INTRODUCTION
The quest for understanding is indivisible. Physics cannot be separated from philosophy. Unfortunately, I dozed off after waves and electromagnetism. This post attempts to expand my little brain by tracing physics from classical to quantum.
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TENETS OF CLASSICAL PHYSICS (1899)
The physical world is all that exists (naturalism) and its existence is independent of and uninfluenced by the observer (realism).
The universe is static and closed, and entropy is constantly increasing.
Space and time are independent and absolute.
Physical reality is created by the interactions of hard, irreducible, separate, and ontological particles of matter in space and time (atomism).
The behavior of a complex system can be completely described as the sum of the behaviors of all its parts (reductionism).
Given the initial conditions of all the particles of a system, we can, in principle (Laplace’s demon), predict all past and all future evolutions of the system (determinism).
Mathematics can give us a true and comprehensive description of reality.
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MAJOR DEVELOPMENTS
Planck’s Black-body Radiation (1901)
Classical View: The energy emitted by a black body should increase indefinitely with frequency, leading to an "ultraviolet catastrophe” (Rayleigh-Jeans law)
Counterexample: The spectrum of black body radiation showed that the intensity of radiation peaks at a certain frequency and then drops off.
Resolution: Max Planck argued (1901) that energy is exchanged in discrete amounts called quanta, with energy (E) proportional to frequency (ν) by E=hν, where h is Planck's constant. An energy drop-off exists because the thermal energy available at typical temperatures is not sufficient to excite high-frequency quanta (which have very high energy).
Photoelectric Effect
Classical View: When light illuminates matter and ejects electrons (photoelectric effect), the energy of the ejected electrons should depend on the intensity of the light. There should also be a brief time lag because electrons would need time to absorb enough energy from the continuous wave to be ejected (wave theory of light).
Counterexample: Above a threshold frequency, electrons were instantly ejected, and their energy depended on the frequency of the light. No electrons were ejected below this threshold regardless of intensity.
Resolution: Einstein’s development to a long history of arguments about light proposed (1905) that that while light behaves like waves in as some situations (e.g. interference and diffraction), it others it behaves like a particle (photons). The energy of these discrete packets is given by E = hν. The existence of a threshold frequency is explained by the need for the photon's energy to exceed the work function of the material, and the instant ejection by direct interaction between photon and electron.
Einstein’s Theory of Special Relativity (1905)
Classical View: Space and time are independent and absolute.
Mass particles move according to Galilean transformations in all inertial frames (so a ball thrown 10km/hr from a car moving 50km/hr moves 60km/hr).
Movements of electrical particles follow Maxwell’s equations and produce an electromagnetic field that influences and is influenced by all other electrical particles.
Counterexample: Objects described by Maxwell’s equations do not follow Galilean transformations when their relative speed is close to the speed of light. The Michelson-Morley experiment suggested that the speed of light is constant in all directions and inertial frames
Resolution: Einstein built upon the theories of Hendrik Lorentz, postulating that a) the laws of physics are the same in all inertial frames of reference and b) the speed of light in a vacuum is constant (c) for all observers, regardless of the motion of the source or observer. Time can dilate (slow down) and lengths can contract depending on the relative velocity between observers, reconciling the behavior of objects moving at speeds close to the speed of light with the constancy of the speed of light.
Einstein's Theory of General Relativity (1915)
Spacetime: General Relativity combines the three dimensions of space and one dimension of time into a single four-dimensional continuum known as spacetime.
Curvature: Massive objects cause spacetime to curve, and this curvature dictates the motion of objects. The more massive an object, the greater the curvature it causes in spacetime. This explains Mercury’s orbit precession (1859) due to mass of Sun and is supported by discovering of light bending around Sun (Eddington 1919).
Geodesics: Objects move along the shortest paths in curved spacetime, known as geodesics. In the absence of other forces, these paths represent free-fall trajectories.
Equivalence Principle: The effects of gravity are locally indistinguishable from the effects of acceleration (e.g. observer in a closed elevator cannot distinguish between gravity and acceleration).
Time Dilation: Time runs slower in stronger gravitational fields.
Gravitational Waves: Ripples in spacetime caused by accelerating massive objects, such as merging black holes or neutron stars. First detected by LIGO (2015)
Black Holes: Regions of spacetime where the gravitational field is so strong that nothing, not even light, can escape from it. Supported by discovery of Cygnus X-1 (1964) and EHT's first black hole image (2019).
