In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...
, an invariant is an
observable of a
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
which remains unchanged under some
transformation. Invariance, as a broader term, also applies to the no change of form of
physical laws under a transformation, and is closer in scope to the
mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment.
Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their
symmetries and invariants.
Examples
In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the
conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the
conservation of energy. In general, by
Noether's theorem, any invariance of a physical system under a
continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to anoth ...
leads to a fundamental
conservation law.
In
crystals, the
electron density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial ...
is periodic and invariant with respect to discrete translations by unit cell vectors. In very few materials, this symmetry can be broken due to enhanced
electron correlations.
Another examples of physical invariants are the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
, and
charge and
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
of a particle observed from two
reference frames moving with respect to one another (invariance under a spacetime
Lorentz transformation), and invariance of
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
and
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
under a
Galilean transformation between two such frames moving at low velocities.
Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching coordinate representations from rectangular to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant.
Physical laws are said to be invariant under transformations when their predictions remain unchanged. This generally means that the form of the law (e.g. the type of differential equations used to describe the law) is unchanged in transformations so that no additional or different solutions are obtained.
Covariance and contravariance generalize the mathematical properties of
invariance
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
in
tensor mathematics, and are frequently used in
electromagnetism,
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
, and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
.
See also
*
Casimir operator
*
Charge (physics)
*
Conservation law
*
Conserved quantity
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are ...
*
General covariance
*
Eigenvalues and eigenvectors
*
Invariants of tensors
*
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) s ...
*
Physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
*
Scalar (physics)
In physics, scalars (or scalar quantities) are physical quantities that are unaffected by changes to a vector space basis (i.e., a coordinate system transformation). Scalars are often accompanied by units of measurement, as in "10 cm".
Example ...
*
Symmetry (physics)
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
A family of particular transformations may be ''continu ...
*
Uniformity of nature
*
Weyl transformation :''See also Wigner–Weyl transform, for another definition of the Weyl transform.''
In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor:
:g_\rightarrow e^g_
which produces another ...
References
{{DEFAULTSORT:Invariant (Physics)
Conservation laws
Physical quantities