The magnitude of an electric field surrounding two equally charged (repelling) particles. Brighter areas have a greater magnitude. The direction of the field is not visible.

Oppositely charged (attracting) particles.

In physics, a field is an assignment of a physical quantity to every point in space (or, more generally, spacetime). A field is thus viewed as extending throughout a large region of space so that its influence is all-pervading. The strength of a field usually varies over a region. Image File history File links Field-illustrations-add-magnitude. ... Image File history File links Field-illustrations-add-magnitude. ... Image File history File links Field-illustrations-sub-magnitude. ... Image File history File links Field-illustrations-sub-magnitude. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time and explaining them using mathematics. ... Antonym of psychical. ... In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ...

Fields are usually represented mathematically by scalar, vector and tensor fields. For example, one can model a gravitational field by a vector field where a vector indicates the acceleration a mass would experience at each point in space. Other examples are temperature fields or air pressure fields, which are often illustrated on weather reports by isotherms and isobars by joining up the points of equal temperature or pressure respectively. In mathematics and physics, a scalar field associates a scalar to every point in space. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ... An isotherm is a line of equal or constant temperature on a graph, plot, or map; an isopleth of temperature. ... The word isobar derives from the two ancient Greek words, ισος (isos), meaning equal, and βαρος (baros), meaning weight. In meteorology, an isobar is a line of equal or constant pressure on a graph, plot, or map; an isopleth of pressure. ...

Field theory

Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other components of the field. Usually this is done by writing a Lagrangian or a Hamiltonian of the field, and treating it as the classical mechanics (or quantum mechanics) of a system with an infinite number of degrees of freedom. The resulting field theories are referred to as classical or quantum field theories. There are two types of field theory in physics: Classical field theory, the theory and dynamics of classical fields. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Fig. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

In modern physics, the most often studied fields are those that model the four fundamental forces. A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ...

Classical fields

There are several examples of classical fields. The dynamics of a classical field are usually specified by the Lagrangian density in terms of the field components; the dynamics can be obtained by using the action principle. This article is in need of attention from an expert on the subject. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

Michael Faraday first realized the importance of a field as a physical object, during his investigations into magnetism. He realized that electric and magnetic fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy. Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed significantly to the fields of electromagnetism and electrochemistry. ... Magnetic lines of force of a bar magnet shown by iron filings on paper In physics, magnetism is one of the phenomena by which materials exert an attractive or repulsive force on other materials. ... It has been suggested that optical field be merged into this article or section. ... Current (I) flowing through a wire produces a magnetic field () around the wire. ...

These ideas eventually led to the creation, by James Clerk Maxwell, of the first unified field theory in physics with the introduction of equations for the electromagnetic field. The modern version of these equations are called Maxwell's equations. At the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime. James Clerk Maxwell (13 June 1831 – 5 November 1879) was an important mathematician and theoretical physicist. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... In electromagnetism, Maxwells equations are a set of equations, developed in the latter half of the nineteenth century by James Clerk Maxwell. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ...

Einstein's theory of gravity, called general relativity, is another example of a field theory. Here the principal field is the metric tensor, a symmetric 2nd-rank tensor field in spacetime. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...

Quantum fields

It is now believed that quantum mechanics should underlie all physical phenomena, so that a classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding quantum field theory. For example, quantizing classical electrodynamics gives quantum electrodynamics. Quantum electrodynamics is arguably the most successful scientific theory; experimental data confirm its predictions to a higher precision (to more significant digits) than any other theory.^{[citation needed]} The two other fundamental quantum field theories are quantum chromodynamics and the electroweak theory. These three quantum field theories can all be derived as special cases of the so-called standard model of particle physics. General relativity, the classical field theory of gravity, has yet to be successfully quantized. Fig. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... Classical electromagnetism is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... In the scientific method, an experiment (Latin: ex-+-periri, of (or from) trying), is a set of actions and observations, performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ... In general, data consist of propositions that reflect reality. ... In Wikipedia, precision has the following meanings: In engineering, science, industry and statistics, precision characterises the degree of mutual agreement among a series of individual measurements, values, or results - see accuracy and precision. ... Significant figures (also called significant digits and abbreviated sig figs or sig digs, respectively) is a method of expressing errors in measurements. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research. Elasticity of materials, fluid dynamics and Maxwell's equations are cases in point. Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... In electromagnetism, Maxwells equations are a set of equations, developed in the latter half of the nineteenth century by James Clerk Maxwell. ...

