Science2be

News for Budding Scientists and Engineers!

Hello everyone.

I’m pleased to share a few new improvements that will benefit all of our aspiring scientists and engineers, especially those just starting out at university. Our dedicated GitHub repository will provide valuable information for students and learners.

Introducing our GitHub Resource Hub: https://github.com/learnig-scientist/mecwave

This repository is designed to be a foundational tool to help you grasp the practical aspects of mechanics and wave physics. Whether you’re tackling homework assignments, preparing for exams, or simply curious about how theoretical concepts apply in real-world scenarios, these Python notebooks are just what you need.

What You Can Expect:

Beginner-Friendly Codes: The codes are crafted with beginners in mind, ensuring that even those new to programming can follow along and learn effectively.
Interactive Learning: Each code is interactive, allowing you to tweak variables and see the outcomes in real-time, which is a fantastic way to learn and understand complex concepts.
Open for Collaboration: Feel free to fork the repository, suggest improvements, or develop your own projects based on what you learn from these codes.

Why Use These Resources?

The transition from high school to university can be challenging, especially when it comes to subjects like physics and engineering. By providing you with tools that bridge the gap between theory and practice, we aim to make your learning journey smoother and more enjoyable.

Get Started Today!

Dive into the repository at Learnig-Scientist/MecWave (Turning “learnig” mistakes into learning opportunities) and start exploring. The world of mechanics and waves (we will add Electromagnetism and Optics with time) is vast and fascinating, and these tools are your key to unlocking its mysteries.

We believe these resources will be incredibly valuable for your studies, and we’re excited to see how you use them to further your understanding and spark your creativity.

Stay curious and happy coding!

Comments Leave a Comment
Categories Academia, Education and Science, Electromagnetism and Optics, Mechanics and Waves, Python notebook

PhyThematics 20 – Pontryagin’s maximum principle

12 Apr

In optimal control theory, Pontryagin’s maximum principle is used to determine the optimal control strategy for transitioning a dynamical system between states, particularly when input or state controls are constrained. It says that any optimum control plus the optimal state trajectory are required to solve the two-point boundary value issue known as the “Hamiltonian system,” plus a maximum condition of the control Hamiltonian. Under specific convexity criteria on the goal and constraint functions, these required conditions become sufficient [1].

REFERENCES:

[1] https://en.wikipedia.org/wiki/Pontryagin%27s_maximum_principle

[2] For those interested in diving deeper into the intricacies of optimal control problems, I highly recommend an in-depth course offered by MIT through their OpenCourseWare. This course, titled ‘Principles of Optimal Control (Spring 2008),’ covers a comprehensive range of topics from the foundational theories to practical applications in control systems. You can explore the full syllabus and course materials here: MIT’s Principles of Optimal Control. It’s an invaluable resource for anyone looking to expand their knowledge and expertise in this fascinating area of study.

NB-“PhyThematics” is meant to be a short calligraphic text containing calculus or content related to science, aiming to maximize understanding of physics and natural processes and help science benefit society. For the fun of it. Remember, for better learning:

Write notes in books
Add unique visual notes to digital documents
Stick with paper and a pen for creative pursuits.

Comments Leave a Comment
Categories Academia, Advanced Physics Topics, Aerospatial engineering, Education and Science, Pontryagin's maximum principle

Basics of String Theory

12 Apr

In this series, our primary aim is to demystify and popularize intricate mathematical structures, making them accessible to the everyday individual, both aspirational learners and mindful engineers. Even though mathematics is seen as mysterious and abstract, it is full of deep symmetry, profound truths, and beautiful things. Gaining mathematics literacy improves our ability to solve problems and creates new opportunities across many industries. A mathematically literate society is more powerful and capable of making the most use of equations and numbers to improve living standards and get a better understanding of the universe.

Introduction

The development of string theory, a cornerstone of modern theoretical physics, began in the late 1960s as a collaborative effort among several physicists, notably Gabriele Veneziano. Veneziano, in his quest to understand the strong nuclear force that binds the constituents of atomic nuclei, proposed a mathematical formula in 1968 that accurately described the scattering amplitudes of hadrons (particles like protons and neutrons) at high energies. This formula, known as the Veneziano amplitude, surprisingly mirrored the characteristics of Regge theory—a framework developed in the 1950s by Italian physicist Tullio Regge to analyze the angular momentum and properties of particles involved in scattering processes.

Regge theory introduced the concept of Regge poles, which are complex angular momenta corresponding to the resonances observed in particle physics. These poles explained the behavior of scattering amplitudes at high energies and were pivotal in understanding the dynamics of hadrons. However, the physical interpretation of Veneziano’s formula and its connection to Regge poles remained elusive until it was realized that the formula could be derived from a model where particles are not point-like but are instead envisioned as one-dimensional “strings.”

This insight led to the foundation of string theory, initially called dual resonance models, where the fundamental objects are strings whose vibrational modes correspond to different particles. The resonance phenomena, explained by Regge theory, naturally emerged from the vibrational patterns of these strings, providing a unified description of hadrons that could potentially include the graviton—the hypothetical quantum of gravity.

The initial excitement over string theory’s ability to describe the strong force waned as quantum chromodynamics (QCD) emerged as a more accurate theory for that purpose. However, string theory found new life in the 1970s and 1980s as a promising candidate for a quantum theory of gravity and a unified framework for all fundamental interactions. The introduction of supersymmetry, leading to superstring theory, further enriched the theoretical landscape, setting the stage for string theory’s evolution into a leading approach for addressing some of the most profound questions in physics.

Thus, the development of string theory, from its inception to address the issue of Regge poles and the strong nuclear force, to its current status as a potential theory of everything, illustrates a remarkable trajectory of scientific innovation and interdisciplinary collaboration.

Gabriele Veneziano made a groundbreaking contribution to theoretical physics in the late 1960s when he presented the Veneziano formula, which provided a fresh perspective on hadron scattering processes. This formula supplied the groundwork for the development of string theory in addition to fitting in with Regge theory, a framework developed to explain the angular momentum characteristics of particles. We discuss the Veneziano formula in this article, along with how it relates to Regge poles and how it makes the conceptual jump to string theory.

The Veneziano Formula: The Veneziano amplitude provided a solution to the puzzle of hadron scattering amplitudes, encapsulating the duality observed in these processes. It is expressed as:

$A(s,t) = \frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))}{\Gamma(-\alpha(s)-\alpha(t))},$

where $A(s,t)$ represents the amplitude for scattering, $\Gamma$ is the Gamma function, and $\alpha(s)$ denotes the Regge trajectory, a linear function given by:

$\alpha(s) = \alpha_0 + \alpha' s.$ Here, $\alpha_0$ is the intercept, and $\alpha'$ is the slope of the trajectory.

Let us see what it means with a concrete example.

Recall the Veneziano amplitude: $A(s,t) = \frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))}{\Gamma(-\alpha(s)-\alpha(t))}$ , where the Regge trajectory $\alpha(s)$ is linear in $s$ , the square of the total energy in the center-of-mass system: $\alpha(s) = \alpha_0 + \alpha' s$ . For this example, let’s assume values for the intercept $\alpha_0$ and the slope $\alpha'$ that are typical in discussions of string theory, though in real-world applications these would be determined by experimental data:

Let $\alpha_0 = 0.5$ , a common intercept for light mesons.
Let $\alpha' = 1\ \text{GeV}^{-2}$ , a typical value for the slope of Regge trajectories.
Example: Calculating the Amplitude for Specific $s$ and $t$ . Suppose we want to calculate the amplitude for specific values of s and t to understand the interaction at a certain energy and momentum transfer. Let’s choose:
$s = 2\ \text{GeV}^2$ (total energy squared),
$t = -1\ \text{GeV}^2$ (momentum transfer squared, negative because it’s spacelike).
First, calculate the values of $\alpha(s)$ and $\alpha(t)$ :
$\alpha(s) = 0.5 + 1 \cdot 2 = 2.5,$ $\alpha(t) = 0.5 + 1 \cdot (-1) = -0.5.$
Then, substitute these into the Veneziano amplitude: $A(2, -1) = \frac{\Gamma(-2.5)\Gamma(0.5)}{\Gamma(-2)}.$ Using the Gamma function properties and values (recall that $\Gamma(n) = (n-1)!$ for positive integers, and $\Gamma(0.5) = \sqrt{\pi}$ ): $A(2, -1) \approx \frac{\Gamma(-2.5)\sqrt{\pi}}{\Gamma(-2)}.$ The Gamma function for negative non-integer values would typically require numerical computation, but for illustrative purposes, let’s focus on the structure of the calculation rather than the exact numerical result.

A computation of the Veneziano amplitude for a given set of s and t values is only one piece of a much broader puzzle, and it may not reveal all about particle physics or string theory. Physicists can develop and improve theories that seek to explain the basic principles underlying the world by having a better understanding of the relationship between these computations and physical facts.

