Field (physics)

Encyclopedia

In physics

, a

associated with each point of spacetime

. A field can be classified as a scalar field

, a vector field

, a spinor field

, or a tensor field

according to whether the value of the field at each point is a scalar

, a vector, a spinor

(e.g., a Dirac electron) or, more generally, a tensor

, respectively. 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

respectively.

A field may be thought of as extending throughout the whole of space. In practice, the strength of every known field has been found to diminish to the point of being undetectable. For instance, in Newton's theory of gravity

, the gravitational field strength is inversely proportional to the square of the distance from the gravitating object. Therefore the Earth's gravitational field quickly becomes undetectable (on cosmic

scales).

Defining the field as "numbers in space" shouldn't detract from the idea that it has physical

reality

. “It occupies space. It contains energy. Its presence eliminates a true vacuum.” The vacuum

is free of matter

, but not free of field. The field creates a "condition in space"”

If an electrical charge is moved, the effects on another charge do not appear instantaneously. The first charge feels a reaction

force, picking up momentum

, but the second charge feels nothing until the influence, traveling at the speed of light

, reaches it and gives it the momentum. Where is the momentum before the second charge moves? By the law of conservation of momentum it must be somewhere. Physicists have found it of "great utility for the analysis of forces" to think of it as being in the field.

This utility leads to physicists believing that electromagnetic fields actually exist, making the field concept a supporting paradigm

of the entire edifice of modern physics. That said, John Wheeler

and Richard Feynman

have entertained Newton's pre-field concept of action at a distance

(although they put it on the back burner because of the ongoing utility of the field concept for research in general relativity

and quantum electrodynamics

).

"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".

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.

In modern physics, the most often studied fields are those that model the four fundamental forces which one day may lead to the Unified Field Theory

.

. 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

.

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.

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.

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.

In a general setting, classical fields are described by sections of fiber bundle

s and their dynamics is formulated in the terms of jet manifolds

(covariant classical field theory

).

In BRST theory one deals with odd fields, e.g. ghosts. There are different descriptions of odd classical fields both on graded manifold

s and supermanifold

s.

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; experiment

al data

confirm its predictions to a higher precision

(to more significant digits) than any other theory. 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.

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.

, 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 field

s are indexed sets of random variable

s; 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

.

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; but the finite values can be associated with the functions 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 number

s.

. The terms used in this classification are —

, the color

symmetry of the interaction of quark

s is an example of an internal symmetry of the strong interaction

, as is the isospin

or flavour

symmetry.

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

) of the apparent position of the source of a gravitational or electric static field when the source moves with constant velocity, the static field "effect" may seem at first glance to be "transmitted" faster than the speed of light. A static field always points to the instantaneous direction of the source

of the source must still give the correct direction of the field as seen by the observer. However, no information is transmitted (propagated) from source to receiver/observer by a static field, even if the true and instantaneous correct direction to the source is maintained at constant relative velocity. The reason is that the direction of the field toward the true position of the emitter at all distances, with no speed-of-light delay, is not maintained in any other circumstances than constant-velocity source motion. If the source of the field does accelerate from its constant velocity, then its static field at a distance still behaves for a time, as though the source had continued with its former constant-velocity (this is now incorrect, as the direction of the field farther way from this distance now point in the wrong direction, and not exactly at present instantaneous position of the source). The correct "update" in the static field due to a source-acceleration, moves outward from the source only at the speed of light. Unlike the static field, such waves are capable of carrying information, but they carry it only at the speed of light.

For example, the direction of the static gravitation field from the Sun points almost exactly at the Sun's current position, and is not corrected by the 8.3 minutes of travel time that light takes between Earth and Sun. There is thus no almost no aberration for static gravity, which may be mistaken for the idea that the gravitational influence moves faster than light. Light from the Sun, as a wave, does show annual solar aberration

, and the optical image of the Sun, as seen in Earth telescopes, shows the position of the Sun as it was in the sky, 8.3 minutes before. Thus, the direction of the Sun's pull on the Earth and direction of sunlight, are from slightly different directions.

