The Relation between the Uncertainty Principle and the Gravitation within the Irreconcilable Theories Enigma

The First Steps towards an Overall Vision

One of the persistent and demanding challenges in the realm of Physics is grappling with the inquiry of whether the Gravitational Force exhibits inherent quantum characteristics and, if so, how to construct a comprehensive quantum framework for Gravity that sidesteps conceptual dilemmas while maintaining predictive efficacy across all Energy scales. As the pursuit of amalgamating gravitational and quantum phenomena advances, it prompts the consideration of whether the foundational tenets of Quantum Mechanics necessitate reevaluation within the domain of Quantum Gravity (further information in here, section 1).

The integration of quantum and gravitational effects within a unified framework introduces nuanced complexities. Numerous quantum gravity models postulate a minimum length scale at the Planck Scale, indicating a fundamental limitation in the resolution of Space - Time. This Planck Length, lp serves as a natural threshold beyond which Space - Time is hypothesized to exhibit a granular, foamy structure due to inherent quantum fluctuations. Consequently, several studies advocate for the modification of the Heisenberg Uncertainty Principle (further information in here, section 2) at the quantum gravity scale to accommodate this fundamental length.

Extending the Uncertainty Concept...

It is widely recognized that a cornerstone of quantum mechanics resides in the Heisenberg Uncertainty Principle (HUP). However, it’s important to note that there isn’t a predetermined quantum limit on the precision of individual position or linear momentum measurements; theoretically, arbitrarily short distances can be probed using exceedingly high energy probes, and conversely. One prevalent generalization, known as the Generalized Uncertainty Principle (GUP), is expressed as:

[math]\LARGE{\delta x \delta p \geq \frac{\hbar}{2} \pm 2|\beta|^2 {{l_{p}}^2} \frac{\delta p}{\hbar} = \frac{\hbar}{2} \pm 2|\beta| \hbar \frac{ \delta p}{{m_{p}}^2}}[/math]

[math]\normalsize{\delta x \delta p \geq \frac{\hbar}{2} \pm 2|\beta|^2 {{l_{p}}^2} \frac{\delta p}{\hbar} = \frac{\hbar}{2} \pm 2|\beta| \hbar \frac{\delta p}{{m_{p}}^2}}[/math]

Equation 1.   The GUP Equation

where [math]\normalsize{\delta{x}}[/math] and [math]\normalsize{\delta{p}}[/math] are the position and linear momentum uncertainties, respectively; [math]\normalsize{\hbar}[/math] is the Planck constant and [math]\normalsize{m_{p}}[/math] is the mass of particle.

Here, the sign [math]\Large{\pm}[/math] denotes positive or negative values of the dimensionless Deformation Parameter, [math]\Large{\beta}[/math] [1] typically assumed to be of order unity in certain quantum gravity models, such as String Theory (further information in here, section 1). However, alternative derivations and experimental inquiries scrutinize the phenomenological implications of this fundamental parameter. The consideration that it could near zero leads to the recovery of standard quantum mechanics, implying that modifications to the HUP become significant only at the Planck scale. Furthermore, for mirror-symmetric states [2] (i.e. [math]\Large{\hat{p} = 0}[/math]), a one can derive the following modified commutator relation.

[math]\LARGE{[\hat{x}, \hat{p}] = i\hbar \left( 1 \pm |\beta| \Biggl(\frac{\hat{p}^2}{{m_{p}}^2} \right)\Biggl)}[/math]

[math]\large{[\hat{x}, \hat{p}] = i\hbar \left( 1 \pm |\beta| \Biggl(\frac{\hat{p}^2}{{m_{p}}^2} \right)\Biggr)}[/math]

Equation 2.   Modified Generalized Commutation Relation expression

[math]\Large{\hat{x}}[/math] and [math]\Large{\hat{p}}[/math] are, in the order, the position and linear momentum operators.

While the assumption of [math]\normalsize{\beta}[/math] being of order unity enjoys widespread acceptance and empirical support from various contexts beyond String Theory, the debate over the sign persists. Arguments advocating for a negative [math]\normalsize{\beta}[/math] suggest compatibility with scenarios featuring a lattice-like structure underlying the Universe, or alignment with observational constraints like the Chandrasekhar Limit [3] for White Dwarfs.

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Has a Black Hole a Quantum Aspect?

Corpuscular Gravity (CG), offers an alternative framework, describing Black Holes as Bose-Einstein Condensates [4] of Gravitons at the critical point of a Quantum Phase Transition [5]. By linking GUP black hole thermodynamics with corpuscular gravity, researchers aim to reconcile these two disparate theories. To do so, let’s examine the GUP-modified expressions of the Emission Rate of Black Holes, expanded up to the order O ([math]\normalsize{1/M^4}[/math]).

[math]\LARGE{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{GUP}}} = -\frac{1}{60(16)^2 \pi} \Biggl(\frac{m^4_p}{\hbar M^2} \pm |\beta| \frac{m^6_p}{\hbar M^4}\Biggr)}[/math]

[math]\normalsize{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{GUP}}} = -\frac{1}{60(16)^2 \pi} \Biggl(\frac{m^4_p}{\hbar M^2} \pm |\beta| \frac{m^6_p}{\hbar M^4}\Biggr)}[/math]

Equation 3.   The GUP Emission Rate equation

While, for CG theory:

[math]\LARGE{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{CG}}} = -\frac{1}{60(16)^2 \pi} \Biggl[\frac{m^4_p}{\hbar M^2} +}[/math] O [math]\LARGE{\Biggl(\frac{m^6_p}{\hbar M^4}\Biggr)}\Biggr][/math]

[math]\normalsize{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{CG}}} = -\frac{1}{60(16)^2 \pi} \Biggl[\frac{m^4_p}{\hbar M^2} +}[/math] O [math]\normalsize{\Biggl(\frac{m^6_p}{\hbar M^4}\Biggr)}\Biggr][/math]

Equation 4.   The CG Emission Rate equation

in which [math]\small{M}[/math] is the Black Hole Mass; [math]\small{t}[/math] is the time coordinate.

As said before, and specifically in the case of Black Holes, the deformation parameter can be positive or negative valued. However, it's possible to prove, at least up to the first order, that the corrections induced by these two theories exhibit the same functional dependence on the black hole mass.

Again, since the coefficient in front of the correction is predicted to be of order unity, numerical consistency between the GUP and CG expressions automatically leads to
[math]\normalsize{\beta \sim}[/math] ([math]1[/math]), which is in agreement with predictions of other models of quantum gravity. Therefore, despite their completely different underlying backgrounds, the GUP and CG approaches are found to be compatible with each other.

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The Importance of Deformation Parameter in the Unification of Quantum Theory and Gravity

Further research is needed to elucidate the precise nature of the GUP deformation parameter, and its implications for the behavior of black holes and quantum gravity. Exploring alternative scenarios, such as [math]\normalsize{\beta}[/math] as a function rather than a constant, promises to deepen our understanding of the intricate interplay between gravity and quantum mechanics.

  1. ResearchForLife7 (revisited from IOPscience). "A Discussion on Deformation Parameter Features"https://httpsresearchforlife7.com/wp-content/uploads/2024/04/A-Discussion-on-Deformation-Parameter-Features.pdf

  2. National Institutes of Health (NIH). "Mirror simmetry breaking at the molecular level" https://www.ncbi.nlm.nih.gov/pmc/articles/PMC38075/pdf/pnas01525-0160.pdf

  3. Space.com. "The Chandrasekhar limit: Why only some stars become supernovas" https://www.space.com/chandrasekhar-limit#:~:text=What%20is%20the%20Chandrasekhar%20limit,the%20mass%20of%20the%20sun.

  4. ScienceDaily. "First quasiparticle Bose-Einstein condensate" https://www.sciencedaily.com/releases/2022/10/221025120127.htm

  5. Quantamagazine. "Physicists Observe ‘Unobservable’ Quantum Phase Transition" https://www.quantamagazine.org/physicists-observe-unobservable-quantum-phase-transition-20230911/

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