Quantum Mechanics for Mathematicians
so the dual representation is given by conjugating all elements of the matrix. The same calculation as in the real case here giv ...
can always be factored into linear factors. For an arbitrarynbyncomplex ma- trix there will bensolutions (counting repeated eige ...
For matrices in the subgroupSU(2),λ 1 =−λ 2 =λ, so in diagonal form an SU(2) matrix will be ( eiλ 0 0 e−iλ ) which is the expone ...
Chapter 5 Lie Algebras and Lie Algebra Representations In this chapter we will introduce Lie algebras and Lie algebra representa ...
the tangent space at a point of manifolds likeG, but we do not want to enter here into the subject of differential geometry in g ...
a more subtle example of this:SU(2) andSO(3) are different groups with the same Lie algebra. We haveG⊂GL(n,C), andX∈M(n,C), the ...
Definition(Lie bracket). The Lie bracket operation ongis the bilinear anti- symmetric map given by the commutator of matrices [· ...
5.2 Lie algebras of the orthogonal and unitary groups The groups we are most interested in are the groups of linear transformati ...
5.2.1 Lie algebra of the orthogonal group Recall that the orthogonal groupO(n) is the subgroup ofGL(n,R) of matrices Ω satisfyin ...
is NOT a space of realnbynmatrices. It is the space of skew-Hermitian matrices, which in general are complex. While the matrices ...
algebras aregl(n,R) =M(n,R) andgl(n,C) =M(n,C). These are the vector spaces of allnbynmatrices, with Lie bracket the matrix comm ...
More concretely, given a basisX 1 ,X 2 ,...,Xdof a Lie algebragof dimension dwith structure constantscjkl, a representation is g ...
of the map at the identity. For general Lie groupsG, something similar can be done, showing that a representationπofGgives a rep ...
Recall 5.1: [X,Y] = d dt (etXY e−tX)|t=0 so π′([X,Y]) =π′ ( d dt (etXY e−tX)|t=0 ) = d dt π′(etXY e−tX)|t=0 (by linearity) = d ...
Note that this linear mapad(X), which can be written as [X,·], can be thought of as the infinitesimal version of the conjugation ...
5.5 Complexification Conventional physics discussion of Lie algebra representations proceed by as- suming complex coefficients a ...
withv 1 in the first copy ofV,v 2 in the second copy. Then the rule for mul- tiplication by a complex number comes from the stan ...
If the original representation was on a complex vector spaceV, the extended one will act on the same space. If the original repr ...
Chapter 6 The Rotation and Spin Groups in 3 and 4 Dimensions Among the basic symmetry groups of the physical world is the orthog ...
This can be written as an exponential,R(θ) =eθL= cosθ 1 +Lsinθfor L= ( 0 − 1 1 0 ) HereSO(2) is a commutative Lie group with Lie ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf