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18 November 7, 2013
which yields the result
e = 5.3843836 10 −14 kg1/2 m 3/2
sec
.
0.3.2 Conventions
By convention, the Minkowski metric has the form 13
g µν = g µν = diag(1, −1, −1, −1)
and the totally antisymmetric Levi-Civita symbol is defined by
This implies the following identities :
ɛ 0123 = + 1 hence ɛ 0123 = − 1 .
ɛ µ1µ 2µ 3µ 4
ɛ ν1ν2ν3ν4 = − ∑
(α 1 , α 2 , α 3 , α 4 ) =
P(µ 1 , µ 2 , µ 3 , µ 4 )
ɛ µ1µ 2µ 3µ 4
ɛ µ1ν2ν3ν4 = − ∑
(α 2 , α 3 , α 4 ) =
P(µ 2 , µ 3 , µ 4 )
ɛ µ1µ 2µ 3µ 4
ɛ µ1µ2ν3ν4 = −2 ∑
(α 3 , α 4 ) =
P(µ 3 , µ 4 )
ɛ µ1µ 2µ 3µ 4
ɛ µ1µ2µ3ν4 = −6 δ ν4 µ 4
,
ɛ µ1µ 2µ 3µ 4
ɛ µ1µ2µ3µ4 = −24 ,
δ ν1 α 1
δ ν2 α 2
δ ν3 α 3
δ ν4 α 4
,
δ ν2 α 2
δ ν3 α 3
δ ν4 α 4
,
δ ν3 α 3
δ ν4 α 4
,
where P stands for all signed permutations 14 of the arguments, and where the
Kronecker symbol is defined by
δ α µ =
{
1 if α = µ
0 if α ≠ µ
.
A subtlety : the contravariant partial derivative contains a somewhat surprising
minus sign :
∂ µ =
∂ ( )
1 ∂
=
∂x µ c ∂t , −⃗ ∇ . (1)
This explains why in nonrelativistic quantum mechanics the momentum operator
is ⃗p = −i¯h ⃗ ∇ whereas in the relativistic theory we use p µ = i¯h ∂ µ .
13 In many textbooks the metric tensor is introduced as a diagonal matrix. This is of course
misleading since the covariant metric tensor has only lower indices, whereas a matrix has one
upper and one lower index. Unfortunately, the ‘correct’ matrix form of the metric, which
would be g µ ν , equals the identity matrix whatever the metric !
14 Even permutations occur with a +, and odd permutations with a – sign.