Group Theory and Its Application to Particle Physics

Non-Abelian Groups

The elements in a non-abelian group do not commute as shown in Figure 01d where the two consecutive 90^o rotations R1 R2 generates a completely different result from R2 R1 with the order reversed. In general, there are three degrees of (rotation) freedom corresponding to rotation about the x, y, and z-axis respectively. This 3-dimensional rotation group in real space is called SO(3). The equivalent 3-dimensional rotation group with two complex numbers and three "phase angles" (parameters) associated to the three non-commuting generators is called SU(2). The precise relation between SO(3) and SU(2) is that each 360^o rotation in three dimensions corresponds to two distinct elements of SU(2). It needed a rotation of 2 x 360^o in the "complex number space" to return to the

Figure 01d Non-Abelian Rotation [view large image]

original form (don't try to visualize such abstract operation).

Mathematically, the SO(3) group is represented by the matrices R_x, R_y, and R_z operating on the vector r:

---------- (2)

For example, the rotations portrayed pictorially in Figure 01d can be expressed as R_x operating on r given r' = R_xr, and then r" = R_zr'. It can be shown that the length of the vector: r² = r'² = r"² = x² + y² + z² is invariant under these operations. In the specific case as shown in Figure 01d, all the angles of rotation are 90^o the matrices can be reduced to:

---------- (2a)

where the vector r is pointing along the negative x axis with length L. The two different sequence of rotations can be expressed explicitly as:

R_zR_xr =

R_xR_zr =

These simple formulas demonstrate that matrices are the mathematical tools to express non-Abelian or non-commutative operations.

In classical physics, spin is the rotation of rigid body around an axis with certain angular momentum uniquely defined by its magnitude and direction. In quantum theory, the angular momentum is quantized and always comes in multiples of a basic unit, which is equal to

/2. Spin in quantum theory is not well-defined; it is in superposition of different spin states. Figure 01e

	shows the geometrical property of the different kinds of spin. A spin 0 particle looks the same from all directions (A). A spin 1 particle looks the same when it is rotated through a full 360^o (B). A spin 2 particle only needs 180^o to regain its original form (C). Spin 1/2 particles (spinor) must go through two complete rotations before they look the same (D).
Figure 01e Different Kinds of Spin [view large image]