Far down the Quantum Chemistry: how to Intoduce an “Entangled” Electron Correlation


Far down the Quantum Chemistry: how to Introduce an “Entangled” Electron Correlation

Exceeding the Old Methods

In Quantum Chemistry, the Hartree–Fock (HF) Method a, which describes interacting Fermion systems using an effective Single-Particle Model [1], is widely used to approximate the Electronic Structure of atoms and molecules.

However, this method neglects Electron Correlation, which results from the inherent interactions between Electrons in quantum systems. The Correlation Energy [2], defined as the energy difference between the HF limit and the exact solution (Equation 1) of the nonrelativistic Schrödinger Equation (further information in here, Section 2), is one measure of this correlation.

[math]\LARGE{E_{c} = |E^{Exact} - E^{HF}| }[/math]

[math]\normalsize{E_{c} = |E^{Exact} - E^{HF}| }[/math]

Equation 1.   The Electronic Energy in HF Method

Nevertheless, other metrics, such as Statistical Correlation Coefficients [3] and Shannon Entropy, have been proposed to quantify electron correlation. Despite these methods, electron correlation remains challenging to calculate accurately for Large Systems [4]. 

Quantum Entanglement (further information in here, Ref. 2), a fundamental concept in quantum mechanics, offers a promising alternative for measuring electron correlation. Unlike traditional measures, entanglement is directly observable and represents a non-classical correlation between quantum systems.

A New Definition of Quantum Entanglement

To quantify entanglement, one can consider a pure two-electron state in a 2m-dimensional Spin-Orbital space, represented by fermionic Annihilation and Creation Operators (further information in here, Section 2), [math]\normalsize{c_{a}}[/math] and [math]\normalsize{{c^{\dagger}}_{a}}[/math], with [math]\small{|0 \rangle}[/math] as the vacuum state. The general form of a two-electron state, [math]\small{| \Psi \rangle}[/math] can be written as:

[math]\LARGE{| \Psi \rangle = \sum_{a, b \in \{1, 2, \ldots, 2m\}} \omega_{a, b} c^\dagger_a c^\dagger_b |0 \rangle}[/math]

[math]\normalsize{| \Psi \rangle = \sum_{a, b \in \{1, 2, \ldots, 2m\}} \omega_{a, b} c^\dagger_a c^\dagger_b |0 \rangle}[/math]

Equation 2.   The form of the Electronic Wavefunction for a two-electron state

where [math]\normalsize{x_{a, b}}[/math] represents the Antisymmetric Expansion Coefficient Matrix [5], satisfying [math]\normalsize{x_{a, b} = - x_{b, a}}[/math]. Using this representation, we can derive a Reduced Density Matrix (further information in here, Section 2), [math]\normalsize{\rho}[/math] by tracing out all but one Spatial Orbital, resulting in a [math]\normalsize{4 \times 4}[/math] matrix. Then one can define the Von Neumann Entropy, like below.

[math]\LARGE{S(\rho) = - \operatorname{Tr}(\rho \log_2 \rho)}[/math]

[math]\normalsize{S(\rho) = - \operatorname{Tr}(\rho \log_2 \rho)}[/math]

Equation 3.   The Von Neumann Entropy equation

With [math]\normalsize{\rho}[/math] of the following form:

[math]\LARGE{\rho = \operatorname{Tr}|\Psi \rangle \langle \Psi|}[/math]

[math]\normalsize{\rho = \operatorname{Tr}|\Psi \rangle \langle \Psi|}[/math]

Equation 4.   The expansion of the Reduced Density Matrix

This approach provides a measure of the entanglement for atomic and molecular systems, focusing on the von Neumann entropy of the reduced density matrix.

Is it an Efficient Model?

Using the derived reduced density matrix, the entanglement for various systems, focusing on the Hydrogen Molecule ([math]\small{H_{2}}[/math]) [6], as an example, can be calculated. The entanglement is evaluated as a function of the Interatomic Distance, R.

These calculations show that entanglement (Equation 6) and electron correlation exhibit similar trends, with maximum entanglement occurring at specific interatomic distances. This behavior aligns with previous findings, indicating that entanglement can be an effective metric for measuring electron correlation.

Additionally, if one explores a model system of two spin-1/2 electrons with an Exchange Coupling Constant a, [math]\small{J}[/math] and a transverse Magnetic Field strength, B. The general Hamiltonian for this system is given by:

[math]\LARGE{H = - \frac{J}{2} (1 + \gamma) \sigma^x_1 \sigma^x_2 - \frac{J}{2} (1 - \gamma) \sigma^y_1 \sigma^y_2 - B \sigma^z_1 \otimes I_2 - B I_1 \otimes \sigma^z_2}[/math]

[math]\small{H = - \frac{J}{2} (1 + \gamma) \sigma^x_1 \sigma^x_2 - \frac{J}{2} (1 - \gamma) \sigma^y_1 \sigma^y_2 - B \sigma^z_1 \otimes I_2 - B I_1 \otimes \sigma^z_2}[/math]

Equation 5.   The Hamiltonian form for a two-electron state

where the subscripts, [math]\small{1}[/math] and [math]\small{2}[/math], are the two electrons, respectively; [math]\normalsize{\sigma^{x}}[/math], [math]\normalsize{\sigma^{y}}[/math] and [math]\normalsize{\sigma^{z}}[/math] are Pauli Matrices, and [math]\normalsize{\gamma}[/math] the Degree of Anisotropy ([math]\small{I}[/math]). This model provides a simplified framework for examining entanglement in a quantum system.

