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 ↓


Supersymmetry and extra dimensions


The quest to understand the fundamental structure of the universe has led physicists to search for theories beyond the standard model of particle physics. Two of the most interesting concepts in this quest are the notion of supersymmetry (SUSY) and extra dimensions. These ideas promise to address some of the limitations of the standard model and provide a path towards a more unified understanding of fundamental forces and particles.

Supersymmetry

Supersymmetry, often abbreviated as SUSY, is a theoretical framework that proposes symmetry between fermions and bosons. The idea is that every particle in one of these groups has a counterpart in the other, known as a superpartner. For example, if the electron is a fermion, there exists a bosonic superpartner called the selectron. Other pairings include quarks and squarks, photons and photino, etc.

Motivation for supersymmetry

One of the main motivations behind introducing supersymmetry is to solve the hierarchy problem, which is related to the large difference between the weak force scale and the gravitational scale. In the absence of SUSY, quantum corrections to the Higgs boson mass can be very large, requiring fine-tuning to reach the observed value. However, in a supersymmetric model, these corrections are naturally cancelled out by the contributions of superpartners.

Σ(Δm_h^2) = Σ(Δm_fermion^2) + Σ(Δm_boson^2) = 0

Another advantage of supersymmetry is that it provides a candidate for dark matter. In many SUSY models, the lightest supersymmetric particles (LSPs) are stable and weakly interacting, fulfilling the properties required for dark matter.

Visualization of supersymmetry

Electron Selectron Susie

This simple visual example demonstrates the concept of supersymmetry, where an electron pairs with its superpartner, the selectron.

Experimental discoveries

As of now, there is no direct experimental evidence for supersymmetry. Large particle accelerators such as the Large Hadron Collider (LHC) are currently attempting to find traces of supersymmetric particles. If discovered, they would be a strong indication that SUSY is a viable extension of the Standard Model.

Extra dimensions

The concept of extra dimensions arises in string theory and modifies our understanding of space-time. Traditional physics considers three dimensions of space and one dimension of time. The notion of extra dimensions suggests that there may be more spatial dimensions, although they are compact and not directly observable.

Why the extra dimensions?

The inclusion of extra dimensions makes it possible to obtain a unified theory of all fundamental forces, including gravity. For example, string theory naturally requires extra dimensions for mathematical consistency.

Visualizing extra dimensions

To understand what extra dimensions might be like, imagine a garden hose. From a distance, the hose looks like a one-dimensional line. However, if you look at it up close, you discover that it has an extra dimension, a spherical surface around its length. Similarly, extra dimensions are thought to be compact and can be 'wrapped' in such a way that we are unable to see them on a large scale.

1D Line 2D surface

Implications of extra dimensions

Extra dimensions affect the strength and behavior of the fundamental forces. Forces can propagate through extra dimensions, changing their apparent strength in our familiar four-dimensional spacetime. Additionally, gravity can be understood in a new way, which could potentially explain why it is weaker than the other fundamental forces.

The total action in a high-dimensional model can be represented as:

S = ∫d^4x d^ny √(-g) L

where d^ny denotes integration over the extra dimensions, and L is the Lagrangian density.

Search for extra dimensions

Experimental approaches to searching for extra dimensions include searching for deviations from the inverse square law of gravity and looking for missing energy and particles in high energy collisions. These efforts are ongoing, and new techniques may uncover evidence of extra dimensions in the future.

Exciting opportunities and challenges

Both supersymmetry and extra dimensions have the potential to revolutionise our understanding of the universe. They provide coherent frameworks that resolve many problems in the standard model as well as predict new physics. However, their primary challenge lies in experimental verification. As particle accelerators and other experimental approaches advance, the scientific community is eagerly awaiting potential discoveries that could validate these theories.

In conclusion, although these theories are yet to be proven, their beauty and depth provide promising insights and keep the field of physics full of fascinating possibilities for future exploration.


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Supersymmetry and extra dimensions

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