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Group Theory and Its Application to Particle Physics


Abelian Groups
Non-Abelian Groups
Lie Groups
Unitary Groups
Local Gauge Symmetry
Global Gauge Symmetry

Abelian Groups

    A group is defined as a collection of elements (a set), which are labelled a, b, c, ... and so on, and which are related to one another by the following rules:

  • If a and b are both members of the group G, then their product, c = ab is also a member of the group G.
  • The process is associative, i.e., a(bc) = (ab)c.
  • There must be an element, called the unit element and usually denoted by I, defined so that
    aI = a, bI = b, and so on for all elements in the group.
  • Each element has an inverse, written as a-1, b-1 and so on, defined so that aa-1 = I and so on.

Figure 01a Parity Operation
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A group for which ab = ba is an Abelian group. The set of ordinary integer numbers ( ... -3, -2, -1, 0, 1, 2, 3, ...) under "addition" is a simple example of an Abelian group, where "0" is the unit element and the inverse is the same number with opposite sign.
Abelian 2D Rotation Another example is the two-dimensional rotation shown in Figure 01b. It consists of an infinite number of elements in the form of continuously varying parmeter, and is known as continuous group or Lie group. The parameter in this case is the angular displacement c, which is the sum of the rotations a and b. The unit element is a rotation of 0o, and the inverse is a rotation in the opposite direction. For better comprehension of the mathematical formula, Figure 01c shows the coordinate system rotates by an angle a about the origin O instead of rotating the point P. The transformation formula between (x,y) and (x',y') can be written as:

Figure 01b 2-D Rotation
[view large image]

Figure 01c Coordinate Rotation [view large image]

x' = [cos()] x + [sin()] y ---------- (1a)
y' = -[sin()] x + [cos()] y ---------- (1b) or in matrix notation :
----- (1c)

The fact that the laws of physics are unchanged by circular motion, implies conservation of the z component of angular momentum. In general, whenever the physical law is invariant under the operation of a symmetry group, there must be some conserved quantity associated with that operation. This is sometimes referred to as Noether's theorem, and is a useful feature of group theory, which can be used to provide physical insights into the behavior of interactions and particles such as whether the process is permissible or if an entity is missing without knowing the detailed dynamic. When this rotation group is generalized to an equivalent complex space, and applied to the "internal space" in quantum electro-dynamics, the global operation (same amount of rotation everywhere) leads to the conservation of electric charge; while the local operation (amount of rotation depends on location) associates a gauge boson (the photon) with the electromagnetic interaction. The rotation group in two-dimensional real space is called SO(2); while the one with the complex number is called U(1). Further detail about the two dimensional rotation is provided in the followings :

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