LOG#089. Group theory(IX).

220px-Sophus_Lie

Definition (36). An infinite group (G,\circ) is a group where the order/number of elements \vert G\vert is not finite. We distinguish two main types of groups (but there are more classes out there…):

1) Discrete groups: their elements are a numerable set. Invariance under a discrete group provides multiplicative conservation laws. Elements are symbolized as g_i \forall i=1,\ldots,\infty for a discrete group.

2) Continuous groups: their elements are not numerable, since they depend “continuously” on a finite number of parameters (real, complex,…):

g=g(\alpha_1,\alpha_2,\ldots)

Note that the number or paraters can be either finite or infinite in some cases. The number of parameters define the so-called “dimension” of the group. Please, don’t confuse group order with its dimension. Group order is the number of elements, group dimension is the number of parameters we do need to characterize/generate the group! Invariance under a continuous group has some consequences (due to the Noether’s theorems):

1) Invariance under a finite dimensional r-parametric continuous group provides conservation laws.

2) Invariance under an infinite dimensional continuous group (parametrized by some set of “functions”) provides some relationships between field equations called “dependencies” or “noether identities” in modern language.

Definition (37). Composition rule/law for a group. Let G be a continuous group and two elements g(\alpha_1),g(\alpha_2)\in G, then

g(\alpha_1)\circ g(\alpha_2)=g(\alpha_3)

and we define the composition law of a continuous  group as the function that gives \alpha_3=f(\alpha_1,\alpha_2) and similarly

g(\alpha_2)=g^{-1}(\alpha_1)

so

\alpha_1=f^{-1}(\alpha_2)

Theorem (Lie). Every continuous group is a Lie Group. It means that whenever you have a group where the composition rule is given, as the inverse element, then the group elements are differentiable functions (analytic in the complex case) on its arguments.

Some examples of Lie groups (some of them we have already quoted in this thread):

1) The euclidean real space \mathbb{R}^n or the hermitian complex space \mathbb{C} with ordinary vector addition form (in any of that two cases) a n-dimensional noncompact abelian Lie group.

2) The general linear (Lie) group of non-singular matrices over the real number or the complex numbers is a Lie group GL_n(\mathbb{R}) or GL_n(\mathbb{C}).

3) The special linear group SL(n,\mathbb{R}) or the complex analogue SL(n,\mathbb{C}) of square matrices with determinant equal to one.

4) The orthogonal group O(n) over the real numbers, n\times n matrices with real entries is a n(n-1)/2 dimensional Lie group.

5) The special orthogonal group SO(n,\mathbb{R}) is the subgroup of the orthogonal group whose matrices have determinant equal to one.

6) The unitary group U(n,\mathbb{C}) of complex n\times n unitary matrices, UU^+=U^+U=\mathbb{I}_n. Its dimension is equal to n^2 over the complex numbers. SU(n) is the n^2-1 dimensional subgroup formed by unitary matrices with determinant equal to one.

7) The symplectic group \mbox{Sp}(2n,\mathbb{R}).

8) The group of upper triangular matrices n\times n is a group with dimension n(n+1)/2.

9) The Lorentz group and the Poincaré group. The are non-compact Lie groups (Poincaré is non-compact due to the fact that the Lorentz subgroup is non-compact). Their dimensions in 4D spacetime are 6 and 10 dimensions respectively.

10) The Standard Model “gauge” (Lie) group U(1)\times SU(2)\times SU(3) is a group formed with direct group (in the group sense) of three groups and it has dimension 1+3+8=12. The dimensions of the gauge groups in the Standard Model is in direct correspondence with the numbers of gauge bosons: 1 massless photon, 3 vector bosons for the electroweak interactions, and 8 gluons for the quantum chromodynamics (QCD).

11) The exceptional Lie groups \mathcal{G}_2,\mathcal{F}_4, E_6, E_7, E_8, the so called Cartan exceptional groups. Their dimensions are respectively 14, 52, 78, 133 and 248.

The continuous group made of matrices (finite and infinite matrices/operators) play an important role in Physics. Moreover, as Lie groups depend continuously on their arguments AND their dependence is generally differentiable, it makes sense to take derivatives in the group elements. In fact, this fact allow us to define the idea of group generator.

