Graduate
  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 → Nuclear and Particle Physics ↓


Beyond the Standard Model


Particle and nuclear physics under the "Standard Model" has been one of the greatest achievements of modern physics. It describes how the basic building blocks of matter interact with fundamental forces. However, it is recognized that the Standard Model is not a complete theory of fundamental interactions. There are several reasons why physicists are eager to explore "beyond the Standard Model" (BSM) physics. In this document, we will discuss the limitations of the Standard Model, the motivations for BSM physics, and some of the key theories that aim to enhance our understanding.

Limitations of the Standard Model

The Standard Model is a beautiful and successful theory, yet it has several limitations:

  • Gravitation: The Standard Model does not include gravity as one of the four fundamental forces. The gravitational force is described separately by general relativity, but a unified theory that incorporates gravity with the other forces remains elusive.
  • Dark matter and dark energy: Observations show that about 85% of the mass of the universe is dark matter, and about 70% of the energy content of the universe is dark energy. The standard model does not take these components into account.
  • Neutrino mass: In the standard model, neutrinos are massless, yet experiments have shown that they have a mass slightly less than zero.
  • Baryon asymmetry: The observable universe is composed primarily of matter, not antimatter. The standard model cannot explain this matter-antimatter asymmetry.

Motivations beyond Standard Model physics

To address these limitations, physicists are exploring new concepts. Here are some of the major inspirations:

  1. Unification: There is a desire to unify all fundamental forces into a single theoretical framework. This would incorporate gravity into particle physics and possibly explain all forces as manifestations of a single interaction.
  2. Hierarchy problem: The Standard Model requires fine-tuning to explain the Higgs boson mass. The hierarchy problem refers to the question of why the Higgs mass is so much lighter than expected from the Planck scale without extensive fine-tuning.
  3. Quantum gravity: The discovery of the theory of quantum gravity, including the graviton, the hypothetical quantum particle of gravity, is important for understanding high-energy astrophysical phenomena and the early universe.

Examples of theories beyond the Standard Model

Several possible theories have been proposed to extend beyond the Standard Model:

String theory

String theory suggests that instead of point particles, the fundamental objects of the universe are tiny, vibrating strings. This is promising because it inherently includes gravity, thus providing a possible theory of everything.

Each type of vibration corresponds to a different particle. Imagine this:

Sorry, your browser does not support inline graphics.

In this illustration, the string vibrates in different ways to represent different particles, such as electrons or quarks.

Supersymmetry (SUSY)

Supersymmetry proposes a symmetry between fermions, the particles that make up matter, and bosons, the particles that mediate forces. For each particle in the standard model, there is a corresponding "superpartner."

If this is true, these superpartners could be responsible for dark matter.

Extra dimensions and Kaluza–Klein theory

These theories suggest that there are additional spatial dimensions in the universe beyond the familiar three dimensions. The Kaluza–Klein theories originally attempted to unify electromagnetism with gravity by postulating a fifth dimension.

Extra dimensions Sorry, your browser does not support inline graphics.

In this diagram, a circle may represent an additional dimension, which is compressed and invisible in everyday experience.

Grand Unified Theories (GUTs)

GUTs aim to unify the strong, weak, and electromagnetic forces into a single force. These theories generally predict that protons can decay, a hypothesis that is still subject to experimental investigation.

Technicolor Principle

This proposal presents an alternative to the Higgs mechanism, introducing new strong interactions that dynamically generate particle masses.

Mathematical formulation and implications

Besides the formalism and philosophy behind BSM, several mathematical tools are used:

Theories on spacetime and gravity

Many BSM models investigate the nature of spacetime on very small scales. For example, string theory suggests a spacetime fabric made of branes.

// Simplified Lagrangian example L = - 1/4 F μν F μν + ψ|D|ψ − (1/2) m²Φ² + ...

In this Lagrangian, different terms represent fields and their interactions. For example, F μν refers to the strength of the gauge field, which represents the unification of forces.

Physics beyond the collider

For example, the Large Hadron Collider (LHC) continues to test BSM predictions with the search for superpartners or Higgs couplings. Future colliders aim to investigate these ideas in greater depth.

Experimental and Observational Tests

Detecting dark matter such as WIMPs (weakly interacting massive particles), observing neutrino behavior, and gravitational waves all provide data that can guide BSM physics.

Computational Methods

Simulating BSM scenarios often requires substantial computational power to handle complex models and analyze collision events. Quantum computers may one day play a role in the study of quantum gravity.

Visualizing the Universe: Conceptual View

Beyond the equations, visualizing BSM concepts can make them more accessible:

Universe AUniverse B Sorry, your browser does not support inline graphics.

These circles symbolize possible "bubble universes" within a larger multiverse framework, which is sometimes considered in BSM theories.

Conclusion

The landscape of physics beyond the Standard Model is vast and exciting. Whether through new mathematical frameworks or experimental innovations, physicists strive to answer fundamental questions about the nature of the universe. Advances in technology and computation offer the potential to uncover new phenomena and deepen our understanding of the universe, moving beyond the known limits set by the Standard Model.


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Beyond the Standard Model

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