Determinants and Their Applications in Mathematical Physics

266 6. Applications of Determinants in Mathematical Physics

c. Dt(φij)=φi 0 φj 2 −φi 1 φj 1 +φi 2 φj 0 −

1

2

(φi+3,j+φi,j+3).

Proof. Putfr=−gr=b

2

r

in identity (D).

(b

2

i−b

2

j)A

ij

=

∑

r

∑

s

[

δrs(b

2

r−b

2

s)er+2(br−bs)

]

A

is

A

rj

=0+2

∑

r

∑

s

(br−bs)A

is

A

rj

=2

∑

s

A

is

∑

r

brA

rj

− 2

∑

r

A

rj

∑

s

bsA

is

=2(ψ 0 iψ 1 j−ψ 0 jψ 1 i).

It follows that if

Fij=2ψ 0 iψ 1 j−b

2

i

A

ij

,

then

Fji=Fij.

Furthermore, ifGijis any function with the property

Gji=−Gij,

then

∑

i

∑

j

GijFij=0. (6.7.17)

The proof is trivial and is obtained by interchanging the dummy suffixes.

The proof of (a) can now be obtained by expanding the quadruple series

S=

∑

p,q,r,s

(b

i

pb

j

r−b

j

pb

i

r)bsA

pq

A

rs

in two different ways and equating the results.

S=

∑

p,q

b

i

pA

pq

∑

r,s

b

j

rbsA

rs

−

∑

p,q

b

j

pA

pq

∑

r,s

b

i

rbsA

rs

=φi 0 φj 1 −φj 0 φi 1 ,

which is identical to the left side of (a). Also, referring to (6.7.17) with

i, j→p, r,

S=

∑

p,r

(b

i

pb

j

r−b

j

pb

i

r)

∑

q

A

pq

∑

s

bsA

rs

=

∑

p,r

(b

i

p

b

j

r

−b

j

p

b

i

r

)ψ 0 pψ 1 r

=

1

2

∑

p,r

(b

i

p

b

j

r

−b

j

p

b

i

r

)(Fpr+b

2

p

A

pr

)