# Quantum Gravity If Non-Locality Is Fundamental

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Remarks

- In General Relativity, time is a dimension. In Quantum Mechanics, time flows. One central issue is how to bridge this difference. I propose a means to do so below.
- Einstein remained concerned with “Now”. For Newton, “Now” is a moving point on a one-dimensional line. In General Relativity, where time is a dimension, “Now” vanishes. General Relativity has replaced Newton. “Now” has vanished from classical physics.
- What is “Now”? Can “Now” be real? This is a major issue. See L. Smolin’s Time Reborn [7]. I propose a solution below.
- General Relativity concerns only Actuals that obey Aristotle’s law of the Excluded Middle or Law of Non-Contradiction. Quantum Mechanics has features, superpositions, that do not obey the law of the Excluded Middle or Law of Non-Contradiction. By contrast, the results of actualization obey both laws. One interpretation of Quantum Mechanics is real res potentia, ontologically real possibles, and res extensa, ontologically real Actuals linked by actualization [8,9]. This is Heisenberg’s 1958 interpretation of Quantum Mechanics, where the quantum states are “potentia” [10]. “Potentia” are neither true nor false [8,9]. It is of fundamental importance that General Relativity cannot be interpreted in terms of “potentia”. All variables of General Relativity are true or false. Thus, in addition to the linearity of QM and the non-linearity of GR, the two theories may be ontologically distinct [8]. Any attempt at uniting the non-Aristotelian variables in Quantum Mechanics in Hilbert space and the Aristotelian variables in General Relativity must show how this ontological distinction can be reconciled. By taking quantum non-locality as fundamental, I propose a concrete way to do so below.
- General Relativity, famously, has not only black holes, but singularities within black holes where the theory simply fails. The current theory of the Big Bang must deal with an “initial singularity” just where General Relativity fails. One hope is that a theory of quantum gravity might address this. It is familiar to hope that on the Planck scale, spacetime is discrete and might obviate singularities. Can a theory of quantum gravity find other ways to eliminate singularities in addition to and perhaps beyond discrete spacetime at a smallest scale? I propose new testable ideas about eradicating singularities below.
- Must a theory of quantum gravity “constitute” General Relativity? Might General Relativity be emergent from but not reducible to QM? I propose below that non-Aristotelian quantum variables create, by successive quantum actualization events, the Aristotelian spacetime among these events in which General Relativity operates.

## 3. Starting with Non-Locality

## 4. The Central Postulate: Decoherence Decreases Non-Locality, So Increases Locality

## 5. Deriving a Distance Metric in Hilbert Space via von Neumann Entropy

## 6. The Emergence of Discrete Classical, i.e., Actual, Space-Time by Episodic Actualizations

## 7. A Simpler Version of Remember: No Adjustable Parameters

^{N}bases to “choose one basis” by collective actualizations among N-entangled variables. An emerging basis shared among the entangled variables can also decay “slowly”. A shared presence of non-zero amplitude for a specific basis among N-entangled variables could be a basis for “remember” and vanish for each variable when it actualizes and is no longer entangled.

## 8. Relation to Causal Set Theory

## 9. Now

## 10. A New Account of Non-Locality

## 11. Spacetime Is Lorentzian

## 12. Views and Leibnitz’ Identity of Indiscernibles

## 13. A Quantum Actualization Arrow of Time Recorded in the Spacetime Constructed

## 14. A Role for Records

## 15. Past and Future Light Cones

## 16. Percolation Theory and an Inevitable UV Cutoff

## 17. An Emergent Metric and Ricci Tensor

- Propose that spacetime becomes a set of four-dimensional tetrahedra, each edge of which is 0 to ln N in length. Thus, there is no unique shape to the atoms of spacetime.
- Because actualization events occur at random moments, each possible length of each edge is equiprobable. Thus, the distribution of volumes, lengths, and areas is uniform and stationary.
- We can define the mean and variance and standard deviation of the 4D volumes in this spacetime, so also the lengths, areas and 3D volumes.
- We can define a Geodesic:
- Preliminary definition: A “straight line” between two events is the shortest distance between the two events, 1 and 2, passing from 1 to 2 by a succession of steps along paths from 1 to 2 via near neighboring events. The shortest path from 1 to 2 is a “straight line”. The length of that path is the sum of the 4D distances between each successive step along the path from 1 to 2.
- A construction to show that a shortest path between two events exists that is defined as the shortest path between each pair of events along the multistep path from event 1 to event 2:
- Start at event 1 and encapsulate it and several other neighboring events in a 4D Euclidian sphere of radius r. Choose r large enough such that each sphere centered on one event encapsulates several other events. Color the sphere blue. (2) For each event in the first blue sphere, again create a 4D Euclidian sphere centered on the first neighbor event and a few others close to it. Color all these second-generation spheres blue. (3) Iterate for N generations of blues spheres, each generation, N, further from event 1 than the preceding N − 1 set of blue spheres.
- Similarly create a set of successive pink spheres, M, around event 2.
- At some point, some one or more of the blue spheres will contain some events that are also in pink spheres.
- At that value of N and M, one or more connected paths between event 1 and event 2 exist along the blue then pink spheres.
- Consider the set of non-identical pathways between event 1 and event 2. Each pathway has some length defined by the mapping from von Neumann entropy distances to real spacetime distances. Among this set of “quasiminimal path- ways” from event 1 to event 2, one is the shortest between each adjacent pair of events between event 1 and event 2. Call this shortest pathway from event 1 to event 2 the “geodesic between event 1 and event two”. The length of the geodesic is given as the sum of the minimal distances between each adjacent pair of events between event 1 and event 2.
- An alternative is to shrink the radius, r, of all the blue and pink spheres connecting events 1 and 2 and stop shrinking r just before the pathway from event 1 to 2 becomes disconnected. Define the geodesic from event 1 to event 2 as the shortest total pathway from 1 to 2.
- Note that given a percolation threshold, Vc, as the radius, r shrinks, the set of spheres, each centered on a single event, becomes disconnected. Local connected patches of spheres at that small radius, r, are present. Further, shrinking r, corresponds to decreasing the maximum wavelength that can fit into the sphere.