Electromagnetic fields may have some mixed component of static field, depending on the ratio of electric field E to magnetic field B. When this ratio is not the same as the ratio characteristic of electromagnetic waves propagating in free space far from the source, then the electromagnetic field has some static component. The difference between these components in antenna theory is discussed in the difference between the near and far field

of the antenna. The reactive (closest part) of the near-field of antennas is heavily influenced by static electric fields from charges in the antenna, and also the magnetic induction effect of currents in the antenna. Both of these effects die away with distance, leaving a radiative electromagnetic field of the kind associated with classical electromagnetic radiation.

In quantum mechanics

, static fields are transmitted by virtual particle

s, which may have speeds that exceed c. When physicist Richard Feynman

was once asked by a questioner how gravity could escape the event horizon of a black hole

, he replied simply that a static gravitational field would be carried by virtual graviton

s, which have no trouble traveling faster than light. More mundanely, static electric field effects show the same lack of light speed limitations, and electric fields would also "escape" the influence of a black hole

. Thus, black holes may be electrically charged.

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a

**field**is a physical quantityPhysical quantity

A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...

associated with each point of spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. A field can be classified as a scalar field

Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

, a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

, a spinor field

Spinor field

In particle physics, a spinor field of order 2s describes a particle of spin s, where s is an integer or half-integer. Therefore, a spinor of order 2s contains as much information as a tensor of order s...

, or a tensor field

Tensor field

In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

according to whether the value of the field at each point is a scalar

Scalar (physics)

In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

, a vector, a spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

(e.g., a Dirac electron) or, more generally, a tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

, respectively. For example, the Newtonian gravitational field

Gravitational field

The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

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 operatorsOperator (physics)

In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

respectively.

A field may be thought of as extending throughout the whole of space. In practice, the strength of every known field has been found to diminish to the point of being undetectable. For instance, in Newton's theory of gravity

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

, the gravitational field strength is inversely proportional to the square of the distance from the gravitating object. Therefore the Earth's gravitational field quickly becomes undetectable (on cosmic

COSMIC

Constellation Observing System for Meteorology, Ionosphere, and Climate is a program designed to provide advances in meteorology, ionospheric research, climatology, and space weather by using GPS satellites in conjunction with low Earth orbiting satellites...

scales).

Defining the field as "numbers in space" shouldn't detract from the idea that it has physical

Physical property

A physical property is any property that is measurable whose value describes a physical system's state. The changes in the physical properties of a system can be used to describe its transformations ....

reality

Reality

In philosophy, reality is the state of things as they actually exist, rather than as they may appear or might be imagined. In a wider definition, reality includes everything that is and has been, whether or not it is observable or comprehensible...

. “It occupies space. It contains energy. Its presence eliminates a true vacuum.” The vacuum

Vacuum

In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

is free of matter

Matter

Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

, but not free of field. The field creates a "condition in space"”

If an electrical charge is moved, the effects on another charge do not appear instantaneously. The first charge feels a reaction

Reaction (physics)

The third of Newton's laws of motion of classical mechanics states that forces always occur in pairs. Every action is accompanied by a reaction of equal magnitude but opposite direction. This principle is commonly known in the Latin language as actio et reactio. The attribution of which of the two...

force, picking up momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

, but the second charge feels nothing until the influence, traveling at the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, reaches it and gives it the momentum. Where is the momentum before the second charge moves? By the law of conservation of momentum it must be somewhere. Physicists have found it of "great utility for the analysis of forces" to think of it as being in the field.

This utility leads to physicists believing that electromagnetic fields actually exist, making the field concept a supporting paradigm

Paradigm

The word paradigm has been used in science to describe distinct concepts. It comes from Greek "παράδειγμα" , "pattern, example, sample" from the verb "παραδείκνυμι" , "exhibit, represent, expose" and that from "παρά" , "beside, beyond" + "δείκνυμι" , "to show, to point out".The original Greek...

of the entire edifice of modern physics. That said, John Wheeler

John Wheeler

John Wheeler may refer to:* John Wheeler , American Emmy Award-winning audio/video engineer* John Wheeler , Union officer in the Civil War; killed at Gettysburg* John Wheeler John Wheeler may refer to:* John Wheeler (audio/video technologist) (born 1957), American Emmy Award-winning audio/video...

and Richard Feynman

Richard Feynman

Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

have entertained Newton's pre-field concept of action at a distance

Action at a distance (physics)

In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant...

