Fundamental Physics PDF

Chapter 3 The “Standard” Models in Fundamental Physics 3.1 Fundamental interactions in nature 3.1.1 Four fundamental forces The four fundamental forces, i.e. forces that cannot be reduced to simpler interactions in the current Universe, and which govern everything that happens in the Univer...

Chapter 3 The “Standard” Models in Fundamental Physics 3.1 Fundamental interactions in nature 3.1.1 Four fundamental forces The four fundamental forces, i.e. forces that cannot be reduced to simpler interactions in the current Universe, and which govern everything that happens in the Universe are the strong force, the weak force, the electromagnetic force, and the gravitational force. An active area of research in physics is working on the gran reunification, a problem that consist of explaining some/all the fundamental forces by using a unifying set of principles. A gran-unification has been theorized for the strong, weak and electromagnetic forces, and the unification of the electromagnetic and weak interactions has been observed at very high energy into a so called electroweak interaction. We will limit ourselves to the standard model that makes use of four fundamental forces. Gravitation This force describes the interaction of all objects with a mass. It is responsible for the attraction of celestial bodies, for example stars in galaxies, galaxies in clusters, and planets in planetary systems. It is a long-range force and the weakest of the four fundamental forces. It becomes dominant when dealing with very massive objects. Electromagnetism This force describes the interaction of charged particles. This force includes attraction of opposite charged particles, repulsion of particles with the same charge and magnetic interaction. It has a long range and is a weak force, but it is stronger than gravity. It plays a significant role in the behaviour of atoms and molecules and on a macroscopic scale is responsible for electricity and magnets. Weak interaction This force is responsible for the radioactive decay of elements. An unstable form of atom will decay in a more stable one through the weak interaction. It has a short range - subatomic distance and is involved in processes such as nuclear fusion and fission.. 15 16 CHAPTER 3. THE “STANDARD” MODELS IN FUNDAMENTAL PHYSICS Strong interaction This force holds together protons and neutrons together in the atomic neu- clei. It is the strongest of the four forces but has the shortest range (within the nucleus radius). 3.1.2 Four fundamental forces in a space context On subatomic scales, gravitation is the weakest of the fundamental forces, but on cosmic scales, it becomes dominant due to its cumulative nature : mass is always positive, and more mass leads to stronger gravitational forces. Unlike the strong and weak nuclear forces, which only act at subatomic distances, and electromagnetic forces, which tend to cancel out over larger distances, gravity governs the large-scale structure of the Universe. Newton’s laws of motion (Sec. 3.3) and gravitation remain crucial for understanding the movement of celestial bodies, helping us measure masses and even revealing the presence of dark matter. Gravitation not only structures matter on large scales but also drives the formation of cosmic substructures through gravitational instability, where denser regions pull in more matter, enhancing density contrasts and leading to the creation of stars, planets, and galaxies. While gravity dominates the large-scale dynamics, it is through electromagnetic radiation that we observe the Universe. Electromagnetic processes are far more powerful than gravitational ones, and photons travel vast distances relatively undisturbed, allowing us to observe stars and galaxies. The fact that we can detect electromagnetic signals from these distant objects confirms the universality of physical laws. The strong nuclear force, though acting at short distances, plays a critical role in cosmic physics by powering stars through nuclear fusion, which releases energy and drives stellar evolution. Fusion occurs when lighter nuclei combine to form more stable, bound states, releasing energy in the process. This is the primary energy source in stars, enabling them to maintain equilibrium by countering gravitational collapse with nuclear pressure. On Earth, nuclear fission is used in reactors, but fusion in stars is more energetic due to the extreme conditions of heat and pressure required to overcome electrostatic repulsion between nuclei. The evolution of the Universe and stars explains the observed abundances of elements. Initially, only elementary particles existed, but as the Universe cooled, protons and neutrons formed, leading to hydrogen and helium nuclei through early nucleosynthesis. Stars then fuse hydrogen into helium and heavier elements like carbon and oxygen, producing the elemental composition we see today. However, fusion ceases to be favorable beyond iron (see Fig. 3.1), posing challenges for the formation of heavier elements. The collapse of stellar cores in supernova explosions releases vast amounts of energy, facilitating the production of elements beyond iron and expelling lighter elements into space. Weak interactions, despite their name, play a crucial role in the Universe, as they affect nearly all particles. Neutrinos, which interact only via the weak force and gravity, are believed to be the most abundant particles in the Universe, though they are incredibly difficult to detect. In stars, including the Sun, weak interactions are critical for initiating nucleosynthesis, allowing protons to transform into neutrons, which help stabilize atomic nuclei by overcoming electrostatic repulsion between protons. In the early Universe, conditions were too energetic for stable atoms or nuclei to exist. As the Universe expanded and cooled, weak interactions produced the first neutrons, though this process lasted only a few seconds. This allowed for the formation of basic atomic nuclei like hydrogen and helium, but heavier elements did not form until hundreds of millions of years later when stars were born. In stellar nucleosynthesis, the slow, weak interaction that converts protons to neutrons is again vital, as it gives stars their long lifespans. This extended stellar lifespan is what ultimately allowed for the production of complex elements and, eventually, life, including humans. 