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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "KAHLER 2-form MAtti Pitkan
en" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "restart: with(liesymm):with(l
inalg):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new defin
ition for close" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definitio
n for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for
 trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "setup(x,y,z,t,r,t
heta,Phi,Psi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "defform(
x=0,y=0,z=0,t=0,r=0,theta=0,Phi=0,Psi=0,a=const,b=const,c=const,k=cons
t,mu=const,m=const,epsilon=const):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "dR:=[d(x),d(y),d(z),d(t)]:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 6 "Matti:" }}{PARA 0 "" 0 "" {TEXT -1 63 "I am having troub
le reproducing the equations in your appendix." }}{PARA 0 "" 0 "" 
{TEXT -1 106 "I understand that you start with a special 6 dimensional
 space (C3) and add two constraints to get to PC2." }}{PARA 0 "" 0 "" 
{TEXT -1 75 "The 6 D space is special for is it presumed to support a \+
complex structure." }}{PARA 0 "" 0 "" {TEXT -1 47 "There are several w
ays to do these projections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 76 "But let me start with your definitions. using t
he dimensionless coordinates " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E1:=z+I*t;E2:=x+I*y;E1C:=z-I
*t;E2C:=x-I*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G,&%\"zG\"\"\"*
&%\"IGF'%\"tGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G,&%\"xG\"\"
\"*&%\"IGF'%\"yGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E1CG,&%\"zG
\"\"\"*&%\"IGF'%\"tGF'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E2CG
,&%\"xG\"\"\"*&%\"IGF'%\"yGF'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 40 "See Equation 29.21 in your TGD appendix." }}{PARA 0 "" 0 "" 
{TEXT -1 39 "Next compute the 1-forms given by 29.27" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 65 "E1E2:=evalc(E1*d(E2)-E2*d(E1));E1E2C:=eva
lc(E1*d(E1C)+E2*d(E2C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%E1E2G,,
*&%\"zG\"\"\"-%\"dG6#%\"xGF(F(*&%\"tGF(-F*6#%\"yGF(!\"\"*&F,F(-F*6#F'F
(F2*&F1F(-F*6#F.F(F(*&%\"IGF(,**&F.\"\"\"F)F=F(*&F'F=F/F=F(*&F1F=F4F=F
2*&F,F=F7F=F2F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&E1E2CG,,*&%\"z
G\"\"\"-%\"dG6#F'F(F(*&%\"tGF(-F*6#F-F(F(*&%\"xGF(-F*6#F1F(F(*&%\"yGF(
-F*6#F5F(F(*&%\"IGF(,**&F-\"\"\"F)F<F(*&F'F<F.F<!\"\"*&F5F<F2F<F(*&F1F
<F6F<F>F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sigma1:=eval
c(Im(E1E2))/r^2;sigma2:=evalc(-Re(E1E2))/r^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'sigma1G*&,**&%\"tG\"\"\"-%\"dG6#%\"xGF)F)*&%\"zGF)-F
+6#%\"yGF)F)*&F2F)-F+6#F/F)!\"\"*&F-F)-F+6#F(F)F6\"\"\"*$)%\"rG\"\"#F:
!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G*&,**&%\"zG\"\"\"-%
\"dG6#%\"xGF)!\"\"*&%\"tGF)-F+6#%\"yGF)F)*&F-F)-F+6#F(F)F)*&F3F)-F+6#F
0F)F.\"\"\"*$)%\"rG\"\"#F:!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "sigma0:=evalc(Re(E1E2C))/r^2;sigma3:=-evalc(Im(E1E2C))/r^2;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%'sigma0G*&,**&%\"zG\"\"\"-%\"dG6#F(F)F)*&%\"tGF)-F+6#F.F)F)*&%\"xGF
)-F+6#F2F)F)*&%\"yGF)-F+6#F6F)F)\"\"\"*$)%\"rG\"\"#F9!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G,$*&,**&%\"tG\"\"\"-%\"dG6#%\"zGF*
F**&F.F*-F,6#F)F*!\"\"*&%\"yGF*-F,6#%\"xGF*F**&F7F*-F,6#F4F*F2\"\"\"*$
)%\"rG\"\"#F;!\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "sigma1, sigma2 and sigma3 are (wi
thout the 1/r^2 factor) standard coomponents of the " }}{PARA 0 "" 0 "
" {TEXT -1 74 "Hopf map from S3 to S2.  See http://www22.pair.com/mapl
e/hopfmap.html for " }}{PARA 0 "" 0 "" {TEXT -1 82 "a discussion of th
e Hopf map and all the differential topology associated with it." }}
{PARA 0 "" 0 "" {TEXT -1 9 "*********" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "However at this stage it is not clear what you mean by r.
