_{1}

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In 2012, the author submitted an article to the Prespacetime Journal based upon the premise of inquiry as to the alleged vanishing of disjoint open sets contributing to quantum vector measures no longer working, i.e. the solution in 2012 was that the author stated that quantum measures in 4 dimensions would not work, mandating, if measure theory were used, imbedding in higher dimensions was necessary for a singularity. The idea was to use the methodology of String Theory as to come up with a way out of the impasse if higher dimensions do not exist. We revisit this question, taking into account a derived HUP, for metric tensors if we look at Pre-Planckian space-time introducing a pre-quantum mechanical HUP which may be a way to ascertain a solution not mandating higher dimensions, as well as introducing cautions as to what will disrupt the offered solution. Note that first, measurable spaces allow disjoint sets. Also, that smooth relations alone do not define separability or admit sets Planck’s length, if it exists, is a natural way to get about the “bad effects” of a cosmic singularity at the beginning of space-time evolution, but if a development is to be believed, namely by Stoica in the article, about removing the cosmic singularity as a breakdown point in relativity, there is nothing which forbids space-time from collapsing to a point. Without the use of a Pre Planckian HUP, for metric tensors, the quantum measures in four dimensions break down. We try to ascertain if a Pre Planckian HUP is sufficient to avoid this pathology and also look at if division algebras which can link Octonionic geometry and E8, to Quark spinors, in the standard model and add sufficient definition to the standard model are necessary and sufficient conditions for a metric tensor HUP which may remove this breakdown of the sum rule in the onset of the “Big Bang”.

As stated in 2012 [^{1}. Afterwards, we will examine the role of the Heisenberg uncertainty principle HUP given in [

In this situation, without considering a modified Heisenberg Uncertainty principle, HUP, in Pre Planckian space, as given by [

As stated before in [

Quote:

A proper choice of coordinates is going to involve more than four dimensions and that what is chosen in four-dimensional space-time usually in the Roberson Walker metric will lead to a singularity problem. We claim that this affects the quantum measure problem in four dimensions. The main point of the article is below where we outline how to fix the glaring problems in four-dimensional measures theory which we state as unphysical.

We note here that higher dimensions will, as in String theory, remove this problem. We wish to reconcile the four and higher dimensional examples of co ordinate behavior and reflect upon what the four dimensional representation does to quantum measures, especially if there is a removal of the standard four dimensional representation of a mathematical singularity at the start of inflation. To do this, we will give an argument which will point in the direction of vanishing of disjoint sets in four dimensions leading to a breakup of the quantum measure in four dimensions.

Our initial goal is to show that disjoint sets, are due to separability in a topological sense, and that at a point in space?time, that the very notion of separability breaks down completely [

Separability in a topological sense can be constructed as follows. A topological space X is said to be separable if X has a countable dense subset. In other words, there is a countable subset D of X such that closure (D) = X.

Equivalently, each nonempty open set in X intersects D. The fact is, that if there is a space?time point, that the countable subset D of X is such that the closure (D) = X breaks down completely.

Afterwards, we should note that disjoint sets in a topological space, X, are due to working with X being a Hausdorff space. We then note the properties of Hausdorf spaces can be written follows:

1) If K is a compact subset of X and

2) Every compact subset of X is closed.

3) Any two disjoint compact subsets of X have disjoint open neighborhoods, i.e. if C and D are compact disjoint subsets of X, then there exist open sets

Note that when one has a point in space time, the there is not a comparable construction to closure (D) = X or

This lack of having at a point in space-time a topological set X with open subsets with these constructions dooms having these properties. I.e. if one does not have a Hausdorff space, one is going to find it impossible to form disjoint sets in a separable X if X is itself a point.

When one does not have separable sub sets, at a single point, then the construction used for quantum measures breaks down. We review in Appendix A what happens due to Stoic’s treatment [

While the existence of the pathological singularity can be treated by use of Planck’s length, which can be used to construct disjoint sets, if Stoica is believable, this Planck’s length is no longer essential, which brings up interesting questions so far avoided by main stream cosmologists. This paper merely brings up that issue, and asks what can be done to correct for it, at the point of the big bang. To do this, we later revisit what happened in Surya’s paper [^{2}.