Connection to Regge Poles: Regge theory introduces the concept of Regge trajectories and poles, fundamental to understanding the scattering amplitudes at high energies. The Veneziano formula’s incorporation of these trajectories highlights the theory’s predictive power and its elegant mathematical structure. The linear relationship in the Veneziano formula mirrors the linear Regge trajectories observed in particle physics.

In Regge theory, the trajectory of a particle is a function that relates the angular momentum $J$ of the particle to its mass squared $m^2$ , symbolized as $J = \alpha(m^2)$ . This function is called a Regge trajectory, and it is observed to be approximately linear for many families of particles, which can be expressed as: $J = \alpha(m^2) = \alpha_0 + \alpha' m^2,$ where $\alpha_0$ (the intercept) and $\alpha'$ (the slope) are constants. The Veneziano amplitude can be written as: $A(s,t) = \frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))}{\Gamma(-\alpha(s)-\alpha(t))},$ where s and t are the Mandelstam variables representing the square of the total energy and the square of the momentum transfer in the center-of-mass frame, respectively {1}. The function $\alpha(s)$ here represents a Regge trajectory, with $s$ playing the role of $m^2$ (mass squared of the exchanged particle in the t-channel). By substituting the linear form of the Regge trajectory into the Veneziano amplitude, we explicitly see the connection: $\alpha(s) = \alpha_0 + \alpha' s$ . This implies that the resonances (or poles) in the amplitude, which correspond to physical particles, occur at values of $s$ and $t$ where $\Gamma(-\alpha(s))$ and $\Gamma(-\alpha(t))$ have poles, which is when $\alpha(s)$ and $\alpha(t)$ are non-negative integers because $\Gamma(-n)$ has poles for non-negative integers $n$ .

Illustration with an Example: Suppose a Regge trajectory with $\alpha_0 = 0.5$ and $\alpha' = 0.9\ \text{GeV}^{-2}$ . For a given resonance with $J = 2$ (angular momentum of the exchanged particle), we can solve for its mass squared: $2 = 0.5 + 0.9 m^2,$ , $m^2 = \frac{2 - 0.5}{0.9} \approx 1.67\ \text{GeV}^2.$ This shows how a specific point on the Regge trajectory corresponds to a resonance with a certain mass and angular momentum.

The Veneziano formula, through its incorporation of Regge trajectories, provides a powerful framework for predicting the properties of resonances in particle physics. The elegance of the formula lies in its ability to encapsulate the duality and linear relationship of Regge trajectories, offering insights into the scattering amplitudes at high energies and the spectral organization of particles.

Link to Strings: The Veneziano formula’s implications extend beyond particle physics, providing a gateway to string theory. The formula can be derived from a model in which fundamental entities are one-dimensional strings, rather than point particles. This realization unveiled a new perspective on the fabric of the universe, where the vibrational states of strings correspond to different particles, including the graviton—a hypothetical quantum of gravity naturally emerging from string theory.

Example: Prediction of the Graviton: A notable example of the Veneziano formula’s impact is its implication for quantum gravity. String theory predicts the existence of a massless spin-2 particle, identified as the graviton, through the quantization of string modes. This discovery illustrates the formula’s far-reaching consequences, bridging the gap between the strong nuclear force and the gravitational force within a unified theoretical framework. The journey from the Veneziano formula to string theory encapsulates a significant chapter in the history of theoretical physics. It illustrates how a mathematical formulation intended to describe hadron scattering led to the development of a theory that promises to unify all fundamental forces of nature, showcasing the intricate beauty and interconnectedness of the universe’s fundamental structures.

String Theory Basics, String Action (Polyakov Action):

The action for a string propagating in spacetime is given by:

$S = -\frac{1}{2\pi\alpha'} \int d\tau d\sigma (\partial_{\alpha} X^{\mu} \partial^{\alpha} X_{\mu})$

where the parameter $\alpha'$ is related to the string tension, and $X^{\mu}(\sigma,\tau)$ are the spacetime coordinates of the string. $\sigma, \tau$ are the parameters of the string worldsheet.

In string theory, the parameter α′ plays a crucial role in the dynamics of strings. It is closely related to the tension of the string, which is a fundamental property determining how the string behaves and interacts. α′ is inversely proportional to the tension of the string, T. The tension can be thought of as the energy per unit length of the string. Higher tension means the string is stiffer, as we know from classical vibrations in a guitar. Mathematically, this relationship is expressed as $T=\frac{1}{2 \pi \alpha '}$ .

The value of α′ affects the mass spectrum of the particles represented by string modes. It determines the scale at which stringy effects become significant. Typically, it’s set to a very small value, ensuring that stringy behavior is evident only at very high energies, far beyond the reach of current particle accelerators.

Example with a Known Particle:

Quantization: By quantizing this action (applying quantum mechanics to string theory), we obtain a spectrum of possible vibrational states of the string. Each state can be interpreted as a different particle. Calculating Particle Properties, Mass Spectrum: The mass of the vibrational modes is determined by the level of excitation of the string. For a closed string, the mass formula in the simplest case is given by: $M^2 = \frac{2}{\alpha'}(N-1)$ where $N$ is the number operator, representing the level of string excitation.

Spin and Other Quantum Numbers: The type of vibration (how the string vibrates in different dimensions) determines the spin and other quantum numbers of the particles. For instance, a string vibrating in a particular mode might correspond to a particle with spin-1, like a photon.

Tachyons and Stability: Early models of string theory predicted particles with imaginary mass, known as tachyons, indicating an instability in the theory. This led to the development of superstring theory, which includes fermions and supersymmetry, to address these issues.

General Relativity (Higher-Dimensional Metrics): In higher-dimensional general relativity, the spacetime metric extends beyond four dimensions. For instance, in 5D Kaluza-Klein theory, the metric tensor $g_{AB}$ (where $A, B$ run from 0 to 4) combines the 4D spacetime metric with an extra dimension, potentially unifying gravity and electromagnetism.

Topological Quantum Field Theory (TQFT): In TQFT, one often encounters algebraic structures like tensor categories. An example is the partition function $Z$ which assigns a complex number to a manifold $M$ :
$Z(M) = \int \mathcal{D}\phi e^{iS[\phi]}$ where $S[\phi]$ is the action functional of the field $\phi$ over the manifold $M$ . In higher dimensions, these structures become more complex and are often studied using higher categorical algebra.

Condensed Matter Physics (Topological Insulators): In topological insulators, the Chern number, an integer describing the topology of the band structure, plays a crucial role. It is given by an integral over the Brillouin zone:
$C_n = \frac{1}{2\pi i} \int_{BZ} d^2k \, \text{Tr} \left( F_{12}(k) \right)$
where $F_{12}(k)$ is the Berry curvature in momentum space. In higher-dimensional systems, these integrals extend over more complex manifolds.

String Theory Quantization and Mass Spectrum

The process of deriving the mass spectrum in string theory from the Polyakov action involves several steps, outlined as follows:

Mode Expansion: The string’s spacetime coordinates, $X^\mu(\sigma, \tau)$ , are expanded in terms of mode functions.

Canonical Quantization: Apply quantum mechanics principles, promoting classical fields to quantum operators and imposing commutation relations.

Virasoro Operators: From the quantization, we obtain Virasoro operators $L_m$ , which are central to string theory. These operators are constructed from the energy-momentum tensor derived from the Polyakov action.

The Virasoro operators play a pivotal role in the quantization of string theory, providing an algebraic structure essential for the theory’s consistency. They emerge from the energy-momentum tensor derived from the Polyakov action, encapsulating the dynamics of strings in spacetime.

-Virasoro Operators: The Virasoro algebra consists of an infinite set of operators $L_m$ and their antiholomorphic counterparts $\bar{L}_m$ , where $(m)$ is an integer. These operators satisfy the following commutation relations:

$[L_m, L_n] = (m - n)L_{m+n} + \frac{c}{12}m(m^2 - 1)\delta_{m+n,0},$

$[\bar{L}_m, \bar{L}_n] = (m - n)\bar{L}_{m+n} + \frac{\bar{c}}{12}m(m^2 - 1)\delta_{m+n,0},$

where $c$ and $\bar{c}$ are the central charges, and $\delta_{m+n,0}$ is the Kronecker delta function.

-Physical State Conditions: A state $|\psi\rangle$ in string theory is considered physical if it satisfies the Virasoro constraints:

$(L_m - \delta_{m,0})|\psi\rangle = 0, \quad \text{for } m \geq 0,$

$(\bar{L}_m - \delta_{m,0})|\psi\rangle = 0, \quad \text{for } m \geq 0.$

-These conditions ensure the elimination of negative norm states and the proper quantization of the string’s energy levels.

