In
physicsPhysics 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...
,
action is an attribute of the dynamics of a
physical systemIn physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
. It is a
mathematical functionalIn mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
which takes the
trajectoryA trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
, also called
path or
history, of the system as its argument and has a real number as its result. Action has the dimension of
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
×
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, and its unit is joule-seconds in the
International System of UnitsThe International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
(SI). Generally, the action takes different values for different paths. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more strictly, is
stationaryIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
. The classical equations of motion of a system can be derived from this
principle of least actionIn physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
. The stationary action formulation of classical mechanics extends to
quantum mechanicsQuantum 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...
in the Feynman path integral formulation, where a physical system follows simultaneously all possible paths with amplitudes determined by the action.
If the action is represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system,
the integrand,
, is called the
LagrangianThe 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...
. For the action integral to be well defined the trajectory has to be bounded in time and space.
History
Action was defined in several, now obsolete, ways during the development of the concept.
- Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, Johann BernoulliJohann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
and Pierre Louis MaupertuisPierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great....
defined the action for lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
as the integral of its speed or inverse speed along its path length.
- Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
(and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space.
- Pierre Louis Maupertuis
Pierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great....
introduced several ad hoc and contradictory definitions of action within a single article, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.
Concepts
Physical laws are most often expressed as
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, which specify how a physical quantity varies over infinitesimally small changes in time, position, or other independent variables in its domain. A solution to a differential equation provides the value of the physical variable at any point in its domain (
e.g.,
), given both initial conditions and boundary conditions.
In
analytical dynamicsIn classical mechanics, analytical dynamics, or more briefly dynamics, is concerned about the relationship between motion of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies...
, the
action represents the final form obtained by working backwards from classical Newtonian mechanics to achieve an integral minimization expression in the form of a variational statement. The statement is profound, simple, and elegant but comes at the cost of several simplifying assumptions. The integral form espoused here can only be applied to conservative
holonomicIn mathematics and physics, the term holonomic may occur with several different meanings.-Holonomic basis:A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish:[e_j,e_k]=0 \,...
mechanical systems and to do otherwise can yield incorrect results.
The equivalence of these two approaches is contained in
Hamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
, which states that the differential equations of motion for
any physical system can be re-formulated as an equivalent
integral equationIn mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
. It applies not only to the
classical mechanicsIn 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...
of a single particle, but also to classical
fieldsIn physics, a field is a physical quantity 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 or, more generally, a tensor,...
such as the
electromagneticElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
and gravitational fields.
Hamilton's principle has also been extended to
quantum mechanicsQuantum 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...
and
quantum 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...
.
Mathematical definition
Expressed in mathematical language, using the
calculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, the
evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a
stationary pointIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
(usually, a minimum) of the action.
Several different definitions of 'the action' are in common use in physics:
- The action is usually an integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
over time. But for action pertaining to fieldsIn physics, a field is a physical quantity 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 or, more generally, a tensor,...
, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.
- The evolution of a physical system between two states is determined by requiring the action be minimized or, more generally, be stationary
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
for small perturbations about the true evolution. This requirement leads to differential equations that describe the true evolution.
- Conversely, an action principle is a method for reformulating differential
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
equations of motion for a physical system as an equivalent integral equationIn mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
. Although several variants have been defined (see below), the most commonly used action principle is Hamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
.
- An earlier, less informative action principle is Maupertuis' principle
In classical mechanics, Maupertuis' principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally stated principle of least action...
, which is sometimes called by its (less correct) historical name, the principle of least actionIn physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
.
Disambiguation of action in classical physics
In
classical physicsWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
, the term "action" has at least eight distinct meanings.
Action (functional)
Most commonly, the term is used for a
functionalIn mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
which takes a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of time and (for
fieldsIn physics, a field is a physical quantity 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 or, more generally, a tensor,...
) space as input and returns a
scalarIn 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...
. In
classical mechanicsIn 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...
, the input function is the evolution
of the system between two times
and
, where
represent the generalized coordinates. The action
is defined as the
integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the
LagrangianThe 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...
for an input evolution between the two times
where the endpoints of the evolution are fixed and defined as
and
. According to
Hamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
, the true evolution
is an evolution for which the action
is
stationaryIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
(a minimum, maximum, or a
saddle pointIn mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
). This principle results in the equations of motion in
Lagrangian mechanicsLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
.
Abbreviated action (functional)
Usually denoted as
, this is also a
functionalIn mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
. Here the input function is the
path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path. The abbreviated action
is defined as the integral of the generalized momenta along a path in the
generalized coordinatesIn the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....
According to
Maupertuis' principleIn classical mechanics, Maupertuis' principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally stated principle of least action...
, the true path is a path for which the abbreviated action
is
stationaryIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
.
Hamilton's principal function
Hamilton's principal function is defined by the
Hamilton–Jacobi equationIn mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
s (HJE), another alternative formulation of
classical mechanicsIn 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...
. This function
is related to the functional
by fixing the initial time
and endpoint
and allowing the upper limits
and the second endpoint
to vary; these variables are the
argumentsThe terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
of the function
. In other words, the action function
is the
indefinite integralIn calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
of the Lagrangian with respect to time.
