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 p ...

, an invariant is an

observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...

of a

physical system A physical system is a collection of physical objects under study. The collection differs from a set: all the objects must coexist and have some physical relationship. In other words, it is a portion of the physical universe chosen for analys ...

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 repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...

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 the

conservation of momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the

conservation of energy 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. In the case of a Closed system#In thermodynamics, closed system, the principle s ...

. In general, by

Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...

, 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 Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...

leads to a fundamental conservation law. In crystals, the

electron density 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 variables and is typical ...

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 exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

, and charge and

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

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 vel ...

), and invariance of

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

and

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

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 via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

, and

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

Informal usage

In the field of

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, the

adjective An adjective (abbreviations, abbreviated ) is a word that describes or defines a noun or noun phrase. Its semantic role is to change information given by the noun. Traditionally, adjectives are considered one of the main part of speech, parts of ...

''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. Briefly, a contravariant vecto ...

) is often used informally as a synonym for "invariant". For example, the

Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...

does not keep its written form under the coordinate transformations of

. Thus, a physicist might say that the Schrödinger equation is ''not covariant''. In contrast, the Klein–Gordon equation and the

Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...

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.

References

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

Examples

Informal usage

See also

References