(Precursor for) Quantum Boundary Conditions for Expanding Universe

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1. Introduction

We work with the Wheeler De Witt Equation as given by [1] , as part of the work by Hall and Reginatto, in 2016, where an ordering, called p, is used to link a Wheeler De Witt Equation, as given below, to an inflaton, and the Friedman Walker space-time metric, with the inflaton described by [2] and the Friedman Walker metric given in [2] [3] .

What we are doing is using [1] with its Wheeler De Witt equation to look at the following

(1)

The inflaton, is defined by [2] as given by

(2)

The wave function we use in Equation (1) we will use the ansatz of

(3)

These three sets of equations will be referenced, in our article, and will form the template of the subsequent analysis.

2. Looking at How to Come Up with a Polynomial Equation for

as given in Equation (2) is used to re define the inflaton in Equation (2) as well as a re definition of the potential U, as in Equation (2) with the upshot that

(4)

Now what is unusual about the bottom quadratic equation for the scale factor, as given in Equation (4)? We have that, here we are using Equation (2) in the end to define, here, a inflaton equation in terms of time, not the scale factor version of it, as given in Equation (4). If we use this approach, and constrain ourselves to very small time steps, i.e. of the order of Planck scale time (very small) we get then that the range of quantum effects, from an initial to the boundary of quantum gravity effects, is given by, approximately for small.

(5)

3. Conclusion: We Have Taken the Simplest Case, and It Could Be More Complicated

What we have done is to look at, while using the inflaton expression given in Equation (6) below:

(6)

In

(7)

Were we to insert Equation (6) for the inflaton into Equation (7) we would have a very nonlinear case, for the scale factor equation. One which could only be deciphered by numerical analysis.

If we stick with the above methodology, we still have to consider conditions for which

(8)

Which presumably would be linked to

(9)

Indeed, though, if there is no way we could possibly retrieve Equation (4) above, i.e. we have a numerical problem, one which we will investigate in future papers. In addition, for Equation (4), Equation (7) and Equation (8) we need to remember comes from Equation (3) and its value will need to be considered.

What we have though is based upon [1] and the idea of a quantum ensemble and operator-ordering. In order for the readers to get more insights as to the physics inherent in the choice of p, in Equation (1) the reader is referred to [4] [5] [6] .

Finally, [7] - [12] have issues which need to be reviewed which may in fact, have a ready impact upon Equation (8), and Equation (9) above, i.e. [7] [8] [9] refers to Corda’s work with the foundation of gravity, and if or not Gravity is quantum, or purely due to classical General Relativity. In particular the issue of scalar-tensor gravity needs to be investigated, to see if it falsifies Equation (7) or if it adds new restrictions as to the boundaries.

(10)

The answer, as given by Ng, is that if the volume of space, V, is-, and that is proportional to the wavelength , then due to the situation of how a massive graviton could at least have accelerated mass values, this will allow for the Ng formula, being changed to

(11)

Does Equation (8) and Equation (9) falsify Equation (10) and Equation (11)?

It needs to be answered. And of course all this needs to avoid being in conflict with [14] and the gravity results so derived. Finally, does Equation (8) and Equation (9), not to mention Equation (4) falsify the conditions given in [15] as to massive gravity? This question should also be investigated.

After these questions are entertained, and examined, the last supposition, as mentioned should be investigated, i.e. of a different time variable, delineating the amount of time in a quantum regime for the expansion of the universe. IMO, using

(12)

And if would lead to a time regime for quantum effects, delineated by

(13)

Of course, if, we would have a different power relationship, very different.

I.e. all these questions need to be investigated in the near future.

Acknowledgements

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

References

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