Hamilton's characteristic function
When the total energy
is conserved, the
HJEIn mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
can be solved with the additive separation of variables
,
where the time independent function
is called
Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative
.
This can be integrated to give
,
which is just the abbreviated action.
Other solutions of Hamilton–Jacobi equations
The
Hamilton–Jacobi equationIn mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
s are often solved by additive separability; in some cases, the individual terms of the solution, e.g.,
, are also called an "action".
Action of a generalized coordinate
This is a single variable
in the
action-angle coordinatesIn classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates...
, defined by integrating a single generalized momentum around a closed path in
phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, corresponding to rotating or oscillating motion
The variable
is called the "action" of the generalized coordinate
; the corresponding canonical variable conjugate to
is its "angle"
, for reasons described more fully under
action-angle coordinatesIn classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates...
. The integration is only over a single variable
and, therefore, unlike the integrated dot product in the abbreviated action integral above. The
variable equals the change in
as
is varied around the closed path. For several physical systems of interest,
is either a constant or varies very slowly; hence, the variable
is often used in perturbation calculations and in determining
adiabatic invariantAn adiabatic invariant is a property of a physical system that stays constant when changes occur slowly.In thermodynamics, an adiabatic process is a change that occurs without heat flow, and slowly compared to the time to reach equilibrium. In an adiabatic process, the system is in equilibrium at...
s.
Euler–Lagrange equations for the action integral
As noted above, the requirement that the action integral be
stationaryIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
under small perturbations of the evolution is equivalent to a set of
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s (called the Euler–Lagrange equations) that may be determined using the
calculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. We illustrate this derivation here using only one coordinate,
x; the extension to multiple coordinates is straightforward.
Adopting
Hamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
, we assume that the Lagrangian
L (the integrand of the action integral) depends only on the coordinate
x(
t) and its time derivative
dx(
t)/
dt, and may also depend explicitly on time. In that case, the action integral can be written
where the initial and final times (
and
) and the final and initial positions are specified in advance as
and
. Let
represent the true evolution that we seek, and let
be a slightly perturbed version of it, albeit with the same endpoints,
and
. The difference between these two evolutions, which we will call
, is infinitesimally small at all times
At the endpoints, the difference vanishes, i.e.,
.
Expanded to first order, the difference between the actions integrals for the two evolutions is
Integration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
of the last term, together with the boundary conditions
, yields the equation
The requirement that
be
stationaryIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
implies that the first-order change
must be zero for
any possible perturbation
about the true evolution. This can be true only if
Those familiar with the concept and notation of a
functional derivativeIn mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
will note that the Euler–Lagrange equation is simply the functional derivative of the action integral:
.
The quantity
is called the
conjugate momentum for the coordinate
x. An important consequence of the Euler–Lagrange equations is that if
L does not explicitly contain coordinate
x, i.e.
- if
, then 
is constant.
In such cases, the coordinate
x is called a
cyclic coordinate,
and its conjugate momentum is conserved.
Example: Free particle in polar coordinates
Simple examples help to appreciate the use of the action principle via the Euler–Lagrangian equations. A free particle (mass
m and velocity
v) in Euclidean space moves in a straight line. Using the Euler–Lagrange equations, this can be shown in polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy
in orthonormal (
x,
y) coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time,
t).
In polar coordinates (
r, φ) the kinetic energy and hence the Lagrangian becomes
The radial
r and φ components of the Euler–Lagrangian equations become, respectively
The solution of these two equations is given by
for a set of constants
a, b, c, d determined by initial conditions.
Thus, indeed,
the solution is a straight line given in polar coordinates.
Action principle for single relativistic particle
When relativistic effects are significant, the action of a point particle of mass
m travelling a
world lineIn physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
C parametrized by the
proper timeIn relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
is
.
If instead, the particle is parametrized by the coordinate time
t of the particle and the coordinate time ranges from
t1 to
t2, then the action becomes
where the
LagrangianThe 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...
is
.
Action principle for classical fields
The
action principle can be extended to obtain the equations of motion for fields, such as the
electromagnetic fieldAn 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...
or
gravityThe Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
.
The Einstein equation utilizes the
Einstein-Hilbert actionThe Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
as constrained by a
variational principleA variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...
.
The path of a body in a gravitational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.
Action principle in quantum mechanics and quantum field theory
In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate the
path integralThe path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
, that gives the
probability amplitudeIn quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
s of the various outcomes.
Although equivalent in classical mechanics with Newton's laws, the
action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics.
Richard FeynmanRichard 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...
's
path integral formulationThe path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
of quantum mechanics is based on a stationary-action principle, using path integrals.
Maxwell's equationsMaxwell'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...
can be derived as conditions of stationary action.
Action principle and conservation laws
Symmetries in a physical situation can better be treated with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is
Noether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
, which states that to every
continuous symmetryIn mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
in a physical situation there corresponds a
conservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
(and conversely). This deep connection requires that the action principle be assumed.
Modern extensions of the action principle
The action principle can be generalized still further. For example, the action need not be an integral because
nonlocal actionIn Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...
s are possible. The configuration space need not even be a functional space given certain features such as
noncommutative geometryNoncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
. However, a physical basis for these mathematical extensions remains to be established experimentally.