3.2. THE STANDARD MODEL OF PARTICLE PHYSICS 17 Figure 3.1: Binding energy per nucleon and nuclear stability. Image credit: Ling et al. (2024). 3.2 The standard model of particle physics 3.2.1 Elementary particles Our world can be understood through 12 elementary particles (Fig. 3.2): six quarks and six lep- tons. Among these, only two quarks (“up” and “down”) and one lepton (the electron) are needed to explain the elements in the periodic table, as protons and neutrons are made from quarks and electrons orbit around nuclei. These particles interact through mediating particles specific to each force. For example, gluons mediate the strong force between quarks, W and Z bosons mediate weak interactions, and photons mediate electromagnetic forces. While mass-less photons allow electro- magnetic forces to act over long distances, the mass of W and Z bosons limits weak interactions to short scales. Gravitons, which are theorized to mediate gravity, remain undetected. Over time, experiments at high energies revealed numerous short-lived particles, which led to the discovery that all baryonic matter could be explained using six quarks. Heavier quarks tend to decay into the lighter “up” and “down” quarks, while similarly, the electron has two heavier counterparts, the muon and tau, which are short-lived and decay to the electron. Each lepton also has an associated neutrino, a nearly massless particle. In each family of particles, the lightest member is the most stable, and heavier particles rapidly decay into lighter ones through the weak interaction. Protons and neutrons are held together in atomic nuclei by the strong nuclear force, which counters the electrostatic repulsion between positively charged protons. Neutrons, being neutral, add stability to the nucleus without contributing to this repulsion. However, if there are too many neutrons, some become unstable and decay. Different nuclei with the same number of protons but varying numbers of neutrons, called isotopes, can still be stable. Neutrons help balance the forces in the nucleus, but their excess leads to decay over time, as seen in the transmutation of neutrons to protons. 18 CHAPTER 3. THE “STANDARD” MODELS IN FUNDAMENTAL PHYSICS Figure 3.2: The standard model of physics. 3.2.2 Is the standard model complete? The Standard Model is highly effective at describing the particles and interactions we detect in experiments, yet it leaves many unanswered questions. For example, why are there two families of six particles, and why do we have four fundamental interactions? Additionally, why do some particles have mass while others are massless? Though the discovery of the Higgs boson in 2013 helped answer part of the mass question by confirming the existence of the Higgs field, which gives particles their mass, many mysteries remain. The mass of elementary particles is just part of the story, as much of the mass of protons comes from the energy of quark interactions rather than the quarks themselves. A major focus in particle physics has been attempting to unify the four fundamental forces. While electromagnetism and the weak force have successfully been combined into the electroweak force, unifying this with the strong force in a “grand unified theory” remains a challenge. The ultimate goal is to also include gravity in this unification, but this has proven extremely difficult. Our understanding of the physical world is inherently limited by the energies we can achieve in experiments, and since we can only detect particles that interact through the known fundamental forces, there may be particles that evade detection altogether if they do not interact with these forces. 3.3 Newton’s laws of motion Newtonian mechanics is based on the Newton’s laws of motion: 3.4. MAXWELL’S EQUATIONS 19 Law of inertia Every object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. In other words, an object will maintain its “status quo” unless its state is changed by an external force. Inertia is the tendency of an object to resist changes in its motion: if all external forces on an object cancel out, resulting in no net force, the object will continue moving at a constant velocity and in the same direction. Without a net force, its motion remains unchanged. For example, looking only and the magnitude of its velocity, a satellite placed into orbit around the Earth will continue to move at a constant speed along its orbit due to inertia, unless acted upon by external forces like atmospheric drag (in the LEO) or gravitational perturbations from other celestial bodies. However, looking at the direction of motion, the velocity of a satellite is continuously changing under the gravitational pull of the Earth. This constant change of direction allows to satellity to orbit around the Earth. Equation of motion The acceleration of an object depends on its mass and the amount of force applied. The time rate of change of momentum equals the force (F = dp dt ). A more familiar way to express the force as a product of mass and acceleration: F = ma. For example, the more mass a rocket has, the more thrust is required to launch it. Action and reaction Whenever one object exerts a force on another object, the second object a force on the first so that these forces are equal in magnitude and opposite in direction: F12 = −F21. This is connected to the principle of the conservation of momentum: because the forces are equal and opposite, the changes in momentum balance out, preserving the total momentum of the system. For example, a solar sail spacecraft can use the small but constant pressure from photons of sunlight hitting the sail (action) to propel the craft in the opposite direction (reaction). 