"In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time."
The study of physical fields and their behavior.
Lagrangian Mechanics: This is the foundation of classical field theory, where the Lagrangian of a system is defined as the difference between its kinetic and potential energy. The equations of motion for fields are derived from the action principle, where the action is the integral of the Lagrangian over time.
Hamiltonian Mechanics: This is an alternative formulation of classical mechanics that is equivalent to Lagrangian mechanics. It involves the use of Hamilton's equations, which describe the evolution of the state of a dynamical system over time.
Calculus of Variations: This is a mathematical tool used in Lagrangian mechanics to find the minimum of a functional. It is used to derive the equations of motion for fields.
Tensor Calculus: This is a mathematical tool used to describe the geometry of spacetime in relativity theory. It is used extensively in classical field theory to describe the structure of fields and their interactions.
Field Equations: These are the fundamental equations that describe the behavior of fields. There are several different field equations, depending on the type of field being described (e.g., electromagnetic, gravitational, etc.).
Electromagnetic Fields: This refers to the study of electric and magnetic fields and their interactions. The theory of electromagnetism is based on Maxwell's equations, which describe the behavior of electric and magnetic fields.
Special Relativity: This is the theory of relative motion between observers moving at constant velocities. It is used extensively in classical field theory to describe the behavior of fields in moving reference frames.
General Relativity: This is the theory of gravity and the curvature of spacetime. It is used to describe the behavior of gravitational fields and their interactions with other fields.
Quantum Field Theory: This is the theory of fields at the quantum level. It is used to describe the behavior of particles and their interactions at the smallest scales.
Symmetry: This is a fundamental concept in physics, where symmetries in the behavior of fields and particles lead to conservation laws. The study of symmetry is essential in understanding the behavior of fields and particles.
General Relativity: It is the theory of gravity that describes the curvature of spacetime caused by the presence of mass and energy.
Electromagnetism: It is the theory of the electromagnetic field that describes the interactions between electrically charged particles and their fields.
Classical mechanics: It is the study of motion and force using principles of mass, acceleration, and energy.
Quantum field theory: It is the study of the quantum mechanics of fields, which includes the behavior of particles that are formed from fields.
Gauge theory: It is the study of the symmetries of physics and their relations to the interactions between particles.
Topological field theory: It is the study of the topological aspects of fields, which involves the study of the properties of objects that remain the same under deformation.
String field theory: It is the study of a hypothetical fundamental particle, the string, that can shape the entire universe's fabric.
"On a weather map, the surface temperature is described by assigning a number to each point on the map."
"A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field."
"Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics."
"The electric field is another rank-1 tensor field."
"Electrodynamics can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field."
"In the modern framework of the quantum theory of fields, even without referring to a test particle, a field occupies space, contains energy, and its presence precludes a classical 'true vacuum'."
"This has led physicists to consider electromagnetic fields to be a physical entity, making the field concept a supporting paradigm of the edifice of modern physics."
"In practice, the strength of most fields diminishes with distance, eventually becoming undetectable. For instance, the strength of many relevant classical fields, such as the gravitational field in Newton's theory of gravity or the electrostatic field in classical electromagnetism, is inversely proportional to the square of the distance from the source (i.e., they follow Gauss's law)."
"A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively."
"A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field somewhere else."
"For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point."
"Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators, respectively."
"In this theory, an equivalent representation of a field is a field particle, for instance, a boson."
"A field is a physical quantity, represented by a scalar, vector, or tensor."
"The temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change."
"This has led physicists to consider electromagnetic fields to be a physical entity, making the field concept a supporting paradigm of the edifice of modern physics."
"In practice, the strength of most fields diminishes with distance, eventually becoming undetectable."
"Specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point."
"The fact that the electromagnetic field can possess momentum and energy makes it very real... a particle makes a field, and a field acts on another particle, and the field has such familiar properties as energy content and momentum, just as particles can have."