PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Quantum field theory → Standard model of particle physics ↓


Electroweak theory


The Standard Model of particle physics is an extraordinary theory that describes the fundamental forces that govern the universe, as well as the basic building blocks that make up all matter. Among its most profound components is the electroweak theory, which unifies two of the four known fundamental forces: the electromagnetic force and the weak nuclear force. In this explainer, we'll go deep into the electroweak theory, exploring its origins, concepts, mathematical formulation, and importance in our understanding of particle physics. This will require us to navigate the landscape of quantum field theory, which underlies the Standard Model.

Historical background

The quest for unification is not new in physics. In the 19th century, James Clerk Maxwell unified electricity and magnetism into a single theory of electromagnetism. This was an important milestone because it established that electric and magnetic fields are different aspects of the same phenomenon. After this great achievement, physicists tried to unify other forces.

The journey of the electroweak theory began with pioneering work on particle interactions in the early 20th century. The weak nuclear force, responsible for processes such as beta decay in atomic nuclei, initially seemed unrelated to electromagnetism because of its short-range effects and different behaviour. However, in the 1960s, Sheldon Glashow, Abdus Salam and Steven Weinberg independently contributed to the electroweak theory. Their work was awarded the Nobel Prize in Physics in 1979.

The basic idea of the electroweak theory

The main idea of the electroweak theory is based on the premise that at high energies, the electromagnetic and weak nuclear forces merge into a single electroweak force. This unification occurs due to a phenomenon called spontaneous symmetry breaking, which appears as the universe cools from the high-energy conditions that existed just after the Big Bang. The electroweak theory successfully predicts that at high enough temperatures, such as those found in the early universe, the electromagnetic and weak forces are unified. As the universe cools, this symmetry breaks, separating the forces as we see them today.

Mathematical formulation

The mathematical backbone of the electroweak theory lies in quantum field theory, which extends quantum mechanics to fields. This theory uses the concept of gauge symmetry to describe interactions mathematically.

The electroweak theory is codified in a gauge theory called SU(2) x U(1). The structure of the theory is as follows:

SU(2) x U(1) → U(1)_EM

Let us analyse these words:

  • SU(2): It exhibits weak isospin symmetry, governs the weak interactions, and contains the W and Z bosons.
  • U(1): This corresponds to the weak hypercharge symmetry.
  • U(1)_EM: This is the residual symmetry after symmetry breaking, corresponding to electromagnetic interactions mediated by photons.

The Lagrangian of the electroweak theory describes how these fields interact. For simplicity, here is a non-detailed form:

𝓛 = 𝓛_gauge + 𝓛_Higgs + 𝓛_fermion
  • 𝓛_gauge: This contains the kinetic terms for the gauge bosons that describe the interactions of the W, Z, and photon fields.
  • 𝓛_Higgs: represents the dynamics of the Higgs field, which is responsible for spontaneous symmetry breaking.
  • 𝓛_fermion: Covers interactions with the fundamental matter particles, fermions.

One of the remarkable predictions of the electroweak theory is the existence of the W and Z bosons. Unlike the photon, these particles are massive, and it is their associated mass that gives the weak interactions their short range. The masses of the W and Z bosons, which were predicted and later confirmed by experiments at CERN, support the validity of the electroweak theory.

Spontaneous symmetry breaking and the Higgs mechanism

The concept of spontaneous symmetry breaking is important in electroweak theory. It involves a change in the state of the system that breaks the fundamental symmetry of the governing equations. In this context, the symmetry between the electromagnetic and weak forces is broken, leading to separate forces.

The Higgs mechanism plays a key role here. Consider the following analogy: Imagine a flat sphere that represents the symmetric state in its simplest state before symmetry breaking. However, underneath this sphere is a Mexican hat-shaped potential energy surface. When the system is perturbed, it finds a minimum energy state away from the peak, breaking the original symmetry.

The possibility of a Mexican hat

In the electroweak theory, this transition is induced by the Higgs field. As the Higgs field attains a non-zero value in its lowest energy state (called the vacuum expectation value), the W and Z bosons gain mass. This phenomenon was confirmed with the discovery of the Higgs boson in 2012.

Visual representation in particle interaction

We can visualize interactions within the electroweak theory using Feynman diagrams. These diagrams simplify complex mathematical expressions into graphical forms. Below is an illustration of a process involving the weak force, represented by the exchange of W bosons:

W

Such diagrams allow one to visualise the interactions of particles, showing initial and final states and exchanged particles such as W or Z bosons.

Experimental verification

The electroweak theory is one of the most experimentally tested and validated theories in physics. The prediction and discovery of the W and Z bosons at CERN's Super Proton Synchrotron in 1983 were important milestones that strengthened the credibility of the theory. Subsequently, the discovery of the Higgs boson at the Large Hadron Collider in 2012 provided further confirmation.

Experiments have tested the predictions of the electroweak theory with ever-increasing precision. The observations align exceptionally well with the theoretical mathematical formulation, demonstrating the robustness of the theory and our growing experimental capabilities.

Implications and significance

The electroweak theory has far-reaching implications beyond its success in particle physics. Its unification of the electromagnetic and weak forces serves as a cornerstone for models such as the Grand Unified Theory, which attempts further unification with the strong force. Such attempts aim to obtain a theory of everything – a final model describing all fundamental interactions.

Additionally, the electroweak theory contributes to understanding the conditions of the early universe, such as the events immediately following the Big Bang, and the processes associated with baryogenesis and the matter–antimatter asymmetry observed today.

Conclusion

The electroweak theory remains a landmark achievement in physics. Bridging the gap between electromagnetism and the weak nuclear forces, it provides a coherent framework contained within the Standard Model. Its implications extend to cosmology, particle physics, and beyond, continually spurring exploration and discovery. With ongoing experimental progress, the potential for further breakthroughs is enormous, moving us closer to a comprehensive understanding of the workings of the universe.


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Electroweak theory

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