Mathematical Physics Vol 1

Chapter 4. Field theory

150

Let us first determine the coefficients α 1 , β 1 , γ 1 . By multiplying equation (4.180) alternately with i , j , k , we obtain

i ′ · i = α 1 = ℓ 11 , i ′ · j = β 1 = ℓ 12 , i ′ · k = γ 1 = ℓ 13 . We have thus proved the first required equation i ′ = ℓ 11 i + ℓ 12 j + ℓ 13 k .

The remaining two equations can be proved analogously. Note that for a clearer representation of these relations, the dependencies can be tabulated. i 1 j 2 k 3 i ′ 1 ℓ 11 ℓ 12 ℓ 13 j ′ 2 ℓ 21 ℓ 22 ℓ 23 k ′ 3 ℓ 31 ℓ 32 ℓ 33

Exercise 78

Prove that

ℓ pm ℓ pn =

3 ∑ p = 1

1 , 0 ,

for m = n , for m̸ = n ,

m and n take the values 1 , 2 , 3.

Solution Similarly as in Example 77, it can be proved that

i = ℓ 11 i ′ + ℓ 21 j ′ + ℓ 31 k ′ , j = ℓ 12 i ′ + ℓ 22 j ′ + ℓ 32 k ′ , k = ℓ 13 i ′ + ℓ 23 j ′ + ℓ 33 k ′ ,

and it thus follows that

i · i = 1 =( ℓ 11 i ′ + ℓ 21 j ′ + ℓ 31 k ′ ) · ( ℓ 11 i ′ + ℓ 21 j ′ + ℓ 31 k ′ )= = ℓ 2 11 + ℓ 2 21 + ℓ 2 31 = 3 ∑ p = 1 ℓ p 1 ℓ p 1 , i · j = 0 =( ℓ 11 i ′ + ℓ 21 j ′ + ℓ 31 k ′ ) · ( ℓ 12 i ′ + ℓ 22 j ′ + ℓ 32 k ′ )= = ℓ 11 ℓ 12 + ℓ 21 ℓ 22 + ℓ 13 ℓ 32 = 3 ∑ p = 1 ℓ p 1 ℓ p 2 , i · k = 0 =( ℓ 11 i ′ + ℓ 21 j ′ + ℓ 31 k ′ ) · ( ℓ 13 i ′ + ℓ 23 j ′ + ℓ 33 k ′ )= = ℓ 11 ℓ 13 + ℓ 21 ℓ 23 + ℓ 31 ℓ 33 = 3 ∑ p = 1 ℓ p 1 ℓ p 3 .