Determinants and Their Applications in Mathematical Physics

196 5. Further Determinant Theory

∑

r,s

†

crcsE

rs

=−

∑

r,s

†

(c

2

r+c

2

s)E

rs

=−

1

3

∑

r,s,t

E

rst,rst

, (5.4.25)

∑

r,s

crcsE

rs

=

∑

r

c

2

rE

rr

−

1

3

∑

r,s,t

E

rst,rst

, (5.4.26)

∑

r,s

(c

2

r+c

2

s)E

rs

=2

∑

r

c

2

rE

rr

+

1

3

∑

r,s,t

E

rst,rst

, (5.4.27)

∑

r,s

†

c

2

rE

rs

=

1

6

∑

r,s,t

E

rst,rst

+

∑

r,s

crE

rs,rs

, (5.4.28)

∑

r,s

†

c

2

sE

rs

=

1

6

∑

r,s,t

E

rst,rst

−

∑

r,s

crE

rs,rs

. (5.4.29)

To prove (5.4.20), apply the second equation of (5.4.4) and (5.4.16).

Epr,ps=

∂Ers

∂xp

.

Multiply by (cr−cs) and sum overrands:

∑

r,s

(cr−cs)Epr,ps=

∂

∂xp

∑

r,s

(cr−cs)Ers

=

∂

∂xp

∑

r,s

Ers,rs

=

∑

r,s

Eprs,prs,

which is equivalent to (5.4.20). The application of the fifth equation in

(5.4.4) with the modification (i, p, r, q, s)→(u, r, t, s, t) to (5.4.20) yields

(5.4.21).

To prove (5.4.22), sum (5.4.11) overiandj, change the dummy variables

as indicated

∑

i,j

(c

2

i−c

2

j)E

ij

=F−G

where, referring to (5.4.6) and (5.4.7),

F=





∑

i,s

E

is









∑

r,j

crE

rj



+





∑

r,j

E

rj









∑

i,s

csE

is





=

∑

i

E

ii









∑

r,j

(j→s)

crE

rj

+

∑

i,s

(i→r)

csE

is