Our contribution is to examine quantum measures assuming a non-string theory treatment of cosmology. And to argue that the breakdown of a quantum measure in four dimensions necessitates use of higher dimensional embedding of the start of cosmological inflation.

End of quote of [

This concludes our introduction as to what is done if the Heisenberg Uncertainty principle HUP of [

What we will do next is to elaborate the HUP which is cited in [

In order to do this we will be looking at the following construction. From [

Iliac. Examining what happens in Pre Planckian Space time

We will be looking at the value of Equation (1) if

If we use the following, from the Roberson-Walker metric [

Following Unruth [

Then, if

This Equation (14) is such that we can extract, up to a point the Heisenberg Uncertainty principle HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [

Then [

Then,

How likely is

We next will then go into a description of what Equation (7) will do to the issue of if not the quantum measure breaks down to that we will work from the stand point of what a traditional quantum measure will do and to put in what the situation is, if we use Equation (7) and if we do NOT assume Equation (7), in terms of a local HUP.

We use, in part a page and a half of information directly from the initial paper [

From [

As given in [

The use of finite additivity of ^{nd} half of this paper, as compared to the situation in [

We assert that if Planck’s length is mandatory due to space-time evolution from a HUP as done later, then there is no question that Equation (8) holds. Furthermore, we assert that the 2^{nd} half of this paper due to the HUP does satisfy Equation (8) so the Surya hypothesis as given in [

The difference in what we are proposing in the 2^{nd} half of this paper is that unlike the situation from [

Assuming no HUP, as is done later, we will present reference [

If there is a HUP for space-time metrics, then the main problem, of applying Equation (8) lies in insuring the existence of disjoint sets at a point of space-time. If we can get through the HUP, the existence of a minimum space-time length allows us to state, there can be disjoint sets, and then the math construction of Surya [

We recite the results of [

Classical relativity theory though will not allow applying Equation (8). In our 2^{nd} half of the paper will claim that Equation (8) holds. We claim that this is due to the disappearance of the Pathology given in page 5 of Reference [

Now, let us go through what we did in [

To do this, let us review the notion of a quantum measure.

The precondition for a quantum measure

We then come to our main result. In the case of NO Heisenberg Uncertainty principle HUP as we talk about later in the 2^{nd} part of the paper, then Equation (8) is replaced by the inequality, i.e. the elimination of disjoint sets will lead to.

Not being able to have a guarantee of having n disjoint sets

So what do we call the quantum measure generally? In Equation (10) the existence of the quantum measure is defined by the supremum argument.

Now what about the inequality of Equation (9) in the case we eliminate disjoint measures, and how would that lead a change in Equation (8) to Equation (9)?

This is what becomes very problematic if [

Our claim is as follows. The Supremum argument as of Equation (10) in the case of NO HUP for metric tensors is no longer valid. If there is no HUP for metric tensors, then the elimination of disjoint partitions means that we have then the elimination of the equality as given in Equation (10).

Why would this break down if we do not apply a metric Heisenberg Uncertainty principle HUP?

We claim that the existence of a singularity will not allow us to analyze disjoint partitions. We furthermore claim that the result we are referencing is that Equation (10) DOES INDEED hold if a metric tensor Heinsenberg Uncertainty principle HUP exists which gives geometric existence to space-time permitting the existence of disjoint sets.

We furthermore claim that if we do not have a metric tensor Heinsenberg Uncertainty principle HUP, that we eliminate disjoint sets. Then we go to Equation (11) below.

If we do NOT apply a HUP which removes the presumed singularity, then Equation (10) is now an inequality written as:

Our point as to what happens if an inequality holds, is due to a breakdown of the notion of “unconditional convergence of the vector measure.”

