PHY 4460 RELATIVITY

1
Lecture 9
Gauge Theory IIICurvature Change of basis
2
Contents

• Connection and curvature
• Curvature in non-Abelian gauge theory
• Change of basis

3
Connection and curvature
4
B

• connects V(x) for x along a path

5
B

• If the connection is independent of path,
• then all the V(x) can be patched together
• consistently, V(x) V. No complication

• If not, there is curvature.

6
Equivalent statement

• Does parallel transport around a closed loop
  reproduce the original vector?

7
Check for rectangular loop
8
(No Transcript)
9
(No Transcript)
10
N N matrix

• R has 4 indices
• ? 2 like i , j
• ? 2 like µ,

11
Back to EM
zero
12
(No Transcript)
13
(No Transcript)
14

• Vector
• potential
• Connection
• 1-form

physics math (geometry)

• EM field
• curvature

15

• There is nontrival E , B
• The complex planes
• fail to patch together
• trivially

16
Non-Abelian gaugetheory
17
Example SU(N)Say N3 , color
18
(No Transcript)
19

• N N anti-hermitian matrix

Recall EM case
Now
20
There are N2-1 photonsIn this case call gluons
21
Curvature
22
(No Transcript)
23
The field is nonlinear in the potential
24
g
g2
25

• Non-Abelian gauge theory is a
• non-linear, interacting theory
• Gluons interact with each other
• No principle of superposition

26
Change of basis (e.g. rotation)
27
Change of basis

• Connection
• Curvature
• Check consistency

28
Connection
29
What if we change basis
30
(No Transcript)
31
(No Transcript)
32
Connection transforms nonlinearly
nonlinear
Because connection relates two points
33
Case of EM
34

• This is just gauge
• transformation of the
• potential in EM

35
Curvature
Linear transformation relates to closed loop
36
Check

• Take transformation of connection
• Directly derive transformation of curvature