[math]\LARGE{S = \frac{1}{2} \log_2\left( \frac{1}{4} + \frac{1}{4 + {\lambda^2}} + \right) + \frac{1}{\sqrt{4 + \lambda^{2}}} \log_2 \frac{( \sqrt{4 + \lambda^{2}} - 2} {\sqrt{4 + \lambda^{2}} + 2}}[/math]

[math]\normalsize{S = \frac{1}{2} \log_2\left( \frac{1}{4 + {\lambda^2}} + \frac{1}{4} + k^2 \right) + \frac{1}{\sqrt{4 + \lambda^{2}}} \log_2 \frac{( \sqrt{4 + \lambda^{2}} - 2} {\sqrt{4 + \lambda^{2}} + 2}}[/math]

Equation 6.   The Quantum Entanglement form expressed via Von Neumann Entropy equation

[math]\large{\lambda}[/math] represents the Eigenvalue for the Hamiltonian of the two spin system (further information in here, Section 3).

Figure 1.   A Pictorial Representation of a Wormhole Interior

Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

Figure 2.   Illustration of the Hologram for a Sphere

A Cross-cutting Solution with no Traces of Complexity...

The results demonstrate that quantum entanglement can serve as an effective measure of electron correlation in quantum chemistry. The use of von Neumann entropy allows for a more observable and intuitive understanding of electron correlation, without relying on traditional methods that require complex wave function calculations.

This approach has implications for larger atomic and molecular systems and can be extended to other quantum systems, offering a robust alrternative for evaluating electronic structures in atoms and molecules. Future work will explore the application of this method to more complex systems and its potential for advancing quantum chemistry calculations.

  1. Springer. "Euclidean Path Integrals"https://link.springer.com/chapter/10.1007/978-1-4612-0009-3_14

  2. Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates" https://inspirehep.net/literature/676262

  3. arXiv.org. "Editorial: New frontiers in holographic duality" https://arxiv.org/abs/2210.03315

  4. Big Think. "Are we living in a baby universe that looks like a black hole to outsiders?" https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/

  5. Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for Brain-Inspired Competitive Learning" https://onlinelibrary.wiley.com/doi/full/10.1002/aisy.202100114

  6. Nature. "Strongly enhanced effects of Lorentz symmetry violation in entangled Yb+ ions" https://www.nature.com/articles/nphys3610

  7. ResearchForLife7 (revisited from arXiv). "Discussion on Consistent Truncations: Uplifting the GPPZ Solutions" https://httpsresearchforlife7.com/wp-content/uploads/2024/05/Discussion_on_Consistent_Truncations__Uplifting_the_GPPZ_Solutions.pdf

  8. Inspire HEP. "Fermion Zero Modes and Topological-charge on a Domain Wall of the D-brane-like Dot" https://inspirehep.net/literature/1353725


An Investigation in Strings Theory to outline Supersymmetric Wormholes Nature

Low-Dimensions vs High-Dimensions Theories

In recent years, significant progress has been made in understanding the Euclidean Path Integral (further information in here, Ref. 3) [1] for low-dimensional quantum gravity theories like Jackiw–Teitelboim (JT) Gravity. These insights often involve summing over Saddle Points with varying Topology(ies). In contrast, for higher-dimensional theories like those involving Einstein - Hilbert Gravity coupled with matter, the situation is less straightforward.

What About Wormholes?

Wormholes (Figure 1), which are Geometrical Structures connecting separate regions of Space - Time, are key to understanding certain quantum gravity aspects. However, their presence in the Euclidean path integral can lead to puzzles regarding Unitarity and Non-Factorization [2] of Correlation Functions. This raises questions about the consistency of the Holographic Duality (Figure 2) [3] in such cases.

Additionally, wormholes suggest the possibility of Baby Universes [4], impacting the Swampland Program's guidelines. To address these issues, a deeper exploration of higher-dimensional Euclidean wormholes within String Theory (further information in here, Section 1) is needed.

Constructing Euclidean Wormholes in Supergravity

Building Euclidean wormhole geometries generally requires a source of negative Euclidean Energy [5]. In string theory, this is achieved using axion (further information in here, Section 2) fields. Analytical continuation of a Lorentzian Theory [6] with axions to the Euclidean Regime can yield a negative energy-momentum tensor (further information in here, Section 1), providing the necessary conditions for wormhole formation.


