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 List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...

, an invariant is an observable
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...

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 (systems), environment. The environment is ignored except ...

which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical law
Scientific laws or laws of science are statements, based on reproducibility, repeated experiments or observations, that describe or prediction, predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, a ...

s 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 theconservation of momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...

, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. This law, first proposed and tested by Émilie du Chât ...

. In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...

.
In crystals
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...

, the electron density 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 relativity, special theory of relativity, is ...

, and charge and mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...

of a particle observed from two reference frames moving with respect to one another (invariance under a spacetime Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...

), and invariance of time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

and acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ...

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 in tensor mathematics, and are frequently used in electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

, special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...

, and general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

.
See also

* Casimir operator * Charge (physics) *Conservation law
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...

* Conserved quantity
* General covariance
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 List of natural phenomena, natural phenomena. This is in con ...

* Eigenvalues and eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

* 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 criterion, Cartan's criteria (criterion of solvability and criterion of se ...

* 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 (mathematics), constant value in time. It is contrasted with a ...

* Scalar (physics)
In physics, scalars (or scalar quantities) are Physical quantity, physical quantities that are unaffected by changes to a Basis (linear algebra), vector space basis (i.e., a coordinate system transformation). Scalars are often accompanied by units ...

* Symmetry (physics)
* Uniformity of nature
Uniformitarianism, also known as the Doctrine of Uniformity or the Uniformitarian Principle, is the assumption that the same natural laws and processes that operate in our present-day scientific observations have always operated in the universe in ...

* Weyl transformation
References

{{DEFAULTSORT:Invariant (Physics) Conservation laws Physical quantities