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 → Quantum Chromodynamics ↓


Lattice QCD


Lattice QCD is a powerful computational technique used in quantum chromodynamics (QCD), the theory of the strong force that binds quarks and gluons into protons, neutrons and other hadrons. In simple terms, it provides a way to study the interactions of quarks and gluons using a grid-like structure in space-time. This approach is important because unlike electromagnetism or gravity, the strong force becomes much stronger when quarks are separated, making it extremely difficult to resolve using conventional analytical methods.

Basics of quantum chromodynamics

Before diving into Lattice QCD, it is essential to understand the basics of quantum chromodynamics. QCD is a part of the Standard Model of particle physics, and it describes the interactions of quarks and gluons. Quarks are the basic constituents of protons and neutrons, while gluons are the force carriers that bind quarks together. The interaction between quarks and gluons is mediated by a property called color charge, which is similar to electric charge in electromagnetism.

Quarks, gluons and color charge

In QCD, there are six types of quarks: up, down, charm, strange, top and bottom. Each quark has a colour charge: red, green or blue. Gluons are massless particles that mediate the strong force between quarks, and they themselves carry a combination of colour and anti-colour charge. The force between quarks becomes stronger when they are separated, a property known as colour confinement, which ensures that quarks are never found in isolation.

Role of strong force

The strong force is one of the four fundamental forces in nature and is responsible for holding atomic nuclei together. Unlike the electromagnetic or gravitational forces, the strong force exhibits a remarkable feature called asymptotic freedom: quarks and gluons interact more weakly at short distances or high energies. At greater distances, however, the force increases, causing quarks to become confined within hadrons.

Introduction to Lattice QCD

Lattice QCD provides a numerical approach to solving QCD by discretizing space-time into a finite grid, or lattice, where quarks occupy lattice points and gluons exist on the links connecting these points. This discretization allows numerical simulations to study non-perturbative aspects of QCD that are inaccessible through perturbative methods, providing insights into quark confinement, hadronization, and the mass spectrum of hadrons.

Lattice formulation

In lattice QCD, space-time is represented as a hyper-cubic lattice with a finite number of lattice points. The lattice spacing, a, determines the distance between neighbouring points, while the entire lattice has a finite size determined by the number of points in each dimension. This setup allows complex quantum systems to be treated through calculations on discrete variables.

Grid example:
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Action and path integral formulations

The path integral formulation is central to lattice QCD. The fundamental object is the action, S, which in the case of QCD is given by the Yang-Mills action combined with the fermionic term for the quarks. In lattice terms, this involves a sum over links and plaquettes (elementary sections of the lattice) representing the gluons and over lattice sites for the quarks. The partition function of the system, is given by:

Z = ∫ D[U] D[ψ] D[ψ̄] e^(-S)

The path integral is approximated by summing over all possible configurations of the quark and gluon fields, weighting these by the exponent of the negative action -S, thus capturing non-perturbative events.

Simulating QCD on a lattice

Numerical simulations in lattice QCD are typically performed using Monte Carlo algorithms, which randomly sample field configurations according to their probability in the path integral formulation. This approach enables the calculation of various observables, such as hadron masses, decay constants, and coupling strengths, directly from first principles.

The rotational symmetry of space is replaced by a discrete cubic symmetry on the lattice, which simplifies calculations but can introduce discretization errors. Sophisticated techniques such as the Siemensik correction are used to reduce these errors.

Challenges and computational effort

Lattice QCD calculations are computationally demanding, requiring significant resources even with modern supercomputers. The finite size of the lattice and the distance between the lattices must be increased to the continuum limit, where physical predictions can be compared with experiments. Additionally, dealing with fermions on the lattice, known as the fermion doubling problem, requires careful handling with various discretization techniques such as Wilson, staggered, and overlap fermions.

Applications of lattice QCD

Lattice QCD plays a key role in many areas of nuclear and high energy physics, enhancing our understanding of fundamental interactions and guiding experimental programs. Some of the key points are as follows:

  • Hadron spectroscopy: Lattice QCD predicts the masses and decay constants of various hadronic states, and provides information about the structure and interactions of particles such as mesons and baryons.
  • Quark–gluon plasma: The study of finite temperature QCD on a lattice helps to explore the properties of matter under extreme conditions, such as those that existed shortly after the Big Bang.
  • Weak interactions: Accurate calculations of the hadronic contributions to weak decays, such as those involving kaons and B-mesons, aid in the determination of parameters in the CKM matrix associated with quark flavor transitions.

Future prospects

The future of lattice QCD looks promising with advances in computational techniques and hardware. Advanced algorithms and increased processing power are expected to lead to more accurate predictions of hadronic structures and interactions, impacting a wide range of theoretical and experimental discoveries in particle physics.

Conclusion

Lattice QCD is a powerful tool that provides deep insights into the non-perturbative aspects of QCD. Through space-time discretization, it allows researchers to simulate and study the complex quark and gluon interactions that define the structural dynamics of the universe at the smallest scales. As computational power continues to increase, lattice QCD remains a frontier for exploring the complex properties of the strong force, providing predictions and understanding that complement experimental findings in high-energy physics.


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Lattice QCD

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