Group Theory
Definition: Group
A group is a set of elements ${g_i}$ and a multiplication rule $g_i\times g_j = g_k$, which always gives another element in the group. The multiplication rule defines the group independent of how you represent the individual group elements. The multiplication needs to fulfill the group axioms:
- Associativity: $(g_i\times g_j)\times g_k = g_i\times (g_j\times g_k)$
- Identity element: $\exists ,1 \in {g_i}: 1\times g_i = g_i\times 1 = g_i ; \forall ,g_i$
- Inverse element: $\forall ,g_i: ;\exists, g^{-1}_i \in {g_i}: g^{-1}_i\times g_i = 1$
A group has Representations which embed the group elements into operators acting on a vector space. In physics often the vectorspace these operators act on is often called the representation which is all objects in the represenation space that mix under the linear transformations of the representation.
Definition: Faithful representation
A faithful representation is a representation where every group elements has a distinct linear map.
Every group does e.g. have the trivial representation where every element $g_i$ corresponds to the identity $1$, and is therefore not faithful. It fulfills all group axioms but is not particularly interesting.
Example: Lorentz Group
The Lorentz Group contains transformations that leave the minkowski metric invariant $\Lambda^Tg\Lambda = g$. Here $\Lambda$ are matrices in the four vector representation.
Remember the $\Lambda$s are made of rotations ans boosts.
However this is only one possible representation. The Group itself is only defined by the relations between the group elements, thus there might be many other representations. The Lorentz Group itself is a mathematical object independent of its representation
Definition: Group generators
Every element $g$ in a group $G$ can be written as $$g=\exp(ic_j\lambda_j),$$ for some $j$. The $\lambda_j$ are called generators of the group.
The generators of a group form a Lie algebra
Definition: Algebra
In a nutshell an algebra is a vector space equipped with a multiplication and an addition. In contrast a group only has a multiplication.
Definition: Lie Group
A Lie Group is a Group that is also a smooth manifold. In a nutshell it has infinite elements and finite generators and the expression in terms of generators is well behaved (smooth).
Definition: Lie Algebra
The generators of a Lie group form a Lie Algebra. In a Lie Algebra the multiplication is defined by a Lie bracket. A Lie bracket $[.,.]$ fulfills
- Bilinearity: $[ax+by,z]=a[x,z]+b[y,z]$
- Jacobi Identity:
- $[x,x]=0 \forall x$
For Matrices the Lie bracket is just the commutator $[A,B]=AB-BA$.
Since all elements in a Lie Algebra can be written as a linear combination of the generators a Lie Algebra is fully defined by the commutation relations of its generators
Example: Lorentz Algebra
The elements of the Lorentz group can be written in terms of six generators. in the simple $4\times 4$ matrix representation these would e.g. be $$J_1=i\begin{pmatrix}0&&&\\ &0&&\\ &&0&-1\\ &&1&0\\end{pmatrix},\quad J_2=i\begin{pmatrix}0&&&\\ &0&&1\\ &&0&\\ &-1&&0\\end{pmatrix},\quad J_3=i\begin{pmatrix}0&&&\\ &0&-1&\\ &1&0&\\ &&&0\\end{pmatrix},$$ $$K_1=i\begin{pmatrix}0&-1&&\\ -1&0&&\\ &&0&\\ &&&0\\end{pmatrix},K_2=i\begin{pmatrix}0&&-1&\\ &0&&\\ -1&&0&\\ &&&0\\end{pmatrix},K_3=i\begin{pmatrix}0&&&-1\\ &0&&\\ &&0&\\ -1&&&0\\end{pmatrix},$$ so that a general group element of the Lorentz group can we written as $$\Lambda = \exp(i\Theta_iJ_i + i\beta_iK_i).$$
Since a Lie Algebra is defined by the Lie bracket, the Lorentz algebra is defined by the commuation relations of the generators, i.e. $$[J_i,J_j]=i\epsilon_{ijk}J_k,$$ $$[J_i,K_J]=i\epsilon_{ijk}K_k,$$ $$[K_i,K_j]=-i\epsilon_{ijk}J_k.$$ These commutation relations define the Lorentz algebra $so(1,3)$.
