PlanetPhysics/Lie Algebras

Lie algebras in quantum theories

Continuous symmetries often have a special [[../Bijective/|type]] of underlying continuous [[../TrivialGroupoid/|group]], called a Lie group . Briefly, a Lie group $G$ is generally considered having a (smooth) $C^{\mathrm{\infty }}$ [[../NoncommutativeGeometry4/|manifold]] structure, and acts upon itself smoothly. Such a globally smooth structure is surprisingly simple in two ways: it always admits an Abelian [[../SingularComplexOfASpace/|fundamental group]], and seemingly also related to this global property, it admits an associated, unique--as well as finite--Lie algebra that completely specifies locally the properties of the Lie group everywhere. There is a finite Lie algebra of quantum commutators and their unique (continuous) Lie groups. Thus, Lie algebras can greatly simplify quantum [[../LQG2/|computations]] and the initial problem of defining the form and symmetry of the quantum [[../Hamiltonian2/|Hamiltonian]] subject to [[../PiecewiseLinear/|boundary]] and initial conditions in the quantum [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] under consideration. However, unlike most [[../CoIntersections/|regular]] [[../PAdicMeasure/|abstract algebras]], a [[../TopologicalOrder2/|Lie Algebra]] is not associative, and it is in fact a [[../NormInducedByInnerProduct/|vector space]]. It is also perhaps this feature that makes the Lie algebras somewhat compatible, or consistent, with [[../LQG2/|quantum logics]] that are also thought to have non-associative, non-distributive and [[../AbelianCategory3/|non-commutative]] lattice structures.

General Lie algebra definition and Examples

A Lie algebra over a [[../CosmologicalConstant/|field]] $k$ is a vector space $𝔤$ together with a bilinear map Failed to parse (syntax error): {\displaystyle [\ ,\] : \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}} , called the Lie bracket and defined by the association $(x,y)\mapsto [x,y]$. The bracket is subject to the following two conditions:

1. $[x,x]=0$ for all $x\in 𝔤$.
2. The Jacobi [[../Cod/|identity]]: $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z\in 𝔤$.

{\mathbf Examples:}

Any vector space can be made into a Lie algebra simply by setting $[x,y]=0$ for all [[../Vectors/|vectors]] $x,y$. Such a Lie algebra is an Abelian Lie algebra.

If $G$ is a Lie group, then the tangent space at the identity forms a Lie algebra over the real numbers.

${ℝ}^3$ with the [[../VectorProduct/|cross product]] [[../Cod/|operation]] is a [[../AbelianCategory3/|non-Abelian]] three dimensional (3D) Lie algebra over $ℝ$.

Consider next the annihilation [[../QuantumOperatorConcept/|operator]] $a$ and the creation [[../QuantumOperatorConcept/|operator]] $a†$ in [[../QuantumOperatorAlgebra5/|quantum theory]]. Then, the Hamiltonian $H$ of a harmonic quantum oscillator, together with the [[../QuantumOperatorAlgebra4/|operators]] $a$ and $a†$ generate a 4--dimensional ([[../Curved4DimensionalSpace/|4D]]) Lie algebra with [[../Commutator/|commutators]]: Failed to parse (syntax error): {\displaystyle [H, a] = −a} , $[H,a†]=a†,$ and $[a,a†]=I$. This Lie algebra is solvable and generates after repeated application of $a†$ all of the eigenvectors of the [[../LieAlgebraInQuantumTheory/|quantum harmonic oscillator]].