## CHEAP WHEELER-DEWITT WAVEFUNCTION FOR SPHERICAL STATIC UNIVERSE

March 13, 2017 by zulfahmed

In theory there is an exact solution of the Wheeler-deWitt equation on a static Einstein universe that is a function of the radius 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 and the constants , $\gamma=7.6\times 10^{81}$ 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 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 and with 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.

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