Example: Massless Scalar Field: To illustrate the application of Virasoro operators, consider a bosonic string in flat spacetime. The mode expansion of $X^\mu$ leads to quantized oscillatory modes, and the Virasoro operators are expressed as:

$L_m = \frac{1}{2} \sum_{n=-\infty}^{\infty} :\alpha_{m-n} \cdot \alpha_n:$ ,

where $\alpha_n^\mu$ are the oscillator modes, and the normal ordering is denoted by $:.:$ . For a massless scalar field, consider the state created by acting on the vacuum with $\alpha_{-1}^\mu$

$|\psi\rangle = \alpha_{-1}^\mu |0;k\rangle.$

This state represents a massless particle, satisfying the physical state conditions, including the crucial $L_1$ constraint: $k_\mu \alpha^\mu_1 |0;k\rangle = 0,$ indicating that $k^2 = 0$ , where $k$ is the momentum of the state, identifying it as a massless scalar field in the spectrum of string theory. The Virasoro operators are instrumental in defining the physical states of string theory, ensuring the theory’s consistency by eliminating unphysical states and determining the properties of the particles in the string spectrum.

The plot above illustrates a Regge trajectory, which is a linear relationship between the angular momentum J and the mass squared (s) of particles. This concept is pivotal in understanding the dynamics of hadrons in particle physics. Each point on the trajectory represents a different vibrational state of a string, corresponding to different particles (e.g., mesons, baryons, tetraquarks, pentaquarks) with increasing mass and angular momentum. This visualization encapsulates the essence of how string theory’s vibrational modes relate to the physical properties of particles, echoing the foundational principles laid out by the Veneziano formula and Regge theory.

Level Matching Condition: For closed strings, the left-moving and right-moving modes of vibration must be equal, leading to a level matching condition $L_0 = \bar{L}_0$ .

Mass Formula: The mass-squared operator in string theory is related to these Virasoro operators. For a closed string, the mass-squared is given by:
$M^2 = \frac{4}{\alpha'} (L_0 - 1)$
for the left-moving modes and similarly for the right-moving modes with $\bar{L}_0$ .

But the main interest for this text is: can string theory be helpful for the society, for anyone beyond the limited and rarefied atmosphere of the savants? The answer is, yes.

The formalism and ideas of string theory, while fundamentally rooted in the quest to understand the universe’s smallest constituents and forces, also inspire innovative approaches across various fields, including medicine, engineering, and finance. Here are some specific examples where the formalism and ideas of string theory might find application outside of pure physics:

Medicine: Understanding Genetic Diseases

Genomic Sequences as Strings: In bioinformatics, sequences of DNA and RNA can be thought of as strings of nucleotides. Analogous to how string theory considers vibrating strings as fundamental elements that compose particles, genomic sequences can be analyzed using string theory formalisms to understand genetic variations and mutations. This perspective can aid in the development of gene therapies and precision medicine by modeling the interactions between different genetic elements in complex diseases.

Engineering: Design of Meta-materials

Vibration and Resonance in Materials: String theory’s focus on vibration and resonance has parallels in the engineering of meta-materials, which are artificial materials engineered to have properties not found in naturally occurring materials. By applying the concepts of vibration modes from string theory, engineers can design meta-materials with unique electromagnetic or acoustic properties, useful in developing cloaking devices, superlenses, and highly efficient energy transmission systems.

Finance: Modeling Market Dynamics

Complex Systems and Entanglement: String theory’s treatment of entangled states, where particles remain connected across vast distances, offers a metaphorical framework for understanding complex financial systems where global markets are deeply interconnected. By borrowing mathematical tools from string theory, financial analysts could model market dynamics under the lens of entanglement, potentially offering new insights into how information and trends propagate through global financial networks, affecting asset prices and market stability.

Cross-disciplinary: Network Theory and Machine Learning

Topology and Connectivity: The study of Calabi-Yau manifolds and other complex topologies in string theory can inspire new algorithms in network theory and machine learning. Understanding how these shapes encode information about extra dimensions could parallel how information is structured and flows in complex networks, leading to innovative ways to analyze connectivity patterns in social networks, the brain, or the internet.

Quantum Computing

Quantum Information Processing: String theory’s exploration of higher-dimensional spaces and quantum gravity might inform the development of quantum computing algorithms by providing insights into the nature of quantum entanglement and coherence. As quantum computing seeks to harness these properties for computing power, insights from string theory could guide the design of more efficient quantum algorithms and error correction methods, with applications ranging from cryptography to drug discovery.

While the direct application of string theory’s formalism to these fields may require a creative leap, the underlying mathematical tools and conceptual frameworks offer a rich source of inspiration. The analogies and insights derived from string theory can stimulate innovative approaches to longstanding problems, driving advancements in technology, finance, and healthcare.

Just look at these references to imagine the span of applications of string theory concepts in fields outside of traditional physics:

Data Science Applications to String Theory by Fabian Ruehle (2020) discusses how machine learning and data science techniques can be applied to string theory, including example codes. These approaches could potentially be adapted for use in medicine, engineering, and finance .https://www.sciencedirect.com/science/article/pii/S0370157319303072
Nonlocality in String Theory by G. Calcagni and L. Modesto (2013) explores the concept of nonlocality in string theory, which could have implications for understanding complex systems in engineering and finance. https://iopscience.iop.org/article/10.1088/1751-8113/47/35/355402/pdf
String Field Theory by Harold Erbin provides an introduction to string field theory, which as a field theory, offers a constructive formulation of string theory. This formalism could inspire new computational models in various disciplines. https://arxiv.org/abs/2301.01686
String Theory and Particle Physics: An Introduction to String Phenomenology by L. Ibáñez and Á. Uranga (2011) focuses on how string theory is connected to the real world of particle physics, providing models of physics beyond the Standard Model. The methodologies discussed could find applications in computational biology and complex systems engineering. https://www.cambridge.org/core/books/string-theory-and-particle-physics/7D005A97DA657F6675C2A62E449FC62E
Introduction to String Theory by Samarth Parekh (2022) offers an overview of the basic concepts of string theory, including its implications for quantum field theory, gravitational physics, and the nature of spacetime. These concepts could potentially influence advanced computational techniques in medicine and finance. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4009963
Focus Issue on String Cosmology by V. Balasubramanian and P. Moniz (2011) appraises recent applications of string-theoretic and string-inspired ideas to cosmology. The discussion on alternative models and dynamics could inspire novel approaches in data analysis and predictive modeling in finance. https://iopscience.iop.org/article/10.1088/0264-9381/28/20/200301

POSTSCRIPT:

The strong nuclear force appears to be described by Leonhard Euler’s beta function, as Gabrielle Veneziano inadvertently found in 1968. Because it is a function of the binomiale distribution, the beta function gets its name. As previously mentioned, it may be connected to the gamma function. In 1969, Veneziano discovered that Euler’s beta function was a suitable option when searching for a function that satisfies the basic postulates for the scattering amplitudes of elementary particles in strong contact. String theory was launched by this initial step, despite the fact that Veneziano was searching for a theory of strong interactions.

According to string theory, the cosmos is made up of vibrating energy filaments, or strings, as it presents us with a new understanding of the universe. The presence of additional physical things known as branes is also predicted by this scientific theory. String theory is a quantum field theory that describes the existence of particles and the generation of forces based on the compactification—the process of condensing hidden dimensions into extremely small sizes—at the level of high-energy theoretical physics.

The hypothesis was created in 1968 in an effort to explain hadron behaviour, but it was quickly shelved until the middle of the 1980s since it required more hidden dimensions. It developed into the M-theory, a more complex theory, in the 1990s [FN1]. Not only does string theory bring all particles together, but it also brings together the forces that govern their interactions. The electromagnetic field is a force that kind of spreads over space, enabling object interaction, and it fits the quantum theory rather well. Gravity doesn’t, though. And why? primarily because space is warped by gravity [FN2].

Additional References:

[FN1] The other leading alternative is known as loop quantum gravity. According to this theory space consists of not ordinary atoms, but extremely small chunks of space (like knots in a carpet).

[FN2] Quantum theory of gravity try to handle with this.

{1} The variables s and t are Mandelstam variables, key components in the description of particle collisions. They are defined in the context of a scattering process involving two incoming particles and two outgoing particles.

The Mandelstam variable s is defined as the square of the total energy in the center-of-mass frame. It reflects the overall energy available for the interaction. In mathematical terms, for particles with four-momenta p1 and p2, it’s defined as $s=(p1+p2)^2$ . It gives an indication of the energy level at which the scattering process occurs.
The Mandelstam variable t, on the other hand, represents the square of the momentum transfer between the incoming and outgoing particles. It’s defined as $t=(p1-p3)^2$ , where p3 is the four-momentum of one of the outgoing particles. This variable measures how much the direction of motion of the particles has changed due to the scattering process.

The function α(s) and α(t) are Regge trajectories, which relate the spin of a particle to its mass squared, essentially describing how the properties of exchanged particles in the interaction vary with s and t. These trajectories are central in reggeon field theory, a framework used before QCD to describe the strong interaction at high energies.

The Veneziano amplitude thus combines these elements to predict the likelihood (amplitude) of various outcomes of particle collisions at different energy levels (s) and angles (t). Its form, invoking the Gamma function, ensures the amplitude has poles at integer values of −α(s) and −α(t), corresponding to the masses of the resonances involved in the scattering process. This was an attempt to explain the pattern of hadron masses and their interactions in the era before quarks and gluons were established as the fundamental constituents of hadrons.

Acknowledgment: Part of this post was based on the course materials provided by MIT’s course, “String Theory for Undergraduates” available in Spring 2007. More information can be found at: https://ocw.mit.edu/courses/8-251-string-theory-for-undergraduates-spring-2007/

Comments Leave a Comment
Categories Academia, Education and Science, Engineering, Medicine, Quantum computers, String Theory

Future-Scaping: The Social Bases of Cognitive Development

12 Apr

Are we on the verge of a paradigm shift in human evolution, surpassing the age of artificial intelligence in the imagined epoch of “human consciousness singularity”? This change, if made possible by leaders (political or of another kind), would herald the reawakening of latent human potential, providing pathways to increased awareness, empathy, and comprehension. It demands a paradigm shift from the existing chaos-driven, materialist world order to one based on enlightenment, ethical principles, and integrated growth. Unlocking the entire potential of humanity requires not only technological growth but also an expansion of the human spirit, which is represented by this singularity.

Introduction

We explore a deep idea in this investigation: the path mankind is taking to become a solitary species with superior intelligence. This voyage goes beyond the capabilities of artificial intelligence (AI), presenting it as only a minor footnote to the larger story of human evolution. We are on the cusp of tapping into dormant capacities, unlocking new realms of consciousness, awareness, and knowledge. To make that happen, though, our societies must work with the following conditions: fair economic growth and wealth distribution; unwavering defense of cultural freedom; harmonious integration of various nationalities; astute and honest public administration; an assembly of the bright in a congenial setting; and educational institutions where instructors are proprietors of their own profession. These circumstances are not unheard of; in fact, they were previously met by the Republic of Venice, which, as a result, attained unparalleled financial strength and international renown. Therefore, we hypothesize that the countries that adopt these values will undoubtedly remain ahead of the rest, which will be mired in anarchy, egotistical desire for the amusement of a select few, and corrupted institutions with extremely restrictive constraints (religious or authoritative).

Was the Republic of Venice purified from sin? Of course not.

The Republic of Venice, known for its longevity and success as a maritime power, indeed had its share of notable achievements. However, like any significant historical entity, it also had its negative aspects:

Oligarchic Rule: Despite being a republic, Venice was effectively governed by a small, elite group of wealthy merchant families. This oligarchy concentrated power and wealth in the hands of a few, often at the expense of wider democratic participation.
Imperial Expansion and Control: Venice’s expansionist policies led to the control and domination of numerous territories across the Mediterranean. This often involved military conflicts, subjugation of local populations, and colonization.
Trade Monopolies and Economic Manipulation: The Venetian economy was largely based on trade monopolies and the manipulation of markets. This sometimes led to economic exploitation of other regions and unethical trade practices.
Involvement in the Slave Trade: Like many powers of the time, Venice was involved in the slave trade. This aspect of their economic practice is often overshadowed by their more celebrated achievements in trade and commerce.
Internal Repression: The Venetian state had a well-known secret service and a system of informants, which it used to control its population and maintain power. The Doge’s Palace had notorious prisons, and the state did not hesitate to quash dissent.
Class Divide and Social Inequity: There was a significant class divide in Venetian society, with a large underclass who did not enjoy the wealth and power of the elite merchant classes.

Regarding their alliances and ethical stance in finances and administration, Venice was primarily pragmatic. Their alliances were often based on trade interests, political expediency, and the balance of power in the region rather than a quest for ethical congruence. Venice was known for its shrewd diplomacy and often played larger powers against each other to maintain its independence and trade dominance.

Though they had in place methods to guard against the possibility of corruption among their ruling classes, for example, the very convoluted process in the election of the Doge, all these methods have been part and parcel in maintaining stability and status quo rather than allowing themselves to pursue broader ethical alliances {1}. Venice was a commercial empire at its core, and its primary interests lay in maintaining and expanding its trade networks and political influence. But throughout history, there have been several republics and nations that have been recognized for their high values, successful governance, and the general happiness of their citizens. It’s crucial to remember, though, that measuring “happiness” and “success” can be arbitrary and frequently dependent on historical circumstances. Here are a few noteworthy instances:

The Athenian Democracy (5th to 4th century BCE):
- Ancient Athens is often cited as the birthplace of democracy. During its Golden Age, particularly under the leadership of Pericles, Athens experienced significant cultural and intellectual growth. Citizens (albeit a limited group excluding women, slaves, and non-citizens) were actively involved in governance.
The Roman Republic (509–27 BCE):
- Before becoming an empire, Rome was a republic with a complex system of governance involving checks and balances. It achieved significant military and cultural successes, and its legal system influenced many modern legal codes.
The Republic of Venice (7th century–1797 CE):
- Known for its economic prosperity and stability, Venice was a major maritime power and an important center of commerce and art. Despite being an oligarchy, it maintained relative internal stability and prosperity for centuries.
The Iroquois Confederacy (Haudenosaunee):
- Pre-dating European colonization of North America, the Iroquois Confederacy was a union of six Native American nations. Known for its sophisticated system of governance, it inspired some elements of the U.S. Constitution.
Bhutan in Contemporary Times:
- Bhutan is known for its unique approach to measuring success through Gross National Happiness (GNH) rather than Gross Domestic Product (GDP). This Himalayan nation prioritizes the well-being of its citizens over economic growth.
The Nordic Countries (contemporary):
- Nations like Denmark, Sweden, Norway, and Finland are often cited for their high standard of living, strong social welfare systems, and democratic governance. They consistently rank high in global happiness and quality of life indices.
The United Provinces of the Netherlands (17th century)
- During the Dutch Golden Age, the Netherlands was a republic characterized by religious tolerance, a flourishing economy, a strong navy, and significant contributions to art and science.
The Portuguese Empire, during its Golden Age in the 15th and 16th centuries, stands out as a remarkable example of exploration, trade, and cultural exchange. This period, often referred to as the Age of Discovery, was marked by Portugal’s pioneering role in global maritime exploration. Maritime Achievements: Under the leadership of visionaries like Prince Henry the Navigator, the Portuguese developed advanced navigation techniques and ship designs, such as the caravel. This facilitated unprecedented exploration, including Vasco da Gama’s historic voyage to India and the discovery of Brazil by Pedro Álvares Cabral; Global Trade Networks: Portugal established a vast trade network spanning from Africa to Asia, bringing considerable wealth to the empire. They were instrumental in the global spice trade and played a significant role in establishing early modern global commerce; Cultural Exchanges and Influence: The Portuguese Empire was not just about commerce and navigation; it also led to significant cultural and linguistic exchanges. Portuguese became an important lingua franca in Asia and Africa, and the cultural fusion can still be seen in the architecture, cuisine, and customs of these regions; Colonial Administration: The empire’s administration ranged from direct colonial rule to more complex systems of alliances and vassalage with local rulers. This variety in governance models showcased a certain level of adaptability in their colonial administration; Impact on the Modern World: The Portuguese Empire left a lasting impact on the modern world. The age of exploration they spearheaded marked the beginning of globalization and significantly shaped the subsequent course of world history.

The reasons for each of these instances’ success or contented residents varied widely, and each had its own intricacies and difficulties. Key components of these societies have included things like social welfare programmes, cultural accomplishments, economic success, involvement in government, and an emphasis on the welfare of people in general. But it’s also critical to acknowledge that historical situations were distinct from our own, and that what was deemed “happy” or “successful” may not have the same connotations now.

Singularity of Human Intelligence—Going Beyond AI The idea of a “singularity” in human intellect refers to a critical juncture in human history, when cognitive capacities significantly surpass existing constraints. This expansion aims to awaken latent innate human potentials rather than merely boosting human intellect through machine learning. This human-centric singularity highlights the distinctive characteristics of human consciousness, in contrast to the singularity sometimes addressed in the context of AI, when computers transcend human cognition.

Unlocking Dormant Capacities: Humans have untapped mental and cognitive potential that, if realised, might lead to entirely new realms of knowledge and competence. These include improved intuition skills, stronger empathy, and a more thorough comprehension of the cosmos. This progress may be aided by deeper research into the workings of the human mind as much as by technological advancements.

Environment and State Organization’s Role: The correct atmosphere and state structure are essential for such a historic shift to take place. It is imperative that we have a culture that develops these innate talents, based on moral principles and values that advance overall development and wellbeing. Against this ideal lies our existing world system, which is primarily motivated by chaos and financial goals. It frequently impedes the full development of human potential by emphasising monetary rewards almost exclusively at the expense of more profound intellectual and spiritual development.

Transitioning from Materialism to Holistic Development: The world is beset with environmental catastrophes and inequality as a result of the unrelenting quest of material wealth and power. Not only has this materialistic attitude stunted human growth, it has put our world in jeopardy (possibly in the form of a nuclear conflict). Humanity’s future depends on a paradigm change away from materialism and towards a balanced approach to growth, which prioritises intellectual, spiritual, and emotional development.

The Ethics of Human development: The ethical ramifications of human development become apparent at this pivotal moment. The pursuit of a singularity of human intellect must be steered by a moral compass that prioritises the development of the person as well as the welfare of mankind as a whole. The foundations of this new period will be ethics in everyday life, education, and policymaking.

It’s interesting to think about how the customs and beliefs of ancient cultures, such those led by the Druids, would have been in line with the early aspirations to achieve a kind of cognitive or awareness singularity. Respected in their communities, the Druids served as both religious leaders and repository of learning and information. Their responsibilities as counsellors and judges, their profound comprehension of natural events, and their thorough memory of oral traditions all point to a highly developed cognitive ability. One may see the Druids’ interaction with nature, their rituals, and their function as intermediaries between the material and spiritual worlds as an early example of the pursuit of a higher level of consciousness. One may argue that their activities were intended to help them develop a deep awareness and connection with the world around them. This could be interpreted as an endeavour to attain the pinnacle of human cognitive and conscious capacity, similar to the concept of a “Human Cognitive Singularity.” From this angle, we may see the Druids’ significance in their communities not only in terms of their religious and cultural practices, but also as pioneers in the development of awareness and cognition. Though their methods and settings were very different, the Druids’ desire for wisdom and knowledge reflects the current interest in comprehending human consciousness and cognitive capacity, providing a historical parallel to current debates over the development of human intelligence and awareness.

Conclusion: The concept of the world as will and representation, as envisaged by philosophers like Schopenhauer, takes on new meaning in this context. It’s not just about interpreting the world through our sensory experiences and mental faculties, but about reshaping these faculties to experience and understand the world in ways previously unimaginable. By embracing this path towards the singularity of human intellect, we open doors to new realms of existence, relegating AI to a supporting role in the rich history of humanity. This trip is about the evolution of our basic nature as human beings, not merely about technology growth.

*The image was generated freely by Dall-3*

The word Human Cognitive Singularity highlights the evolution’s human-centric aspect, emphasising the amplification and awakening of innate human abilities above the advancement of artificial intelligence. It alludes to a paradigm shift in human evolution, one in which a profound change in our perception of the self, the world, and ourselves results from the growth of awareness and cognitive capacities.

To sum up, the term “Human Consciousness Singularity” or “Human Cognitive Singularity” accurately characterises this anticipated shift in human development, setting it apart from the machine-oriented notion of AI singularity.

COMMENTS:

{1} – The procedure was created to reduce the level of corruption in the governing classes. It was elaborate and multifaceted, with many selection phases to guarantee that no one person or group could simply influence the result. This painstaking process was a component of Venice’s larger initiatives to preserve political stability and avoid power consolidation. The convoluted electoral procedure had unexpected repercussions even while it helped preserve the status quo and served as a safeguard against corruption. Venice’s capacity to pursue more expansive moral endeavours or form new alliances may have been constrained by its intense concentration on stopping the abuse of power among her governing classes. Put another way, Venice may have found it more difficult to adjust or maintain stability due to the very systems that guaranteed internal integrity and stability.

Comments Leave a Comment
Categories Academia, Education and Science, Enlightenment, History of Science, Science and Philosophy

Overcoming the Gambler’s Ruin: Techniques for Optimising Winning Circumstances

11 Apr

The Gambler’s Ruin problem, which simulates a gambler’s path from victory to failure in a game of chance, provides an intriguing look into the mathematical foundations of gambling. In essence, the concept offers a fair game in which a player wagers until they either lose all of their money or hit a specified financial target. Framed within a Markov chain, this scenario provides valuable insights into the optimal circumstances that allow a gambler to increase their odds of winning. Here, we investigate these factors and their effects on the result of such probabilistic efforts. This is the principle of the method:

Starting Strong: Higher Initial Fortune: One of the most direct pathways to increasing your odds of winning, according to the model, is by beginning the game with a larger initial fortune. The probability of reaching your financial goal before depleting your resources grows as your starting point ( $X_0=k$ ) moves closer to that goal ( $N$ ), clearly illustrating the advantage of a strong start.
Setting Realistic Goals: Smaller Target Amounts: Conversely, setting a smaller goal for your gambling endeavor can significantly boost your chances of success. A smaller gap between your initial state and your financial target reduces the journey’s complexity, making victory more attainable within the model’s parameters.
Knowing When to Stop: Limited Play: While the play limitation tactic isn’t explicitly addressed in the traditional Gambler’s disaster trouble, implementing a stopping point based on existing earnings or losses can be a useful way to prevent disaster. This tactic recognises that gambling is an unpredictable activity and stresses the value of risk management in actual betting situations.
Managing Risk Wisely: Adjusting your betting strategy based on your current standing and the distance to your goal can help manage the inherent risk of gambling. While the Gambler’s Ruin assumes equal probability bets, practical application might involve varying bet sizes to protect your stake as you progress.
The Edge of External Factors: Ideal conditions might also include leveraging games where the odds slightly favor the gambler or utilizing insights that offer a probabilistic edge. These factors, however, introduce variables beyond the simple model of the Gambler’s Ruin, hinting at the complex nature of real-world gambling.

The Gambler’s Ruin issue provides practical insights into the nature of risk, reward, and techniques that might tip the odds in a gambler’s favour, going beyond a mere theoretical investigation of gambling dynamics. But it’s important to keep in mind that there are a lot of other elements that affect real-world gambling, such as psychological pressures, the house edge, and the thrilling, uncertain nature of chance. Therefore, even though the model offers a framework for comprehending and raising one’s odds, there are always inherent hazards associated with gambling’s unpredictability. The lessons from the Gambler’s Ruin provide insightful viewpoints on the fine line between risk and return in the gambling industry, regardless of your level of experience as a bettor or your interest in mathematics. Building on the fundamental ideas discussed in the Gambler’s Ruin argument, let’s examine the probabilistic equations and the optimisation procedure that can inform a gambler’s approach in this theoretical framework.

*The illustration depicts the Gambler’s Ruin problem’s delicate risk-reward ratio, which is supported by the probabilistic equations that control the gambler’s path.*

The Probabilistic Framework

At the heart of the Gambler’s Ruin problem lies a set of probabilistic outcomes defined by simple yet powerful equations. The primary equation, $P[X_\tau=N]=Nk$ , succinctly captures the gambler’s chances of reaching their goal (N) from an initial state (k). Because of its intrinsic linearity, this probability shows a clear correlation between the beginning luck and the chance of success. This equation is elegant because it may be used to measure hope by converting starting conditions into a quantifiable probability of success.

Furthermore, the expected time until the game concludes, denoted by $E[\tau]=k(N-k)$ , provides insight into the duration of the gambler’s endeavor. This equation reflects the dual impact of starting proximity to the goal and the potential depth of the journey towards ruin. It underscores an optimization challenge: balancing the ambition of one’s goal against the pragmatic considerations of risk and duration.

Optimization in the Gambler’s Ruin

Using these equations as a guide, one may optimise the Gambler’s disaster framework to increase the likelihood of success while lowering the chance of disaster. The gambler must strike a careful balance between caution (the risk of going to zero) and ambition (the aim N).

Initial Fortune Management: Increasing the initial fortune (k) is the most straightforward method to improve winning odds. However, this is often outside the gambler’s control. Therefore, optimization might involve saving or accumulating a “war chest” before embarking on the gambling journey, aligning with the principle of starting strong.
Goal Setting: Setting a realistic and attainable goal (N) can significantly enhance the likelihood of success. This involves a self-aware assessment of one’s resources and constraints, optimizing for a target that balances ambition with the practical reality of the gambler’s situation.
Risk Adjustment: Adapting the betting strategy to manage the size and frequency of bets in response to wins and losses can help mitigate risk. While the simple model assumes equal bets, real-world application could involve reducing bet sizes as one approaches the goal or after significant losses, optimizing the preservation of capital.
Utilizing Stopping Points: Introducing predefined stopping points based on profit or loss thresholds allows the gambler to exit the game before extreme outcomes occur. This strategy optimizes the engagement in gambling by setting rational limits that protect against the natural variance of chance.
Exploring Favorable Conditions: Seeking out gambling opportunities where the odds are slightly in favor, or at least not heavily against the gambler, can also form part of an optimization strategy. While true “fair games” are rare, being selective about where and how to gamble can improve overall outcomes.

Concrete Example of Optimization in the Gambler’s Ruin Problem

To illustrate the optimization process within the Gambler’s Ruin framework, let’s consider a practical example of a gambler named Alice, who steps into a casino with a certain strategy based on her initial conditions and the equations that govern her probabilities and expected duration of play.

Initial Setup:

Alice starts with an initial fortune of $50 (k=50).
Her goal (N) is to reach $100 before she goes broke.
The game she plays is a fair coin toss where she bets $10 on each toss.

