232
8. Elementary Processes in Scalar and Spinor Electrodynamics
where we have made the matrix indices explicit, and summation on all repeated
matrix indices is understood. In particular, note that every matrix index is
repeated, so that each one is in fact summed over: there are no ‘spare’ indices.
Now, since we can reorder matrix elements as we wish, we can bring the u δ
to the front of the expression, and use the same trick to perform the second
spin sum:
∑
′
′
′
u δ (k , s
′ )¯ u α (k , s
′ ) = (k / + m) δα .
(8.64)
s '
Thus L
μν takes the form of a matrix product, summed over the diagonal
elements:
′
L
μν
=
=
1
2
1
2
(k / + m) δα (γ
μ ) αβ (k / + m) βγ (γ
ν ) γδ
(8.65)
∑ ′
[(k / + m)γ
μ (k / + m)γ
ν ] δδ
(8.66)
δ
where we have explicitly reinstated the sum over δ. The right-hand side of
(8.66) is the trace (i.e. the sum of the diagonal elements) of the matrix formed
by the product of the four indicated matrices:
′
L
μν =
1
2 Tr[(k / + m)γ
μ (k / + m)γ
ν ].
(8.67)
Such matrix traces have some useful properties which we now list. Denote
the trace of a matrix A by
∑
TrA =
A ii .
(8.68)
i
Consider now the trace of a matrix product,
∑
Tr(AB) =
A ij B ji
(8.69)
i,j
where we have written the summations in explicitly. We can (as before) freely
exchange the order of the matrix elements A ij and B ji , to rewrite (8.69) as
∑
Tr(AB) =
B ji A ij .
(8.70)
i,j
But the right-hand side is precisely Tr(BA); hence we have shown that
Tr(AB) = Tr(BA).
(8.71)
Similarly it is easy to show that
Tr(ABC) = Tr(CAB).
(8.72)
We may now return to (8.67). The advantage of the trace form is that we
can invoke some powerful results about traces of products of γ-matrices. Here
8. Elementary Processes in Scalar and Spinor Electrodynamics
where we have made the matrix indices explicit, and summation on all repeated
matrix indices is understood. In particular, note that every matrix index is
repeated, so that each one is in fact summed over: there are no ‘spare’ indices.
Now, since we can reorder matrix elements as we wish, we can bring the u δ
to the front of the expression, and use the same trick to perform the second
spin sum:
∑
′
′
′
u δ (k , s
′ )¯ u α (k , s
′ ) = (k / + m) δα .
(8.64)
s '
Thus L
μν takes the form of a matrix product, summed over the diagonal
elements:
′
L
μν
=
=
1
2
1
2
(k / + m) δα (γ
μ ) αβ (k / + m) βγ (γ
ν ) γδ
(8.65)
∑ ′
[(k / + m)γ
μ (k / + m)γ
ν ] δδ
(8.66)
δ
where we have explicitly reinstated the sum over δ. The right-hand side of
(8.66) is the trace (i.e. the sum of the diagonal elements) of the matrix formed
by the product of the four indicated matrices:
′
L
μν =
1
2 Tr[(k / + m)γ
μ (k / + m)γ
ν ].
(8.67)
Such matrix traces have some useful properties which we now list. Denote
the trace of a matrix A by
∑
TrA =
A ii .
(8.68)
i
Consider now the trace of a matrix product,
∑
Tr(AB) =
A ij B ji
(8.69)
i,j
where we have written the summations in explicitly. We can (as before) freely
exchange the order of the matrix elements A ij and B ji , to rewrite (8.69) as
∑
Tr(AB) =
B ji A ij .
(8.70)
i,j
But the right-hand side is precisely Tr(BA); hence we have shown that
Tr(AB) = Tr(BA).
(8.71)
Similarly it is easy to show that
Tr(ABC) = Tr(CAB).
(8.72)
We may now return to (8.67). The advantage of the trace form is that we
can invoke some powerful results about traces of products of γ-matrices. Here