three kinds of Euler angular unitary matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg117383] three kinds of Euler angular unitary matrices
• From: Roger Bagula <roger.bagula at gmail.com>
• Date: Wed, 16 Mar 2011 06:31:07 -0500 (EST)

```These were inspired by the Cabbibo CKM matrix of flavor mixing.
They are a subset of SL(3) as unitary rotation matrix sets of three.

http://en.wikipedia.org/wiki/Neutrino_oscillations

(*Elliptic:(Real)*)
Clear[s, a]
s[1] = {{Cos[a[1]], Sin[a[1]], 0},
{-Sin[a[1]], Cos[a[1]], 0},
{0, 0, 1}};
s[2] = {{Cos[a[2]], 0, Sin[a[2]]},
{0, 1, 0},
{-Sin[a[2]], 0, Cos[a[2]]}};
s[3] = {{1, 0, 0},
{0, Cos[a[3]], Sin[a[3]]},
{0, -Sin[a[3]], Cos[a[3]]}};
ExpandAll[s[1].s[2].s[3]]
FullSimplify[%]
Table[FullSimplify[Det[s[i]]], {i, 1, 3}]

(*hyperbolic 1 :*(Real))
Clear[s, a]
s[1] = {{Cosh[a[1]], Sinh[a[1]], 0},
{Sinh[a[1]], Cosh[a[1]], 0},
{0, 0, 1}};
s[2] = {{Cosh[a[2]], 0, Sinh[a[2]]},
{0, 1, 0},
{Sinh[a[2]], 0, Cosh[a[2]]}};
s[3] = {{1, 0, 0},
{0, Cosh[a[3]], Sinh[a[3]]},
{0, Sinh[a[3]], Cosh[a[3]]}};
ExpandAll[s[1].s[2].s[3]]
FullSimplify[%]
Table[FullSimplify[Det[s[i]]], {i, 1, 3}]

(*hyperbolic 2 :Complex)*)
Clear[s, a]
s[1] = {{Cosh[a[1]], I*Sinh[a[1]], 0},
{-I*Sinh[a[1]], Cosh[a[1]], 0},
{0, 0, 1}};
s[2] = {{Cosh[a[2]], 0, I*Sinh[a[2]]},
{0, 1, 0},
{-I*Sinh[a[2]], 0, Cosh[a[2]]}};
s[3] = {{1, 0, 0},
{0, Cosh[a[3]], I*Sinh[a[3]]},
{0, -I*Sinh[a[3]], Cosh[a[3]]}};
ExpandAll[s[1].s[2].s[3]]
FullSimplify[%]
Table[FullSimplify[Det[s[i]]], {i, 1, 3}]

The result isn't the classical three mixing angles of Euler,
but 9 angles total.
They form a unitary group in analogy to U(1)*SU(3).

The problem is to find the Real {a[1],a[2],a[3]} angle set for:
M={{0.97428, 0.2253, 0.00347},
{0.2252, 0.97345, 0.0410},
{0.00862, 0.0403, 0.999152}}
In any of the sets of three as the Cabbibo angle set
for quark interactions.

Clear[s, a, m0, m, t]
s[1] = {{Cos[a[1]], Sin[a[1]], 0},
{-Sin[a[1]], Cos[a[1]], 0},
{0, 0, 1}};
s[2] = {{Cos[a[2]], 0, Sin[a[2]]},
{0, 1, 0},
{-Sin[a[2]], 0, Cos[a[2]]}};
s[3] = {{1, 0, 0},
{0, Cos[a[3]], Sin[a[3]]},
{0, -Sin[a[3]], Cos[a[3]]}};
m0 = s[1].s[2].s[3];
m = {{0.97428, 0.2253, 0.00347},
{0.2252, 0.97345, 0.0410},
{0.00862, 0.0403, 0.999152}};
t = m0 - m;
Flatten[Table[t[[n, m]] == 0, {n, 1, 3}, {m, 1, 3}]]
Solve[Flatten[Table[t[[n, m]] == 0, {
n, 1, 3}, {m, 1, 3}]], {a[1], a[2], a[3]}]

Respectfully, Roger L. Bagula
11759 Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :