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/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix 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 : http://www.google.com/profiles/Roger.Bagula alternative email: roger.bagula at gmail.com