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Group Theory and Its Application to Particle Physics

Lie Groups

Let's consider the phase transformation of a complex function (with real and imaginary parts):

f' = eia f ---------- (3)
where a (representing the elements of a continuous group) is either a constant for a global transformation or a function of (x,y,z,t) for local transformation. It is known as the phase angle because if f is a wave represented by f = foei(kx-t), then a is just the phase shift of the wave. The phase shift interpretation provides another representation other than the rotation in the complex plane as introduced in the followings.

If f and f ' are decomposed into the real and imaginary parts: f = fR + i fI ---------- (4)

Then Eq.(3) can be re-written as:

f 'R = [cos(a)] fR - [sin(a)] fI ---------- (5)
f 'I = [sin(a)] fR + [cos(a)] fI ---------- (6)

Rotation in Complex Plane which is in the same form as Eqs.(1a,b) for the 2-dimensional rotation, but in the (fR, fI) complex plane with the real and imaginary parts of the function as coordinates (Figure 01f). This plane is called an internal space, and the associated symmetry an internal symmetry, if the mathematical formulation is unchanged after the operation. For example, fR and fI can be identified to the two components of the scalar field in the Klein-Gordon Equation, e.g., . The internal rotation just presents different aspect of the field components in the complex plane. The Klein-Gordon Equation is invariant (unchanged) under such internal rotation, which is another way to say that the equation is symmetrical with respect to such operation. It is obvious from Figure 01f that length of the arrow has not changed by merely rotating the complex plane. It is something like renaming a person from John to Jack - that person remains the same although there may be some legal or financial consequences. The analogy in Figure 01f can be taken with the "arrow" as the person, f and f ' as the different names.

Figure 01f Complex Plane Rotation [view large image]

The Klein-Gordon Equation can be used to demonstrate the invariance under such so-called U(1) Global Gauge transformation (the 2 components could represent the + and - charge state respectively).

This demonstration is for a = constant, the case with dependence on x and/or t is more complicated (see "Local Gauge Symmetry").

Two successive transformations such as: f f ' f " ---------- (7)
where f " = eib f ', can be written in the form:
f " = ei(a+b) f = eic f ---------- (8)
with c = a + b. This is a transformation to the same form as the original one satifying one of the definitions for a group. The inverse elements are -a, -b, ..., and the identity operator is a 0. Since the phase angle can be infinitesimally small, this kind of group also belongs to the Lie group (continuous group) which is important in studying physical theories such as the Noether's theorem or local gauge transformation.

In general a Lie Group is specified by a finite set of continuous parameters ai (with i running from 1, 2, ... to Na) and there are well defined derivatives of the group elements g's with respect to all the parameters. We will simplify the discussion by the example of a group with a single parameter a = a1. According to the rules for a Lie Group, we can define the generator:
t = -i dg/da|a=0 ---------- (9)
where the i is included so that if the representation D(a) is unitary, the t will be hermitian operator. The identity representation corresponds to a = 0 is D(0) = 1. Using the Taylor series expansion for small increment from a = 0, D can be expressed as:
D() 1 + it
If we apply this operation n times with a = n, then
D(a) = (1 + iat/n)n = eiat as n
which is very similar to the U(1) group transformation in Eq.(3).

The Lie algebra is defined by the commutator relation:
[ta, tb] = i fabctc
where fabc is the structure constant that can be computed in any representation. It plays a role similar to the multiplication law for the finite group.

Sophus Lie E8 Lie Group Sophus Lie (1842-1899, Figure 02a) showed how the generators can actually be defined in the abstract group without mentioning representations at all. Thus, t = 1 for the U(1) group and the Pauli matrices (see Eq. (10)) are the three generators for the SU(2) group, ... Beside generating the displacement for the ai, the generators are important for forming a vector space. This means two generators can be added together to obtain a third generator, and the generators can be multiplied by a scalar and still remain generators for the group. A complete vector space can be used as a basis for representing other vector spaces, hence the

Figure 02a Sophus Lie and Symmetry [view large image]

Figure 02b E8 Lie Group [view large image]

generators of a group can be used to represent other vector spaces, e.g., the Pauli matrices can be used to describe any 2x2 matrix.

The E8 Lie group is the most complicated mathematical structure consisting of 240 vectors in an 8-dimensional space. These vectors are the vertices (corners or dots in Figure 02b) of an 8-dimensional object called the Gosset polytope 421. The diagram is a 2-dimensional representation of the Gosset polytope 421. There are 8 concentric circles of 30 vertices each. The lines in the picture connect adjacent vertices in the polytope, with colors chosen according to the length of the 2-dimensional projection. Since the picture is a 2-dimensional projection of an 8-dimensional object, it captures only some of the symmetries of the Gossett polytope.

E<sub>8</sub> Representations According to the eccentric physicist Garrett Lisi (who has no academic affiliation, spends much of the day either surfing (Summer in Hawaii) or snowboarding (Winter in Colorado), and all the while lives out of a van with a girlfriend), every point in the E8 represents a particle. The scheme would account for all the elementary particles including gravity and much more. This is very similar to the game of numerology by assigning meanings to numbers (Figure 02c). Under the best of circumstances, the scheme cannot be the "Theory of Everything" as claimed since the dynamics of such system is altogether left out.

Figure 02c E8 Representations [view large image]

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