey Z.J. Zheng, *Variant Construction from Theoretical Foundation to Applications*, **Springer Nature** 2019 https://link.springer.com/book/10.1007/978-981-13-2282-2

[3] 郑智捷，*变值体系理论及其应用*，第1册：*理论基础及其应用* , **科学出版社** 2020年11月

Zheng Zhijie*, Theory and Application of Variant Construction, Volume 1: The Theoretical Foundation and Its Application*, **Science Press**, November 2020.

[4] John von Neumann, *Mathematical Foundation of Quantum Mechanics*, **Princeton University Press** 1955

[5] A. Einstein, B. Podolsky, N. Rosen. Can Quantum-mechanical Description of Physical Realitybe Considered Complete? Phys. Rev. 47 770-780 (1935).

[6] Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in Quantum Theory, Physical Review 115: 485-491 (1959)

[7] J.S.Bell, On theEinstein-Podolsky-Rosen paradox. Physics 1,195(1964) https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[8] 张永德，*量子信息物理原理*，**科学出版社**2009, 94页 (图4.2)

Zhang Yongde, *Principles of Quantum Information Physics*, *Science Press* 2009, 94 pages (Figure 4.2)

[9] L.Hardy, Quantum mechanics, local realistic theories and Lorentz-invariant realistic theories.Phys. Rev. Lett. 68, 2981 (1992)

[10] E.G. Beltrametti, G. Cassinelli, *The Logic of Quantum Mechanics*, **Addison-Wesley Publishing Company** 1981

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