WheelerDeWittEquation

The Wheeler-DeWitt equation plays a central role in the attempt to unify quantum mechanics with general relativity. It's part of a framework known as canonical quantum gravity and arises in the quantization of the Hamiltonian constraint in general relativity.

The classical Hamiltonian constraint is a central part of the Einstein-Hamilton equations, reflecting the invariance of general relativity under reparametrizations of time. When attempting to quantize gravity, this constraint leads to a rather unusual feature: the absence of time in the fundamental description.

The Wheeler-DeWitt equation can be more precisely written as:

$$ \hat{H}\Psi[h_{ij}(x), \phi(x)] = 0 $$

Here, $h_{ij}(x)$ represents the 3-metric on a spatial slice, $\phi(x)$ represents matter fields, and $\Psi[h_{ij}(x), \phi(x)]$ is the wave function of the universe. This equation is essentially a functional differential equation for the wave function $\Psi$, and it incorporates both the dynamics of the gravitational field and the matter fields.

The Question of Time

This absence of time makes the interpretation of the Wheeler-DeWitt equation challenging. It raises deep questions about what it means for a system to evolve or for an observation to occur at a specific time.

Some approaches to solving or interpreting the Wheeler-DeWitt equation involve introducing a notion of time through additional structures or conditions, such as the introduction of a "clock" field within the system or the use of semiclassical approximations where classical time re-emerges. These attempts are misguided and doomed to fail because our universe lives in a specific surface that expanded from one of the roots of the conformally transformed Hardy Z function $X_a(t)=\tanh(\ln(1+a*Z(t)^2))$ where a is a scale factor related to the 'age' of the universe to which it corresponds; there would be a specific index that indicates all the parameters of the StandardModel via the impending solutions to the YangMillsEquations using this infinite sequence of geometric structures stemming from the fact that the zero curve of the real part of $T(t)=\tanh(\ln(1+t^2))$ where t is regarded as a complex variable, is a LemniscateOfBernoulli. Take the inverse equation

$$ Re(X_1(t))=b $$

where the scale is taken to be $a=1$ then as $b$ ranges from $0$ to $-1$ it transforms from a closed region bounded by a Bernoullian lemniscate centered at each root of $X_1(t)=0$ to a single point of no volume at the location of the root approximately $\sqrt{2}$ north of each root of Z(t), and whose deviation from being 90 degrees is dependent upon the local curvature properties of the root to which it corresponds of $X_1(t)=0$ introduced by the singularities of the hyperbolic tangent function. It shall become obvious later that the Timeless feature the Wheeler-DeWitt equation is related to a metric that corresponds to a surface related to Re(X(x+iy)) where $t=x+iy$ and which is associated with a set of stochastic processes which provide for the evolution of the metrics passed the singularities involved and that resulting metrics share many similar features of the Göedel metric.