Continuous symmetries often have a special type
of underlying continuous group, called a Lie group. Briefly, a Lie group
is generally considered having a (smooth)
manifold
structure, and acts upon itself smoothly. Such a globally smooth structure is surprisingly simple in two ways: it always admits an Abelian 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 computations
and the initial problem of defining the form and symmetry of the quantum Hamiltonian
subject to boundary
and initial conditions in the quantum system
under consideration. However, unlike most regular
abstract algebras, a Lie algebra is not associative, and it is in fact a vector space. It is also perhaps this feature that makes the Lie algebras somewhat compatible, or consistent, with quantum logics
that are also thought to have non-associative, non-distributive and non-commutative
lattice structures.
Examples:
Any vector space can be made into a Lie algebra simply by setting
for all vectors
. Such a Lie algebra is an Abelian Lie algebra.
If
is a Lie group, then the tangent space at the identity forms a Lie algebra over the real numbers.
with the cross product
operation
is a non-Abelian
three dimensional (3D) Lie algebra over
.
Consider next the annihilation operator
and the creation operator
in quantum theory. Then, the Hamiltonian
of a harmonic quantum oscillator, together with the operators
and
generate a 4-dimensional (4D) Lie algebra with commutators:
,
and
. This Lie algebra is solvable and generates after repeated application of
all of the eigenvectors of the quantum harmonic oscillator.
As of this snapshot date, this entry was owned by bci1.