Solution of the WheelerDeWitt Equation
Solution of the WheelerDeWitt Equation
Hontas Farmer
Presenting a solution to the WheelerDeWitt Equation in the framework of the theory of Lagrangian Euclidian Space. Using the unified theory published as a pre print previously a version of the WheelerDeWitt equation was derived. Namely the Hamiltonian constraint equation will be solved for ψ. It will be shown that in the framework of this unified theory the equation resembles that of a harmonic oscillator. Inverting a coordinate transformation to recover normal space time results in a graph of the wave function and potential which resembles the CMB in at least a qualitative sense. NOTE:This is all work in progress.
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Introduction
Introduction
The basic coordinates in LEST are Lagrangians. The Lagrangian of the standard model ( l ) the Einstein Hilbert (L) and another arbitrary Lagrangian meant o stand in for the dark sector (O). It should be assumed that these have been multiplied by appropriate constants so as to render them dimensionless. Insert a brief review of papers I have written on this already The universal Lagrangian is given by equation one as shown in
U=(L+lω(O+L))ExpL(lωO)
1
2
(
1
)Theory
Theory
Evaluating the required functional integrals for the generating functional is beyond the scope of the present notebook. This is detailed in the above references. The einstein Hilbert action (s), the standard model action (S) and the action of the dark sector (). The unified action (). Those appear and allow a simple expression of the action of the hypothesized unified field theory. Z=Exp[ ]
=(s+Sω(+s))Exp[L(lωO)]
(
2
)L(lωO)
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In the above that sum of actions is the unified current it is a result of integrating U by parts.
=s+Sω(+s)
(
3
)The exponential is the unified field.
ℱ = Exp[L (l  ω O)]
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= Exp[L (l  ω O)]
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Series[, {L, 0, 3}]
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+(lOω)L+++
1
2
2
(lOω)
2
L
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6
3
(lOω)
3
L
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O[L]
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This allows the writing of a simple beautiful partition function that includes all possible interactions valid at all energy scales.
Z
∞
∑
n=0
n
(ℱ)
n!
1
Z
∞
∑
n=0
n
(ℱ)
n!
Z=
∞
∑
n=0
n
(ℱ)
n!
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ℱ
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With the generating functional in place we can compute the full interaction due to this unified field ℱ and the conserved current, the unified action .
〈0ℱℱ0〉=Z=Z
1
Z
δ
δ
δ
δ

J=0
1
Z
2
ℱ

J=0
⟹Exp[ℱ]Exp[ℱ]=
2
ℱ

J=0
2
ℱ
Eigenvalues an Ladder operators
Eigenvalues an Ladder operators
WDW Equation
WDW Equation
Let ℏ = c = m =. .. = 1
Now to solve the equation for the ground state of the UNIVERSE!
Sol=DSolveℱψ0[ℱ]+D[ψ0[ℱ],{ℱ,1}]0,ψ0[0]1,ψ0[ℱ],ℱ
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2
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ψ0[ℱ]

2
ℱ
2
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Now to find the normalization constant for this wave function.
Then plot the normalized wave function.
The nth State.
The nth State.
This is just the first Hermite polynomial.
Therefore from now on Mathematica’s built in symbol can be used. A manipulateable plot will be used here.
Substituting for the function ℱ. ℱ = Exp[L (l  ω O)] I will set ω = 0 basically ignoring the dark sector in this 3 D plot.
Substituting the Ricci scalar from the Friedmann–Lemaître–Robertson–Walker solution makes sense in this situation because this is a solution to the Wheeler DeWitt equation. A curvature that applies to the universe on a large scale makes sense. For the quantum field I will just brutally and roughly approximate all of that with a oscillatory function of the FLRW invariant interval
Where a is the scale function k and c are constants.
In real space  time
In real space  time
I am given pause by the error messages but by and large I don't think they are a show stopping problem.
References
References