Group Theory in Particle Physics

Transcription

1 Group Theory in Particle Physics Joshua Albert Phy 205

2 Where Did it Come From? Group Theory has it's origins in: Algebraic Equations Number Theory Geometry Some major early contributers were Euler, Gauss, Lagrange, Abel, and Galois.

3 What is a group? A group is a collection of objects with an associated operation. The group can be finite or infinite (based on the number of elements in the group. The following four conditions must be satisfied for the set of objects to be a group...

4 1: Closure The group operation must associate any pair of elements T and T' in group G with another element T'' in G. This operation is the group multiplication operation, and so we write: T T' = T'' T, T', T'' all in G. Essentially, the product of any two group elements is another group element.

5 2: Associativity For any T, T', T'' all in G, we must have: (T T') T'' = T (T' T'') Note that this does not imply: T T' = T' T That is commutativity, which is not a fundamental group property

6 3: Existence of Identity There must exist an identity element I in a group G such that: TI=IT=T For every T in G.

7 4: Existence of Inverse For every element T in G there must exist an inverse element T such that: 1 T T 1 = T 1 T = I These four properties can be satisfied by many types of objects, so let's go through some examples...

8 Some Finite Group Examples: Parity Rotations of an Equilateral Triangle Representable by {1, 1}, {+, }, {even, odd} Clearly an important group in particle physics Representable as ordering of vertices: {ABC, ACB, BAC, BCA, CAB, CBA} Can also be broken down into subgroups: proper rotations and improper rotations The Identity alone (smallest possible group).

9 Some infinite Group Examples: The set of integers under addition The set of real numbers under addition or multiplication The set of all real 3 vectors under addition The set of all rotations in 3 space Can be broken into the set of proper rotations and improper rotations

10 Abelian vs. Non Abelian An abelian group is a group where all the group elements commute. That is: T T' = T' T for all T, T' in G A non abelian group has elements which do not necessarily commute. Of the previous examples, only the rotations in 3 space group was non abelian. Most of the really interesting groups are non abelian.

11 Representations To make groups more manageable, and to see relations between group, we can assign each element in a group an n x n matrix to represent it. These matrices must have the same multiplication relations as the original group elements had. It is possible for one group to have more than one representation.

12 An Example A representation (far from the only one) of the equilateral triangle symmetry group is shown below.

13 Lie Groups Lie Groups are continuous groups whose elements are described by one or more smooth parameters (differentiable manifold). Sophus Lie, with his beard

14 Lie Group Examples Rotations in 3 space Described by 3 parameters: Euler Angles are one possibility. Equivalent to the group O(3), of all orthogonal 3 x 3 matrices. Non abelian. Translations in Euclidean Space An abelian group, represented by x, y, z.

15 More Lie Group Examples Lorentz Group SO(n) Group of all rotations and Lorentz boosts Parameterized by 3 rotation parameters, 3 boost parameters. Group of all orthogonal n x n matrices of determinant 1 SU(n) Group of all unitary n x n matrices with det = 1

16 Lie Algebras Lie Algebras are the generators of Lie groups. The Algebras represent what is effectively an infinitesimal transformation. By exponentiating the representations of the algebras, we generate group elements. These Lie Algebras do not necessarily form a group!

17 Example: SU(2) Initially a 2 x 2 complex matrix has 8 degrees of freedom. The conditions UU =I and Det(U)=1 reduce this to three free parameters. The three generators are the Pauli matrices.

18 Approximate SU(3)... SU(3) has generators which can be given by the eight Gell Mann matrices. Around 1960, physicists struggled to categorize the new particles...

19 Flavor! Each particle corresponds to one Algebra!

20 Group Theory Success! In 1962, Murray Gell Mann predicted the existence of the In 1964, a particle with the predicted properties was discovered.

21 And Beyond... The strong force is associated with SU(3), the gluons correspond to SU(3) algebras... The electroweak interaction is united under U(1) SU(2). Perhaps a grander unification?

22 References from the paper