- Hence, shrinking r corresponds to increasing the minimum energy of a quantum variable, say photon, that can fit into the sphere.
- For volumes above Vc, using geodesics, define triangles in 2D subspaces of this 4D space. As the lengths of each side, L, of the triangle increase relative to l, the variance, and the standard deviation of these lengths, L, falls off as the square root of L/l. For triangles with 10,000 l per side, L, the standard deviation in lengths is very small.
- The sum of the interior angles of each triangle is increasingly well defined as L/l increases. This sum is less than 180 degrees, 180 degrees or greater than 180 degrees, thus assessing if this local region of spacetime is negatively curved, flat, or positively curved.
- This is definable for 3D and 4D tetrahedra, as L/l increases. The sum of the interior angles assesses negative, flat or positive curvature.
- As L decreases to l, the standard deviation in all lengths increase and “angles” lose clearly defined values.
- We can now define a Ricci Tensor on an induced metric created on large L/l 4D tetrahedra.

## 18. Hilbert Space Has a Metric Structure

## 19. Endogenous Sources of 0 von Neumann Entropy and Possible Dark Energy

## 20. A Quantum Creation of Spacetime, QCS

## 21. Dark Matter?

## 22. A Purely Quantum Creation of Spacetime Might Eliminate the Singularities in Black Holes

## 23. A Possible Union of a Quantum Creation of Spacetime by Matter with General Relativity

## 24. Tentative Supportive Observational and Possible Experimental Evidence

- I.
- Observational Evidence on the Granularity of Space Below the Planck Scale

^{–48}m, or 13 orders of magnitude less than the Planck scale” [52].

^{–48}m, but on what grounds? Why not 10

^{–67}m? Sub-Planckian photons cast doubt on any fixed UV Energy Cutoff.

- II.
- The High-Energy Gamma Ray Spectrum: A Statistical UV Energy Cutoff

^{–21}m. For shorter wavelengths and of higher energy than the knee, the power law slope steepens to a steady −3.1 [53]. Five orders of magnitude further, at 10

^{18.5}eV and a wavelength of 10

^{–26}m, there is an ankle, where the slope increases slightly to approximately −3.0 [54].

^{–21}m. This is far above the Planck scale and sets the length scale of the present theory that maps von Neumann entropy distances to real spacetime distances on a maximum length of 10

^{–21}m. Evidence that may be consistent with something special approximately 10

^{−21}m is discussed just below.

^{–26}m, the ankle arises, and the slope decreases slightly from −3.1 to approximately −3.0. It is conceivable that the ankle reflects the 0 Entropy input into entangled systems helping to drive Dark Energy, hence an increased abundance of very tiny volumes, so an increase in the slope from −3.1 to −3.0? If confirmed, we might have evidence for the basis of Dark Energy.

- III.
- Possible Experimental Avenues via the Casimir Effect

^{2/3}. Here, L is the distance between the plates, Lp is the Planck length.

^{–20}m. For lower energy photons, the distance will shrink approximately 10

^{–21}m or less. This suggests that there may be something special approximately a length scale of 10

^{–21}m.

^{–21}m is also the energy of the gamma ray knee, 0.4 PeV, and thus it is also the proposed length scale, far above the Planck length scale, at which a transition from continuous to discrete classical spacetime occurs. In short, if m is the length scale below which classical spacetime becomes discontinuous, classical spacetime connected patches should be power law distributed at each length scale and have smaller connected patches as length scale decreases. If we propose that photons of any wavelength can only be emitted from continuous spacetime patches large enough to hold that wavelength, the number of emitted photons should decrease as a power law in decreasing wavelengths below a phase transition to discontinuous spacetime patches. If the knee is the discontinuity, at 0.4 PeV, this is now testable with the extant data confirming the observed onset at the knee of a steeper power law rate of decreasing numbers of photons observed ever further below the knee. With a better quantitative theory, this should predict the precise increase slope of the power law to −3.1 below the knee.

## 25. Disproof of the Theory

## 26. Future Directions

## 27. Conclusions

- Fully non-local entangled coherent quantum variables.
- The onset of locality via decoherence.
- A metric in Hilbert space among entangled quantum variables by the sub-additive von Neumann entropies between pairs of variables.
- Mapping from distances in Hilbert space to a classical spacetime by episodic actualization events.
- Discrete spacetime is the relations among these discrete events.
- Now is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously.
- The discrete episodic irreversible actualization events constitute a quantum arrow of time.
- The arrow of time history of these events is recorded in the very structure of the spacetime constructed.
- Actual Time is a succession of two or more actual events.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Kauffman, S.A.
Quantum Gravity If Non-Locality Is Fundamental. *Entropy* **2022**, *24*, 554.
https://doi.org/10.3390/e24040554

**AMA Style**

Kauffman SA.
Quantum Gravity If Non-Locality Is Fundamental. *Entropy*. 2022; 24(4):554.
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**Chicago/Turabian Style**

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2022. "Quantum Gravity If Non-Locality Is Fundamental" *Entropy* 24, no. 4: 554.
https://doi.org/10.3390/e24040554