Pauli Exclusion Principle (1925)
Previous View: Electrons were thought to be distinguishable particles that could occupy the same quantum state within an atom.
Development: Wolfgang Pauli proposed that no two electrons in an atom can have the same set of quantum numbers (n, l, m_l, m_s), each must occupy a unique quantum state.
Principal (n): energy level and relative distance from the nucleus.
Azimuthal (l): shape of the electron's orbital.
Magnetic (m_l): orientation of the orbital in space.
Spin (m_s): electron's spin direction.
Significance: The periodic table, our understanding of chemical behavior, chemical bonds, astrophysics (Chandrasekhar Limit), fermions, and semiconductors all build upon the exclusion principle.
Schrödinger and Born’s Unification (1926)
Previous View: Electrons were thought to orbit the nucleus in fixed paths (Bohr model).
Counterexample: Experimental data, including spectral lines and electron scattering, indicated that electrons did not follow precise orbits but exhibited wave-like behavior.
Schrödinger’s Approach:
Wave Mechanics: Schrödinger's equation treats particles as wave functions ψ, representing the particle's state in terms of probability amplitudes where Ĥ is the Hamiltonian operator, ℏ is the reduced Planck's constant, and E is the energy of the system.
Time-Dependent Equation: which describes how the quantum state of a physical system changes over time is iℏ(∂t/∂ψ)=Ĥψ
Time-Independent Equation: that describes the stationary states of a system, where the energy is constant is Ĥψ=Eψ
Born’s Approach:
Matrix Mechanics: Developed in 1925 with Werner Heisenberg and Pascual Jordan, matrix mechanics describes quantum systems where observables (A) such as position, momentum, and energy are represented as matrix operators and 𝐻 is the Hamiltonian matrix.
Heisenberg Equation of Motion: governs the evolution of these observables over time: (dt/dA)=(i/ℏ)[H, A]+(∂A/∂t)
Probabilistic Interpretation: Schrödinger initially aimed for a deterministic description, where ψ provided complete description of a particle’s state. Born proposed in 1926 that |ψ(x, t)|^2 represents the probability density of finding a particle at position x and time t.
Unification: Schrödinger showed that both approaches were mathematically equivalent, laying the foundations for modern quantum theory.
Heisenberg’s Uncertainty Principle (1927)
Classical View: Given perfect instruments and precise measurements, we can, in principle, determine all past and future evolutions of a system.
Development: There is a fundamental limit to the precision with which certain pairs of physical properties (e.g. position and momentum) can be simultaneously known. The more accurately one property is measured, the less accurately the other property can be known.
Complementarity Principle (1927)
Classical View: Particles and waves are distinct entities with separate behaviors.
Complementarity: Niels Bohr argued that both wave and particle aspects are necessary for a complete description of quantum phenomena, but only one aspect can be observed at a time.
Copenhagen Interpretation: Builds upon complementarity, uncertainty, and probability, to argue that the act of measurement “collapses the wave function” and determines the observed outcome. This is one possible explanation for the double-slit experiment (Young 1801) and its variations.
Dirac Equation (1928)
Problem: The Schrödinger equation could not accurately describe particles moving at relativistic speeds
Resolution: Combined quantum mechanics with special relativity to describe the behavior of electrons. Dirac accounted for the electron's spin and predicted the existence of the antimatter, specifically the positron (later discovered 1932)
Hubble’s Discovery of the Expanding Universe (1929)
Gödel’s Incompleteness Theorems (1931)
Classical View: A complete and consistent set of axioms can be formulated to prove all mathematical truths.
Gödel’s First Theorem: In any consistent formal system that is rich enough to include arithmetic, there are true statements about the natural numbers that cannot be proven within the system.
Gödel’s Second Theorem: Such a system cannot demonstrate its own consistency.
Dark Matter (Zwicky 1933)
Classical View: The mass of galaxies and galaxy clusters is accounted by visible matter alone—stars, gas, and dust that emit, absorb, or reflect light.
Counterexample: Fritz Zwicky observed that the Coma galaxy cluster's visible mass was insufficient to explain its gravitational effects.
