MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

27

In this notation,

c-X-id-X -e-X-i~f-x\__ f c-X-id-X -e-X~if-X\

x

e-X-if-X c-X + id-X ) \e-X-if-X c • X + id • X l

The mapping

(wi, W2, W3, w±) i— W

w\ — IW2 —^3 — IW4

W3 — IW4 w\ + iw2

is an isometric isomorphism of standard Euclidean H 4 into a subspace W of the 4 x 4 complex

matrices endowed with the inner product (Wj., W2) = \ trace (WfW2). Furthermore, the

action of SU(2) x SU(2) on W given by W 1—• W = U2 W U^1 is isometric on this space.

Since SU(2) x SU(2) is connected, it follows that the corresponding action on Euclidean IR4

is given by

(l, w2, w& W4) 1— O i , W2, w3, m) = ( i , w2, ws, w\)Ou^uv

where Ojjlu2 G 50(4). The mapping (C/i, ^2) '—• ^E/1,^2 ls a g r o u P homomorphism. By

explicit calculation, the differential of this map takes the generators of the Lie algebra su{2) x

su(2) onto the generators of the Lie algebra so(4). Thus, the map is a local isomorphism of

the Lie groups. SU{2) x SU{2) is connected and simply connected, and 50(4) is connected.

It follows that the mapping is a covering map, and therefore is surjective (In fact, it is two-

to-one with kernel { (J, / ) , (—i", —/)}.) [47]. If (c d e / ) denotes the 4 x 4 matrix whose

columns are the vectors c, d, e, and / , then we have

(c ~d e ~f) = (c d e f) 0VliUv

where OJJ1JJ2 can be taken to be any element of 50(4) if U\ and U2 are chosen properly.

Our claims are thus proved if we can show that any invertible matrix A = (c d e f) has

the property that it maps some orthonormal basis (the columns of Oux u2) into non-zero,

mutually orthogonal vectors { c, d, e, / } . To show that this is the case, we choose the

orthonormal basis { v\, V2, ^3, U4 } to be an orthonormal basis in which the real symmetric

matrix A* A is diagonal. Such bases always exist, and one can always arrange for OJJX u2 —

(v\ V2 ^3 U4 ) to be in S0(4). Then for i / j , (Av{, AVJ ) = (V{, A*AVJ ) = fij (V{, Vj ) —

0. This proves the claim.