Re: Quaternion problem--> Gell-Man version as dual quaternion
- To: mathgroup at smc.vnet.net
- Subject: [mg68132] Re: Quaternion problem--> Gell-Man version as dual quaternion
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Tue, 25 Jul 2006 04:01:32 -0400 (EDT)
- References: <e7ap5q$952$1@smc.vnet.net> <e7lf1d$3l9$1@smc.vnet.net> <e7no3j$97f$1@smc.vnet.net> <e9ri0i$pa1$1@smc.vnet.net> <e9sukq$rl6$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
I made up an SU(3) quantum Rodigues that appears to work too...
Clear[a, m, m1, qr, dqr, qrs, dqrs, v, w, x, y, z, t, q, i, j, k]
s[1] = {{1, 0, 0}, {0, -1, 0}, {0, 0, 0}}
s[2] = {{1, 0, 0}, {0, 1, 0}, {0, 0, -2}}/Sqrt[3]
s[3] = {{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}
s[4] = {{0, -I, 0}, {I, 0, 0}, {0, 0, 0}}
s[5] = {{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}
s[6] = {{0, 0, I}, {0, 0, 0}, {-I, 0, 0}}
s[7] = {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}
s[8] = {{0, 0, 0}, {0, 0, I}, {0, -I, 0}}
q[t_, x_, y_, z_, t1_, x1_, y1_, z1_] := s[1]*t + x*
s[3] + y*s[5] + s[7]*z + s[1]*t1 + s[3]*s1 + s[5]*y1 + s[7]*z1;
(* SU(3) as Lorentzian Rodrigues dual quaternion : quantum number
functional
definition : r0 radial distance/ Bessel - like,
r, r1 time angle, {t, p, t1, p1} polar sphere angles*)
qr = Exp[l*r0]*q[Cos[m*r],
Sin[m*r]*Sin[n*p]*Cos[o*t], Sin[m*r]*Sin[n*p]*Sin[0*t], Sin[
m*r]*Cos[n*p], Cos[m*r],
Sin[m1*r1]*Sin[n1*p1]*Cos[o1*t1], Sin[m1*r1]*Sin[n1*p1]*Sin[01*t1], Sin[m1*
r1]*Cos[n1*p1]];
qrs = Exp[l*r0]*q[Cos[m*r], -Sin[m*r]*Sin[n*p]*Cos[o*t], -Sin[m*r]*Sin[
n*p]*Sin[0*t], -Sin[m*r]*Cos[n*p],
Cos[m1*r1], -
Sin[m1*r1]*Sin[n1*p1]*Cos[o1*t1],
-Sin[m1*r1]*Sin[n1*p1]*Sin[01*t1], -
Sin[m1*r1]*Cos[n1*p1]];
(* Linear vector differential definition : r0 used twice to get 8 variable*)
v = {r0, r, p, t, r0, r1, p1, t1};
(* space - time polar differential factors defined*)
w = {1/Exp[2*r0],
1/Exp[2*r0], 1/(Exp[2*r0]*Sin[t]^2*Sin[r]^2),
1/(Exp[2*r0]*Sin[r]^2),
1/Exp[2*r0], 1/Exp[2*r0], 1/(Exp[2*r0]*Sin[t1]^2*
Sin[r1]^2), 1/(Exp[2*r0]*Sin[r1]^2)};
dqr = Table[FullSimplify[w[[i]]*D[qr, {v[[i]], 2}]], {i, 1, 8}]
dqrs = Table[FullSimplify[w[[i]]*D[qrs, {v[[i]], 2}]], {i, 1, 8}]
Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]], {j, 1, 8}]
Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]*MatrixPower[qr.qrs, -1]], {
j, 1, 8}]
Roger Bagula wrote:
>I did Jacobians and metrics way back in Mathematica using
>a Jacobian matrix form from Theoretical Mechanics ,Ames and Murnagham,
>Dover Books.
>I stuck the Rodrigues formula in that to get both the correct
>differential factors
>and the metric. I just had a feeling that something was wrong with the
>equations this morning!
>
>Right factors are:
>w = {1/Exp[2*r0], 1/Exp[2*r0], 1/(Exp[2*r0]*Sin[r]^2*Sin[t]^2),
>1/(Exp[2*r0]*Sin[d]^2)};
>
>It still gives the Klein-Gorden plate q^4 quantum result with angular
>terms for the y and z variables.
>Suppose we think of this as the Weinberg U(1)*SU(2) Weak -Electromagnetic
>as an Quaternion equation, then we have four quantum numbers in
>Leptons in the quaternion shape.