Development: Zwicky proposed dark matter, an invisible form of matter detectable through gravitational effects. Its existence is supported by ’s the rotation curves of spiral galaxies (Rubin and Ford 1970s), CMB studies (1990s), and perhaps direct detection experiments (e.g. DAMA/LIBRA, XENON, and LUX)
Quantum Electrodynamics (QED) (1940s)
Classical View: Electromagnetic interactions are described by Maxwell's equations. Light is a wave and electromagnetic fields are continuous.
Development: Feynman, Schwinger, and Tomonaga showed that electromagnetic interactions are the result of photon exchange, incorporating quantum mechanics and special relativity. When two charged particles come close, they can exchange a virtual photon. This exchange exerts a force on the particles, repulsing them if they have like charges and attracting them if they have opposite charges.
Feynman Diagrams: Help visualize and calculate interactions between particles in a intuitive way. with lines depicting particles and wavy lines depicting photons.
Electroweak Theory (1960s-1970s)
Previous View: Electromagnetism and the weak nuclear force are separate forces.
Development: Glashow, Salam, and Weinberg united the two forces into a single framework at high energies. The Higgs mechanism explains how these unified forces differentiate at lower energies.
Chaos Theory (1963)
Classical View: Deterministic systems, given precise initial conditions, evolve predictably over time.
Counterexample: Edward Lorenz discovered that minute changes in the starting state of a deterministic system (e.g. initial state of a weather model) led to vastly different outcomes, making long-term prediction impossible.
Quark Model (1964)
Previous View: Hadrons (like protons and neutrons) are elementary particles without internal structure.
Development: Murray Gell-Mann proposed that hadrons are composed of quarks, explaining the patterns and properties of these particles.
Higgs Mechanism (1964)
Classical View: Mass is an intrinsic property of matter.
Problem: There was no explanation for why some particles have mass (e.g. W and Z boson) and others (e.g. photon) do not.
Higgs Mechanism: Higgs-Englert-Brout introduced the Higgs Field, a scalar field that permeates all of space and gives mass to particles.
Sombrero Model: Imagine the Higgs field as a sombrero-shaped potential energy field. It is symmetric at its peak and has a circular valley at the bottom of the sombrero (with some constant and non-zero vacuum expectation value v).
Symmetric State: At high energies (like the early universe), the Higgs field is at the center of the sombrero and has no preferred direction. In this state, electroweak holds and W and Z bosons are massless.
Spontaneous Symmetry Breaking: As the universe cools, the Higgs field "rolls down" into the circular valley of the sombrero potential. It now chooses a specific direction in this valley and its new position represents the stable state (lowest energy configuration) at value v.
Mass Gain: Similar to how walking through tall grass (valley) is different from walking on land (peak), particles like the W and Z bosons now have to "move through" the Higgs field. This resistance is what gives them mass.
Massless Movement: The photon is like a light particle that doesn't interact with the tall grass. The U(1) gauge symmetry for the photon remains unbroken, meaning the photon does not acquire mass.
Yukawa Couplings: The strength of the interaction between fermions and the Higgs field, known as the Yukawa coupling, determines the mass of the fermion.
Bell's Theorem (1964) and the EPR Paradox (1964-1982)
Classical View: Local realism holds. Information cannot travel faster than light (locality) and physical properties exist independently of observation (realism).
Einstein-Podolsky-Rosen Paradox: Imagine two particles, A and B, that are entangled and then separated by a large distance. If measuring A instantly gives you information about B, either:
Locality Is Violated: Measuring A instantly affects B.
Realism Holds: Both particles had pre-determined properties (hidden variables) from the start, so measuring A simply reveals these pre-existing properties in B.
Methodology: John Bell derived inequalities that local hidden variables theories must satisfy. Quantum mechanics predicts that the correlations between entangled particle measurements can violate Bell's inequalities.
Experiments: Alain Aspect’s experiments on entangled photons (1980s) and later quantum teleportation experiments like Zeilinger (1997) and China’s Micius satellite (2017) showed that correlations could exceed the limits set by Bell's inequalities. This means one (or both) of locality and realism is false.
String Theory (Veneziano et al. 1968)
Aim: Unify all fundamental forces, including gravity, into a single theoretical framework.
Theory: Proposes that the fundamental constituents of the universe are not point-like particles but one-dimensional "strings." These strings can vibrate at different frequencies, with each vibration mode corresponding to a different particle.
