Re: Quaternion problem--> Jacobian derivation/ Metric
- To: mathgroup at smc.vnet.net
- Subject: [mg68089] Re: Quaternion problem--> Jacobian derivation/ Metric
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Sat, 22 Jul 2006 06:24:12 -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>
- Sender: owner-wri-mathgroup at wolfram.com
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}]
>
>
>
>>
>>
>>
>>
>
>
>