group theory in particle physics
TRANSCRIPT
Group Theory in Particle PhysicsJoshua 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 3vectors under addition● The set of all rotations in 3space
– Can be broken into the set of proper rotations and improper rotations
Abelian vs. NonAbelian● 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 nonabelian group has elements which do not necessarily commute. Of the previous examples, only the rotations in 3space group was nonabelian.
● Most of the really interesting groups are nonabelian.
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 3space
– 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 GellMann matrices.
● Around 1960, physicists struggled to categorize the new particles...
Flavor!
● Each particle corresponds to one Algebra!
Group Theory Success!● In 1962, Murray GellMann predicted the
existence of the ● 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