One distinguishes different parts of the Lorentz group, which mainly differ in ex- or including time reversal $T$ and parity $P$:
- Lorentz Group $O(1,3)$: $\det(\Lambda)=1$, includes $P$ and $T$.
- Proper Lorentz Group $SO(1,3)$: $\det(\Lambda)=+1$, excludes $T$ and $P$, but includes e.g. $TP$.
- Orthochronous Lorentz Group $O^+(1,3)$: $\Lambda^0_0\geq 0$, excludes time reversal $T$.
- Proper orthochronous Lorentz Group $SO^+(1,3)$: $\det(\Lambda)=+1$ and $\Lambda^0_0\geq 0$, excldues all $T$, $P$ and $TP$
The Proper orthochronous Lorentz Group $SO^+(1,3)$ is also called restricted Lorentz Group. It contains all elements that can be connected to the identity by a continous line in group space.
The excluded elements $T$ and $P$ are disconnected.
All the Lorentz groups have the same Algebra, but O(1,3) contains the disconnected elements which cannot be generated by the algebra.
Example: Representations of the Lorentz Group
Using the generators of the Lorentz alebra, one can redefine: $$J^+_i=\frac{1}{2}(J_i+iK_i),\quad J^-_i=\frac{1}{2}(J_i-iK_i).$$ This is a different basis for the generators and it satisfies the defining commutation relations: $$[J^+_i,J^+_j]=i\epsilon_{ijk}J^+_k$$ $$[J^-_i,J^-_j]=i\epsilon_{ijk}J^-_k$$ $$[J^+_i,J^-_j]=0$$ Since $J^+$ and $J^-$ decouple and the commutation relation for both correspond to the relations fo $su(2)$, we can conclude that the Lorentz algebra $$so(1,3)=su(2)\oplus su(2)$$ We can therefore understand the representations of the Lorentz Group based on the representations of $su(2)$.
The irreducable representations of $su(2)$ are generated by the pauli matrices with each representation characterized by a half integer $j$. Each representation $j$ acts on a vector space of $2j+1$ elements.
Therefore the irreducable representations of the Lorentz group are characterized by two half integers $A$ and $B$. The representation $(A,B)$ has $(2A+1)(2B+1)$ degrees of freedom.
Since the 3D rotations $SO(3)$ are a subgroup of the Lorentz group every representations of the Lorentz group also has to contain a representation of $SO(3)$ (representation has to comply group structure of $SO(3)$).
In fact the Lorentz group representation $(A,B)$ contains multiple $SO(3)$ representations. Since $SO(3)\approx SU(2)$ (they are pretty much the same, but every element in $SU(2)$ corresponds to two elements in $SO(3)$ - double cover), and $so(3)=su(2)$ (the algebras are the same) one can think of it like the addition of spins: Every Representation of $su(2)\oplus su(2), (A,B)$ contains the $so(3)$ representations with $j=A+B, A+B-1, …, |A-B|$ E.g.
- $(A,B)\to j$
- $(0,0)\to 0$
- $(\frac{1}{2},0)\to \frac{1}{2}$
- $(0,\frac{1}{2})\to \frac{1}{2}$
- $(\frac{1}{2},\frac{1}{2})\to 1\oplus 0$
- $(1,0)\to 1$
Here the $SO(3)$ representations correspond to the representations of the Little Group (which characterize particles). Therefore certain Lorentz group representations give rise to different particles.
E.g. $(\frac{1}{2},\frac{1}{2})$ is a representation of the Lorentz Group. It corresponds to $A_\mu(x)$. However from the above reasoning we can see that it contains two physical particles, Spin $1$ and Spin $0$.