Author: Dr. Jonathan Kenigon, FRSA*
Institution: Athanasian Hall, Cambridge, LTD
The Standard Model of Physics is a Theory of Particles and Forces that forms the basis for understanding the behavior of the universe. It is one of the most successful scientific theories of all time, having been tested to an unprecedented level of accuracy and precision. The Standard Model attempts to combine Quantum Mechanics and Relativity to describe the behavior of particles and forces at the subatomic level [1]. The compass of the subatomic theory is divided into three parts: the Electromagnetic, Weak and Strong forces. The Standard Model also explains the behavior of the subatomic particles, including their mass and charge, as well as the interactions between them. It is a highly successful theory, and though it cannot yet explain all phenomena in the universe, it is an invaluable tool for scientists in their search for a deeper understanding of the workings of the universe [1]. Quantum Gravity is an attempt at a more fundamental unification of Gravity, Electromagnetism, and the Nuclear Forces under a single aegis [2]. Superstring Theory is a promising candidate for such unification, but numerous mathematical dilemmas continue to beset the theory [2]. These challenges are mathematically formidable and shall be the subject of a future submission.
The Standard Model both presupposes and predicts the existence of distinct classes of particles. One could more properly assert that the Standard Model hazards a classification of such particles via Chromodynamics (the theory of color) and Spin (rotational invariance) [1]. Hadron is a term used in particle physics to describe a type of composite particle made up of two or more Quarks, where the Quarks interact via the strong nuclear force. Hadrons are the fundamental particles of the Standard Model of particle physics, which is the current accepted theory for explaining how the subatomic world works. Hadrons are composed of Quarks and are held together by the strong nuclear force, a fundamental force of nature [3]. This force acts over short distances and is mediated by the exchange of gluons, which are particles that carry the strong nuclear force. Hadrons have mass and can interact with other particles and fields, such as electromagnetic and gravitational fields [3]. Examples of Hadrons include protons, neutrons, and mesons, which are all made up of three Quarks each. In turn, Quarks are subdivided by mass (flavor) and color. In most classical treatments of the Standard Model, Quarks are the fundamental building blocks of matter, and they come in a variety of flavors and colors. Quarks come in six flavors: up, down, charm, strange, top, and bottom, which are determined by a particle’s electrical charge and its interaction with the strong force of chromodynamics. Each Quark has a distinct mass, which also affects its behavior. The heaviest Quark is the top Quark, which has a mass about 200 times larger than the up Quark. Quarks also come in three colors: red, green, and blue. These colors correspond to the three forces of the strong nuclear force, which is what holds the Quarks together in an atom. The combination of Quark flavors and colors allows for a huge variety of combinations and particles—and thus, matter.
The Standard Model demands the existence of force-carrying particles (Bosons) and matter-building particles (Fermions). A Boson is a type of subatomic particle with integer spin. It is one of two main classes of particles, the other being fermions. Bosons are known for their role in carrying force particles, such as Photons, which mediate electromagnetic interactions [4]. They can also be found in the nucleus of atoms, where they help hold the nucleus together. Bosons are made of Quarks, which are held together by the strong nuclear force. Bosons are one of two types of Hadrons, the other being Baryons [4]. The distinction between the two is largely based on their spin, with Bosons having integer spin and Baryons having half-integer spin. The Boson definition is important to understand in order to understand how these particles work together to form atoms and molecules. Understanding Bosons is also important for the study of particle physics, as many theories in particle physics rely on the properties of Bosons [4]. A critical aspect of this understanding is derived from the Hilbert Formalism, which permits descriptions of the disorder (Entropy) of quantum systems and the dynamics of Bosonic interactions via the theory of Fock Spaces [5].
Fock Spaces may be developed naturally from the classical spaces of Functional Analysis. Banach Spaces and Hilbert Spaces are two different types of spaces that are used when studying mathematics. A Banach Space is a complete topological vector space, while a Hilbert Space is an inner product space. The main difference between the two is that a Banach Space is defined using the norm of a vector, whereas Hilbert Space is defined using the inner product of two vectors [6]. Banach Spaces are used to study linear functional analysis and operator theory, while Hilbert Spaces are used to study quantum mechanics and signal processing [6]. Both are important for understanding mathematical theory, and both have their own unique applications. In general, a Banach Space is better suited for tasks that require a greater degree of precision, while Hilbert Space is better suited for tasks that require a more intuitive approach. Depending on the task, either space can be used effectively. Informally, a Hilbert Space is a type of mathematical space used in quantum mechanics where one is required to measure angles and directions, or “vector” quantities. It is a generalization of the space of Euclidean vectors, and it allows for the development of a mathematical framework for studying quantum systems [6][7]. It is named after the German mathematician David Hilbert, who published the first axiomatic formulation in the early 1900s. Hilbert Spaces are used to define abstract quantum states, such as wave functions and other operators. It is also used to define the Fock space, which is used to describe the behavior of interacting particles. It can also be used to describe multi-particle entangled states, which are essential for studying quantum systems [7]. The Fock Space is used to determine the entanglement and entropic states of Bosons via notions of Thermodynamic Disorder (Entropy) [5]. One can imagine heating a glass in a microwave, where incident Photons excite matter via Planck Radiation and increase the Entropy (heat) of the items being warmed.