Continuous random fields

Classical fields as above, such as the electromagnetic field, are usually infinitely differentiable functions, but they are in any case almost always twice differentiable. In contrast, generalized functions are not continuous. When dealing carefully with classical fields at finite temperature, the mathematical methods of continuous random fields have to be used, because a thermally fluctuating classical field is nowhere differentiable. Random fields are indexed sets of random variables; a continuous random field is a random field that has a set of functions as its index set. In particular, it is often mathematically convenient to take a continuous random field to have a Schwartz space of functions as its index set, in which case the continuous random field is a tempered distribution. This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... In mathematics, generalized functions are objects generalizing the notion of functions. ... In probability theory, let S = {X1, ..., Xn}, with the Xi in {0,1,...,G-1}, be a set of random variables on the sample space Ω={0,1,...,G-1}n, a probability measure π is a random field if . There exist several types of random fields, such as Markov... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, Schwartz space is the function space of rapidly decreasing functions. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

As a (very) rough way to think about continuous random fields, we can think of it as an ordinary function that is almost everywhere, but when we take a weighted average of all the infinities over any finite region, we get a finite result. The infinities are not well-defined, the last sentence is nonsense to a mathematician, but the finite values can be associated with the functions we supposedly used as the weight functions to get the finite values, and that can be well-defined. We can define a continuous random field well enough as a linear map from a space of functions into the real numbers. In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the real numbers may be described informally in several different ways. ...

Symmetries of fields

Main article: Symmetry in physics

A convenient way of classifying fields (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types: This article or section does not cite its references or sources. ...

Spacetime symmetries

Main article: Spacetime symmetries

Fields are often classified by their behaviour under the symmetry transformations of spacetime. The terms used in this classification are — The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ... In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ...

• Scalar fields (such as temperature) whose values are given by a single variable at each point of space. This value does not change under transformations of space.
• vector fields (such as the magnitude and direction of the force at each point in a magnetic field) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves as usual under rotations in space.
• tensor fields, (such as the stress tensor of a crystal) specified by a tensor at each point of space. The components of the tensor transform between themselves as usual under rotations in space.
• spinor fields are useful in quantum field theory.

In relativity, a similar classification holds, except that scalars, vectors and tensors are defined with respect to the Poincaré symmetry of spacetime. In mathematics and physics, a scalar field associates a scalar to every point in space. ... Fig. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... Current (I) flowing through a wire produces a magnetic field () around the wire. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... This article is in need of attention from an expert on the subject. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Two-dimensional analogy of space-time distortion described in General Relativity. ... It has been suggested that this article or section be merged with Poincaré group. ...

Internal symmetries

Main article: internal symmetries

Fields may have internal symmetries in addition to spacetime symmetries. For example, in many situations one needs fields which are a list of space-time scalars: (φ₁,φ₂...φ_N). For example, in weather prediction these may be temperature, pressure, humidity, etc. In particle physics, the color symmetry of the interaction of quarks is an example of an internal symmetry of the strong interaction, as is the isospin or flavour symmetry. Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In quantum chromodynamics (QCD), color or color charge refers to a certain property of the subatomic particles called quarks. ... These are the 6 quarks and their most likely decay modes. ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). ... Isospin (isotopic spin, isobaric spin) is a physical quantity which is mathematically analogous to spin. ... Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ...

If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an internal symmetry. One may also make a classification of the charges of the fields under internal symmetries.

See also

• Elasticity
• Electromagnetic field
• Fluid dynamics
• Gauge theory
• General relativity
• Maxwell's equations
• Particle physics
• Quantum field theory
• Standard model
• Symmetry in physics
• Scalar field theory

Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In electromagnetism, Maxwells equations are a set of equations, developed in the latter half of the nineteenth century by James Clerk Maxwell. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... This is a detailed description of the standard model (SM) of particle physics. ... This article or section does not cite its references or sources. ... It has been suggested that quartic interaction be merged into this article or section. ...

References

• Landau, Lev D. and Lifshitz, Evgeny M. (1971). Classical Theory of Fields (3rd ed.). London: Pergamon. ISBN 0-08-016019-0. Vol. 2 of the Course of Theoretical Physics.

Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... Evgeny Mikhailovich Lifshitz (Russian: ; February 21, 1915 – October 29, 1985) was a notable Soviet physicist. ...