Application of the Probabilistic Framework and Optimization Strategies:

Initial Fortune Management:
- Alice starts with an initial fortune of \$50, which gives her a probability of reaching her \$100 goal as: $P[X_\tau = N] = \frac{k}{N} = \frac{50}{100} = 0.5 \text{ or } 50\%.$ To optimize her chances, Alice could consider starting with more money or choosing a lower target to increase this probability.
- Goal Setting: Alice’s target of \$100 is realistic given her initial fortune, but she might decide to adjust her goal lower (say \$80 instead of \$100) to increase her success probability to: $P[X_\tau = N] = \frac{k}{N} = \frac{50}{80} = 0.625 \text{ or } 62.5\%.$
Risk Adjustment:
- Initially betting $10 per toss, Alice should consider reducing her bet size if her fortune decreases. For instance, if her fortune drops to $30, reducing her bet to $5 can decrease the risk of ruin and extend her play, giving her more chances to reach her goal.
Utilizing Stopping Points:
- Alice decides in advance that if she drops to $20, she will stop playing to avoid total ruin, or if she reaches $90, she will also stop and cash out, securing her winnings and avoiding major losses close to her goal.
Exploring Favorable Conditions:
- Alice scouts the casino for games that might offer better than fair odds or looks for promotions that give players an edge, such as doubling their first win or providing a rebate on losses.

Optimization Play-through:

Alice begins playing, and during her session, she adjusts her bets according to her predefined strategy.
When her funds increase close to her adjusted goal of $80, she lowers her bet to minimize risk.
If Alice reaches a point where her funds are $20, she stops as per her stopping rule, minimizing losses.

Conclusion

The intricate relationship between chance, strategy, and human ambition is shown via the mathematical and strategic analysis of the Gambler’s Ruin dilemma. People may negotiate the inherent uncertainties of gambling with more knowledge and control by learning and using the concepts of probabilistic outcomes and optimisation tactics. But it’s important to understand that theoretical models can’t fully account for all the factors present in real-world gambling situations, such as the psychological effects and unpredictable nature of luck.

Acknowledgment: The problem discussed was inspired by the MIT course on Introduction to Probability, available online at https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/resources/gamblers-ruin/

Comments Leave a Comment
Categories Academia, Education and Science, Learning science

Yeomen and the Roots of Scientific Revolution: Rethinking the Heroic Theory of Progress

9 Apr

The conventional narrative of the scientific revolution, dominated by singular moments of breakthrough by heroic figures, has been a longstanding staple in our understanding of scientific progress. However, Peter Turney’s insights [1] invite us to challenge this perspective, shedding light on the broader socio-economic foundations that have historically underpinned major scientific advancements. My analysis extends this challenge further by spotlighting the pivotal role of the yeoman class in fostering an environment conducive to scientific innovation.

Exploring the lives of notable figures like Copernicus, Leibniz, Newton, Wallace, and Darwin reveals a pattern: their contributions to science were not made in isolation nor solely driven by individual genius. Instead, they were embedded within a class of independent thinkers and workers—yeomen—who, motivated by the competition and the intellectual ferment of their times, pushed the boundaries of knowledge. This class, often operating outside the traditional patronage systems that constrained much of intellectual life, was instrumental in creating the conditions for scientific breakthroughs.

The historical context of these scientific pioneers underscores the significance of economic independence and access to education and resources as catalysts for innovation. The yeoman class, with its relative autonomy and stability, provided a fertile ground for intellectual pursuits that questioned established dogmas and explored new frontiers of thought.

Furthermore, the consideration of public policy thinkers from Martin Luther King Jr to Charles Murray, who advocate for a universal citizens’ dividend, echoes the transformative potential of ensuring that all citizens have the economic security to pursue innovation and intellectual growth. Such proposals not only aim to address contemporary economic challenges but also to reinvigorate the spirit of inquiry and progress that marked the scientific revolution. This form of universal basic income (UBI), where all citizens receive a regular, unconditional sum of money from the government, would ensure that everyone has enough money to cover their basic needs, promoting economic security. This status would enable citizens to pursue innovation (creating new ideas, goods, services, and processes) and intellectual growth (expanding knowledge, education, and critical thinking). This proposal of a universal citizens’ dividend would solve or mitigate current economic issues, such as poverty, inequality, and unemployment. Economic security for all – instead the unsteady economic situation we live since decades – could lead to an era of intellectual vitality and innovation. That is what humanity needs.

Let us hope that this analysis invites us to broaden our understanding of the forces that drive scientific innovation. By recognizing the critical role of socio-economic structures, such as the yeoman class, in supporting the intellectual freedom and curiosity of history’s great scientists, we can better appreciate the complex interplay of individual genius, economic conditions, and societal values in shaping the trajectory of scientific progress.

*Yeomen: The Unsung Heroes of the Scientific Revolution*

REFERENCES:

[1] https://jimbowery.blogspot.com/2007/01/yeomen-as-foundation-of-scientific.html

Acknowledgments: This post was inspired in the blogue https://jimbowery.blogspot.com/2007/01/yeomen-as-foundation-of-scientific.html

Comments Leave a Comment
Categories Academia, Economic progress, Education and Science, Freedom, History of Science, Universal basic income (UBI)

Investigating the Tensorial Cross Product Modulation (TCPM): Applications in Engineering and Finance

1 Apr

In mathematics, manipulating and comprehending complicated structures is crucial to addressing real-world issues, especially in the domains of engineering and finance. The Tensorial Cross Product Modulation (TCPM) is one such mathematical procedure that shows how abstract mathematical ideas may be used in real-world situations. This operation provides a sophisticated method for analysing and interpreting multidimensional data and interactions by extending the concept of the cross product into the tensorial domain and incorporating modulation by the Levi-Civita symbol.

Mathematical Foundation of TCPM

The TCPM operation involves the calculation of a matrix that effectively represents a generalized cross product between a vector $\vec{A}$ and a dyadic tensor $\overline{\overline{D}}$ . This matrix is modulated by the Levi-Civita symbol, which introduces the antisymmetric property characteristic of cross products, and is further articulated through the directional components provided by the unit vectors of the standard basis in $R^3$ .

The mathematical definition of TCPM is given by the equation:

$\overline{\overline{B}}=\sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \delta_i (\sum_{l=1}^3 \epsilon_{ljk} \delta_l) D_{ij} A_k$
where $i,j,k=1,2,3$ in cyclic order and $\epsilon_{ljk}$ is the Levi-Civitta tensor and the $\delta_i$ are the unit vectors.

import sympy as sp

# Define symbols
x, y, z = sp.symbols('x y z')
D_11, D_12, D_13, D_21, D_22, D_23, D_31, D_32, D_33 = sp.symbols('D_11 D_12 D_13 D_21 D_22 D_23 D_31 D_32 D_33')

# Symbolic representation of D (as p)
D = sp.Matrix([
    [D_11, D_12, D_13],
    [D_21, D_22, D_23],
    [D_31, D_32, D_33]
])

# Levi-Civita tensor (epsilon_ijk) for three dimensions
def epsilon(i, j, k):
    if (i, j, k) in [(1, 2, 3), (2, 3, 1), (3, 1, 2)]:
        return 1
    elif (i, j, k) in [(3, 2, 1), (1, 3, 2), (2, 1, 3)]:
        return -1
    else:
        return 0

# Symbolic representation of D and A
D = sp.Matrix([
    [D_11, D_12, D_13],
    [D_21, D_22, D_23],
    [D_31, D_32, D_33]
    ])

A = sp.Matrix([
    [x],
    [y],
    [z]
    ])

# Unit vectors (standard basis in R^3) as column matrices
delta = [sp.Matrix([1, 0, 0]), sp.Matrix([0, 1, 0]), sp.Matrix([0, 0, 1])]

# Initialize the resulting matrix B_bar_bar as a zero matrix
B_bar_bar = sp.zeros(3, 3)

# Perform the operation to calculate B_bar_bar according to the formula
for i in range(3):
    for j in range(3):
        for k in range(3):
            # Reset sum_over_l for each new combination of i, j, k
            sum_over_l = sp.zeros(3, 1)  # Transpose delta[l] here, initializing as a row vector
            for l in range(3):
                # Apply transpose to delta[l] directly
                sum_over_l += epsilon(l+1, k+1, i+1) * delta[l] * D[i, j] * A[k]
            # Multiply by delta_i on the left without transposing the sum_over_l
            B_bar_bar += sum_over_l * delta[j].transpose() # delta[i] as a column vector, sum_over_l already a row vector

B_bar_bar

Notice that $\vec{A} \times \overline{\overline{D}} \neq \overline{\overline{D}} \times \vec{A}$ .

