Ausgabe bis 06.07.07, Besprechung 16.07.07

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Quantenfeldtheorie II SS 2007 Prof. Jan Plefka

Ubungsblatt 6 ¨

Ausgabe bis 06.07.07, Besprechung 16.07.07

Aufgabe 1: Non-Abelian gauge fields (10 Punkte)

Consider non-Abelian gauge fields whose field strengths are given by F

_µν

= ∂

_µ

A

_ν

− ∂

_ν

A

_µ

− ig [A

µ

, A

_ν

].

a) Show the following Bianchi identity by a direct calculation ε

^µνλσ

D

_ν

F

_λσ

= 0 .

b) Show that ε

^µνλσ

Tr

F

_µν

F

_λσ

= ∂

_µ

K

^µ

, i.e., is a total derivative.

Aufgabe 2: Adjoint representation (5 Punkte)

Consider a set of Lie algebra generators T

^a

, a = 1 , · · · , dim( G ) whose com- mutation relations are given by

[ T

^a

, T

^b

] = if

^abc

T

^c

. (1) a) Show that the Jacobi identity [T

^a

, [T

^b

, T

^c

]]+ [T

^b

, [T

^c

, T

^a

]]+ [T

^c

, [T

^a

, T

^b

]] = 0 leads to the following algebraic relations between the structure constants:

f

^ade

f

^bcd

+ f

^bde

f

^cad

+ f

^cde

f

^abd

= 0 . (2) b) The representation matrices in the adjoint representation R are given by the structure constants as follows

( T

_R^b

)

ac

= if

^abc

.

Using this definition and Eq.(2), show that T

_R^a

satisfies the Lie algebra (1):

[ T

_R^a

, T

_R^b

] = if

^abc

T

_R^c

.

1

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Aufgabe 3: Gell-Mann matrices: Problem 15.1 in Peskin/Schroeder (5 Punkte) The standard basis for the fundamental representation of SU (3) is

t

¹

= 1 2





0 1 0 1 0 0 0 0 0



 , t

²

= 1 2





0 − i 0

i 0 0

0 0 0



 , t

³

= 1 2





1 0 0

0 − 1 0

0 0 0



 ,

t

⁴

= 1 2





0 0 1 0 0 0 1 0 0



 , t

⁵

= 1 2





0 0 − i 0 0 0 i 0 0



 ,

t

⁶

= 1 2





0 0 0 0 0 1 0 1 0



 , t

⁷

= 1 2





0 0 0 0 0 − i 0 i 0



 , t

⁸

= 1 2 √

3 



1 0 0

0 1 0

0 0 − 2



 .

a) Explain why there are exactly eight matrices in the basis.

b) Evaluate all the commutators of these matrices, to determine the strucu- ture constants of SU (3). Show that, with the normalization used here, f