Equation (10) depends upon having [

Equation (11) is what we have if no Heisenberg Uncertainty principle HUP exists, and that the main result of our paper is that Equation (11) no longer holds, and that we instead have Equation (10) if a metric tensor Heinsenberg Uncertainty Principle HUP exists. We detail the reasons for that statement in Section V below.

In order to reformulate the conclusions of Equation (11), we will be examining if the existence of Equation (7) stops physical disruptions of a disjoint partition, i.e. what we will be examining if we have an effective way to examining disjoint partitions, as showing up for why in the case of not using Equation (7) we had effectively removed the singularity at the beginning of space-time.

Usually as given by Penrose-Hawking singularity theorems [

Quote

The strategy of the sketched proof presented was to assume that null geodesics were complete, proving that then the boundary of the future of the closed trapped surface is compact.

End of quote.

Strategy here, is to remove the caveat of compactness. Compactness, according to [

We now should go to a new version of the modified HUP, and it will be stated as approximately as that unless an inflaton field exists in the Pre Planckian space-time so that [

Then by [

Perforce, the enormity of the change in energy, will remove the possibility of a closed surface, in the Pre Planckian space-time, i.e. in doing so, the change in energy disrupts conditions as given in [

We shall next go to the division algebra results and gravitons, to give more structure to applying Equation (13) above.

In [

Quote:

“Each of the four Division Algebras R, C, H, and O can also be viewed as a spinor space”, and later “The mathematics linking these pairs is an SU (2) group.”

i.e. so what permits the existence of a spinor space in a non-Compact domain? To whit the existence of gravitons, in a non-compact space-time, and we will state then that there are theoretical arguments that a massless spin-2 particle has to be a graviton. The basic idea is that massless particles have to couple to conserved currents, and the only available one is the stress-energy tensor, which is the source for gravity. If a graviton is massless, a given as given by [

We can of course, make a simple identification of h_ (+) and h_(x) with the Up and Down states of Equation (14), due to the fact that in the massless case, there are only 2 helicity states. We could as an example, make a simple relations

As stated by [

According to [

In the case of Equation (16), [

Quote:

“Complexification of Quaternions’ and is equivalent to a pair of Paul Spinors, and if we form a Column matrix of two such elements we get a pair of Dirac spinors.”

End of quote.

Furthermore from page 47-48 of [

Quote:

“It is very interesting that the set of all unit quaternions is a copy of SU(2) (Since H is 4 dimensional, the set of unit quaternions is topologically the same as the set of all points in 4 space a distance 1 from the origin, which is the 3-sphere, one of our Parallelizable spheres).”

End of quote.

This is the language of what [

In addition,

Is called in [

Quote:

“P and T spinors are SU (2) doublets, so that leaves us with the reduced group, SO (1, 3) X U (1) X SU (2) X SU (3).”

The summary of what we are looking at is [

Each element of Equation (18) is given by

This refers to 1, 2, 4, and 8 “dimensional” spheres, and 1, 2, 4, 8 are the Caley numbers. This construction should be seen as a way of quantifying, as an example, Equation (18) as a direct construction of 4 SU (2) “spinors, and 2 Quaternion spinors”. The moral being that we can build up a systematic algebra this way, which can use the set of spin ½ wave function eigenvalue entries to build up through Equation (17), Equation (18) and Equation (19) a linkage of 1, 3 spinors as given by a proper interpretation of Equation (16) as with comparison with 1, 9 spheres given in Equation (17), Equation (18) and Equation (19). This construction though, and a linkage to massless versions of the Graviton, works well, if we wish to tie in the usual construction which may be appropriate for the interpretation of the 2 graviton polarization states as having a time in, via Equation (14) and Equation (15) with the UP and Down basis spinors of SU (2) and by extension the buildup of the spinors of the Octonian as alluded to in Equation (18).

The relevance, in terms of space-time, in the case of massless gravitons is as follows, namely that if we can make the identification of Equation (18) and link that to the idea of Equation (15). Then the following situation occurs, namely.

The change in geometry is occurring when we have first a pre quantum space time state, in which, in commutation relations [

Equation (20) is such that even if one is in flat Euclidian space, and i= j, then

In the situation when we approach quantum “octonion gravity applicable” geometry, Equation (20) becomes

Equation (22) is such that even if one is in flat Euclidian space, and i= j, then

Also the phase change in gravitational wave data due to a change in the physics and geometry between regions where Equation (21) and Equation (22) hold will be given by a change in phase of GW, which may be measured inside a GW detector.

The simplest way to consider what may be involved in alterations of geometry is seen in the fact that in pre octonionic space time regime (which is Pre Planckian), one would have [

This Pre Octonionic space-time behavior should be seen to be separate from the flatness condition as referred to in [

Whereas in the octonion gravity space time regime where one would have Equation (22) hold that for enormous temperature increases Equation (22), then by [

Here,

We argue that Equation (24) holds, and that the Stoica non pathological singularity is removed, if there is a sufficiently large energy flux given by change in energy, in Equation (13), but this requires that it not be infinite.

We shall next go to the conclusions and to first review the conclusions made if we do not have the modifications due to an ultra large change in initial energy due to Equation (13).

First of all, the question we need to ask is, “Is the existence of a Planck length, as a minimum length mandatory as to space-time?” If it is, the problem of the existence of disjoint intervals is solved, i.e. we need not worry, even if it is 10^-35 meters in length. If this minimum length exists, Equation (8) holds everywhere.

If a mandatory minimum non-zero space-time interval is necessary, then there is nothing which forbids the existence of (8) above. If such an interval does not exist, then (1) breaks down. Furthermore, the space of all infinitely differentiable functions is also separable, and a fundamental sequence is the sequence of all powers of x. This is shown by Taylor Series and Weierstrass Theorem [

It should be noted that Connes [

In essence, for making a consistent cosmology, our results argue in favor of a string theory style embedding of the start of inflation and what we have argued so far is indicating how typical four dimensional cosmologies have serious mathematical measure theoretic problems. These quantum measure theoretic problems are unphysical especially in light of the Stoica findings [

We argue in this second case, that then the problems are consistent with regards to the sort of measure theory as advocated by Tao [

A general comment should be raised, here, that Reference [

In doing this, note that what is implied in our document, if [

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

Beckwith, A.W. (2017) Reconsidering “Does the Sum Rule Hold at the Big Bang?” with Pre Planckian HUP, and Division Algebras. Journal of High Energy Physics, Gravitation and Cosmology, 3, 539-557. https://doi.org/10.4236/jhepgc.2017.34041

This is straight from reference [

If Equation (13) is not used, which removes, this construction, we state that Stoica [

Furthermore we also have the accelleration equation given by

Using the re-scaling of [

We then re-scale the density and also the pressure as follows:

This will lead to

The upshot is, as explained in [

So then the acceleration equation and Friedman equation vanish at

This is straight from reference [

We introduce the formalism by appealing to the concept of spatial diffeomorphism [

We submit that the difficulties in terms of consistency of Equation (8) of this document. In terms of initial causal structural breakdown-which we claim leads to Equation (8) being re written as an inequality-one has to come up with a different way to embed quantum measures within a superstructure, as noted in the conclusions of this paper. Spatial diffiomorphisms as stated in [

The author’s main point is that there is a breakdown of measurable structure, starting with definitions given in [

Let

This is done for a cylinder set , where

This probability would be a quantum probability which would not obey the classical rule of Kolmogrov [

The actual probability used would have to take into account quantum interference. That is due to Equation (1b) and Kolmogrov probability no longer applying, leading to

Here,

A quantum vector measurment is defined via

Where

Also

with a Hilbert space

So then the following happens,

This is for all

The claim associated with Equation (b1) above is that since

The main point of the formalism for Equation (b13) is of bi-additivity of D leading to the finite addivity of

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