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f' = e

where

If f and f ' are decomposed into the real and imaginary parts: f = f

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

f '

f '

which is in the same form as Eqs.(1a,b) for the 2-dimensional rotation, but in the (f_{R}, f_{I}) 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, f_{R} and f_{I} 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.
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## 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). See next section for SU(2), SU(3) groups containing more particle field components. The number of phase angles corresponds to the number of gauge bosons. |

Two successive transformations such as: f f ' f " ---------- (7)

where f " = e

f " = e

with

In general a Lie Group is specified by a finite set of continuous parameters

where the i is included so that if the representation D(

D() 1 + i

If we apply this operation n times with

D(

which is very similar to the U(1) group transformation in Eq.(3) with

The Lie algebra is defined by the commutator relation:

[

where f

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 a_{i}, 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
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## Figure 02a Sophus Lie and Symmetry [view large image] |
## Figure 02b [view large image] E |
vector spaces, hence the 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. |

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 E_{8} 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.
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## Figure 02c E |

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