" }}{PARA 0 "" 0 "" {TEXT -1 56 "I do not know why you want to multipl
y the sigma by r^2." }}{PARA 0 "" 0 "" {TEXT -1 59 "If I use the defin
itions of 29.22  and 29.21 I deduce that " }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 2 "= " }
{XPPEDIT 18 0 "x^2+t^2+z^2+y^2;" "6#,**$%\"xG\"\"#\"\"\"*$%\"tG\"\"#F'
*$%\"zG\"\"#F'*$%\"yG\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "VOL4:=r^8*factor(sigma0&^sigma1&^sigma2&^sigma3);" }}{PARA 0 "" 
0 "" {TEXT -1 73 "When I do not divide by the r^2 factor you get the c
lassic Hopf formulas." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VOL4G*&),*
*$)%\"tG\"\"#\"\"\"\"\"\"*$)%\"zGF+F,F-*$)%\"yGF+F,F-*$)%\"xGF+F,F-F+F
,-%#&^G6&-%\"dG6#F6-F;6#F3-F;6#F0-F;6#F*F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 67 "If you divide by r instead of r^2 you get the unit volume
 element ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "NOw use the formulas you give in 29.22 an
d evaluate x,y,z,t" }}{PARA 0 "" 0 "" {TEXT -1 43 "(They are denoted b
y capital letters below)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 
"X:=r*cos((Psi-Phi)/2)*sin(theta/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"XG*(%\"rG\"\"\"-%$cosG6#,&%$PsiG#!\"\"\"\"#%$PhiG#F'F/F'-%$sinG6
#,$%&thetaGF1F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Y:=r*sin
((Psi-Phi)/2)*sin(theta/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG,
$*(%\"rG\"\"\"-%$sinG6#,&%$PsiG#!\"\"\"\"#%$PhiG#F(F0F(-F*6#,$%&thetaG
F2F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Z:=r*cos((Phi+Psi
)/2)*cos(theta/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG*(%\"rG\"
\"\"-%$cosG6#,&%$PhiG#F'\"\"#%$PsiGF-F'-F)6#,$%&thetaGF-F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "T:=r*sin((Phi+Psi)/2)*cos(theta/2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG*(%\"rG\"\"\"-%$sinG6#,&%$Ph
iG#F'\"\"#%$PsiGF-F'-%$cosG6#,$%&thetaGF-F'" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 63 "Substitute thes coordinates into the basic formulas to \+
compute:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e3:= -wcollect(
(T*d(Z)-Z*d(T)+Y*d(X)-X*d(Y))):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 77 "Now compute the r^2sigma3 from 29.27.  no
te that e3 above is your r^2 sigma3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 99 "e3A:=-simplify(factor(getcoeff(e3&^d(Phi))),trig):e3B:=simplify(
factor(getcoeff(e3&^d(Psi))),trig):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "VE3:=wcollect(factor(e3A*d(Psi)+e3B*d(Phi))/(r*F));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$VE3G,&*&*(%\"rG\"\"\",&*$)-%$cosG
6#,$%&thetaG#F)\"\"#F3\"\"\"F3!\"\"F)F)-%\"dG6#%$PhiGF)F4%\"FG!\"\"F2*
&*&F(F4-F76#%$PsiGF)F4F:F;F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 61 "THis formula does not a
gree with your result for e3 in 29.30." }}{PARA 0 "" 0 "" {TEXT -1 36 
"The other vierbiens seem consistent." }}{PARA 0 "" 0 "" {TEXT -1 88 "
NOW I substitute your formula for F, and compute the Pfaff sequence, B
, dB, B^dB, dB^dB." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "P1:=factor(e3A)/(r*F);P2:=factor(e3B/(r*F))
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Action:=P1*d(Psi)+P2*d(Phi);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,$*&%\"rG\"\"\"%\"FG!\"\"#\"\"
\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G,$*&*&%\"rG\"\"\",&*$
)-%$cosG6#,$%&thetaG#F)\"\"#F3\"\"\"F3!\"\"F)F)F4%\"FG!\"\"F2" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ActionG,&*&*(%\"rG\"\"\",&*$)-%$cos
G6#,$%&thetaG#F)\"\"#F3\"\"\"F3!\"\"F)F)-%\"dG6#%$PhiGF)F4%\"FG!\"\"F2
*&*&F(F4-F76#%$PsiGF)F4F:F;F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "
Thes formulas are not the same as 29.40" }}{PARA 0 "" 0 "" {TEXT -1 
