In physics, a field is the presence of a physical quantity at every point in space (or, more generally, spacetime). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A physical Quantity is a physical property that can be quantified SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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.
Fields are usually represented mathematically by scalar, vector and tensor fields. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity 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. A gravitational field is a model used within Physics to explain how gravity exists in the universe 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.
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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 Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons For information on degrees of freedom in other sciences see 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. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms In Physics, a unified field theory is a type of Field theory that allows all of the Fundamental forces between Elementary particles to be written
Classical fields
There are several examples of classical fields. A classical field theory is a Physical theory that describes the study of how one or more physical fields interact with matter 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. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation
Michael Faraday first realized the importance of a field as a physical object, during his investigations into magnetism. Michael Faraday, FRS ( September 22 1791 – August 25 1867) was an English In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. 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. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges
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. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. The electromagnetic field is a physical field produced by electrically charged objects. The modern version of these equations are called Maxwell's equations. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric At the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. The electromagnetic field is a physical field produced by electrically charged objects. 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. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Here the principal field is the metric tensor, a symmetric 2nd-rank tensor field in spacetime. This article discusses metrics in General relativity, for a discussion of metrics in general see Metric tensor.
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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In quantum field theory (QFT the forces between particles are mediated by other particles For example, quantizing classical electrodynamics gives quantum electrodynamics. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently Quantum electrodynamics ( QED) is a relativistic Quantum field theory of 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. In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or Debt AIDS Trade in Africa (or DATA) is a Multinational non-government organization founded in January 2002 in London by U2 's The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy See precision tests of QED. Quantum electrodynamics ( QED) a relativistic quantum field theory of electrodynamics is among the most stringently tested theories in Physics. The two other fundamental quantum field theories are quantum chromodynamics and the electroweak theory. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the These three quantum field theories can all be derived as special cases of the so-called standard model of particle physics. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them General relativity, the classical field theory of gravity, has yet to be successfully quantized. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916
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. A material is said to be elastic if it deforms under stress (e Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric
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. The electromagnetic field is a physical field produced by electrically charged objects. In contrast, generalized functions are not continuous. In Mathematics, generalized functions are objects generalizing the notion of functions There is more than one recognised theory 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. In Mathematics, the Weierstrass function is a pathological example of a real -valued function on the Real line. 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. A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real but can instead be a multidimensional vector space or even a manifold A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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. 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 weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness 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 numbers. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the real numbers may be described informally in several different ways
Symmetries of fields
A convenient way of classifying fields (classical or quantum) is by the symmetries it possesses. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these Physical symmetries are usually of two types:
Spacetime symmetries
Fields are often classified by their behaviour under the symmetry transformations of spacetime. Spacetime symmetries refers to aspects of Spacetime that can be described as exhibiting some form of Symmetry. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS The terms used in this classification are —
- Scalar fields (such as temperature) whose values are given by a single variable at each point of space. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature 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. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges 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. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity Stress is a measure of the average amount of Force exerted per unit Area. The components of the tensor transform between themselves as usual under rotations in space.
- spinor fields are useful in quantum field theory. 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 In quantum field theory (QFT the forces between particles are mediated by other particles
In relativity, a similar classification holds, except that scalars, vectors and tensors are defined with respect to the Poincaré symmetry of spacetime. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime
Internal symmetries
Fields may have internal symmetries in addition to spacetime symmetries. In Physics, a field is a Physical quantity associated to each point of Spacetime. For example, in many situations one needs fields which are a list of space-time scalars: (φ1,φ2. . . φN). 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. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Particle physics, color charge is a property of Quarks and Gluons which are related to their Strong interactions in the context of Quantum In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and In Physics, and specifically Particle physics, isospin ( isotopic spin, isobaric spin) is a Quantum number related to the In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their
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
References
- Landau, Lev D. and Lifshitz, Evgeny M. (1971). A material is said to be elastic if it deforms under stress (e The electromagnetic field is a physical field produced by electrically charged objects. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In quantum field theory (QFT the forces between particles are mediated by other particles For a basic description see the article on the Standard Model. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these For the pseudoscientific "scalar field theory" see " Scalar field theory (pseudoscience " In Theoretical physics, Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 Evgeny Mikhailovich Lifshitz (Евгений Михайлович Лифшиц February 21 1915 &ndash October 29 1985) was a leading Soviet Classical Theory of Fields (3rd ed. ). London: Pergamon. ISBN 0-08-016019-0. Vol. 2 of the Course of Theoretical Physics.
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