3.4 Maxwell’s equations The Maxwell’s equations describe how electric and magnetic fields work and interact with each other and, together with the Lorentz force law, are the basis of classical electromagnetism: Gauss’s law Electric charges create electric fields. The more charge there is in an area, the stronger the electric field, which points away from positive charges and towards negative charges. For example, this laws helps understand how charges distribute across the surface of a spacecraft. Gauss’s law for magnetism No isolated magnetic charges (also called magnetic monopoles) exist: magnetic field lines always loop around and don’t start or stop anywhere. For example, the Sun’s magnetic field lines form loops and never terminate. Faraday’s law of induction Changing magnetic fields can create electric fields: if a magnetic field changes over time, it causes electricity to flow. For example, changes in the Sun’s magnetic field creates electric fields that can cause geomagnetic storms on Earth, damaging satellites, disrupting GPS and communication signals. Ampere’s law Electric currents and changing electric fields create magnetic fields. For example, when electricity flows through a wire, it creates a magnetic field around the wire. 20 CHAPTER 3. THE “STANDARD” MODELS IN FUNDAMENTAL PHYSICS 3.5 Special relativity The theory of special relativity, which describes the relationship of space and time, is based on two postulates: The principle of relativity: the laws of physics are the same in all frames of reference1 ; The principle of light speed invariance: the speed of light, c, in vacuum is the same for all observers, approximately equal to 3 × 108 meters per second. Below we briefly discuss four interesting consequences of special relativity. The equivalence of mass and energy is represented by Einstein’s famous equation E = mc2. This formula indicates that energy (E) and mass (m) are interchangeable; they are two forms of the same physical quantity. The equation implies that a small amount of mass can be converted into a large amount of energy, given that c2 is a very large number. Special relativity makes a distinction between rest mass and relativistic mass. While the rest mass of an object remains constant regardless of its velocity, its relativistic mass increases with speed, approaching infinity as it approaches the speed of light. This increase in relativistic mass requires more energy to accelerate the object further and prevents any object with a mass to reach the speed of light in the vacuum. Time dilatation refers to the difference in elapsed time measured by two clocks depending on their relative motion (or the strength of a gravitational field). In other words, when an object moves very fast (close to the speed of light) or is in a strong gravitational field, time appears to slow down for that object compared to an observer at rest or in a weaker gravitational field. For example, a light clock (illustrated on Fig. 3.3) consists of two mirrors facing each other with a beam of light bouncing between them. For someone using the light clock while moving at high speed, the light has to travel a longer diagonal path between the mirrors, taking more time to complete a bounce compared to someone using the same clock while at rest. As a result, the moving person’s clock ticks slower relative to the stationary observer’s clock, illustrating time dilation. This concept has been popularised in the twin-travel thought experiments and has been widely adapted in science fiction. Figure 3.3: An example of a stationary (left) and a moving light clock (right). 1A frame of reference is an abstract coordinate system, where the position and motion of objects is determined. 3.6. GENERAL RELATIVITY 21 Figure 3.4: An example of an object contracting in the direction of movement. Length contraction refers to how objects appear shorter when they move at very high speeds, close to the speed of light, relative to an observer at rest (as illustrated on Fig. 3.4). When observing a fast-moving object, like a rocket, it seems to shrink in the direction it’s moving compared to how it looks when it’s at rest. The faster the rocket moves, the more it appears to contract. 3.6 General relativity Unlike special relativity, general relativity does deal with gravity. This theory explains how massive objects, such as stars, galaxies, black holes, etc, bend the fabric of space and time around them (e.g. like on Fig. 3.5). The curvature of spacetime caused by mass and energy in general relativity is represented by the Einstein tensor, Gµν. Figure 3.5: An illustration of Earth and the Sun bending the curvature of spacetime. General relativity plays a crucial role in understanding gravitational lensing, a phenomenon where the light from a distant object, such as a galaxy or a star, is bent around a massive object (like another galaxy or a black hole) that lies between the observer and the distant source. This effect can lead to several observational consequences, such as multiple appearances of the same object, arcs, or Einstein rings (Fig. 3.6), depending on the alignment of the observer, the massive 22 CHAPTER 3. THE “STANDARD” MODELS IN FUNDAMENTAL PHYSICS object, and the source. Gravitational lensing allows to determine the mass of the lensing object and the properties of the source. Figure 3.6: Einstein Rings imaged by the HST and the Sloan Digital Sky Survey.