Transcript isacker_a2

Scale invariance
Object of which a detail when enlarged becomes
(approximately) identical to the object itself. Condition of
self-similarity leads to properties defined in fractal
dimensions.
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Examples of scale invariance
Snow flakes
Trees
Lungs
Neurons
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Symmetry in art
Symmetric patterns are present in the artistry of all
peoples.
Symmetry of Ornaments (Speiser, 1927): analysis of
group-theoretic structure of plane patterns.
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Symmetries of patterns
Four (rigid) transformations in a plane:
Reflections
Rotations
Translations
Glide-reflections
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One-dimensional patterns
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Two-dimensional patterns
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Group theory
Group theory is the mathematical theory of symmetry.
Group theory was invented (discovered?) by Evariste
Galois in 1831.
Group theory became one of the pillars of mathematics
(cfr. Klein’s Erlangen programme).
Group theory has become of central importance in
physics, especially in quantum physics.
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The birth of group theory
Are all equations solvable algebraically?
Example of quadratic equation:
b  b 2  4ac
ax  bx  c  0  x1,2 
2a
2

Babylonians (from 2000 BC) knew how to solve quadratic
equations in words but avoided cases with negative or no
solutions.
Indian mathematicians (eg. Brahmagupta 598-670) did
interpret negative solutions as `depths’.
Full solution was given in 12th century by the Spanish Jewish
mathematician Abraham bar Hiyya Ha-nasi.
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The birth of group theory
No solution of higher equations until dal Ferro, Tartaglia,
Cardano and Ferrari solve the cubic and quartic
equations in the 16th century.
ax 3  bx 2  cx  d  0 & ax 4  bx 3  cx 2  dx  e  0

Europe’s finest mathematicians (eg. Euler, Lagrange,
Gauss, Cauchy) attack the quintic equation but no
solution is found.
1799: proof of non-existence of an algebraic solution of
the quintic equation by Ruffini?
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The birth of group theory
1824: Niels Abel shows that general quintic and
higher-order equations have no algebraic
solution.
1831: Evariste Galois answers the solvability
question: whether a given equation of degree n
is algebraically solvable depends on the
‘symmetry profile of its roots’ which can be
defined in terms of a subgroup of the group of
permutations Sn.
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The insolvability of the quintic
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The axioms of group theory
A set G of elements (transformations) with an operation
 which satisfies:
1.
2.
3.
4.
Closure. If g1 and g2 belong to G, then g1g2 also belongs
to G.
Associativity. We always have (g1g2)g3=g1(g2g3).
Existence of identity element. An element 1 exists such
that g1=1g=g for all elements g of G.
Existence of inverse element. For each element g of G, an
inverse element g-1 exists such that gg-1=g-1g=1.
This simple set of axioms leads to an amazingly rich
mathematical structure.
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Example: equilateral triangle
Symmetry transformations are
- Identity
- Rotation over 2/3 and
4/3 around ez
- Reflection with respect to
planes (u1,ez), (u2,ez),
(u3,ez)
Symmetry group: C3h.
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Groups and algebras
1873: Sophus Lie introduces the notion of the
algebra of a continuous group with the aim of
devising a theory of the solvability of
differential equations.
1887: Wilhelm Killing classifies all Lie algebras.
1894: Elie Cartan re-derives Killing’s
classification and notices two exceptional Lie
algebras to be equivalent.
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Lie groups
A Lie group contains an infinite number of elements
characterized by a set of continuous variables.
Additional conditions:
Connection to the identity element.
Analytic multiplication function.
Example: rotations in 2 dimensions, SO(2).
cos sin  
gˆ    

sin  cos

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Lie algebras
Idea: to obtain properties of the infinite number of
elements g of a Lie group in terms of those of a finite
number of elements gi (called generators) of a Lie
algebra.
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

Lie algebras
All properties of a Lie algebra follow from the
commutation relations between its generators:
r
k
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
g
,
g

g

g

g

g

c
 i j  i j j i  ij gˆ k
k1
Generators satisfy the Jacobi identity:
gˆ ,gˆ , gˆ  gˆ ,gˆ , gˆ  gˆ ,gˆ , gˆ  0
i
j
k
j
k
i
k
i
j
Definition of the metric tensor or Killing form:
gij 
r
l k
c
 ikc jl
k,l1
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
Classification of Lie groups
Symmetry groups (of projective spaces) over R, C and
H (quaternions) preserving a specified metric:
x  Rn :
x  Cn :
x  Hn :
xx  1, det  1 SOn
xx  1, det  1 SUn
xx  1 Spn
The five exceptional groups G2, F4, E6, E7 and E8 are
similar constructs over the normed division algebra of
the octonions, O.
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Rotations in 2 dimensions, SO(2)


Matrix representation of finite elements:
cos sin  
gˆ    

sin  cos
Infinitesimal element and generator:
1 0 0 1
lim gˆ    
  
 eˆ  gˆ1


 0
0 1 1 0
Exponentiation leads back to finite elements:
 1 
  n
gˆ    lim eˆ  gˆ1  exp eˆ  gˆ1  exp
 gˆ  

n 
n 
 1
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
Rotations in 3 dimensions, SO(3)
Matrix representation of finite elements:
cos  3 sin  3 0cos 2 0 sin  2 1
0
0 




sin

cos

0
0
1
0
0
cos

sin

3
3
1
1




0
1
 0

sin  2 0 cos  2 

0 sin 1 cos1 

Infinitesimal elements and associated generators:
0 0 0 
0 0 1
0 1 0






gˆ1  0 0 1, gˆ 2  0 0 0, gˆ 3  1 0 0



0 1 0 

1 0 0

0 0 0

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

Rotations in 3 dimensions, SO(3)
Structure constants from matrix multiplication:
3
gˆ k , gˆ l    c klm gˆ m
with
c klm  klm Levi  Civita 
m1
Exponentiation leads back to finite elements:
3


gˆ 1, 2 , 3   expeˆ   k gˆ k 
 k1

Relation with angular momentum operators:
 
3
lˆk  igˆ k  lˆk , lˆl  iklm lˆm
m1
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
Casimir operators
Definition: The Casimir operators Cn[G] of a Lie algebra
G commute with all generators of G.
The quadratic Casimir operator (n=2):
Cˆ 2 G 
r
ij
g
 gˆ igˆ j with
r
3
jk
2
ˆ
ˆ
g
g


.
Ex
:
C
SO
3

l


 k
 ij
ik
2
i, j1
i, j1
k1
The number of independent Casimir operators (rank)
equals the number of quantum numbers needed to
characterize any (irreducible) representation of G.
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Symmetry rules
Symmetry is a universal concept relevant in
mathematics, physics, chemistry, biology, art…
Since its introduction by Galois in 1831, group theory
has become central to the field of mathematics.
Group theory remains an active field of research, (eg.
the recent classification of all groups leading to the
Monster.)
Symmetry has acquired a central role in all domains of
physics.
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