(although they put it on the back burner because of the ongoing utility of the field concept for research in general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

and quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

).

"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".

## 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 LagrangianLagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

or a Hamiltonian

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

of the field, and treating it as the classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

(or quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

) of a system with an infinite number of degrees of freedom

Degrees of freedom (physics and chemistry)

A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

. The resulting field theories are referred to as classical or quantum field theories.

In modern physics, the most often studied fields are those that model the four fundamental forces which one day may lead to the Unified Field Theory

Unified field theory

In physics, a unified field theory, occasionally referred to as a uniform field theory, is a type of field theory that allows all that is usually thought of as fundamental forces and elementary particles to be written in terms of a single field. There is no accepted unified field theory, and thus...

.

### Classical fields

There are several examples of classical fieldsClassical field theory

A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....

. The dynamics of a classical field are usually specified by the Lagrangian density

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

in terms of the field components; the dynamics can be obtained by using the action principle

Action (physics)

In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...

.

Michael Faraday

Michael Faraday

Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....

first realized the importance of a field as a physical object, during his investigations into magnetism

Magnetism

Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

. He realized that electric

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

and magnetic

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy.

These ideas eventually led to the creation, by James Clerk Maxwell

James Clerk Maxwell

James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

, of the first unified field theory in physics with the introduction of equations for the electromagnetic field

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

. The modern version of these equations are called Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

. At the end of the 19th century, the electromagnetic field

Electromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

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.

Einstein's theory of gravity, called general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, is another example of a field theory. Here the principal field is the metric tensor

Metric tensor (general relativity)

In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...

, a symmetric 2nd-rank tensor field in spacetime.

In a general setting, classical fields are described by sections of fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

s and their dynamics is formulated in the terms of jet manifolds

Jet bundle

In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...

(covariant classical field theory

Covariant classical field theory

In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented by sections of fiber bundles and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the...

).

In BRST theory one deals with odd fields, e.g. ghosts. There are different descriptions of odd classical fields both on graded manifold

Graded manifold

Graded manifolds are extensions of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra. Graded manifolds are not supermanifolds though there is a certain correspondence between the graded manifolds and the DeWitt supermanifolds. Both graded manifolds and...

s and supermanifold

Supermanifold

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.- Physics :...

s.

### Quantum fields

It is now believed that quantum mechanicsQuantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

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

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

. For example, quantizing

Quantization (physics)

In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...

classical electrodynamics gives quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

. Quantum electrodynamics is arguably the most successful scientific theory; experiment

Experiment

An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

al data

Data

The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

confirm its predictions to a higher precision

Accuracy and precision

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

(to more significant digits) than any other theory. The two other fundamental quantum field theories are quantum chromodynamics

Quantum chromodynamics

In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

and the electroweak theory. These three quantum field theories can all be derived as special cases of the so-called standard model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

. General relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, the classical field theory of gravity, has yet to be successfully quantized.

Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research. Elasticity

Elasticity (physics)

In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

of materials, fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

and Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

are cases in point.

### Continuous random fields

Classical fields as above, such as the electromagnetic fieldElectromagnetic field

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

, 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 field

Random field

A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....

s are indexed sets of random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s; 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

Distribution (mathematics)

In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

.

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

Infinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

over any finite region, we get a finite result. The infinities are not well-defined; but the finite values can be associated with the functions 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 number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s.

