Group Theory and Its Application to Particle Physics
The transformation in Eq.(3) is of unit length, the set of all such transformations forms what mathematicians call a unitary group. In this case U(1) = eia means the group of all "unitary" 1-dimensional matrices. A unitary matrix U is one that satisfies
UU = UU = I
where I is the identity (unity matrix) and U is the conjugate of U such that the average value *|U| = |U*|*; it is called Hermitean if U = U, i.e., the average value is a real number; it is called unitary if U = U-1. An unitary transformation leaves the normalization of the state vectors unchanged. Specifically, unitary operator preserves the probabilities for quantum transition. It plays a special role in quantum theory. The U(1) group have the simple property that it does not matter in what order they are performed, i.e., a + b = b + a. Thus U(1) = eia = eia(1) = eiaI is called an Abelian group in which different transformations commute.
In SU(2) the phase angle a = aI in U(1) is replaced by at = a1t1 + a2t2 + a3t3 - there are now three phase angles
a = (a1, a2, a3), and three 2x2 non-commuting matrices (generators) for t, which are 1/2 of the Pauli matrices:
where the factor of 1/2 is responsible for the curious property that it takes a 2 x 360o rotation to return to the original form.
|| ---------- (10)|
For non-Abelian unitary groups, the number of phase angles (parameters) is determined by the formula Na = n2 - 1, where n is the dimension of the internal space, e.g., Na = 3 for n = 2 in SU(2). The rank of SU(n) is n - 1. It gives the number of diagonal matrix representation for the generators. Thus, the rank of SU(2) is 1 as shown in Eq.(10).
It can be shown that the absolute square for the determinant of U is 1, i.e., |detU|2 = 1 (which implies the trace of the generators equal to zero). Thus |detU| can be either +1 or -1. The "S" in "SU" means the transformations are "Special" satisfying |detU| = +1, which implies continuous roatations without reflections. Therefore SU(2) designates the group of special, unitary 2x2 matrices.
SU(2) also has other representations for different kinds of spin in 3-dimensional space (see Figure 01e):
The rotation in internal space has nothing to do with angular momentum. In SU(2), the two complex fields can be identified with the electron and neutrino fields respectively in analogy to the spin up and spin down states. It is called "isospin" or "weak isospin" or weak charge (for the left-handed electron and neutrino in weak interaction, which actually cannot tell the difference between the two kinds and lumps them together as leptons) to distinguish from the ordinary "spin". For example, in terms of the Pauli matrices, the phase angle , and the complex fields :
- For the trivial (identity) representation all the generators ti 0. It can be identified to particle with spin 0 such as the Higgs boson. In this representation, the particle looks the same from every direction.
- The fundamental representation of SU(2) is for leptons with spin 1/2. This is the one that has to rotate 2 x 360o to return to its original form.
- There is the representation of SU(2) by the usual 3-dimensional rotations (the SO(3) group) acting on three dimensional vectors. The generators can be produced from the definition in Eq.(9), for example:
This is the spin 1 representation and can be used to describe gauge particles such as photon, vector boson, and gluon.
a SU(2) transformation of can be expressed as:
' = ei(/2)1 = [cos(/2)1 + i sin(/2)1]
It can be shown that the magnitude of the complex fields: u*u + v*v is conserved under such operation.
In SU(3), the three complex fields can be identified to the three quark fields, which can be thought of as the three directions at 120o to each other. It is called colour isospin or colour spin or colour charge.
In SU(3) Na = 8, there are eight phase angles and eight 3x3 non-commuting matrices (generators) operating on three complex functions (representing the u, d, s quarks as shown in Eq.(20)):
| ---------- (11)|
The SU(2) and SU(3) transformations are more like the non-commuting 3-dimensional rotations of real space because of the non-commuting generators, e.g., ei(at1+ bt2) ei(bt1+ at2). These groups are called non-Abelian for this reason.
If we define t2 = t12 + t22 + t32 + ... ---------- (12)
It can be shown that the phase transformations are characterized by the eigenvalue of t2 and t3 with eigenvector |t,m:
t2 |t,m = t(t+1) |t,m ---------- (13)
t3 |t,m = m |t,m ---------- (14)
where t can be determined by the formula n = 2t +1 (n is the dimension of the generator) and for a given t, the value of m can be -t, -t+1, ..., t-1, t; i.e., there are 2t+1 degenerate states for a given t. For example, in the case of SU(2) n = 2, t = 1/2, and m = -1/2 or +1/2 (an isodoublet). The eigenvalue t is often referred to as isospin to indicate that it is similar to spin but in an abstract space. The eigenvalue for t3, is sometimes referred to as weak charge (when applied to weak interaction). For SU(3), n = 3, t = 1, and m = +1, 0, -1 forming an isotriplet.
A casimir operator is a nonlinear function of the generators that commutes with all of the generators. The number of casimir operators is equal to the rank of the group. For example, there is only 1 casimir operator in SU(2), e.g., t2 = (3/4) I, which obviously commutes with all the ti. Since the casimir operator is proportional to the identity, this constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group). It is usually related to the mass or (iso)spin. The proportional constant in the example is the "square" of total (iso)spin T2 = t (t+1) for t = 1/2.