Lie algebras | Physics Library

1 Lie Algebras

1.1 Lie Algebras in Quantum Theories

Continuous symmetries are often described by Lie groups. A Lie group G is a group that is also a smooth manifold, with multiplication and inversion given by smooth maps. Associated with every finite-dimensional Lie group is a finite-dimensional Lie Algebra, obtained from the tangent space at the identity together with the Lie bracket.

The Lie algebra captures the local, infinitesimal structure of the Lie group. Different Lie groups can share the same Lie algebra, so the Lie algebra does not in general determine the global topology of the group uniquely. It does, however, determine the connected simply connected Lie group associated with that infinitesimal structure.

Lie algebras are especially useful in quantum mechanics because commutators of operators naturally define Lie brackets. Symmetry generators, conserved quantities, and hamiltonians can therefore often be organized using Lie-algebra methods. A Lie algebra is a vector space equipped with a bilinear bracket; the bracket operation is generally not associative.

1.2 General Lie Algebra Definition and Examples

Definition 1.1. A Lie algebra over a field k is a vector space 𝔤 together with a bilinear map

[ , ] : 𝔤 × 𝔤 − → 𝔤,

called the Lie bracket, satisfying

$[x, x] = 0 for all x ∈ 𝔤,$
the Jacobi identity, $[x, [y,z ]] + [y,[z,x]] + [z,[x,y]] = 0$
for all x,y,z ∈ 𝔤.

Examples.

Any vector space can be made into a Lie algebra by setting

[x,y] = 0

for all vectors x and y. Such a Lie algebra is called Abelian.

If G is a Lie group, then the tangent space at the identity element, equipped with the induced Lie bracket, forms the Lie algebra of G.

The vector space ℝ³ with the cross product as its bracket,

[x, y] = x × y,

is a non-Abelian three-dimensional Lie algebra over ℝ.

Consider the annihilation operator a and creation operator a^† of the quantum harmonic oscillator. If

( 1 ) H = ℏω a†a + -I , 2

then

† † † [H, a] = − ℏω a, [H, a ] = ℏω a , [a,a ] = I.

Thus the vector space spanned by

{I,H, a,a†}

is closed under commutators and forms a four-dimensional Lie algebra. In units where ℏω = 1, the first two commutators become

[H, a] = − a, [H, a†] = a†.

This Lie algebra is solvable. Repeated application of a^† generates the excited oscillator eigenstates from the ground state, up to normalization.

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