# A Quantum Gravity problem: Exploring a Continous Early Universe?

## The Singularity Role in Cosmological Research

The __ Big Bang__

__(__

**Singularity**__Figure 1__) poses a significant challenge within

**standard cosmological**and

__(ies). The__

**Inflationary Theory****divergence**of

**tensors**in the

**Einstein Equations**near the

**singularity**suggests that classical

**General Relativity**may not be applicable under such extreme conditions.

__s likely become__

**Quantum Gravitational Effect****relevant**in proximity to the singularity, prompting the exploration of quantum theories of

**Gravity**. Various approaches, including

__[__

**String Theory**__1__] and

__[__

**Loop Quantum Gravity**__2__], have been pursued to address this issue. A quantum theory of gravity emerges from the unification of General Relativity and

**Quantum Mechanics**, potentially introducing a fundamental minimal length and a non-zero minimal

**Uncertainty**in position measurements.

## Fundation of Quantum World

To extend the __ Uncertainty Principle__ (

__Equation 1__) and incorporate a non-zero minimal uncertainty, [math]\normalsize{x_{0}}[/math] in Position, [math]\normalsize{x}[/math] the

**Commutation relation**between

**Position**([math]\normalsize{\hat{X}}[/math]) and

**Linear Momentum**([math]\normalsize{\hat{P}}[/math])

__s is modified. This modification yields a__

**Operator****generalized uncertainty relation**, leading to novel consequences in the

__of free particle systems.__

**Statistical Mechanics****Equation 1**. Classical Heinsenberg Uncertainty Principle expression

in which [math]\normalsize{\Delta{x}}[/math] and [math]\normalsize{\Delta{p}}[/math] are, in the order, the Position and Linear Momentum Operators uncertainties; while [math]\normalsize{\hbar}[/math] is the reduced **Planck Constant**.

For instance, the __ Phase Space__ measure is adjusted accordingly for particles
adhering to the generalized uncertainty principle.
Several such generalizations exist, each with its own characteristics. For instance, the

**Kempf - Mangano - Mann**(

**K.M.M.**) [

__3__]

**Deformation Commutation**relation is represented as:

**Equation 2**. Kempf - Mangano - Mann Commutation relation

**Equation 3**. Maggiore's Generalized Commutation relation

where [math]\normalsize{\lambda}[/math] denotes an extremely small **Length Parameter**; [math]\normalsize{m}[/math] is the **Mass** and [math]\normalsize{c}[/math] is the **Light Speed**.

Both commutation relations yield the
generalized uncertainty relation,

**Equation 4**. Maggiore's and K.M.M. Commutation Relations combination

with the **second relation**
converging to this form in a **suitable limit**. A three-dimensional extension preserving **rotational
symmetry** is proposed, leading to **noncommutative geometry**.

## The Statistical Approach

The **Statistical Mechanics** of **free ultra-relativistic particle**s subject to the **Kempf-Mangano-Mann deformation** are investigated within the __ Grand Canonical Ensemble__ approach. In traditional
quantum statistical mechanics, where the parameter [math]\normalsize{\lambda}[/math] equals zero, the

**Heisenberg Uncertainty Principle**partitions phase space into cells of volume [math]\normalsize{h^3}[/math], with [math]\normalsize{h}[/math] representing the Planck constant. Calculating the

**Grand Canonical Partition Function**^{1}for a

**non-relativistic**

**Ideal Gas**^{2}in conventional quantum mechanics involves rewriting the sum over

**one-particle states**in terms of

**integrals**. In the

**high energy - temperature limit**, characterized by

**negligible particle mass**and

**inter - particle forces**, particles behave akin to free ultra-relativistic particles, obeying

__. Consequently, the__

**Maxwell - Boltzmann Statistics****grand canonical partition function**takes the following form

**Equation 5**. Grand Canonical Partition Function at High Energy - Temperature conditions

Contrary to conventional quantum statistical mechanics, this analysis reveals that the **entropy** and
**internal energy** of the system attain **finite values** as **temperature** approaches **infinity**. Furthermore,
the possibility of **negative temperatures** and **pressures** in such systems is explored, elucidating
differences from __ Spin__ systems.

**Figure 1**. A geometric horizontal Section of a Space - Time Fabric (white grid-like lattice) deformation with a Singularity (black part at the bottom)

## No more Big Bang Singularities

The **modified Equations of State** due to the generalized uncertainty principle are considered in the context of the **early Universe dynamics** [__5__] (__Figure 2__). Additionally, the study suggests that negative temperatures may lead to alternative solutions for the __ Friedmann Equations__ (

__Equation 6__), potentially altering the

**history of the Universe**[

__6__]. It is proposed that these modifications alone can potentially resolve the big bang singularity by ensuring a

**constant Entropy**, thus avoiding singularities in the dynamical equations. This relation implies constant entropy, consistent with

**Reversibility assumptions**[

__7__].

**Equation 6**. The two (a and b label) Friedmann Equations

[math]\Large{a}[/math] is a scalar factor (single and double upper dots represent, in the order, the first and second derivatives with respect to proper time); [math]\Large{\rho}[/math] is the **Mass Density** of Universe and [math]\Large{P}[/math] is the **Pressure**.

Substituting the equations of state in the Friedmann equations, it leads to following result.

**Equation 7**. The reformuled Friedmann Equation

Importantly, the **minimum scale
factor** is **greater than zero**, contrary to conventional cases. The singularity occurs at **infinite
temperature** or [math]\normalsize{x = 0}[/math], but the term [math]\normalsize{a^3}[/math] does not tend to zero as [math]\normalsize{x \rightarrow 0}[/math], indicating **finite
Entropy** at [math]\normalsize{x = 0}[/math], a **non-trivial consequence** of the **minimal uncertainty principle**.

**Figure 2**. Pictorial representation of a Growing Newborn Universe

## A very Important Principle

So the **potential** of the **generalized uncertainty principle** to address
fundamental cosmological singularities, such as the **big bang singularity**, has been highlighted. By integrating this principle
into the **statistical mechanics** of particle systems and considering its **implications** for the **early universe**,
novel insights into the **nature of Space - Time** and the dynamics of **cosmic evolution** emerge. Further
exploration of these concepts promises to deepen our understanding of the **fundamental nature** of
the **Universe**.

- American Scientist. "Is String Theory Even Wrong?"
__https://www.americanscientist.org/article/is-string-theory-even-wrong__ - Nature. "Experimental simulation of loop quantum gravity on a photonic chip"
__https://www.nature.com/articles/s41534-023-00702-y__ - ResearchForLife7 (revisited from arXiv). "K.M.M. Overview on Minimal Uncertainty Length"
__https://httpsresearchforlife7.com/wp-content/uploads/2024/03/K.M.M.-Overview-on-Minimal-Uncertainty-Length.pdf__ - ResearchForLife7 (revisited from ScienceDirect). "A Depth on the Maggiore's Commutation Relations"
__https://httpsresearchforlife7.com/wp-content/uploads/2024/03/A-Depth-on-the-Maggiore-s-Commutation-Relations.pdf__ - IOPscience. "Critical dynamics in the early universe"
__https://iopscience.iop.org/article/10.1088/0264-9381/10/S/009__ - Scientific American. "Origin of the Universe"
__https://www.jstor.org/stable/26001524__ - ResearchGate. "Time Reversibility and the Logical Structure of the Universe
"
__https://www.researchgate.net/publication/236616749_Time_Reversibility_and_the_Logical_Structure_of_the_Universe__