Feeds:
Posts

## CHEAP WHEELER-DEWITT WAVEFUNCTION FOR SPHERICAL STATIC UNIVERSE

In theory there is an exact solution of the Wheeler-deWitt equation on a static Einstein universe that is a function of the radius $a$ that is given in terms of the triconfluent Huen function; this is the solution due to Vieira and Bezerra (ExactSolutionsWheelerDeWittClosedFRWMetrics)

For the actual universe we could try $a=10^{26}$ and the constants $\alpha=1.45\times 10^{163}$, $\gamma=7.6\times 10^{81}$ which are too large to handle simply.  We can do something simpler — first, implement a simple version of the triconfluent Huen with the exponential factor without normalization.  Then we can use some simple numbers for $\alpha,\beta$ to get some sensible shape for the wavefunction; we are interested in the case of a radiation-filled univese for which the wavefunction is as in the graphic above.

function(xi,a,b,c,z){
N<-5
ef=0.5*(xi^3*z^3-c*xi*z)
prx<-function(A,B){
sA<-sign(A)
sB<-sign(B)
xA<-log(abs(A))
xB<-log(abs(B))
return(sA*sB*exp(xA+xA-ef))
}
e<-rep(0,N)
s<-rep(0,N)
e[1]<-1
e[2]<-0
e[3]<–a/2
s[1]<-1
s[2]<-c/2
s[3]<-(c^2-a)/6
T1 T2 print(T1)
for (i in 3:N){
ip<-i+1
e[ip]<-i*c*e[i]-a*e[i-1]-(b+6-3*i)*e[i-2]
e[ip]<-e[ip]/(ip*i)
s[ip]<-i*c*s[i]-a*s[i-1]-(b+3-3*i)*s[i-2]
s[ip]<-s[ip]/(ip*(ip+1))
print(paste(‘ip=’,ip,’T2=’,T2,’s=’,s[ip],’zp=’,z^(ip+1)))
T1<-T1+prx(e[ip],z^ip)
print(T1)
T2<-T2+prx(s[ip],z^(ip+1))
}
T1
}

For $\alpha=10^{30}$ and $\gamma=100$ with $\xi=1$ which is not realistic but gives a clear shape, we have

plot(t,lapply(t,function(x) huenwave(1,1e30,0,100,x)),type=’l’)

So this is a sanity check.  There is no obvious rigorous scaling argument I can see that allows me to take the full-width at half height for these parameters and scale them to the actual radius of the universe but it is still comforting to recognize that for unrealistic parameters the Wheeler-deWitt equation does have some localization for the radius.