Casimir invariant

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra

Universal enveloping algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.To understand the basic idea of this construction, first note that...

 of a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

. A prototypical example is the squared angular momentum operator

Angular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry...

, which is a Casimir invariant of the three-dimensional rotation group

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R³ under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves...

.

Suppose that is an -dimensional semisimple Lie algebra

Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras whose only ideals are {0} and itself. It is called reductive if it is the sum of a semisimple and an abelian Lie algebra....

.
Let
be any basis

Basis

Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis future, the value differential between a future and the spot price...

 of , and
be the dual basis of with respect to a fixed invariant bilinear form (e.g. the Killing form

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. In fact, the Killing form was actually invented by Élie Cartan, whereas the Cartan matrix is due to Wilhelm Killing.- Definition...

) on . The Casimir element is an element of the universal enveloping algebra given by the formula
Although the definition of the Casimir element refers to a particular choice of basis in the Lie algebra, it is easy to show that the resulting element is independent of this choice.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra

Universal enveloping algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.To understand the basic idea of this construction, first note that...

 of a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

. A prototypical example is the squared angular momentum operator

Angular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry...

, which is a Casimir invariant of the three-dimensional rotation group

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R³ under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves...

.

Definition

Suppose that is an -dimensional semisimple Lie algebra

Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras whose only ideals are {0} and itself. It is called reductive if it is the sum of a semisimple and an abelian Lie algebra....

.
Let
be any basis

Basis

Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis future, the value differential between a future and the spot price...

 of , and
be the dual basis of with respect to a fixed invariant bilinear form (e.g. the Killing form

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. In fact, the Killing form was actually invented by Élie Cartan, whereas the Cartan matrix is due to Wilhelm Killing.- Definition...

) on . The Casimir element is an element of the universal enveloping algebra given by the formula
Although the definition of the Casimir element refers to a particular choice of basis in the Lie algebra, it is easy to show that the resulting element is independent of this choice. Moreover, the invariance of the bilinear form used in the definition implies that the Casimir element commutes with all elements of the Lie algebra , and hence lies in the center of the universal enveloping algebra

Given any representation  of on a vector space V, possibly infinite-dimensional, the corresponding Casimir invariant is , the linear operator on V given by the formula
A special case of this construction plays an important role in differential geometry and global analysis. Suppose that a connected Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 G with the Lie algebra acts on a differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

 M, then elements of are represented by first order differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .There are certainly reasons not to restrict...

s on M. The representation is on the space of smooth functions on M.
In this situation the Casimir invariant is the G-invariant second order differential operator on M defined by the above formula.

More general Casimir invariants may also be defined, commonly occurring in the study of pseudo-differential operator

Pseudo-differential operator

In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory....

s in Fredholm theory

Fredholm theory

In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm...

.

Properties

The Casimir operator is a distinguished element of the center

Center (group theory)

In abstract algebra, the center of a group G, denoted Z, is the set of elements that commute with every element of G. In set-builder notation,....

 of the universal enveloping algebra

Universal enveloping algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.To understand the basic idea of this construction, first note that...

 of the Lie algebra. In other words, it is a member of the algebra of all differential operators that commutes with all the generators in the Lie algebra.

The number of independent elements of the center of the universal enveloping algebra is also the rank

Rank (linear algebra)

The column rank of a matrix A is the maximal number of linearly independent columns of A. Likewise, the row rank is the maximal number of linearly independent rows of A....

 in the case of a semisimple Lie algebra

Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras whose only ideals are {0} and itself. It is called reductive if it is the sum of a semisimple and an abelian Lie algebra....

. The Casimir operator gives the concept of the Laplacian on a general semisimple Lie group; but this way of counting shows that there may be no unique analogue of the Laplacian, for rank > 1.

By definition any member of the center of the universal enveloping algebra commutes with all other elements in the algebra. By Schur's Lemma

Schur's lemma

In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that that if M and N are two finite-dimensional irreducible representations...

, in any irreducible representation of the Lie algebra, the Casimir operator is thus proportional to the identity. This constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

). Physical mass and spin are examples of these constants, as are many other quantum number

Quantum number

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

s found in quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...

. Superficially, topological quantum number

Topological quantum number

In physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations...

s form an exception to this pattern; although deeper theories hint that these are two facets of the same phenomenon.

Example: so(3)

The Lie algebra is the Lie algebra of SO(3), the rotation group

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R³ under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves...

 for three-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...

. It is semisimple of rank 1, and so it has a single independent Casimir. The Killing form for the rotation group is just the Kronecker delta

Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...

, and so the Casimir invariant is simply the sum of the squares of the generators of the algebra. That is, the Casimir invariant is given by
In an irreducible representation, the invariance of the Casimir operator implies that it is a multiple of the identity element e of the algebra, so that
In quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...

, the scalar value is referred to as the total angular momentum. For finite-dimensional matrix-valued representations

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

 of the rotation group, always takes on integer values (for bosonic representation

Boson

In particle physics, bosons are particles which obey Bose–Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein. In contrast to fermions, which obey Fermi-Dirac statistics, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the...

s) or half-integer values (for fermionic representation

Fermion

In particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...

s).

For a given value of , the matrix representation is -dimensional. Thus, for example, the three-dimensional representation for so(3) corresponds to , and is given by the generators
The quadratic Casimir invariant is then

as when . Similarly, the two dimensional representation has a basis given by the Pauli matrices

Pauli matrices

In physics, the Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices . Usually indicated by the Greek letter 'sigma' , they are occasionally denoted with a 'tau' when used in connection with isospin symmetries...

, which correspond to spin

Spin (physics)

In particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....

1/2.