Definition(38).  Group generator. If g=U(\alpha) is a continuous (therefore differentiable; remember that continuity implies differentiability but the converse is not necessarily true), then we define the generators of the group L_i in the following (hermitian) way:

-iL_j=\dfrac{\partial U(\alpha)}{\partial \alpha_j}\bigg|_{\alpha=0}

Theorem (Lie). Let us choose some G=U(\alpha) one-parameter continuous group and K its generator. Then, the following facts hold:

i) K fully determines the group U(\alpha).

ii) Group elements are obtained using “exponentiation” of generators. That is,

U(\alpha)=\exp\left(-iK\alpha\right)

The “proof” involves a group parametrization and an expansion as a series. We have U(0)=1 and U(x+y)=U(x)U(y). Therefore,

\dfrac{dU(x)}{dx}=\dfrac{d}{dy}\left(U(x+y)\right)\vert_{y=0}=\dfrac{d}{dy}(U(x)U(y))\vert_{y=0}

\dfrac{dU(x)}{dx}=U(x)\dfrac{dU(y)}{dy}\bigg|_{y=0}=\dfrac{dU(y)}{dy}\bigg|_{y=0}U(x)=-iKU(x)

so

-iKU(x)=\dfrac{dU(x)}{dx} and U(0)=\mathbb{I}

This differential equation has one and only one solution for every K-value\forall x. The general solution of this equation is the exponential:

U(x)=U(0)\exp\left(-iKx\right)

Taking into account the initial conditions U(0)=\mathbb{I} (elements nearby of any group element are the identity element) we have the desired result for every x=\alpha. Q.E.D.

Theorem (Lie). A multiparametric Lie group (N-dimensional) is a Lie group G with functions g=U(\alpha_j),\forall j=1,2,\ldots, N and generators L_j obtained by exponentiation. That is:

\boxed{\displaystyle{U(\alpha_j)=\prod_{j=1}^N\exp \left(-iL_j\alpha_j\right)}}

Check (Easy simplified proof): Using the previous result, we have to fix only all the parameters \alpha_j\forall j=1,\ldots,N. Then, a simple “empatic mimicry” of the previous one dimensional provides:

U(\alpha_N)=U(0,\alpha_2,\alpha_3,\ldots,\alpha_N)\exp\left(-iL_1\alpha_1\right)

and then

U(\alpha_N)=U(0,0,\alpha_3,\ldots,\alpha_N)\exp\left(-iL_1\alpha_1\right)\exp\left(-iL_2\alpha_2\right)

and finally, iterating the process N-times we get

\displaystyle{U(\alpha_N)=U(0,0,0,\ldots,0)\prod_j^N\exp\left(-iL_j\alpha_j\right)}

The generators of any Lie group satisfy some algebraic and important relations. In the case of dealing with matrix or operator groups, the generators are matrices or operator theirselves. These mathematical relations can be written in terms of (ordinary) algebraic commutators. There is a very important theorem about this fact:

Theorem. First Lie theorem. Lie group generators form a closed commutator algebra under “matrix/operator” products. That is:

\boxed{\left[L_i,L_j\right]=C_{ij}^{k}L_k} or \boxed{\left[L_i,L_j\right]=C^{ijk}L_k}

without distinction of lower and upper “labels”.

There the commutator of two matrices/operators is defined to be \left[A,B\right]=AB-BA and the contants C_{ijk} or C^k_{ij} are the so-called structure constants of the Lie group. The structure constants of a Lie group are:

1) Antisymmetric with respect to the first two indices (or the paired ones, ij, with our notation).

2) Characteristic of the group but they do change, in a particular way, if we form linear combinations of the Lie group generators.

There is a nice formula called Baker-Campbell-Hausdorff identity that relates group exponentials and group commutators. It is specially important in the theory of Lie groups and Lie algebras:

The Baker-Campbell-Hausdorff (BCH) formula. For any matrix/operator A,B, under certain very general conditions, we have:

\exp(A)\exp(B)=\exp\left(A+B+\dfrac{1}{2}\left[A,B\right]+\dfrac{1}{12}\left[\left[A,B\right],B\right]-\dfrac{1}{12}\left[\left[A,B\right],A\right]-\ldots\right)

In the case that the matrices A and B do commute, then we recover the usual ordinary exponentiation of “elements”:

\exp(A)\exp(B)=\exp(A+B)

A beautiful and simple application of the BCH formula is the next feature which allows us to write ANY member of a Lie group as the exponential of a sum of the Lie group generators. Let us write the group elements, firstly, as

g=U(\alpha_j)\forall j=1,2,\ldots,N

and let us write the group generators as L_j. Then, we have

\displaystyle{U(\alpha_j)=\prod_{j=1}^N\exp\left(-i\alpha_jL_j\right)=\exp\left(-i\sum_ {j=1}^N\omega_jL_j\right)}

where the parameters \omega_j are related to the \alpha_j parameters in a simple continuous way

\omega_j=\omega_j(\alpha_k)

The specific form of this realtion can be expanded and computed/calculated term by term using the BCH formula, as given before.

See you in the next blog post of this group theory thread!

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