The result is

Applications in Engineering

The TCPM is useful in engineering applications where it is necessary to analyse forces, moments, and field characteristics in three-dimensional environments. For example, in fluid dynamics, the procedure may be used to examine the circulation and vorticity in a fluid flow, assisting engineers in creating fluid handling systems that are more effective. Analogously, in the field of structural engineering, the TCPM can facilitate the assessment of stress and strain in materials, providing critical information for determining the safety and integrity of the structure.

The TCPM has another interesting use in electromagnetism, where it may be used to calculate the distributions of magnetic and electric fields more easily. This is especially helpful for the design and optimisation of electrical equipment, where it’s important to comprehend how magnetic fields and electric currents interact, such transformers and motors.

Example of application: Torque Calculation using Maxwell Stress Tensor

Consider a flat plate of area A lying in the xy-plane, centered at the origin for simplicity. Incident light is assumed to come in along the z-axis, exerting pressure on the plate due to its electromagnetic field. The Maxwell stress tensor for electromagnetic fields in a vacuum is defined as:

$T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right)$ . Focusing on the electric field component, we simplify our consideration by assuming that the light’s pressure effect is primarily due to its electric field component perpendicular to the surface. The force differential $d\mathbf{F}$ on an elemental area dA due to the stress tensor is:

$d\mathbf{F} = \mathbf{T} \cdot d\mathbf{A}$ . For light incident normally on the xy-plane, $d\mathbf{A}$ can be considered as $dA \hat{z}$ , where $\hat{z}$ is the unit vector in the z-direction. The torque $\mathbf{\tau}$ about the origin is calculated by integrating the cross product of the position vector $\mathbf{r}$ with the force differential over the surface S:

$\mathbf{\tau} = \int_S (\mathbf{r} \times d\mathbf{F})$ , $\mathbf{\tau} = \int_S (\mathbf{r} \times (T_{zz} \hat{z} \cdot dA \hat{z})) = \int_S (\mathbf{r} \times (T_{zz} dA \hat{z}))$ .

Given $T_{zz}$ as a uniform pressure p across the surface, and $T_{zz} = p$ , we have:

$\mathbf{\tau} = \int_S (\mathbf{r} \times (p dA \hat{z}))$

Visual representation of the experimental setup involving a flat plate exposed to incident light, illustrating the electromagnetic interaction between the light and the plate amidst a laboratory environment, created by the author with DALL-E.

For a flat plate centered at the origin, the cross product becomes: $\mathbf{\tau} = \int_S \left( -py dA \hat{x} + px dA \hat{y} \right)$ . Due to the symmetry about the origin and the uniform pressure p, the integral expressions for $x dA$ and $y dA$ over the area S will cancel out, leading to no net torque for a perfectly symmetrical plate. For scenarios involving non-symmetrical plates or non-uniform pressure, detailed numerical integration over the actual geometry and electromagnetic field distribution is required for a precise torque calculation.

Applications in Finance

TCPM has potential in risk management and derivatives pricing, even if its use in finance may not be as straightforward as in engineering. Tensors are a useful tool in the financial markets to describe the behaviour of diverse financial instruments because they can capture the multidimensional character of market movements and correlations among different assets. In this case, modelling the dynamics of complicated derivatives—whose values rely on several underlying variables—can be greatly aided by the TCPM method. Financial analysts may improve risk assessment and hedging strategies by using TCPM to obtain greater insights into the sensitivities and exposures of these instruments.

Additionally, the TCPM can aid in the comprehension of the multifaceted risk-return environment during portfolio optimisation, allowing for the development of portfolios that are more closely linked with the return goals and risk tolerance of investors. The operation is an effective instrument for managing the numerous interdependencies of financial markets because of its capacity to manage complex, multidimensional interactions. In engineering and finance, the Tensorial Cross Product Modulation operation serves as a link between abstract mathematical theory and real-world applications. The TCPM provides a comprehensive method for analysing and understanding multidimensional interactions, such as physical pressures in engineering contexts or market dynamics in finance, by extending standard vector operations into the tensorial domain. It is certain that the applications of these mathematical structures will increase as we learn more about and comprehend their possibilities.

Although the Tensorial Cross Product Modulation (TCPM) may not be widely used or documented in the financial industry, its conceptual foundations offer a rich environment for creative financial modelling, especially when it comes to intricate, multifaceted financial instruments and risk management techniques. Even though the particular mathematical structure isn’t utilised directly, let’s examine how the concepts behind TCPM might be theoretically applied to the banking industry.

Portfolio Optimization and Risk Management

In finance, the complexity of relationships between assets in a portfolio can often mirror the multidimensional interactions captured by tensorial operations. Portfolio optimization, which seeks to balance the trade-off between risk and return across a basket of investments, can benefit from the high-dimensional analysis capabilities that TCPM suggests.

For example, the covariance matrix, a key component in modern portfolio theory, could be extended into a higher-dimensional tensor to capture not only pairwise asset volatilities and correlations but also the influence of external multidimensional factors such as macroeconomic indicators or global financial trends. The principles behind TCPM could inform the development of these tensorial models, providing a more nuanced view of portfolio dynamics and risks.

Derivatives Pricing and Risk Sensitivity Analysis

Financial derivatives, whose values depend on the behavior of one or more underlying assets, can exhibit complex risk sensitivities, known as “Greeks” in finance. These sensitivities can be influenced by a range of factors, including changes in the underlying asset prices, volatility, time decay, and interest rates.

Conceptually, the application of a TCPM-like operation could enhance the modeling of such sensitivities, especially for exotic derivatives that depend on multiple underlying variables in complex ways. By considering the interactions between these variables in a multidimensional tensor framework, analysts could potentially uncover new insights into the derivatives’ behavior and risk profiles.

High-dimensional Financial Data Analysis

The explosion of big data in finance requires analytical methods that can handle multidimensional datasets. Examples of this type of data include sentiment analysis on social media, economic indicators, and data from high-frequency trading. Although tensor operations are not directly related to TCPM, their multidimensional nature and mathematical complexity provide inspiration for methods of analysing such complicated data.

For example, latent features from high-dimensional financial data may be extracted using tensor decomposition techniques, providing insights that may be lost in lower-dimensional analysis. These techniques may help uncover underlying trends and connections in market behaviour, which may result in better investing choices.

The abstract nature of Tensorial Cross Product Modulation (TCPM) and its speculative use in finance allow us to investigate the conceptual effects of tensor-based operations on financial models, especially in high-dimensional data analysis, derivatives pricing, and portfolio optimisation. In the sections that follow, we go deeper into each topic and provide examples of possible applications for tensor operations, where suitable utilising Python for demonstrations. Refer to this reference [1].

Portfolio Optimization

In portfolio optimization, the goal is to allocate assets in a way that maximizes return for a given level of risk, or equivalently, minimizes risk for a given level of return. The covariance matrix, which captures the volatilities of assets and their correlations, is central to this problem. Extending this to incorporate external factors leads to a more complex model that can be represented using tensors.

Python Demonstration:

import numpy as np

# Assuming 3 assets and 2 external factors
# Mock data for asset returns
asset_returns = np.random.rand(3, 10)  # 3 assets, 10 time periods
# Mock data for factor sensitivities of each asset
factor_sensitivities = np.random.rand(3, 2)  # 3 assets, 2 factors
# Mock factor returns
factor_returns = np.random.rand(2, 10)  # 2 factors, 10 time periods

# Calculate asset returns influenced by factors
# This simplifies the interaction but illustrates the concept
influenced_returns = asset_returns + factor_sensitivities @ factor_returns

# Portfolio optimization would then proceed using these influenced returns

Although it streamlines the procedure, this example shows how to include outside variables in asset return computations, much like a higher-dimensional tensor.

Derivatives Pricing

Modelling the “Greeks” or risk sensitivity for financial derivatives entails comprehending how shifts in market circumstances impact the derivative’s value. The interactions between several underlying variables and their cumulative effect on the derivative may be modelled using tensors.

Conceptual Framework:

Consider a derivative where the two underlying assets determine its value. A number of variables, such as interest rates and market volatility, may have an impact on its price sensitivity to changes in these assets (Delta) and volatility (Vega). Tensors might capture these interactions in a multidimensional representation of this intricate connection.

High-dimensional Financial Data Analysis

High-dimensional financial data, such as latent elements in market movements or investor sentiment from social media, may be analysed using tensor decomposition techniques.

Python Demonstration using Tensor Decomposition:

from tensorly.decomposition import parafac
import tensorly as tl
import numpy as np

# Mock high-dimensional data: 3 assets, 4 time periods, 2 external factors
data = np.random.rand(3, 4, 2)

# Decompose the tensor to identify latent factors
factors = parafac(tl.tensor(data), rank=2)

# factors now contains the decomposed tensor representing principal components
</code>

In order to examine high-dimensional data and maybe uncover underlying patterns or variables that affect financial markets, this example makes use of tensor decomposition.

