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Topic 5 of 8

Lie Groups

A Lie group is simultaneously a group and a smooth manifold: its elements can be continuously varied, and both the multiplication and inverse maps are smooth. Lie groups describe continuous symmetries in physics — rotations, Lorentz boosts, gauge transformations — and their dimension (the number of continuous parameters) directly determines how many conserved quantities and gauge bosons arise.

Key Concepts

Lie Group

A group

G

that is also a smooth manifold, with group multiplication

G\times G\to G

and inversion

G\to G

both smooth maps. Examples:

SO(n)

SU(n)

U(n)

GL(n,\mathbb{R})

Dimension

The dimension of

G

as a manifold: the number of real parameters needed to specify a group element. Equals the number of independent generators in the Lie algebra.

Compactness

SO(n)

SU(n)

U(n)

are compact (closed and bounded as matrix groups). Compact groups have finite-dimensional unitary representations.

GL(n,\mathbb{R})

is not compact.

Simply Connected

SU(n)

is simply connected (

\pi_1=0

SO(n)

for

n\geq2

is not — its fundamental group is

\mathbb{Z}_2

, which is why spinors require

SU(2)

rather than

SO(3)

Double Cover

There is a 2-to-1 group homomorphism

SU(2) \to SO(3)

. The two elements

\pm I \in SU(2)

both map to the identity in

SO(3)

. Spin-

\frac{1}{2}

particles require the full

SU(2)

Key Equations

Dimensions of Classical Lie Groups

\dim SO(n) = \frac{n(n-1)}{2}, \quad \dim SU(n) = n^2-1, \quad \dim U(n) = n^2

The most-used dimension formulas in particle physics.

Exponential Map

g(\alpha) = \exp\!\left(i\,\alpha_a\, T^a\right) \in G

Every element near the identity is an exponential of a Lie algebra element. The

T^a

are generators and

\alpha_a

are real parameters.

Worked Example

Dimension of $SO(4)$

Problem

How many independent parameters are needed to specify a rotation in 4-dimensional Euclidean space? In other words, what is $\dim SO(4)$ ?

Solution

$SO(n)$ consists of $n\times n$ orthogonal matrices with det $=1$ . The independent degrees of freedom are the number of independent planes of rotation.

Use $\dim SO(n) = n(n-1)/2$ .

\dim SO(4) = \frac{4 \times 3}{2} = 6

Answer

\dim SO(4) = 6

Practice

Exercises

7 problems

1 of 7

What is $\dim SU(4)$ ? Use $\dim SU(n) = n^2 - 1$ .

Your answer

2 of 7

What is $\dim SO(5)$ ? Use $\dim SO(n) = n(n-1)/2$ .

Your answer

3 of 7

What is $\dim U(4)$ ? Use $\dim U(n) = n^2$ .

Unlock Exercise 3