Quantum Chromodynamics (QCD) (Gross-Wilczek and Politzer 1973)
Problem: Earlier models of particle physics couldn't adequately explain how quarks are bound so tightly inside protons and neutrons.
Development: QCD introduced the concept of color charge and showed that the strong force is mediated by massless gluons, leading to the understanding of quark confinement. Unlike photons in QED, gluons themselves carry color charge, allowing them to interact with each other, leading to the complex behavior of the strong force.
Quark Confinement: Quarks are confined within hadrons (protons, neutrons, mesons) due to the property of the strong force that it becomes stronger as quarks move apart. This increasing force with distance ensures that quarks cannot be isolated and are always found in combination, forming color-neutral particles.
Asymptotic Freedom: At very short distances (high energies), quarks interact more weakly. This phenomenon, known as asymptotic freedom, allows quarks to behave almost as free particles when very close together.
Standard Model of Particle Physics (1975)
Standardize: unify QED, Quark Model, Electroweak, QCD, Higgs, etc.
Particles
Fermions: These particles make up matter. They obey the Pauli exclusion principle and are divided into quarks and leptons.
Quarks
Types: Six flavors: up (u), down (d), charm (c), strange (s), top (t), bottom (b).
Properties: Quarks have fractional electric charges (uct have +2/3 and dsb have -1/3) and come in any of three "colors" (red, green, blue). Antiquarks carry opposite charges and anti-colors (e.g. anti-red).
Combinations: Quarks combine to form composite particles called hadrons, which are color-neutral.
Baryons: Composed of three quarks, with each quark having a different color. Includes protons (uud) and neutrons (udd).
Mesons: Composed of a quark and an antiquark. Includes the positive pion (up + antidown quark).
Leptons
Types: Six flavors: electron (e), muon (μ), tau (τ), and their corresponding neutrinos (ν_e, ν_μ, ν_τ).
Properties: Leptons have integer electric charges (electrons, muons, and taus have -1, while neutrinos have 0).
Bosons: These particles are force carriers, mediating three of four fundamental forces of nature.
Photon (γ): Massless particle that mediates electromagnetic interactions.
W and Z Bosons: W bosons (W^+, W^-) are charged whereas Z boson (Z^0) is neutral. They mediate the weak nuclear force.
Gluon (g): Massless particles that carry color charge and mediate the strong force.
Higgs Boson (H): A particle with mass that is an excitation of the Higgs field.
Fundamental Forces (3 of 4)
Electromagnetic Force: Acts on charged particles, governing interactions like light emission, chemical bonding, and electricity.
Weak Nuclear Force: Responsible for processes like radioactive decay and neutrino interactions. It changes the flavor of quarks and leptons.
Strong Nuclear Force: Is incredibly powerful but acts only at very short ranges. Binds quarks together to form hadrons as well as protons and neutrons in atomic nuclei. Quarks interact via gluons, which carry the strong force.
Other Mechanisms
Higgs Mechanism: The Higgs field acquires a nonzero value in its vacuum state, breaking the electroweak symmetry and giving mass to W and Z bosons.
Annihilation: The process where a particle and its corresponding antiparticle collide and convert their mass into energy. This can occur with quark-antiquark pairs within mesons, electron-positron pairs, and proton-antiproton pairs. The energy released is often in the form of gamma-ray photons or other particles.
Observational Support
Particle Discoveries: top quark (Fermilab 1995), tau neutrino (Fermilab 2000), Higgs boson (CERN 2012), etc.
Electroweak Precision Tests: Measurements of W and Z boson properties at LEP and SLC
QCD Tests: High-energy collisions at particle accelerators confirm the predictions of QCD.
Cornell-Wieman’s Bose-Einstein Condensate (1995)
Classical States of Matter: Matter exists in three primary states: solid, liquid, and gas. Atoms and molecules in these states have distinct, individual behaviors and properties.
Prediction by Bose and Einstein (1920s): Satyendra Nath Bose and Albert Einstein predicted that at extremely low temperatures, a group of bosons could occupy the same quantum state.
Experimental Realization: Eric Cornell and Carl Wieman successfully created a BEC by cooling rubidium-87 atoms to within a few billionths of a degree above absolute zero. At this temperature, the rubidium atoms occupied the same quantum state and behaved as a single quantum entity.
