AI on Causality !
Faysal El Khettabi Ph.D.
Applied Mathematics to Real-World Measurement Problems: Towards Understanding Complex Mathematical Systems and Their Educational Teaching.
Causality is rooted in classical physics and serves as a way to understand the world based on observable phenomena. This perspective is natural because human senses act as measurement tools that can only perceive the relationships between events. However, it is difficult to ignore the fact that all simultaneous changes throughout the universe adhere to the two universal conservation laws and are described by the dynamics of the universal electric field and its associated magnetic field.
Key element of the classical physics perspective, which is that the human senses are measurement instruments that can only detect the mutual relations between the phenomena. This implies that causality is based on the observation of correlations and regularities in nature, and that it does not necessarily imply a deeper or fundamental explanation of why things happen the way they do.
However, this view of causality may not be sufficient or satisfactory for some questions or situations, especially when we encounter phenomena that seem to violate or transcend the classical laws and fields, such as quantum entanglement, quantum tunneling, quantum superposition, etc. These phenomena suggest that there may be more to reality than what we can measure or observe with our senses, and that there may be other principles or mechanisms that govern the behavior of the physical world at the most fundamental level.
Therefore, some physicists and philosophers have proposed alternative or generalized frameworks for causality that can accommodate the quantum phenomena and reconcile them with the classical ones. These frameworks may involve concepts such as retro causality, probabilistic causality, causal emergence, causal asymmetry, causal sets, causal loops, causal histories, causal networks, etc. These concepts aim to capture the essence of causality in a more general and flexible way, without assuming a fixed or predetermined causal structure or order. They also try to account for the role of the observer and the measurement process in determining the causal relations and outcomes.
Mathematical Representations
Mathematical representations can enable us to make predictions about physical systems more accurately than we could through observation alone. For instance, in astrophysics, mathematical formulations of the movement of planets and stars can help us understand how the universe formed and evolved over time. Similarly, mathematical systems of the spread of infectious diseases (price variation or co-movement in a financial system) can help policymakers make decisions about public health measures to stop outbreaks (to sustain the stability of the financial system) .
However, it is important to note that mathematical representations are not always unique or definitive. Depending on the assumptions, methods, and goals of the modeler, there may be different ways of modeling the same physical system or phenomenon. For example, there are different models of gravity, such as Newtonian, Einsteinian, and quantum, that have different levels of accuracy and applicability. In contrast, some physical phenomena, such as turbulence, chaos, or quantum entanglement, are still difficult to model mathematically.
Therefore, mathematical representations are not only tools for understanding and describing physical reality but also for exploring and discovering new aspects of it. By creating and testing different models, we can expand our knowledge and challenge our assumptions about the nature of reality. On the other hand, by encountering the limitations and paradoxes of mathematical models, we can also appreciate the complexity and mystery of physical reality.
In conclusion, while the tangible existence of mathematical representations is uncertain, the mathematical properties of a structure provide a framework for understanding the behavior of physical systems or phenomena. Through the study of mathematical representations, we can uncover fundamental principles and laws that govern the behavior of physical systems, expand our knowledge, and challenge our assumptions about the nature of our reality.
Important Contribution as Work Around:
John C. Baez is a mathematical physicist who has been investigating the role of the continuum in physics. In his work Struggles with the Continuum, he asks some intriguing questions:
He also makes the following remarks:
We have faced many challenging mathematical problems in every major physical theory that presupposes spacetime is a continuum. The continuum leads to infinities that make it difficult to formulate and predict physical phenomena. We can deal with these problems, but it takes a lot of hard work.
Only time will tell the answer. Nature is offering us many clues, but we need patience to interpret them correctly.
These questions and remarks reveal the potential limitations of our current understanding of spacetime as a continuum. The occurrence of infinities in physical theories that depend on a continuous spacetime model raises doubts about their validity and reliability. The possibility that the continuum may be only a crude approximation of a more sophisticated spacetime model demands more exploration and possibly a drastic change in our worldview.
The answers to these questions and solutions to these problems will likely require a lot of creativity and rigor in both theoretical and experimental physics. It is important to pay attention to the clues that nature gives us and to be prepared for new and surprising discoveries. Only time and continued inquiry will show us if the continuum model of spacetime is sufficient or if a more complex and nuanced model is needed.
Baez has also been exploring the connections between the Standard Model and the octonions. The octonions are a type of numbers that extend the complex numbers and the quaternions, but have some unusual properties, such as non-associativity.
In two presentations at a workshop on “Octonions and the Standard Model” at Perimeter Institute (February 8, 2021 to May 17, 2021), Baez has been trying to understand why the Standard Model has the specific gauge group and fermion representations that it does, and whether these can be derived from some reasonable principles. He has been using the octonions and the exceptional Jordan algebra, which is a mathematical structure that involves 3×3 self-adjoint matrices of octonions, to construct the Standard Model gauge group and its action on one generation of fermions.
In his first presentation, he focused on two aspects of this construction:
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The Standard Model gauge group is S(U(2)×U(3)), which is the group of symmetries of an octonionic qutrit that preserve all the structure arising from a choice of unit imaginary octonion i ∈ O and reduce to give symmetries of an octonionic qubit.
