Casimir invariant

In mathematics, the Casimir invariant , or Casimir operator , is a remarkable element of the center of the universal enveloping algebra of a Lie algebra . Named after the Dutch physicist Hendrick Casimir . An example is the square of the angular momentum operator , which is the Casimir invariant of the three-dimensional rotation group . Casimir operators of the Poincare group have a deep physical meaning, since they are used to determine the concepts of mass and spin of elementary particles ^[1] .

Content

1 Definition
2 Properties
3 Example: so (3)
4 See also
5 notes
6 References
7 Literature

Definition

We assume that ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ - $n$ ${\ displaystyle n}$ -dimensional semisimple Lie algebra . Let be $\{X_{i}\}_{i=1}^{n}$ ${\ displaystyle \ {X_ {i} \} _ {i = 1} ^ {n}}$ - any basis ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ , but $\{X^{i}\}_{i=1}^{n}$ ${\ displaystyle \ {X ^ {i} \} _ {i = 1} ^ {n}}$ Is a dual basis constructed from a fixed invariant bilinear form (for example, the Killing form ) on ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ . Casimir element $\Omega$ ${\ displaystyle \ Omega}$ Is an element of the universal enveloping algebra defined by the formula

\Omega =\sum _{i=1}^{n}X_{i}X^{i}.

{\ displaystyle \ Omega = \ sum _ {i = 1} ^ {n} X_ {i} X ^ {i}.}

Despite the fact that the definition of a Casimir element refers to a specific choice of a basis in a Lie algebra, it is easy to show that the element obtained $\Omega$ ${\ displaystyle \ Omega}$ 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 algebra ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ , and therefore lies at the center of the universal enveloping algebra $U({\mathfrak {g}}).$ ${\ displaystyle U ({\ mathfrak {g}}).}$

Any submission $\rho$ ${\ displaystyle \ rho}$ algebras ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ on a vector space V , possibly infinite-dimensional, there corresponds the Casimir invariant $\rho (\Omega )$ ${\ displaystyle \ rho (\ Omega)}$ , a linear operator on V defined by the formula

\rho (\Omega )=\sum _{i=1}^{n}\rho (X_{i})\rho (X^{i}).

{\ displaystyle \ rho (\ Omega) = \ sum _ {i = 1} ^ {n} \ rho (X_ {i}) \ rho (X ^ {i}).}

A special case of this construction plays an important role in differential geometry and general analysis . If a connected Lie group G with a Lie algebra ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ act on a differentiable manifold M , then the elements ${\mathfrak {g}}$ ${\ displaystyle {\ mathfrak {g}}}$ are represented by first order differential operators on M. Performance $\rho$ ${\ displaystyle \ rho}$ acts on the space of smooth functions on M. In such a situation, the Casimir invariant is a G- invariant second-order differential operator on M defined by the above formula. It (depending on the convention, up to a sign) coincides with the Laplace – Beltrami operator on the underlying manifold of the Lie group G with respect to the Cartan – Killing metric .

More general Casimir invariants can also be defined. They are usually found in the study of pseudo-differential operators and Fredholm theory .

Properties

The Casimir operator is a remarkable element of the center of the universal enveloping algebra of a Lie algebra . In other words, it is a member of the algebra of all differential operators that commutes with all generators in the Lie algebra.

The number of independent elements of the center of a universal enveloping algebra is also a rank in the case of a semisimple Lie algebra . The Casimir operator gives the concept of Laplacian on general semisimple Lie groups ; but such a path shows that there may not be a single Laplacian analogue for rank> 1.

In any irreducible representation of a Lie algebra, by Schur's lemma , any member of the center of a universal enveloping algebra commutes with everything and, therefore, is proportional to the unit. This proportionality coefficient can be used to classify representations of a Lie algebra (and, therefore, also its Lie group ). Physical mass and spin are examples of such coefficients as many other quantum numbers used by quantum mechanics . Outwardly, topological quantum numbers are an exception to this model; although deeper theories suggest that these are two facets of the same phenomenon.

Example: so (3)

Algebra Lee ${\mathfrak {so}}(3)$ ${\ displaystyle {\ mathfrak {so}} (3)}$ corresponds to SO (3), the rotation group of 3-dimensional Euclidean space . It is simple of rank 1, and thus it has the only independent Casimir invariant. The Killing form for a rotation group is only a Kronecker symbol , and the Casimir invariant is just the sum of the squares of the generators $L_{x},\,L_{y},\,L_{z}$ ${\ displaystyle L_ {x}, \, L_ {y}, \, L_ {z}}$ given algebra. That is, the Casimir invariant is given by the formula

L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}.

{\ displaystyle L ^ {2} = L_ {x} ^ {2} + L_ {y} ^ {2} + L_ {z} ^ {2}.}

In the irreducible representation, the invariance of the Casimir operator assumes its multiplicity to the identity element e of the algebra, so

L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\ell (\ell +1)e.

{\ displaystyle L ^ {2} = L_ {x} ^ {2} + L_ {y} ^ {2} + L_ {z} ^ {2} = \ ell (\ ell +1) e.}

In quantum mechanics , scalar value $\ell$ ${\ displaystyle \ ell}$ refers to the full moment of momentum. For finite-dimensional matrix-valued representations of the rotation group, $\ell$ ${\ displaystyle \ ell}$ it is always integer (for bosonic representations ) or half-integer (for fermion representations ).

For a given number $\ell$ ${\ displaystyle \ ell}$ matrix representation $(2\ell +1)$ ${\ displaystyle (2 \ ell +1)}$ -dimensionally. So, for example, the 3-dimensional representation of so (3) corresponds to $\ell =1$ ${\ displaystyle \ ell = 1}$ and set by generators

L_{x}={\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix}},L_{y}={\begin{pmatrix}0&0&-1\\0&0&0\\1&0&0\end{pmatrix}},L_{z}={\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}}.

{\ displaystyle L_ {x} = {\ begin {pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \ end {pmatrix}}, L_ {y} = {\ begin {pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \ end {pmatrix}}, L_ {z} = {\ begin {pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \ end {pmatrix}}.}

Then the Casimir invariant:

L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=2{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},

{\ displaystyle L ^ {2} = L_ {x} ^ {2} + L_ {y} ^ {2} + L_ {z} ^ {2} = 2 {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \ end {pmatrix}},}

as $\ell (\ell +1)=2$ ${\ displaystyle \ ell (\ ell +1) = 2}$ at $\ell =1$ ${\ displaystyle \ ell = 1}$ . In the same way, the 2-dimensional representation has a basis defined by the Pauli matrices , which correspond to 1/2 spin .

Notes

↑ Rumer, 2010 , p. 134.

Links

Humphreys, James E. Introduction to Lie Algebras and Representation Theory . - Second printing, revised. Graduate Texts in Mathematics, 9. - New York: Springer-Verlag, 1978. - ISBN 5-9221-0055-6 .

Literature

Rumer Yu. B. , Fet A. I. Group theory and quantized fields. - M .: Librocom, 2010 .-- 248 p. - ISBN 978-5-397-01392-5 .

[_7462f769eeac19ea-1] Rumer, 2010 , p. 134.