Schmidt-Riess and Perlmutter’s Accelerating Universe (1998)
Previous View: Einstein introduced the cosmological constant to allow for a static universe in general relativity, but abandoned it after Hubble’s discovery of the expanding universe.
Development: Discovery of accelerating universe expansion and later Wilkinson Microwave Anisotropy Probe (2003) revitalize research on a mysterious “dark energy” responsible for accelerated expansion.
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CONTEMPORARY UNDERSTANDING (2024)
The fundamental nature of physical reality is quantum. Nature is not continuous but discrete, with a fundamental limit to the divisibility of space, time, and matter (e.g. Planck length). All properties of fields and states (particles) are discrete.
Elementary particles appear in spacetime only when fields are excited or interact with other fields. Particles do not exist as objects, but only as excited states of the homonymous fields (e.g. Higgs boson).
Particles are not subject to “forces” as classical physics describes. Quantum particles interact by exchanging other particles (e.g. gluons).
The universe is expanding at an accelerating pace and entropy is increasing.
Space and time are interconnected.
Locality and realism cannot coexist. Information may travel faster than light and/or properties of physical reality are not independent of observation.
Reductionism is false. Entangled states (e.g. hydrogen atom) resulting from the interaction of two quantum systems can be a structure with entirely new properties than that of its components (electron field and proton field)
Determinism is false; there is not a one-to-one correspondence between an element of theory and an element of reality. We cannot measure two non-commutable variables at the same time, and we cannot even predict where a particle will be, not even in principle. We can only predict the probabilities of finding a particle in a certain region of space.
Physical reality is described by a set of fields subject to the laws of quantum field theory (QFT) and general relativity (GR).
Mathematics cannot give us a comprehensive description of reality.
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LINGERING QUESTIONS
Hard Problem of Consciousness (Chalmers)
Question: Why do physical states or electromagnetic states give rise to subjective experiences, or qualia? For instance, how do physical processes in the brain translate to the experience of smelling an orange or feeling love?
Some Theories: I will discuss the materialist (Crick) and idealist (Faggin) answers along with my own cop-out.
Quantum Entanglement and Collapse of Wave Function
Question: It seems some “collapse” must occur between the quantum system and the measuring apparatus in order to make sense of observations like the double-slit and its variations. But how? And how does entanglement work?
Some Theories
Copenhagen Interpretation: Suggests that particles remain in a superposition of states until measured, at which point the wave function collapses and the entangled state is resolved.
Many-Worlds Interpretation: Suggests that all possible outcomes of a quantum event actually occur in separate, branching universes, thus avoiding the need for wave function collapse.
Pilot-Wave Theory (Bohmian Mechanics): Proposes that particles have definite positions and are guided by a deterministic wave, eliminating the need for collapse.
Ghirardi-Rimini-Weber (GRW) Theory: Proposes spontaneous collapses of the wave function that occur independently of observation.
Penrose's Objective Collapse Model: Links wave function collapse to gravitational effects, suggesting that gravity causes the collapse independently of observation.
Decoherence Theory: The wave function of the system does not collapse in the traditional sense. Instead, the coherence of the system's superposed states is distributed into the environment, making the different outcomes non-interfering and resulting in the appearance of collapse.
Theory of Everything
Question: How can we unite the Standard Model and General Relativity to account for all four fundamental forces?
Theories
String Theory: See under “Major Developments.”
Loop Quantum Gravity: Attempts to quantize spacetime itself, providing a theory where space and time are discrete, potentially reconciling quantum mechanics with general relativity.
Quantum Gravity Theories: Other approaches to quantum gravity, such as asymptotically safe gravity, causal dynamical triangulations, and spin foam models, attempt to unify gravity with quantum mechanics through different mathematical frameworks.
Grand Unified Theories (GUTs): These theories attempt to unify the three fundamental forces of the Standard Model (electromagnetic, weak, and strong) at high energy levels, possibly leading to unification with gravity in a larger framework.
Dark Matter and Dark Energy
The Question: Dark matter and dark energy constitute approximately 95% of the total mass-energy content of the universe, but what is their nature?
Theories: Various candidates for dark matter (like WIMPs or axions) and explanations for dark energy (like the cosmological constant) have been proposed but not definitively proven.
This post is part of Core Reading. Visit the main work to learn more.
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REVISION HISTORY
07.07.2024
First iteration.