Entropy plays an important role in statistical mechanics. In a nutshell, entropy is a measure of the randomness or disorder of a system [8]. It is the quantitative measure of how much energy is unavailable for work. Through the laws of thermodynamics, entropy is linked to the probability of a system’s energy state. The higher the entropy, the higher the probability of the system’s energy state [8]. Entropy is also closely related to the number of microstates a system can have. In other words, entropy is a measure of how many different arrangements of energy a system can have. Entropy can be used to calculate the probability of a system being in a particular state, or the probability of a system undergoing a certain process [9]. The Von Neumann Entropy is a mathematical concept used to study the thermodynamic behavior of systems. It is defined as the entropy of a system that is in equilibrium with its environment and is measured by the average probability distribution over the system’s energy levels. The Von Neumann Entropy is a measure of the disorder in a system and can be used to determine the amount of energy lost to the environment. It also has applications in quantum computing, as it can be used to calculate the entropy of a quantum system [9].
Because the measure of Entropy is related to the measure of heat, it is little wonder that classical Partial Differential Equations in the Elliptic regime bear special mathematical significance. The solutions of some of such systems that also possess the general form of a Normal Probability Distribution are termed “Gauss-Laplace Kernels,” and the study of these Kernels in all areas of applied mathematics has accelerated rapidly in the past decade [10]. Gauss Kernel Expansion is a powerful mathematical tool that is used to solve integral equations. It is based on the Taylor series expansion of the kernel function and is used to approximate the kernel function with a finite number of terms. This technique can be used to solve a wide range of problems in mathematics, engineering, and physics. For example, it can be used to solve boundary value problems, find minimums and maximums of functions, or even calculate numerical solutions to differential equations [10]. The key advantage of using Gauss Kernel Expansion is that it is computationally efficient, as it involves a finite number of operations. Furthermore, the accuracy of the solution can be improved by increasing the number of terms in the expansion.
The original theory of heat was pioneered (most notably) by Fourier. His Transform as series are the bedrocks of basic Separable Partial Differential Equations. In a pointwise sense, Fourier Analysis provides a complete characterization of many well-posed systems in Wave Mechanics as well [11]. The Fourier Transform is an incredibly powerful mathematical tool used in many areas of science and engineering as well. It was developed by the French mathematician Joseph Fourier, who was looking for a way to break down a function into simpler components. The Fourier Transform does just that – it decomposes a function into a series of sine and cosine waves known as a Fourier Series. This series contains all the information necessary to reconstruct the original function [11]. The Fourier Transform is used in many areas of science, including signal processing and image processing. It is also used to filter out unwanted noise, identify patterns in data, and compress large amounts of data. Fourier Analysis has numerous limitations that Gauss-Laplace expansion does not have [11]. Coefficients are nearly impossible to compute analytically and strong conditions on the periodicity of the function being transformed must be explicitly stated. While relaxations are possible, the computational tractability of the more sophisticated Fourier schemes renders them functionally inapplicable to many fundamental problems, even with the advantages of nigh-on unlimited computational power.
While prominent, the Fourier Transform is by no means the only tool at a Mathematical Physicist’s disposal. Mellin Transforms operating upon Complex Elliptic Functions form a second mainstay of the toolkit. The Mellin Transform is used to transform functions which contain singularities and discontinuities, while the Fourier Transform is used to transform functions that are continuous and periodic [12]. The Mellin Transform can also be used to solve certain types of integral equations and is especially useful for computing the Laplace Transform of a function. On the other hand, the Fourier Transform is better suited to analyzing periodic signals, such as those found in music and speech [12]. When combined with Elliptic Functions, Mellin Transforms provide a link between Gauss-Laplace Expansion and the Theory of Elliptic Curves [13]. This synergy produces subtle consequences in Quantum Mechanics, Zeta Analysis, and Algebraic Geometry, among other fields [14]. Briefly stated, Elliptic Functions are types of transcendental functions that are used to solve problems related to elliptic integrals [15]. They form an important part of the field of Number Theory, and they are closely related to Modular Forms. From a classical perspective, Elliptic Functions are used to solve equations of the form y^2 = x^3 + ax + b, and they are also used to calculate areas enclosed by simple curves [15]. In more general modern contexts, the coefficients a and b may be chosen from finite Fields or Commutative Rings and can provide a bridge to Cryptography and Cipher Theory in this way. Elliptic Functions also have applications in physics, such as in the calculation of the acceleration due to gravity or in the study of wave motion. They are used to solve problems involving elliptic curves, which are the basis for many cryptographic algorithms [16]. In short, Elliptic Functions are a powerful tool for solving a wide range of problems in mathematics, physics, and cryptography. Jacobi Elliptic Functions are a set of mathematical functions that are used to solve certain types of problems in the Theory of Transforms. These functions were first studied by Carl Gustav Jacobi and are widely used in fields such as differential equations, complex analysis, number theory, and special functions [17]. They are usually defined in terms of elliptic integrals, which are integrals involving the ratio of two polynomials. The Jacobi Elliptic Functions are used to solve elliptic differential equations, which involve derivatives of two variables that are related by a nonlinear equation [17]. These functions can also be used to evaluate integrals, solve certain types of nonlinear equations, and even calculate the length of an arc on an ellipse [17].
