Lie algebras LieAlgebras 2009-05-01 03:16:38 2026-09-09 22:05:59 Topic added top level section for clean numbering harmonic quantum oscillator finite Lie algebra of quantum commutators Lie group tangent space globally smooth structure Abelian Lie algebra Lie algebra bilinear map non-associative structures harmonic quantum oscillator finite Lie algebra of quantum commutators Lie group tangent space globally smooth structure Abelian Lie algebra % Object 711: Lie Algebras % Minimal pdflatex/make4ht-safe preamble. % % Removed xypic, amscd, latexsym, eucal, and the large unused macro block. \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \subsection{Lie Algebras in Quantum Theories} Continuous symmetries are often described by \emph{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. \subsection{General Lie Algebra Definition and Examples} \begin{definition} A \emph{Lie algebra} over a field $k$ is a vector space $\mathfrak{g}$ together with a bilinear map \[ [\ ,\ ]: \mathfrak{g}\times\mathfrak{g} \longrightarrow \mathfrak{g}, \] called the \emph{Lie bracket}, satisfying \begin{enumerate} \item \[ [x,x]=0 \qquad \text{for all }x\in\mathfrak{g}, \] \item the \emph{Jacobi identity}, \[ [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 \] for all $x,y,z\in\mathfrak{g}$. \end{enumerate} \end{definition} \textbf{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 \emph{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 $\mathbb{R}^3$ with the cross product as its bracket, \[ [x,y]=x\times y, \] is a non-Abelian three-dimensional Lie algebra over $\mathbb{R}$. Consider the annihilation \PMlinkname{operator}{QuantumOperatorConcept} $a$ and creation \PMlinkname{operator}{QuantumOperatorConcept} $a^\dagger$ of the quantum harmonic oscillator. If \[ H=\hbar\omega\left(a^\dagger a+\frac12 I\right), \] then \[ [H,a]=-\hbar\omega\,a, \qquad [H,a^\dagger]=\hbar\omega\,a^\dagger, \qquad [a,a^\dagger]=I. \] Thus the vector space spanned by \[ \{I,H,a,a^\dagger\} \] is closed under commutators and forms a four-dimensional Lie algebra. In units where $\hbar\omega=1$, the first two commutators become \[ [H,a]=-a, \qquad [H,a^\dagger]=a^\dagger. \] This Lie algebra is solvable. Repeated application of $a^\dagger$ generates the excited oscillator eigenstates from the ground state, up to normalization.