In physics, a field is a physical quantity,
represented by a number or tensor, that has
a value for each point in space-time. For
example, on a weather map, the surface wind
velocity is described by assigning a vector
to each point on a map. Each vector represents
the speed and direction of the movement of
air at that point. As another example, an
electric field can be thought of as a "condition
in space" emanating from an electric charge
and extending throughout the whole of space.
When a test electric charge is placed in this
electric field, the particle accelerates due
to a force. Physicists have found the notion
of a field to be of such practical utility
for the analysis of forces that they have
come to think of a force as due to a 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 led physicists to consider electromagnetic
fields to be a physical entity, making the
field concept a supporting paradigm of the
edifice of modern physics. "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." In practice,
the strength of most fields has been found
to diminish with distance to the point of
being 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). One consequence
is that the Earth's gravitational field quickly
becomes undetectable on cosmic scales.
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 unique tensorial character in every
point where 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 fact in this theory an equivalent
representation of field is a field particle,
namely a boson.
== History ==
To Isaac Newton, his law of universal gravitation
simply expressed the gravitational force that
acted between any pair of massive objects.
When looking at the motion of many bodies
all interacting with each other, such as the
planets in the Solar System, dealing with
the force between each pair of bodies separately
rapidly becomes computationally inconvenient.
In the eighteenth century, a new quantity
was devised to simplify the bookkeeping of
all these gravitational forces. This quantity,
the gravitational field, gave at each point
in space the total gravitational acceleration
which would be felt by a small object at that
point. This did not change the physics in
any way: it did not matter if all the gravitational
forces on an object were calculated individually
and then added together, or if all the contributions
were first added together as a gravitational
field and then applied to an object.The development
of the independent concept of a field truly
began in the nineteenth century with the development
of the theory of electromagnetism. In the
early stages, André-Marie Ampère and Charles-Augustin
de Coulomb could manage with Newton-style
laws that expressed the forces between pairs
of electric charges or electric currents.
However, it became much more natural to take
the field approach and express these laws
in terms of electric and magnetic fields;
in 1849 Michael Faraday became the first to
coin the term "field".The independent nature
of the field became more apparent with James
Clerk Maxwell's discovery that waves in these
fields propagated at a finite speed. Consequently,
the forces on charges and currents no longer
just depended on the positions and velocities
of other charges and currents at the same
time, but also on their positions and velocities
in the past.Maxwell, at first, did not adopt
the modern concept of a field as a fundamental
quantity that could independently exist. Instead,
he supposed that the electromagnetic field
expressed the deformation of some underlying
medium—the luminiferous aether—much like
the tension in a rubber membrane. If that
were the case, the observed velocity of the
electromagnetic waves should depend upon the
velocity of the observer with respect to the
aether. Despite much effort, no experimental
evidence of such an effect was ever found;
the situation was resolved by the introduction
of the special theory of relativity by Albert
Einstein in 1905. This theory changed the
way the viewpoints of moving observers were
related to each other. They became related
to each other in such a way that velocity
of electromagnetic waves in Maxwell's theory
would be the same for all observers. By doing
away with the need for a background medium,
this development opened the way for physicists
to start thinking about fields as truly independent
entities.In the late 1920s, the new rules
of quantum mechanics were first applied to
the electromagnetic fields. In 1927, Paul
Dirac used quantum fields to successfully
explain how the decay of an atom to a lower
quantum state lead to the spontaneous emission
of a photon, the quantum of the electromagnetic
field. This was soon followed by the realization
(following the work of Pascual Jordan, Eugene
Wigner, Werner Heisenberg, and Wolfgang Pauli)
that all particles, including electrons and
protons, could be understood as the quanta
of some quantum field, elevating fields to
the status of the most fundamental objects
in nature. That said, John Wheeler and Richard
Feynman seriously considered Newton's pre-field
concept of action at a distance (although
they set it aside because of the ongoing utility
of the field concept for research in general
relativity and quantum electrodynamics).
== Classical fields ==
There are several examples of classical fields.
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.
Some of the simplest physical fields are vector
force fields. Historically, the first time
that fields were taken seriously was with
Faraday's lines of force when describing the
electric field. The gravitational field was
then similarly described.
=== Newtonian gravitation ===
A classical field theory describing gravity
is Newtonian gravitation, which describes
the gravitational force as a mutual interaction
between two masses.
Any body with mass M is associated with a
gravitational field g which describes its
influence on other bodies with mass. The gravitational
field of M at a point r in space corresponds
to the ratio between force F that M exerts
on a small or negligible test mass m located
at r and the test mass itself:
g
(
r
)
=
F
(
r
)
m
.
{\displaystyle \mathbf {g} (\mathbf {r} )={\frac
{\mathbf {F} (\mathbf {r} )}{m}}.}
Stipulating that m is much smaller than M
ensures that the presence of m has a negligible
influence on the behavior of M.