Elliptic functions both define, and are defined by, Modular forms. Modular forms and modular functions are mathematical concepts used to describe functions that are periodic with respect to a certain group of transformations. Modular forms are defined on the upper half-plane, while modular functions are defined on the fundamental domain of the group [18]. Both modular forms and modular functions play an important role in number theory and mathematics in general. For example, modular forms can be used to describe properties of elliptic curves, while modular functions can be used to solve Diophantine equations. Modular forms and functions have also been used in cryptography, algebraic geometry, and many other areas of mathematics [18]. The Riemann Zeta Function, also known as the Riemann Zeta Function of Complex Argument, is a special mathematical function that has been studied and used extensively in mathematics. It is closely related to the Elliptic Function and Mellin Transform, which are both analytical tools. The Riemann Zeta Function is used to determine the zeros of the Zeta function and to prove the Riemann Hypothesis, which is a fundamental theorem in mathematics. It can also be used to calculate certain characteristics of the zeta function, like the critical points. Furthermore, the Riemann Zeta Function is used to study prime numbers, as well as the distribution of prime numbers. Zeta Analysis has nontrivial applications in cryptography and cryptocurrency [19]. The combinatorics of the Zeta Function are thus as intimately related to the distribution of primes and the Theory of Transforms as the ostensibly unrelated field of Standard Model physics and Boson propagation. One begins to appreciate the extent to which Heat Kernel expansions render such seemingly unrelated fields tractable via suitably but naturally applicable means.
Works Consulted.
[1]. Cottingham, W. Noel, and Derek A. Greenwood. An introduction to the standard model of particle physics. Cambridge university press, 2007.
[2]. Schwarz, John H. “Introduction to superstring theory.” Techniques and Concepts of High-Energy Physics. Springer, Dordrecht, 2001. 143-187.
[3]. Shupe, Michael A. “A composite model of leptons and quarks.” Physics Letters B 86.1 (1979): 87-92.
[4]. Olive, Keith A., et al. “Review of particle physics.” Chinese physics C 38.9 (2014): 090001.
[5]. Grassberger, Peter, and M. Scheunert. “Fock‐space methods for identical classical objects.” Fortschritte der Physik 28.10 (1980): 547-578.
[6]. Muscat, Joseph. Functional analysis: an introduction to metric spaces, Hilbert spaces, and Banach algebras. Springer, 2014.
[7]. Debnath, Lokenath, and Piotr Mikusinski. Introduction to Hilbert spaces with applications. Academic press, 2005.
[8]. Zurek, Wojciech H. Complexity, entropy and the physics of information. CRC Press, 2018.
[9]. Petz, Dénes. “Entropy, von Neumann and the von Neumann entropy.” John von Neumann and the foundations of quantum physics. Springer, Dordrecht, 2001. 83-96.
[10]. ter Haar Romeny, Bart M. “The gaussian kernel.” Front-End Vision and Multi-Scale Image Analysis: Multi-Scale Computer Vision Theory and Applications, written in Mathematics (2003): 37-51.
[11]. Elden, Lars, Fredrik Berntsson, and Teresa Reginska. “Wavelet and Fourier methods for solving the sideways heat equation.” SIAM Journal on Scientific Computing 21.6 (2000): 2187-2205.
[12]. Bertrand, Jacqueline, Pierre Bertrand, and Jean-Philippe Ovarlez. “The mellin transform.” (1995).
[13]. Swinnerton-Dyer, H. P. F., and B. J. Birch. “Elliptic curves and modular functions.” Modular functions of one variable IV. Springer, Berlin, Heidelberg, 1975. 2-32.
[14]. Krichever, Igor Moiseevich. “Methods of algebraic geometry in the theory of non- linear equations.” Russian Mathematical Surveys 32.6 (1977): 185.
[15]. Akhiezer, Naum Ilʹich. Elements of the theory of elliptic functions. Vol. 79. American Mathematical Soc., 1990.
[16]. Kapoor, Vivek, Vivek Sonny Abraham, and Ramesh Singh. “Elliptic curve cryptography.” Ubiquity 2008.May (2008): 1-8.
[17]. Weisstein, Eric W. “Jacobi Elliptic Functions.” https://mathworld. wolfram. com/ (2002).
[18]. Koblitz, Neal I. Introduction to elliptic curves and modular forms. Vol. 97. Springer Science & Business Media, 2012.
[19]. Vieira, Paulo. “Blockchain and the Riemann Zeta Function.” International Congress on Blockchain and Applications. Springer, Cham, 2021.
Daily magazine for entrepreneurs and business owners