>Mathematica:
>(* Rodrigues quaternion as a four space Jacobian and metric*)
>(*The hard way: there has got to be a more compact way of doing this!*)
>Exp[r]*{Cos[d], Sin[d]*Sin[p]*Cos[t], Sin[d]*Sin[p]*Sin[t], Sin[d]*Cos[p]}
>x = Exp[r]*Sin[d]*Sin[p]*Cos[t]
>y = Exp[r]*Sin[d]*Sin[p]*Sin[t]
>z = Exp[r]*Sin[d]*Cos[p]
>t0 = Exp[r]*Cos[d]
>xr = D[x, r]
>yr = D[y, r]
>zr = D[z, r]
>tr = D[t0, r]
>xt = D[x, t]
>yt = D[y, t]
>zt = D[z, t]
>tt = D[t0, t]
>xp = D[x, p]
>yp = D[y, p]
>zp = D[z, p]
>tp = D[t0, p]
>xd = D[x, d]
>yd = D[y, d]
>td = D[t0, d]
>h1 = FullSimplify[1/Sqrt[xr^2 + yr^2 + zr^2 + tr^2]]
>h2 = FullSimplify[1/Sqrt[xt^2 + yt^2 + zt^2 + tt^2]]
>h3 = FullSimplify[1/Sqrt[xp^2 + yp^2 + zp^2 + tp^2]]
>h4 = FullSimplify[1/Sqrt[xd^2 + yd^2 + zd^2 + td^2]]
>(*4d metric defined*)
>ds2 = FullSimplify[ExpandAll[dr2/h1^2 + dt2/h2^2 + dp2/h3^2 - dt02/h4^2]]
>rra = Simplify[Expand[1/h1^2]]
>tta = Simplify[Expand[1/h2^2]]
>ppa = Simplify[Expand[1/h3^2]]
>dda = Simplify[Expand[1/h4^2]]
>
>(* Corrected factors for the differential equation from Jacobian factors*)
>Clear[a, m, m1, qr, dqr, qrs, dqrs, v, w, x, y, z, t, q, i, j, k]
><< Algebra`Quaternions`
>i = {{0, 1}, {-1, 0}};
>j = {{0, I}, {I, 0}};
>k = {{I, 0}, {0, -I}};
>e = IdentityMatrix[2];
>q[t_, x_, y_, z_] := e*t + x*i + j*y + k*z;
>(* Rodrigues quaternion : quantum number functional
> definition : r0 radial distance/ Bessel - like,
> r time angle, {t, p} polar sphere angles*)
>qr = Exp[l*r0]*q[Cos[m*r], Sin[m*r]*
> Sin[n*p]*Cos[o*t], Sin[m*r]*Sin[n*p]*Sin[0*t], Sin[m*r]*Cos[n*p]];
>qrs = Exp[l*r0]*q[Cos[m*r], -Sin[m*r]*Sin[n*p]*Cos[o*t], \
>-Sin[m*r]*Sin[n*p]*Sin[0*t], -Sin[m*r]*Cos[n*p]];
>(* Linear vector differential definition *)
>v = {r0, r, p, t};
>(* space - time polar differential factors defined: derived*)
>w = {1/Exp[2*r0], 1/Exp[2*r0], 1/(Exp[2*r0]*Sin[
> r]^2*Sin[t]^2), 1/(Exp[2*r0]*Sin[d]^2)};
>dqr = Table[FullSimplify[w[[i]]*D[qr, {v[[i]], 2}]], {i, 1, 4}]
>dqrs = Table[FullSimplify[w[[i]]*D[qrs, {v[[i]], 2}]], {i, 1, 4}]
>Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]*MatrixPower[qr.qrs, -1]], {
> j, 1, 4}]
>Roger Bagula wrote:
>
>
>
>>Here is an effort to get a space-time polar quantum LaPlacian.
>>Quantum numbers in the q^4 powers result from a hyper-plate
>>type of vibrational equation.
>>The result doesn't appear normalizable in simple terms.
>>Dividing out :
>>MatrixPower[qr.qrs, -1]
>>would give a Klein-Gordon type factor.
>>Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]*MatrixPower[qr.qrs,
>>-1]], { j, 1, 4}]
>>
>>n and o quantum number factors have a complex wave function term which
>>if solves as unity
>>in each case tends to give quantum number angular interdependence.
>>A radial Chladni type vector on the "plate" as: qr0 as the base
>>Rodrigues quaternion
>>psi=(Cos[l*r0]+Cos[m*r]+Cos[n*t]+Cos[o*p])*qr0
>>might represent standing waves for a four quantum model such as this.
>>Mathematica:
>>Clear[a, m, m1, qr, dqr, qrs, dqrs, v, w, x, y, z, t, q, i, j, k]
>><< Algebra`Quaternions`
>>i = {{0, 1}, {-1, 0}};
>>j = {{0, I}, {I, 0}};
>>k = {{I, 0}, {0, -I}};
>>e = IdentityMatrix[2];
>>q[t_, x_, y_, z_] := e*t + x*i + j*y + k*z;
>>(* Rodrigues quaternion : quantum number functional
>> definition : r0 radial distance/ Bessel - like,
>> r time angle, {t, p} polar sphere angles*)
>>qr = Exp[l*r0]*q[Cos[m*r], Sin[m*r]*
>> Sin[n*p]*Cos[o*t], Sin[m*r]*Sin[n*p]*Sin[0*t], Sin[m*r]*Cos[n*p]];
>>qrs = Exp[l*r0]*q[Cos[m*r], -Sin[m*r]*Sin[n*p]*Cos[o*t], \
>>-Sin[m*r]*Sin[n*p]*Sin[0*t], -Sin[m*r]*Cos[n*p]];
>>(* Linear vector differential definition *)
>>v = {r0, r, p, t};
>>(* space - time polar differential factors defined: not derived directly*)
>>w = {1, 1/Exp[2*r0], 1/Exp[2*r0], 1/(Exp[2*r0]*Sin[t]^2)};
>>dqr = Table[FullSimplify[w[[i]]*D[qr, {v[[i]], 2}]], {i, 1, 4}]
>>dqrs = Table[FullSimplify[w[[i]]*D[qrs, {v[[i]], 2}]], {i, 1, 4}]
>>Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]], {j, 1, 4}]
>>
>>
>>
>>
>>
>>>
>>>
>>>
>>>
>>
>>
>>
>>
>
>
>