"A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization."
Introduces the mathematical tools and concepts necessary to describe classical fields, including Lagrangians, Euler-Lagrange equations, and Noether's theorem.
Lagrangian mechanics: A formulation of classical mechanics that uses a single energy-based function to describe the behavior of a system.
Hamiltonian mechanics: A formulation of classical mechanics that uses a different energy-based function, the Hamiltonian, to describe the behavior of a system.
Symmetries and conservation laws: A concept in classical field theory that states that certain properties of a system do not change under specific transformations.
Noether's theorem: A theorem that relates symmetries in a system to the existence of corresponding conservation laws.
Electromagnetic fields: The study of the electromagnetic force and its interactions with particles.
Maxwell's equations: A set of equations that describe the behavior of electromagnetic fields.
Lorentz invariance: A fundamental concept in special relativity that states that the laws of physics should be the same in all inertial reference frames.
Gauge symmetry: A concept in quantum field theory that relates to the choice of different mathematical descriptions of the same physical system.
Path integral formulation: A mathematical technique in quantum field theory that uses a sum over all possible paths to calculate the probability of a particular outcome.
Quantum chromodynamics: The study of the strong nuclear force and the behavior of quarks and gluons.
Feynman diagrams: A pictorial representation of particle interactions in quantum field theory.
Scalar fields: A type of field in quantum field theory that describes particles with spin-0.
Spinor fields: A type of field in quantum field theory that describes particles with half-integer spin.
Canonical quantization: A method for turning classical field theories into quantum field theories.
Renormalization: A mathematical technique in quantum field theory that resolves divergences that arise when calculating physical observables.
Effective field theories: A useful approximation technique in quantum field theory that involves expanding low-energy interactions.
Lattice field theory: A numerical technique used in quantum field theory to simulate particle interactions on a discrete grid.
Supersymmetry: A theoretical framework that proposes a symmetry between bosons and fermions.
String theory: A theoretical framework that unifies gravity and quantum mechanics by describing particles as one-dimensional strings.
Loop quantum gravity: A theoretical framework that attempts to quantize gravity using a background-independent approach.
Electromagnetic field theory: This theory describes the behavior of electric and magnetic fields and their interactions with charged particles.
Gravitational field theory: This theory deals with the description of gravitational fields and their effects on matter and energy.
Thermodynamics: This is the study of the behavior of thermal energy in a system and its transformation into other forms of energy.
Fluid dynamics: This theory deals with the motion of fluids and their applications in engineering and physics.
Quantum field theory: This theory describes the behavior of quantum fields and their interactions with particles.
Relativistic field theory: This theory deals with the description of fields in the context of special and general relativity, and their implications for particle physics and cosmology.
Statistical field theory: This theory is concerned with the behavior of complex systems of interacting particles, such as those found in condensed matter physics.
Scalar field theory: This theory describes the behavior of scalar fields, which are fields that have a single numerical value at each point in space and time.
Gauge field theory: This theory describes the interactions between fields that are invariant under certain symmetries, known as gauge symmetries.
Nonlinear field theory: This theory deals with the behavior of fields that exhibit nonlinear behavior, such as solitons and nonlinear waves.
"Theories that incorporate quantum mechanics are called quantum field theories."
"A classical field theory is specifically intended to describe electromagnetism and gravitation."
"A physical field can be thought of as the assignment of a physical quantity at each point of space and time."
"For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space."
"Each vector represents the direction of the movement of air at that point."
"As the day progresses, the directions in which the vectors point change as the directions of the wind change."
"The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields were developed in classical physics."
"The advent of relativity theory in 1905"
"Had to be revised to be consistent with that theory."
"Classical field theories are usually categorized as non-relativistic and relativistic."
"Modern field theories are usually expressed using the mathematics of tensor calculus."
"A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles."
"...predicts how one or more physical fields interact with matter through field equations..."
"...effects of quantization..."
"...classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature."
"A set of all wind vectors in an area at a given point in time constitutes a vector field."
"The wind velocity during a day over a country is described by assigning a vector to each point in space."
"...field equations..."
"...theories that incorporate quantum mechanics are called quantum field theories."