In
theoretical physics, 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
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
. 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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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 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 ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
. In general, by
Noether's theorem, any invariance of a physical system under a
continuous symmetry leads to a fundamental
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
.
In
crystals, 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, and
charge and
mass of a particle observed from two
reference frames
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
moving with respect to one another (invariance under a spacetime
Lorentz transformation), and invariance of
time and
acceleration under a
Galilean transformation
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
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, and
general relativity.
Informal usage
In the field of
physics, the
adjective ''covariant'' (as in
covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...
) is often used informally as a synonym for "invariant". For example, the
Schrödinger equation does not keep its written form under the coordinate transformations of
special relativity. Thus, a physicist might say that the Schrödinger equation is ''not covariant''. In contrast, the
Klein–Gordon equation and the
Dirac equation do keep their written form under these coordinate transformations. Thus, a physicist might say that these equations are ''covariant''.
Despite this usage of "covariant", it is more accurate to say that the Klein–Gordon and Dirac equations are invariant, and that the Schrödinger equation is not invariant. Additionally, to remove ambiguity, the transformation by which the invariance is evaluated should be indicated.
See also
*
Casimir operator
*
Charge (physics)
*
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
*
Conserved quantity
*
General covariance
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is ...
*
Eigenvalues and eigenvectors
*
Invariants of tensors In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor \mathbf are the coefficients of the characteristic polynomial
:\ p(\lambda)=\det (\mathbf-\lambda \mathbf),
where \m ...
*
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) show ...
*
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, ...
*
Poincaré group
*
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".
Examples o ...
*
Symmetry (physics)
*
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 anothe ...
References
{{DEFAULTSORT:Invariant (Physics)
Conservation laws
Physical quantities