Exercise List #3a

Classical Field Theory

???, 2009

1. Check how the Levi-Civita symbol (defined with 

0123

= 1, 

abcd

= η

ai

η

bj

η

ck

η

dl



ijkl

)

behaves under general Lorentz transformation.

2. Prove the following identities.



ijkl



mnpq

= −










δ

i

m

δ

i

n

δ

i

p

δ

i

q

δ

j

m

δ

j

n

δ

j

p

δ

j

q

δ

k

m

δ

k

n

δ

k

p

δ

k

q

δ

l

m

δ

l

n

δ

l

p

δ

l

q










,

(1)



ijkl



mnpl

= −








δ

i

m

δ

i

n

δ

i

p

δ

j

m

δ

j

n

δ

j

p

δ

k

m

δ

k

n

δ

k

p








,

(2)



ijkl



mnkl

= −2(δ

i

m

δ

j

n

− δ

i

n

δ

j

m

),

(3)



ijkl



mjkl

= −6δ

i

m

,

(4)



ijkl



ijkl

= −24.

(5)