Conclusion

Though its application in finance is currently limited, the Tensorial Cross Product Modulation (TCPM) operation and its theoretical underpinnings—the analysis of complex, multidimensional relationships through the use of advanced mathematical structures—have the potential to inspire innovative thinking in the field of financial modelling and analysis. The use of sophisticated mathematical models, impacted by TCPM activities, will be crucial in handling the opportunities and challenges brought about by the ongoing development and intricacy of financial markets.

REFERENCES:

[1] ACM Transactions on Information SystemsVolume 34Issue 2Article No.: 11pp 1–30https://doi.org/10.1145/2838731

Comments Leave a Comment
Categories Advanced Physics Topics, Education and Science, Electromagnetic Theory, Engineering Applications, Light-Matter Interaction, Optical Physics, Physics Experiments, Research and Development, Research Insights

Louis de Broglie and the necessity of freedom of scientific work

1 Apr

Louis de Broglie’s contribution to the development of scientific knowledge, which naturally upholds the concept of intellectual freedom, is demonstrated by his engagement in the fundamentals of quantum physics as well as his leadership and mentoring roles in the scientific community.

The work of Louis de Broglie, particularly his groundbreaking theories on wave-particle duality, constitutes a breakthrough that called for a great deal of intellectual freedom and investigation within the scientific community. His conviction in creating places where new ideas might be freely investigated and argued is demonstrated by his involvement in founding and directing a scientific school that supported the growth of theoretical physics in France and impacted scientists worldwide.

In addition, the establishment of this school and de Broglie’s teaching efforts at the Sorbonne point to his commitment to information dissemination and inspiring the next generation of physicists to take creative research directions. The significance of academic and scientific independence in promoting innovative research and advancements in the scientific domains is emphasised by this educational heritage.

Comments 1 Comment
Categories Academia, Economic progress, Education and Science, Learning science, Louis de Broglie, Nature

Some relevant works on the Foundations of Relativity Theory

26 Mar

Max von Laue was a pivotal figure in the development of modern physics, particularly known for his groundbreaking work in x-ray crystallography. Born in 1879 in Germany, he was a passionate physicist from a young age. His academic journey took him to several German universities, where he was deeply influenced by notable figures such as Max Planck and became an early advocate for Albert Einstein’s theory of relativity.

Von Laue’s most significant contribution came from his insight that crystals could act as a diffraction grating for x-rays, leading to the first experimental proof of the wave nature of x-rays. This discovery not only earned him the Nobel Prize in Physics in 1914 but also laid the foundation for the field of x-ray crystallography. This new method allowed scientists to determine the atomic structure of materials and later played a crucial role in understanding the structure of DNA and RNA, thereby facilitating the emergence of molecular biology.

During World War I, von Laue applied his expertise to improve communication technologies for the German military. His post-war career was marked by efforts to revive and reintegrate German science into the international community, as well as a staunch defense of intellectual freedom against the Nazi regime’s attempts to politicize science.

His contributions extended beyond his Nobel Prize-winning work. Von Laue was involved in the study of superconductivity and continued to refine x-ray interference techniques. He was a respected figure in the scientific community, known for advocating the freedom of scientific inquiry and expression, and played a significant role in rebuilding German science after World War II.

In terms of the theory of relativity, von Laue was indeed a significant proponent and contributed to its acceptance in the scientific community. Although he was not the precursor of the Lorentz transformations, his early and strong support for Einstein’s work, along with his own contributions to theoretical physics, helped cement the theory’s foundational role in modern physics.

For more detailed insights into Max von Laue’s life and contributions, you can visit the following pages:

Examine M. v. Laue’s groundbreaking work, “Die Relativitätstheorie,” which is essential to comprehending relativity theory. Published in Braunschweig in 1921, this fourth edition establishes the basic ideas and mathematical structures that have influenced contemporary physics. Explore the historical background and perceptive analysis that Laue offers; this work is a priceless tool for both amateurs and academics. Access the full text here on Archive.org to embark on a journey through the evolution of physics.

Comments Leave a Comment
Categories Academia, Education and Science, History of Science, Max von Laue, Theory of relativity

Global Financial Trends revealed: A Look Back at Stocks and Currencies from 2022 to the Present

23 Mar

Following the performance of different asset classes may provide priceless insights into investor mood and global economic trends in the fast-paced world of finance. A fresh examination of financial data going back to the beginning of 2022 offers a clear picture of the state of the world markets. This research delves deeply into a number of categories, illustrating the many paths that these assets have taken. These divisions include world stock indexes, currencies, cryptocurrencies, and commodities. One noteworthy finding is the Nikkei index for Japan, which has been trending upward and appears to be a good indicator of the state of the Japanese market. In the meanwhile, the US Dollar has demonstrated a notable plateau in the context of US, Canadian, and European currencies, suggesting steadiness in the face of volatile market conditions. By dissecting these trends, this piece seeks to provide readers with a clear view of the current financial climate and its implications.

*Image created by the author, visualized using AI technology.*

Each plot in the generated image represents a different category of financial assets, such as:

World Stock Indices: Shows the performance of major global stock indices like the S&P 500, Dow Jones Industrial Average, NASDAQ, Nikkei 225, etc.
US, Canada & Europe Currencies and Bonds: Focuses on the currency exchange rates against the USD and bond yields in these regions.
East Asia: Covers currency exchange rates in East Asian countries.
Southeast Asia and Oceania: Features currencies from Southeast Asia and Oceania.
Crypto Currencies: Tracks the performance of major cryptocurrencies like Bitcoin, Ethereum, Binance Coin, etc.
Oil & Precious Metals: Displays the price changes in commodities such as crude oil, gold, silver, and platinum.

the Japan/Nikkei 225 (labeled as “Japan/Nikkei225”) is trending upwards from the start date of January 1, 2022, to the present, this indicates that the Nikkei 225 index, which is a major stock market index for the Tokyo Stock Exchange in Japan, has experienced an increase in its value over this period. An upward trend means that, on average, the share prices of the companies listed on the Nikkei 225 have increased, suggesting positive investor sentiment, economic optimism, or favorable market conditions in Japan.

This increase could be due to a variety of factors, such as:

Economic Growth: Improvements in Japan’s economic indicators such as GDP growth, employment rates, or consumer spending could contribute to a positive outlook on stocks.
Corporate Earnings: Strong earnings reports from companies within the Nikkei 225 could drive up their stock prices, positively impacting the index.
Monetary Policy: Actions by the Bank of Japan, such as interest rate decisions or quantitative easing, could make investing in stocks more attractive.
Global Market Trends: Sometimes, global trends or market movements can influence local markets. Positive developments in other major economies or markets could also boost investor confidence in Japanese stocks.
Sector Performance: The Nikkei 225 includes companies from various sectors. Strong performance in key sectors like technology, automotive, or manufacturing could lead to an overall increase in the index.

An uptrend in the Nikkei 225 could signal confidence in Japan’s market and economy, making it an attractive option for investors looking to diversify their portfolio with Japanese stocks. However, investors should also consider other factors and conduct thorough research before making investment decisions, as stock markets are influenced by a wide array of factors and can be volatile.

For the “US, Canada, & Europe” category, if the US Dollar (USD) is shown as almost in a “plateau” or leveling off after an upward trend, it indicates that the value of the USD has stabilized after a period of increase. This could be a result of several factors, including monetary policy decisions by the Federal Reserve, economic indicators showing steady growth, or global market conditions that affect the demand for the USD.

Implications

Japan Nikkei 225 Going Up: An upward trend for the Nikkei 225 suggests investor confidence in the Japanese stock market. For investors, this might be a sign of potential profitability in investing in Japanese equities. The economic implications could include increased capital inflow into Japan and potential appreciation of the Japanese Yen if the stock market growth reflects broader economic strength.
USD Plateauing After an Increase: The USD stabilizing after an increase suggests a period of consolidation. This could mean that factors which were driving the USD’s value higher have been fully absorbed by the market, and now it is in a phase of equilibrium. For traders and investors, a stable USD could mean lower forex volatility in the short term, affecting decisions on currency trades or investments in assets priced in USD. Economically, a strong and stable USD impacts global trade, as it can make US exports more expensive and imports cheaper, affecting the trade balance.

Considerations

The strong demand for precious metals and cryptocurrencies has highlighted their place as the cornerstones of investment portfolios. These asset classes have proven resilient in the face of the volatility and unpredictability inherent in global markets, drawing in investors looking for both safety and innovation. With their innovative appeal, cryptocurrencies never fail to enthrall those seeking development and diversification outside of conventional financial instruments. In the meanwhile, precious metals like silver and gold have reiterated their position as safe havens, particularly during uncertain economic times.

When interpreting financial data and trends like these, it’s important to consider the broader context, including geopolitical events, changes in monetary policy, and other economic indicators. Market trends are influenced by a wide range of factors, and understanding these can provide more insight into what such trends might mean for the future.

Acknowledgment – The code provided in the following post was utilized: https://medium.com/tech-talk-tank/run-my-python-code-and-I-can-show-you-the-money-7c948f9f31f4, by Michael Wai, to achieve the above-mentioned interpretations and the graphs displayed below.