PH56 PH563 3 - Gr Grou oup Theo p Theory Meth ry Methods ods - - PowerPoint PPT Presentation

PH56 PH563 3 - Gr Grou

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p Theory Meth ry Methods

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References

1)

Group Theory by Hammermesh

2)

Lie algebra methods in Particle Physics by Georgi

3)

Group Theory for Physicists by Ramadevi & Dubey (to appear soon)

4)

P.Ramadevi, ramadevi@iitb.ac.in Institute Chair Professor, Department of Physics, IIT Bombay Room No- 209F, Extn – 7563

In-Sem + End Sem evaluation

 10 marks Quiz in the fourth week of August  20 marks for assignments  30 marks mid-sem  40 marks end-sem  80% attendance compulsory

Syllabus and Plan

 Discrete Groups (First half of the sem)

Cyclic groups, Permutation groups, Point groups,irreducible representations, Great

• rthogonality theorem, character tables, application

in solid state physics



Continuous Groups (2nd half )

Space translation, time translational, rotational symmetries, Introduction to Lie algebras& Groups- SU(2),SU(3), Lorentz group, applications

What is symmetry

 Take a square and a circle

Rotate by 90 a b d a b d Rotate by any angle c c Continous symmetry Discrete symm

Definition

 What is a Group G:

set : {a,b,c,d…} + group operation

 Satisfying 4 properties:

(1) Closure (2) Identity element (3) Inverse element (4) associative

 Abelian Group (Commutative)  Subgroup , Multiplication table

.

Examples

 Example 1. The set of all integers Z is a group if the group product

is taken to be the usual addition of integers. This group is clearly abelian and has an infinite number of elements.

 Example 2. The set of all complex numbers C is a group under

addition of complex numbers. This group again is abelian and infinite.

 Example 3. The set C − {0} is an infinite abelian group under the

usual multiplication of complex numbers.

 Example 4. The set of all 2×2 matrices with complex entries is an

infinite abelian group under matrix addition.

 What is the nature of the group if matrix multiplication replaces

matrix addition in example 4?

Finite group

 Group with finite number of elements is called finite group.  Order of a group: Number of elements denoted by |G|  Subgroup: subset satisfying all the four axioms of group  Generators: subset of elements whose finite powers give group elements-

(i) cyclic group (ii) symmetric group

generators

 Suppose a group G is generated using generators

a,b,c,d. This means all possible words of various powers of these generators are elements of G

 Cyclic group elements are given by powers of one

• generator. The group is abelian. Order of the generator

is also the order of the cyclic group.

 Last lecture, we discussed a group of order 4

generated using two generators each of order 2 as well as commuting.

Generators of Klein group

a, b are generators satisfying a2=b2=e; ab=ba

Multiplication table

Find another group again generated using a,b but satisfying: a2 = b3 =e ; ab= b2 a- Is this group abelian or non-abelian?

Symmetric group

Subgroups of this Group?

Subgroups

 e and G are trivial subgroups  In the symmetric group, there are four cyclic subgroups H1 = {e,a} ; H2 = {e,b, b2}

;H3 = {e, ab} ; H4 ={e, ab2)

 If a is any element of G and H is a subgroup of G, then Ha is a subset of elements

in G. We call these subsets as left coset of subgroup G. Similarly a H will be right coset.

 Left coset of e is the subgroup H itself  G = H U Ha U….(Disjoint union of left cosets)

 Lagrange’s Theorem- |H| divides |G|

Conjugate groups

 Let H be non-trivial subgroup of G  Then a H a^-1 will also be a subgroup conjugate to H  If aH a-1= H for all choices of a, then H is called

normal subgroup or invariant subgroup.

 The left coset of a subgroup will be same as right coset

• f the subgroup if H is a normal subgroup

 Find the conjugate subgroups of the symmetric group  The set of cosets of a normal subgroup is called factor

group.

Conjugacy class

Order of g and order of its conjugate element g’ will be same. For the symmetric group, the elements can be disjoint union of three conjugacy classes:

Quaternion group

 Generators are i, j, k, s such that  i2=j2=k2= ijk = s ; s2=e  Find the group elements of such a group  Also decompose the group elements into conjugacy class

Dihedral group

 Dn has 2n elements  Generators are r whose order is n and s whose order is 2 . That

is., rn = s2= e

 Further sr = r-1 s  Find the elements, subgroups, normal subgroups

Homomorphism

 Map between two groups  Kernel  Examples

Symmetric Group

 Various ways of writing permutations of n objects  Group structure  Classes and number of elements in the class

SymmetricGroup

In cycle form,

• dd

Order?

• Subgroups. Cayley’s

theorem

 Every group G of order n is isomorphic with a

subgroup H (known as permutation group) of symmetric group

 We know order 4 groups  Find the isomorphism?

class

 Cycle form doesnot change under conjugation  Class will have all elements with same cycle structure  No. of elements with same cycle structure

Platonic Solids

F = 4, E = 6 V = 4 (χ = 2) F = 6, E = 12 V = 8 (χ = 2)

Rotation axes of cube