Index Concordia

Group theory – Week 06

Posted in Group theory by Index Guy on January 4, 2008

Our sixth week marked the transition to continuous groups and Lie algebras. We started things with a historical overview, which served to mention some of the topics we are going to study.

It all started with Felix Klein and Sophus Lie, two students under professor Kronecker in Berlin. Klein was concerned with transformation groups, for example rotations of regular polyhedra and isohedra. Unitary transformations are another example, which leave the inner product

(x, y) = \displaystyle\sum_{i = 1}^{n} x_{i}^{*}y_{i},

invariant under transformations on the vectors x' = Ux and y' = Uy, with U^{-1} = U^{\dagger}. On the other hand, Lie was interested in continuous groups, groups that could be parametrized with continuous variables.

In 1854 Cayle studied finite groups and their representations. By 1878 there were important results by Frobenius, Schur and Burnside, and in 1889 the Killing metric was introduced. Cartan followed with a list of important results. Other important highlights include the work of Dynkin (1947) and his introduction of diagrams to study Cartan’s results

The terrible Ws are acknowledge with introducing group theory to quantum physicist. They were Weyl, Wigner and van der Waerden. Then during the 1960s the quark model was develop and the relevance of group theory for particle physics was recognized. Further works include the renormalization of Yang-Mills theory and all the GUT models.

Now back to present, we start with a discussion of the relation between the groups SU(2) and SO(3). We begin by defining the notion of simply connected. A space is simply connected if a curve can always be contracted to a point. It turns out that the group manifold for SO(3) is not simply connected, while that for SU(2) is. Nevertheless, both of these groups have the same algebra.

As mentioned above, unitary transformations leave the following inner product invariant,

x_{i}^{*}y_{i} = \displaystyle\sum_{i = 1}^{n} x_{i}^{*}y_{i}.

Particularly these will be two-by-two unitary matrices. Recall that the Dirac matrices can be used as components for a matrix. In two dimensions we use the identity and the set of Pauli matrices. Then any unitary matrix can be written as

U = aI + i b^{j}\sigma_{j}.

The unitary condition implies UU^{\dagger} = I which after using our expansion for U we get,

a^{*}a + (b^{ j})^{*}b_{j} = 1 and a^{*}b^{j} - a (b^{j})^{*} + \epsilon_{jkl}b^{k}(b^{l})^{*} = 0.

Taking the second expression above, and multiplying by b^{j} on both sides one gets

a^{*} (b^{j})^{2} - a \left| b^{j} \right|^{2} = 0.

This tells us that both a, b^{j} have the same phase. Requiring unit determinant implies that this phase must be zero. So then we obtain

a^{2} + \sum (b^{j})^{2} = 1.

This is the equation of a hypersphere, or a sphere in four dimensions. We can then see SU(2) as a 3-sphere, the surface of a four dimensional ball. But we can also see SU(2) as a 3-ball, the body of a solid, 3-dimensional ball.

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