45 "Next define B, and compute the Pfaff sequence" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B:=subs(F=1
+r^2,VE3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&*&*(%\"rG\"\"\",
&*$)-%$cosG6#,$%&thetaG#F)\"\"#F3\"\"\"F3!\"\"F)F)-%\"dG6#%$PhiGF)F4,&
F)F)*$)F(F3F4F)!\"\"F2*&*&F(F4-F76#%$PsiGF)F4F:F=F2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "J:=d(B);" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#>%\"JG,(*&*(,&*$)-%$cosG6#,$%&thetaG#\"\"\"\"\"#F2\"\"\"F2!\"\"F1F1,
&F4F1*$)%\"rGF2F3F1F1-%#&^G6$-%\"dG6#F8-F=6#%$PhiGF1F3*$),&F1F1F6F1\"
\"#F3!\"\"#F4F2*&**F8F1F+F1-%$sinGF-F1-F:6$-F=6#F/F?F1F3FDFFF4*&*&F5F3
-F:6$F<-F=6#%$PsiGF1F3*$)FD\"\"#F3FFFG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "TTORSION:=B&^J;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
)TTORSIONG,$*&**)%\"rG\"\"#\"\"\"-%$cosG6#,$%&thetaG#\"\"\"F*F2-%$sinG
F.F2-%#&^G6%-%\"dG6#F0-F96#%$PhiG-F96#%$PsiGF2F+*$),&F2F2*$F(F+F2\"\"#
F+!\"\"#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "TPARITY:
=J&^J;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(TPARITYG*&*,,&!\"\"\"\"\"
*$)%\"rG\"\"#\"\"\"F)F)F,F)-%$cosG6#,$%&thetaG#F)F-F)-%$sinGF1F)-%#&^G
6&-%\"dG6#F,-F;6#F3-F;6#%$PhiG-F;6#%$PsiGF)F.*$),&F)F)F*F)\"\"$F.!\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "From These computations, it \+
would appear that the Topological Parity vanishes on the unit sphere, \+
and places where theta = n*pi, and of course the origin, r = 0." }}
{PARA 0 "" 0 "" {TEXT -1 103 "Hence on the unit 4 sphere, the divergen
ce of the topological torsion is zero.  For initial conditions " }}
{PARA 0 "" 0 "" {TEXT -1 277 "away frome the unit sphere, there is an \+
irreversible decay comp[onent in the direction of the torsion vector, \+
which has only one component in these coordinates,  along r.  The ulti
mate motion winds up on the unit 4 sphere or at the origin of at the v
alues of pi stated above.  " }}{PARA 0 "" 0 "" {TEXT -1 128 "When the \+
evolution reaches the unit sphere, then their exists a unique Hamilton
ian path on the unit sphere that is conservative." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Lets now do the same thing for t
he other vierbeins." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "e1 :
= wcollect(T*d(X)+Z*d(Y)-Y*d(Z)-X*d(T)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "e1A:=-simplify
(factor(getcoeff(e1&^d(Phi))),trig):e1B:=simplify(factor(getcoeff(e1&^
d(theta))),trig):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "VE1:=wcollect(
factor(e1A*d(theta)+e1B*d(Phi))/(r*(F)^(1/2)));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$VE1G,&*&*(%\"rG\"\"\",&*&-%$sinG6#,&%$PhiG#F)\"\"#%$
PsiGF1F)-%$cosG6#,&F3#!\"\"F2F0F1F)F)*&-F-F6F)-F5F.F)F9F)-%\"dG6#%&the
taGF)\"\"\"*$-%%sqrtG6#%\"FGFA!\"\"F8*&*(F(FA,&**F,FA-F56#,$F@F1F)F;FA
-F-FMF)!\"#**F<FAFLFAF4FAFOFAFPF)-F>6#F0F)FA*$-FD6#FFFAFGF8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e1As:=simplify(factor(2/r^2*
e1A),trig);e1Bs:=simplify(factor(2/r^2*e1B),trig);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%e1AsG,&*&-%$sinG6#,&%$PhiG#\"\"\"\"\"#%$PsiGF,F--%
$cosG6#,&F/#!\"\"F.F+F,F-F5*&-F(F2F--F1F)F-F-" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%e1BsG,&**-%$sinG6#,&%$PhiG#\"\"\"\"\"#%$PsiGF,F--%$c
osG6#,$%&thetaGF,F--F(6#,&F/#!\"\"F.F+F,F--F(F2F-F.**-F1F)F-F0\"\"\"-F
1F6F-F:F=F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "VE1:=r*(sin(
Psi)*d(theta)+sin(theta)*cos(Psi)*d(Phi))/(2*F^(1/2));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$VE1G,$*&*&%\"rG\"\"\",&*&-%$sinG6#%$PsiGF)-%\"d
G6#%&thetaGF)F)*(-F-F2F)-%$cosGF.F)-F16#%$PhiGF)F)F)\"\"\"*$-%%sqrtG6#
%\"FGF;!\"\"#F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "25 0 0" 32 }
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