## Symmetries of fields

A convenient way of classifying a field (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types:### Spacetime symmetries

Fields are often classified by their behaviour under transformations of spacetimeSpacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. The terms used in this classification are —

- scalar fieldScalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

s (such as temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

) whose values are given by a single variable at each point of space. This value does not change under transformations of space. - vector fieldVector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s (such as the magnitude and direction of the force at each point in a magnetic fieldMagnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

) 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 fieldTensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

s, (such as the stress tensorStress (physics)In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

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. - spinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

fields are useful in quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

.

### 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: (φ_{1},φ_{2}...φ_{N}). For example, in weather prediction these may be temperature, pressure, humidity, etc. In particle physicsParticle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, the color

Color charge

In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics . Color charge has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD,...

symmetry of the interaction of quark

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s is an example of an internal symmetry of the strong interaction

Strong interaction

In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

, as is the isospin

Isospin

In physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...

or flavour

Flavour (particle physics)

In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...

symmetry.

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.### Propagation of static field effects

Since there is no "retardation" (or aberrationAberration of light

The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

) of the apparent position of the source of a gravitational or electric static field when the source moves with constant velocity, the static field "effect" may seem at first glance to be "transmitted" faster than the speed of light. A static field always points to the instantaneous direction of the source

*as if it continued with the same relative velocity of source and emitter at a previous time calculated by their distance from each other, divided by c.*Thus, static fields from objects moving with constant velocity are always kept "up to date" at great distances from the source with no "signal delay"-- an effect which is permitted by the fact that a change to the reference frameReference frame

Reference frame may refer to:*Frame of reference, in physics*Reference frame , frames of a compressed video that are used to define future frames...

of the source must still give the correct direction of the field as seen by the observer. However, no information is transmitted (propagated) from source to receiver/observer by a static field, even if the true and instantaneous correct direction to the source is maintained at constant relative velocity. The reason is that the direction of the field toward the true position of the emitter at all distances, with no speed-of-light delay, is not maintained in any other circumstances than constant-velocity source motion. If the source of the field does accelerate from its constant velocity, then its static field at a distance still behaves for a time, as though the source had continued with its former constant-velocity (this is now incorrect, as the direction of the field farther way from this distance now point in the wrong direction, and not exactly at present instantaneous position of the source). The correct "update" in the static field due to a source-acceleration, moves outward from the source only at the speed of light. Unlike the static field, such waves are capable of carrying information, but they carry it only at the speed of light.

For example, the direction of the static gravitation field from the Sun points almost exactly at the Sun's current position, and is not corrected by the 8.3 minutes of travel time that light takes between Earth and Sun. There is thus no almost no aberration for static gravity, which may be mistaken for the idea that the gravitational influence moves faster than light. Light from the Sun, as a wave, does show annual solar aberration

Aberration of light

The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

, and the optical image of the Sun, as seen in Earth telescopes, shows the position of the Sun as it was in the sky, 8.3 minutes before. Thus, the direction of the Sun's pull on the Earth and direction of sunlight, are from slightly different directions.

Electromagnetic fields may have some mixed component of static field, depending on the ratio of electric field E to magnetic field B. When this ratio is not the same as the ratio characteristic of electromagnetic waves propagating in free space far from the source, then the electromagnetic field has some static component. The difference between these components in antenna theory is discussed in the difference between the near and far field

Near and far field

The near field and far field and the transition zone are regions of the electromagnetic radiation field that emanates from a transmitting antenna, or as a result of radiation scattering off an object...

of the antenna. The reactive (closest part) of the near-field of antennas is heavily influenced by static electric fields from charges in the antenna, and also the magnetic induction effect of currents in the antenna. Both of these effects die away with distance, leaving a radiative electromagnetic field of the kind associated with classical electromagnetic radiation.

In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, static fields are transmitted by virtual particle

Virtual particle

In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

s, which may have speeds that exceed c. When physicist Richard Feynman

Richard Feynman

Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

was once asked by a questioner how gravity could escape the event horizon of a black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

, he replied simply that a static gravitational field would be carried by virtual graviton

Graviton

In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

s, which have no trouble traveling faster than light. More mundanely, static electric field effects show the same lack of light speed limitations, and electric fields would also "escape" the influence of a black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

. Thus, black holes may be electrically charged.