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  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Quantum mechanics → Quantum field theory ↓


Spontaneous symmetry breaking and the Higgs mechanism


In the field of quantum field theory, which describes the fundamental forces and particles of the universe, spontaneous symmetry breaking and the Higgs mechanism play important roles. By understanding these concepts, we can gain a deeper understanding of the inner workings of nature at its most basic level. Let us understand these topics in a comprehensive manner.

Symmetry in physics

In physics, symmetry refers to a state where a system remains unchanged under certain transformations. These transformations may include rotations, translations, or any other operation that leaves the basic properties of the system unchanged. Symmetries are an essential part of modern physics because they lead to conservation laws, as shown by Noether's theorem.

Visual example: symmetry in an object

There is no change even when the circle is rotated about its centre, which shows rotational symmetry.

Spontaneous symmetry breaking

Spontaneous symmetry breaking occurs when the ground state, or lowest energy state, of a system does not share the symmetry properties of the underlying rules that describe it. Even though the fundamental rules may exhibit symmetry, the system chooses a configuration that breaks this symmetry.

Example: The possibility of a Mexican hat

A common analogy for spontaneous symmetry breaking involves the Mexican hat potential. Consider the potential energy shaped like a "Mexican hat":

V(ϕ) = -μ²|ϕ|² + λ|ϕ|⁴

Here, φ denotes a field, μ² and λ are constants. The potential is symmetric about the vertical axis, and the potential minimum forms a circular valley.

A ball at the top center of the cap is in a perfectly symmetric state, but this state is unstable. The system eventually converges to a stable state within the surrounding valley, breaking the symmetry.

Implications in physics

Spontaneous symmetry breaking explains a variety of phenomena in physics. For example, in ferromagnetism, individual magnetic moments (spins) tend to align in a particular direction despite isotropic microscopic interactions. While the fundamental rules are symmetric, the ground state (aligned spins) breaks this symmetry.

Higgs mechanism

The Higgs mechanism is a mechanism that explains how particles gain mass in gauge theories. To understand this, let's explore gauge symmetries and see how the Higgs field fits into this framework.

Gauge symmetry and mass

Gauge symmetry is a type of symmetry that is important for formulating fundamental interactions in physics. However, gauge theories typically describe massless particles. In order for particles to acquire mass while maintaining gauge symmetry, an additional mechanism is needed: the Higgs mechanism.

Introduction to the Higgs Field

The Higgs field is a scalar field that pervades the entire universe. The interaction between particles and the Higgs field gives rise to mass. The field has a vacuum expectation value different from zero, which spontaneously breaks the symmetry even though the underlying rules remain symmetric.

Field Values Higgs vacuum expectation value

As particles interact with this field, they gain mass. The non-zero expected value of the field selects a specific direction in internal space and breaks the symmetry.

Mass production by the Higgs mechanism

Particles in the Higgs mechanism acquire mass through the following process:

  1. The field with nonzero vacuum expectation value interacts with gauge fields.
  2. The gauge symmetry is spontaneously broken.
  3. The resulting broken symmetry affects the equations governing the gauge fields, and gives them a mass term.

The interaction of gauge fields with the scalar Higgs field leads to modifications to the field equations. These equations describe the propagation of heavy particles rather than massless particles.

Examples in the Standard Model

The Higgs mechanism is the cornerstone of the Standard Model of particle physics. It explains the masses of the W and Z bosons, the fundamental particles that mediate the weak interaction. Without the Higgs mechanism, these particles would be massless, leading to incorrect predictions.

In 2012, the Higgs boson, a particle associated with the Higgs field, was discovered at the Large Hadron Collider, confirming the existence of the Higgs mechanism.

Theoretical framework and mathematical formulation

Let us take a closer look at the theoretical framework and mathematical formulation of spontaneous symmetry breaking and the Higgs mechanism to understand them in greater depth.

Spontaneous symmetry breaking in field theory

Consider a scalar field theory with a Lagrangian:

L = ∂μϕ ∂^μϕ – V(ϕ)

The potential V(φ) can have a form that results in automatic symmetry breaking. For example:

V(ϕ) = -μ²|ϕ|² + λ|ϕ|⁴

This potential has a degenerate ground state, and the field selects a specific vacuum state, causing symmetry breaking. Such potentials usually describe fields with spontaneous symmetry breaking in field theory.

Mathematical description of the Higgs mechanism

In the context of the Higgs mechanism, consider the Lagrangian for a complex scalar field φ:

L = (Dμϕ)*(D^μϕ) - V(ϕ)

Dμ refers to the covariant derivative, which involves the gauge fields. The potential V(φ) has a form that enables spontaneous symmetry breaking:

V(ϕ) = -μ²|ϕ|² + λ|ϕ|⁴

When the Higgs field acquires the vacuum expected value, it breaks the gauge symmetry, providing a mass term for the gauge bosons in the process. This mathematical construction describes how particles acquire mass through the Higgs mechanism.

Concluding remarks

Spontaneous symmetry breaking and the Higgs mechanism are fundamental concepts that provide insights into the workings of the universe at the quantum level. They underpin our understanding of particle masses, fundamental forces and symmetries in physics. The discovery of the Higgs boson validated this theoretical framework, paving the way for a deeper understanding of the fabric of reality.


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Spontaneous symmetry breaking and the Higgs mechanism

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