According to Newton's law of universal gravitation,
F(r) is given by
F
(
r
)
=
−
G
M
m
r
2
r
^
,
{\displaystyle \mathbf {F} (\mathbf {r} )=-{\frac
{GMm}{r^{2}}}{\hat {\mathbf {r} }},}
where
r
^
{\displaystyle {\hat {\mathbf {r} }}}
is a unit vector lying along the line joining
M and m and pointing from m to M. Therefore,
the gravitational field of M 
is
g
(
r
)
=
F
(
r
)
m
=
−
G
M
r
2
r
^
.
{\displaystyle \mathbf {g} (\mathbf {r} )={\frac
{\mathbf {F} (\mathbf {r} )}{m}}=-{\frac {GM}{r^{2}}}{\hat
{\mathbf {r} }}.}
The experimental observation that inertial
mass and gravitational mass are equal to an
unprecedented level of accuracy leads to the
identity that gravitational field strength
is identical to the acceleration experienced
by a particle. This is the starting point
of the equivalence principle, which leads
to general relativity.
Because the gravitational force F is conservative,
the gravitational field g can be rewritten
in terms of the gradient of a scalar function,
the gravitational potential Φ(r):
g
(
r
)
=
−
∇
Φ
(
r
)
.
{\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla
\Phi (\mathbf {r} ).}
=== Electromagnetism ===
Michael Faraday first realized the importance
of a field as a physical quantity, 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 is called
Maxwell's equations.
==== Electrostatics ====
A charged test particle with charge q experiences
a force F based solely on its charge. We can
similarly describe the electric field E so
that F = qE. Using this and Coulomb's law
tells us that the electric field due to a
single charged particle is
E
=
1
4
π
ϵ
0
q
r
2
r
^
.
{\displaystyle \mathbf {E} ={\frac {1}{4\pi
\epsilon _{0}}}{\frac {q}{r^{2}}}{\hat {\mathbf
{r} }}.}
The electric field is conservative, and hence
can be described by a scalar potential, V(r):
E
(
r
)
=
−
∇
V
(
r
)
.
{\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla
V(\mathbf {r} ).}
==== Magnetostatics ====
A steady current I flowing along a path ℓ
will exert a force on nearby moving charged
particles that is quantitatively different
from the electric field force described above.
The force exerted by I on a nearby charge
q with velocity v is
F
(
r
)
=
q
v
×
B
(
r
)
,
{\displaystyle \mathbf {F} (\mathbf {r} )=q\mathbf
{v} \times \mathbf {B} (\mathbf {r} ),}
where B(r) is the magnetic field, which is
determined from I by the Biot–Savart law:
B
(
r
)
=
μ
0
I
4
π
∫
d
ℓ
×
d
r
^
r
2
.
{\displaystyle \mathbf {B} (\mathbf {r} )={\frac
{\mu _{0}I}{4\pi }}\int {\frac {d{\boldsymbol
{\ell }}\times d{\hat {\mathbf {r} }}}{r^{2}}}.}
The magnetic field is not conservative in
general, and hence cannot usually be written
in terms of a scalar potential. However, it
can be written in terms of a vector potential,
A(r):
B
(
r
)
=
∇
×
A
(
r
)
{\displaystyle \mathbf {B} (\mathbf {r} )={\boldsymbol
{\nabla }}\times \mathbf {A} (\mathbf {r}
)}
==== Electrodynamics ====
In general, in the presence of both a charge
density ρ(r, t) and current density J(r,
t), there will be both an electric and a magnetic
field, and both will vary in time. They are
determined by Maxwell's equations, a set of
differential equations which directly relate
E and B to ρ and J.Alternatively, one can
describe the system in terms of its scalar
and vector potentials V and A. A set of integral
equations known as retarded potentials allow
one to calculate V and A from ρ and J, and
from there the electric and magnetic fields
are determined via the relations
E
=
−
∇
V
−
∂
A
∂
t
{\displaystyle \mathbf {E} =-{\boldsymbol
{\nabla }}V-{\frac {\partial \mathbf {A} }{\partial
t}}}
B
=
∇
×
A
.
{\displaystyle \mathbf {B} ={\boldsymbol {\nabla
}}\times \mathbf {A} .}
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.
=== Gravitation in general relativity ===
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. This replaces Newton's law of
universal gravitation.
=== Waves as fields ===
Waves can be constructed as physical fields,
due to their finite propagation speed and
causal nature when a simplified physical model
of an isolated closed system is set. They
are also subject to the inverse-square law.
For electromagnetic waves, there are optical
fields, and terms such as near- and far-field
limits for diffraction. In practice though,
the field theories of optics are superseded
by the electromagnetic field theory of Maxwell.
== 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. The two other fundamental quantum
field theories are quantum chromodynamics
and the electroweak theory.
