Quantum Mechanics for Mathematicians

between quaternionsHand a space of 2 by 2 complex matrices, and work with
matrix multiplication and complex numbers. The Pauli matrices can be used
to give such an isomorphism, taking

1 → 1 =

(

1 0

0 1

)

, i→−iσ 1 =

(

0 −i

−i 0

)

, j→−iσ 2 =

(

0 − 1

1 0

)

k→−iσ 3 =

(

−i 0

0 i

)

The correspondence betweenHand 2 by 2 complex matrices is then given
by

q=q 0 +q 1 i+q 2 j+q 3 k↔

(

q 0 −iq 3 −q 2 −iq 1

q 2 −iq 1 q 0 +iq 3

)

Since

det

(

q 0 −iq 3 −q 2 −iq 1

q 2 −iq 1 q 0 +iq 3

)

=q^20 +q^21 +q^22 +q 32

we see that the length-squared function on quaternions corresponds to the de-
terminant function on 2 by 2 complex matrices. Takingq∈Sp(1), so of length
one, the corresponding complex matrix is inSU(2).
Under this identification ofHwith 2 by 2 complex matrices, we have an
identification of Lie algebrassp(1) =su(2) between pure imaginary quaternions
and skew-Hermitian trace-zero 2 by 2 complex matrices

~w=w 1 i+w 2 j+w 3 k↔

(

−iw 3 −w 2 −iw 1

w 2 −iw 1 iw 3

)

=−iw·σ

The basis 2 i, 2 j,k 2 ofsp(1) gets identified with a basis for the Lie algebra
su(2) which written in terms of the Pauli matrices is

Xj=−i

σj

2

with theXjsatisfying the commutation relations

[X 1 ,X 2 ] =X 3 ,[X 2 ,X 3 ] =X 1 ,[X 3 ,X 1 ] =X 2

which are precisely the same commutation relations as forso(3)

[l 1 ,l 2 ] =l 3 ,[l 2 ,l 3 ] =l 1 ,[l 3 ,l 1 ] =l 2

We now have three isomorphic Lie algebrassp(1) =su(2) = so(3), with
elements that get identified as follows

(

w 12 i+w 22 j+w 3 k 2

)

↔−

i

2

(

w 3 w 1 −iw 2

w 1 +iw 2 −w 3

)

↔





0 −w 3 w 2

w 3 0 −w 1

−w 2 w 1 0