The representation of the Standard Model gauge group on one generation of fermions arises from splitting 10-dimensional Euclidean space into 4+6 dimensions and identifying the 4-dimensional part with the 2×2 self-adjoint octonionic matrices, and the 6-dimensional part with the octonionic qubit.
In his second presentation, he asked more fundamental questions, such as:
He also provided some important observations, such as:
There is no Hilbert space picture of the octonionic qutrit, because the octonions are non-associative and do not form a complex vector space.
The 2×2 self-adjoint octonionic matrices can be identified with 10-dimensional Minkowski spacetime, and pairs of octonions can be identified with left- or right-handed Majorana–Weyl spinors in 10-dimensional spacetime.
To Do
While John C. Baez's work on using octonions and the exceptional Jordan algebra to explain the Standard Model is certainly exciting, there is still a need for a broader mathematical framework to fully understand the implications of these ideas. Baez's specific construction is a starting point, but more work needs to be done to develop a more general understanding of hypercomplex number systems.
This would involve building up a mathematical background that incorporates not just Baez's iterative construction, but other possible methods for constructing hypercomplex number systems.
Furthermore, such a framework would need to be developed in a way that is rigorous and sound. This would involve developing mathematical tools for reasoning about hypercomplex number systems and proving results. These tools would need to be reliable and universally applicable, so that researchers across different domains can use them to explore the implications of these systems.
Overall, developing a more comprehensive mathematical framework for hypercomplex number systems would be a significant undertaking, one that would require substantial creativity and rigor. However, such a framework would be essential for answering some of the fundamental questions that Baez has raised about the relationship between the Standard Model and the octonions. Until this work is done, there will continue to be many unanswered questions about the role of hypercomplex number systems in physics.
Additionally, it would be important to consider the implications of hypercomplex number systems in other areas of physics beyond the Standard Model. For example, could hypercomplex number systems be used to explain the nature of dark matter or dark energy? Could they provide a more comprehensive understanding of quantum mechanics or the behavior of black holes? Therefore, developing a comprehensive mathematical framework for hypercomplex number systems could have far-reaching implications for our understanding of the physical world. It could lead to new insights and discoveries, and fundamentally shift our perspective on the nature of reality.
Quantum Causality Project
Causality is the science of cause and effect. It is the study of how things influence one another and how causes lead to effects. In the classical world we live in, causality comes with a few basic assumptions. First, things have causes. They do not just happen of their own accord. Second, effects follow causes in a predictable, linear manner. Third, big effects grow up from little causes. These assumptions are based on the observation of correlations and regularities in nature, and they do not necessarily imply a deeper or fundamental explanation of why things happen the way they do. However, it is difficult to ignore the fact that all simultaneous changes throughout the universe adhere to the two universal conservation laws and are described by the dynamics of the universal electric field and its associated magnetic field.
In this work, I will discuss how causality is rooted in classical physics and how it is challenged by quantum phenomena. I will also discuss how mathematical representations can help us understand and describe physical reality, as well as explore and discover new aspects of it. I will provide some examples and comparisons to illustrate my points, to be continued ...
In Abstract Form:
This work explores the concept of causality in classical physics and its limitations, especially in phenomena that violate classical laws such as quantum entanglement. It also discusses the importance of mathematical representations in understanding physical systems and the limitations of these representations. The work of John C. Baez on the role of the continuum in physics, and the connections between the Standard Model and the octonions, is also presented. Finally, the need for a more comprehensive mathematical framework to understand hypercomplex number systems, and its potential implications in different areas of physics, is discussed.
Quantum Causality
What makes the world go round and round?
What links the past, the present, and the future?
What tells us why things happen and how they are bound?
Is it causality, the classical ruler?
But what if causality is not so clear and fixed?
What if it can change and bend and twist?
What if it can be quantum and superposed?
What if it can be indefinite and composed?
In the quantum realm, things are not what they seem
Particles can be entangled and communicate instantly
They can tunnel through barriers and exist in many states
They can interfere with themselves and create new fates
How can we make sense of these phenomena?
How can we reconcile them with the classical laws?
How can we find a general framework for causality?
How can we explore the hidden causes and effects?
Some physicists and philosophers have tried to answer these questions
They have proposed new concepts and models for quantum causality
They have challenged the assumptions and implications of classical physics
They have opened new doors and possibilities for understanding reality
Quantum causality is a fascinating and mysterious topic
It invites us to rethink and expand our view of the world
It offers us a chance to learn and discover more about the cosmic
It inspires us to create and imagine more about the quantum
AI poem about Quaternion Manifolds:
Quaternion manifolds are so cool
They are like complex manifolds with a twist
They have four dimensions and a noncommutative rule
They are hard to visualize but hard to resist
Quaternion manifolds have many applications
They can describe physics and geometry
They can model strings and supergravity
They can capture symmetries and singularities
Quaternion manifolds are a rich and beautiful topic
They are full of challenges and surprises
They are worth exploring and studying
They are one of my favorite mathematical devices