In quantum chromodynamics, the color field
lines are coupled at short distances by gluons,
which are polarized by the field and line
up with it. This effect increases within a
short distance (around 1 fm from the vicinity
of the quarks) making the color force increase
within a short distance, confining the quarks
within hadrons. As the field lines are pulled
together tightly by gluons, they do not "bow"
outwards as much as an electric field between
electric charges.These three quantum field
theories can all be derived as special cases
of the so-called standard model of particle
physics. General relativity, the Einsteinian
field theory of gravity, has yet to be successfully
quantized. However an extension, thermal field
theory, deals with quantum field theory at
finite temperatures, something seldom considered
in quantum field theory.
In BRST theory one deals with odd fields,
e.g. Faddeev–Popov ghosts. There are different
descriptions of odd classical fields both
on graded manifolds and supermanifolds.
As above with classical fields, it is possible
to approach their quantum counterparts from
a purely mathematical view using similar techniques
as before. The equations governing the quantum
fields are in fact PDEs (specifically, relativistic
wave equations (RWEs)). Thus one can speak
of Yang–Mills, Dirac, Klein–Gordon and
Schrödinger fields as being solutions to
their respective equations. A possible problem
is that these RWEs can deal with complicated
mathematical objects with exotic algebraic
properties (e.g. spinors are not tensors,
so may need calculus over spinor fields),
but these in theory can still be subjected
to analytical methods given appropriate mathematical
generalization.
== 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 independent physical variables on
which the field depends. Usually this is done
by writing a Lagrangian or a Hamiltonian of
the field, and treating it as a classical
or quantum mechanical system with an infinite
number of degrees of freedom. The resulting
field theories are referred to as classical
or quantum field theories.
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.
It is possible to construct simple fields
without any prior knowledge of physics using
only mathematics from several variable calculus,
potential theory and partial differential
equations (PDEs). For example, scalar PDEs
might consider quantities such as amplitude,
density and pressure fields for the wave equation
and fluid dynamics; temperature/concentration
fields for the heat/diffusion equations. Outside
of physics proper (e.g., radiometry and computer
graphics), there are even light fields. All
these previous examples are scalar fields.
Similarly for vectors, there are vector PDEs
for displacement, velocity and vorticity fields
in (applied mathematical) fluid dynamics,
but vector calculus may now be needed in addition,
being calculus over vector fields (as are
these three quantities, and those for vector
PDEs in general). More generally problems
in continuum mechanics may involve for example,
directional elasticity (from which comes the
term tensor, derived from the Latin word for
stretch), complex fluid flows or anisotropic
diffusion, which are framed as matrix-tensor
PDEs, and then require matrices or tensor
fields, hence matrix or tensor calculus. It
should be noted that the scalars (and hence
the vectors, matrices and tensors) can be
real or complex as both are fields in the
abstract-algebraic/ring-theoretic sense.
In a general setting, classical fields are
described by sections of fiber bundles and
their dynamics is formulated in the terms
of jet manifolds (covariant classical field
theory).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.
=== 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 spacetime. 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. 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 contravariantly
under rotations in space. Similarly, a dual
(or co-) vector field attaches a dual vector
to each point of space, and the components
of each dual vector transform covariantly.
tensor fields, (such as the stress tensor
of a crystal) specified by a tensor at each
point of space. Under rotations in space,
the components of the tensor transform in
a more general way which depends on the number
of covariant indices and contravariant indices.
spinor fields (such as the Dirac spinor) arise
in quantum field theory to describe particles
with spin which transform like vectors except
for the one of their component; in other words,
when one rotates a vector field 360 degrees
around a specific axis, the vector field turns
to itself; however, spinors would turn to
their negatives in the same case.
==== Internal symmetries ====
Fields may have internal symmetries in addition
to spacetime symmetries. 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, that
of the strong interaction. Other examples
are isospin, weak isospin, strangeness and
any other 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 internal symmetry.
One may also make a classification of the
charges of the fields under internal symmetries.
=== Statistical field theory ===
Statistical field theory attempts to extend
the field-theoretic paradigm toward many-body
systems and statistical mechanics. As above,
it can be approached by the usual infinite
number of degrees of freedom argument.
Much like statistical mechanics has some overlap
between quantum and classical mechanics, statistical
field theory has links to both quantum and
classical field theories, especially the former
with which it shares many methods. One important
example is mean field theory.
=== 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 are used, because
thermally fluctuating classical fields are
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.
We can think about a continuous random field,
in a (very) rough way, as an ordinary function
that is
±
∞
{\displaystyle \pm \infty }
almost everywhere, but such that 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 numbers.
== See also ==
Conformal field theory
Covariant Hamiltonian field theory
Field strength
History of the philosophy of field theory
Lagrangian and Eulerian specification of a
field
Scalar field theory
== Notes
