# Mathematician Emmy Noether changed the face of physics https://www.sciencenews.org/article/emmy-noether-theorem-legacy-physics-math # Symmetry, Algebra and the “Monster”

Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.

https://www.quantamagazine.org/symmetry-algebra-and-the-monster-20170817/

Summary:

• Symmetry of a Square
• Isometry (*) or Rigid Motion (刚体运动) = no change in shape and size after a transformation
• What is a Group (群 “CAN I” ) ? = Closure Associative Neutral Inverse
• Monster Group = God ?
• String Theory: Higgs boson (玻色子) aka “God Particles”

Note (*): “保距映射” (Isometry），是指在度量空间 (metric space) 之中保持距离不变的”同构“关系 (Isomorphism) 。几何学中的对应概念是 “全等变换”

# Group is Symmetry

Landau’s book “Symmetry” explains it as follow:

Automorphism = Congruence= 叠合 has
1). Proper 真叠合 (symmetry: left= left, right = right)
2). Improper 非真叠合 (non-symmetry: reflection: left changed to right, vice-versa).
Congruence => preserve size / length
=> Movement 运动 (translation 平移, rotation about O )
= Proper congruence (Symmetry)

In Space S, the Automorphism that preserves the structure of S forms a Group Aut(G).
=> Group Aut(G) describes the Symmetry of Space S.

Hence Group is the language to describe Symmetry.

# Positron Symmetry

Positron is the anti-matter of Electron, born by symmetry of universe.

It disappears the moment it is created, to neutralise to nothingness.

Application in medicine: ‘PET‘ scanning (P= Positron).
Positron is smaller than electron, so can scan smaller tumor than CT Scan machine.

# Maxwell Equation: Symmetry

Maxwell Equation

Maxwell boldy derived from Faraday experimental results by symmetry to get the Maxwell Equation for Electro-Magnetic Fields:

1) rot E = -1/c ∂H/∂t
div H = 0

By symmetry (swap E <-> H )

2) rot H = +1/c ∂E/∂t
div E = 0

Note:

E = Electric Field
H= Magnetic Field
c = Speed of light

# Noether Theorem: Symmetry

Symmetry (hence Group) explains :
1. Conservation of Energy;
2. Conservation of Angular Momentum;
3. Periodic Table;
4. Laws of Thermodynamic.

Emmy Noether Theorem (1918): Conservation Laws owes to Symmetry :
1. In Linear motion
=> Conservation of Momentum

2. In Angular movement
=> Conservation of Angular Momentum

3. In Time
=> Conservation of Energy