# 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 challenge**s 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**__serves as a__

*l*_{p}**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__ (

__), is expressed as:__

**GUP****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**.

**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 Dwarf__s.

## Has a Black Hole a Quantum Aspect?

__ Corpuscular Gravity__ (

__), offers an__

**CG****alternative framework**, describing

__s as__

**Black Hole**__s [__

**Bose-Einstein Condensate**__4__] of

__s at the__

**Graviton****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]).

**Equation 3**. The GUP Emission Rate equation

While, for CG theory:

**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]*O * ([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.

## 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.

- 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__ - 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__ - 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.__ - ScienceDaily. "First quasiparticle Bose-Einstein condensate"
__https://www.sciencedaily.com/releases/2022/10/221025120127.htm__ - Quantamagazine. "Physicists Observe ‘Unobservable’ Quantum Phase Transition"
__https://www.quantamagazine.org/physicists-observe-unobservable-quantum-phase-transition-20230911/__