Group Theory in Particle Physics Joshua Albert Phy 205 http://en.wikipedia.org/wiki/Image:E8_graph.svg 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. 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... 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. 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 3: Existence of Identity ● There must exist an identity element I in a group G such that: – T I = I T = T ● For every T in G. 4: Existence of Inverse ● For every element T in G there must exist an inverse element T -1 such that: – 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... Some Finite Group Examples: ● Parity – Representable by {1, -1}, {+,-}, {even, odd} – Clearly an important group in particle physics ● Rotations of an Equilateral Triangle – 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). 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 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. 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. An Example ● A representation (far from the only one) of the equilateral triangle symmetry group is shown below. Lie Groups ● Lie Groups are continuous groups whose elements are described by one or more smooth parameters (differentiable manifold). Sophus Lie, with his beard http://en.wikipedia.org/wiki/Sophus_Lie 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. More Lie Group Examples ● Lorentz Group – Group of all rotations and Lorentz boosts – Parameterized by 3 rotation parameters, 3 boost parameters. ● SO(n) – Group of all orthogonal n x n matrices of determinant 1 ● SU(n) – Group of all unitary n x n matrices with det = 1 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! 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. 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... Flavor! ● Each particle corresponds to one Algebra! Group Theory Success! ● In 1962, Murray Gell-Mann predicted the existence of the W-. ● In 1964, a particle with the predicted properties was discovered. 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? References from the paper.