The Wheeler de Witt Equation and the Problem of Time

[Mystery111]

The Problem of Time in physics is one born out of the Wheeler de Witt equation. The Wheeler de Witt equation fails to describe the universe with a time evolution because the right hand side of the equation does not contain a time derivative. The Field Equations Einstein derived describe motion in time which arises as a symmetry of the theory, not true time evolution itself [1].

 

I propose that it is a matter of not being a true time evolution in one case where there are pure gravity solutions to Einstein's equation. By allowing to choose two conformal time descriptions in the Wheeler de Witt equation [math]t = \chi[/math] and [math]\tau = \alpha[/math] you can describe two unique fields which we can use under the true definition of what allows the motion of real clocks to be measured.

 

To explain this more accurately, it is conjectured that the diffeomorphism contraints on the Hamiltonian of the Universe which leads to the vanishing of the time derivative should be taken as the analogue of a universe with only objects who's frame of reference ceases to exist because of Time Dilation. Chronological description of evolution of events in a universe where relativity treats the parts as moving in the null trajectory leads to no time description. Such a universe could be one with simply a radiation field and no matter at all. In fact, primordially-speaking this was the fundamental makeup of the universe at one point in it's history. It was just a soup of gamma radiation.

 

Matter Fields and for that matter (fields which defined moving clocks) appeared when a processes of geometrogensis [2] appeared when the universe sufficiently cooled down. Matter at the high energy scale implies the conditions we attribute to the early universe. That means by those calculations, the universe had to cool [math]10^{20}[/math] times less than what it is observed to be today and that high level is the level which quantum gravity is speculated to merge all the forces known to nature. So matter fields are attributed to low energy theories of our universe. In the fundamental theory, Fotini Markopoulou has remarked that geometry will no longer be involved.

 

On this limit, my theory would be advocating the presence of only radiation fields. Some may find this difficult to still understand because essentially we can only model our known field theories on Langrangian's which is measured in a space [math]dx^3[/math], but Fotini may be closer to the truth than we could realize. Perhaps when we are invoking the laws of physics in higher temperatures we are needing to change the view we are looking at the geometrical properties of the vacuum and how matter becomes manifest from this geometry and curvature.

 

The application and understanding of how matter fields could alter the way we view a Hamiltonian for a universe described by the state vector [math]|\psi>[/math] may imply the possibility of there being atleast in description, two different outcomes for when time can be logically applied to the Hamiltonian in the normal theories constraints which traditionally tend to favor the time derivative vanishing. To make this work, you must assume that the interaction has two solutions, one which is physical and the other pure radiation.

 

But before that is even mathematically-presented, anyone familiar with the Wheeler de Witt equation will know it by the form:

 

[math]\hat{H}|\psi> = 0[/math]

 

I will present this another way:

 

[math](H + (\alpha^2 - g^2\alpha^4))|\psi> = \hat{H}|\psi>[/math]

 

This part [math](\alpha^2 - g^2\alpha^4)[/math] plays the interaction of the Hamilonian. [math]\alpha[/math] for now can be known as a scale factor. The full Hamiltonian of the WDW-equation is

 

[math]H_{WDW}=H_{T}=H_{\phi, h_{\mu\nu}} - H_0[/math]

 

This comes from the implicit assumption [3] one imposes conformal time gauge conditions on the Hamiltonian. There is a physical component of the total Hamiltonian, the metric perturbation [math]h_{\mu \nu}[/math] recognized most famously from weak metric limits in General Relativity [4].

 

In a proceedure of seperation of variables [math]|\psi> = |\phi_\alpha>|\chi_{\phi, h}>[/math] leads to two equations

 

[math](H + (\alpha^2 - g^2\alpha^4))|\psi_a> = E|\psi_\alpha>[/math]

 

and

 

[math](H + (\alpha^2 - g^2 \alpha^4 dt))_{\phi, h}|\chi_{\phi, h} > = E|\chi_{\phi, h}>[/math]

 

Where [math]\chi[/math] is a matter field acting as time. In fact, in these equations in the superminispace model you can freely choose between using the scale factor [math]\alpha[/math] or the matter field [math]\chi[/math] as your choice of time coordinate. Interestingly, you may be able to break your Hamiltonian into a matter field [math]\chi[/math] and a radiation field [math]\alpha[/math]. You may be allowed toconstruct a time for a universe where real clocks can be measured, but if the constraint lets [math](\alpha^2 - g^2\alpha^4 dt) \rightarrow 0[/math] determine whether it is a massless radiation field, which cannot be used to measure the motion of clocks due to relativity. This means under certain limits matter fields can vanish.

 

Matter fields are trivial in General Relativity. Trivial scalar fields like a matter field in conformally flat spacetime is in fact a non-gravitating part of the theory. Vanishing matter fields would be akin to pure gravity solutions of General relativity.

 

Assuming the inertial part [math]\chi[/math] of

 

[math](H + (\alpha^2 - g^2\alpha^4 dt))_{\phi, h}|\chi_{\phi, h} > = E|\chi_{\phi, h}>[/math]

 

vanishes when the potential reaches the ground state then you are dealing with two different potentials. You find these potentials as:

 

[math]\alpha^2-g^2\alpha^4 = V(\alpha)[/math]

 

for a radiation field [math]a= \tau[/math] and

 

[math]\chi^2 - g^2\chi^2 = V(\chi)[/math]

 

For your matter field. You retreive these potential terms by treating the wave function in the WDW-equation as being [math]\Psi(\alpha,\chi)[/math]. If there is an inertial part then you have the presence of a matter field, you obtain the equation

 

[math](\frac{1}{4} \frac{\partial^2}{\partial \chi^2})\psi_{\chi}(\chi) = E\psi_{\chi}(\chi)[/math]

 

So in order to solve the Time Problem of Physics resulting from the time derivative vanishing, why don't we just adopt the idea it depends on your choice of restraint and what field you use to try and define your clocks? The only premise that would need to be adopted is that the derivative vanishing as a statement of having your Hamiltonian in two different states: one allowing clocks to be defined in the matter state [math]\psi_{\chi}(\chi)[/math] and for those where relativity naturally does not permit time passing, pure radiation fields [math]\psi_{\alpha}(\alpha)[/math].

 

 

 

[1] - http://www.fqxi.org/...ou_SpaceDNE.pdf

 

[2] - def - http://en.wiktionary...geometrogenesis

 

[3] - http://arxiv.org/PS_...3/9503073v2.pdf

 

[4] - http://ned.ipac.calt...3/Carroll6.html

 

It might be worth noting that the approach is similar to the construction of the Freidmann equation where it is divided into a radiation part and [math]\rho[/math]-matter part. The radiation part of the pressure is usually ignored because the matter part dominates when a universe begins to cool down.

Edited October 21, 2011 by Mystery111