Figure 1.   A Pictorial Representation of a Wormhole Interior

Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

Step by Step from a Five-Dimensional Framework

One can start with a five-dimensional metric, which is written in the form:

[math]\LARGE{ds{_5}{^2} = dr^2 + e^{2A(r)} ds^2_{\mathbf{R}^4}}[/math]

[math]\Large{ds{_5}{^2} = dr^2 + e^{2A(r)} ds^2_{\mathbf{R}^4}}[/math]

Equation 1.   The form of a Five-Dimensional Metric

where [math]\small{ds^2_{\mathbf{R}^4}}[/math] is the Flat Metric on Euclidean Space, [math]\small{\mathbf{R}^4}[/math], and the Metric Function, [math]\small{A}[/math] ([math]\small{r}[/math]) depends solely on the radial coordinate, [math]\small{r}[/math]. The Euclidean gravity path integral compactifies [math]\small{\mathbf{R}^4}[/math] to the Normal Space, [math]\small{T^4}[/math].

Supersymmetry is broken from [math]\small{N = 4}[/math] to [math]\small{N = 1}[/math], with distinct Quantum Field Theory(ies) (QFTs) on each side of the wormhole, differentiated by their Yang-Mills coupling constants and other parameters. A Special Point exists where the wormhole's "neck" shrinks to zero, leading to a Singular Metric resembling the well-known GPPZ Solution [7].

... Here is Supersymmetry!

To ensure supersymmetry, the solutions must satisfy the first-order Bogomol'nyi - Prasad - Sommerfield (BPS) Equations (Equation 2 and Equation 3). These equations guide the construction of the supersymmetric Euclidean wormhole solutions. A key point in this analysis is whether the wormhole solutions dominate over corresponding disconnected geometries with the same Boundary Conditions.

[math]\LARGE{\mathcal{E}{_A} \equiv A' + \frac{1}{3} W = 0}[/math]

[math]\Large{\mathcal{E}{_A} \equiv A' + \frac{1}{3} W = 0}[/math]

Equation 2.   First BPS Equation: the form of Total Energy in the Euclidean Space

in which [math]\small{A'}[/math] is the derivative of the metric function and [math]\normalsize{W}[/math] is a real Supepotential.

[math]\LARGE{\mathcal{E}^{i} \equiv (z^{i})' - \frac{1}{3} K^{i\bar{J}} \partial_{\bar{J}} W = 0}[/math]

[math]\Large{\mathcal{E}^{i} \equiv (z^{i})' - \frac{1}{3} K^{i\bar{J}} \partial_{\bar{J}} W = 0}[/math]

Equation 3.   Second BPS Equation: the Energy for each point on the Euclidean Space

[math]\Large{i}[/math] is a generic Point on Euclidean space; ( [math]\Large{z^{i}}[/math])[math]\Large{'}[/math] is the first derivative of the Scalar Field, [math]\Large{z^{i}}[/math] of each generic point, with respect to the radial coordinate; and [math]\Large{K^{i\bar{J}}}[/math] is the Kähler Metric.

Using a Consistent Truncation7 of maximal five-dimensional Supergravity with an SO(6) Gauge Group, a specific set of BPS equations is derived. A simplified set of field variables and a constant of motion help in solving these equations. A critical aspect is the interpretation of these solutions within the context of the holographic duality and the implications for non-factorization.


Figure 2.   Illustration of the Hologram for a Sphere

Implications of Wormholes Discovery

The existence of Euclidean wormholes and their role in quantum gravity path integrals raise significant questions about unitarity, holographic duality, and the Swampland program. Future work involves uplifting these wormhole solutions to ten dimensions and exploring the implications of imaginary scalars in the Euclidean regime. The factorization puzzle in Holography remains an open question, with possible solutions involving supersymmetry considerations and potential Fermion Zero-Mode [8] effects. This area of research holds great potential for advancing our understanding of quantum gravity and its holographic connections.

  1. Springer. "Euclidean Path Integrals"https://link.springer.com/chapter/10.1007/978-1-4612-0009-3_14

  2. Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates" https://inspirehep.net/literature/676262

  3. arXiv.org. "Editorial: New frontiers in holographic duality" https://arxiv.org/abs/2210.03315

  4. Big Think. "Are we living in a baby universe that looks like a black hole to outsiders?" https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/

  5. Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for Brain-Inspired Competitive Learning" https://onlinelibrary.wiley.com/doi/full/10.1002/aisy.202100114

  6. Nature. "Strongly enhanced effects of Lorentz symmetry violation in entangled Yb+ ions" https://www.nature.com/articles/nphys3610

  7. ResearchForLife7 (revisited from arXiv). "Discussion on Consistent Truncations: Uplifting the GPPZ Solutions" https://httpsresearchforlife7.com/wp-content/uploads/2024/05/Discussion_on_Consistent_Truncations__Uplifting_the_GPPZ_Solutions.pdf

  8. Inspire HEP. "Fermion Zero Modes and Topological-charge on a Domain Wall of the D-brane-like Dot" https://inspirehep.net/literature/1353725


Traveling the Heart of a Supernova Explosion via a Dynamic Stream of Neutrinos

What is a Supernova?

Type II Supernovae (SN), triggered by the Gravitational Collapse of Massive Stars, emit a substantial portion of their energy in the form of Neutrinos (Figure 1). These elusive particles, constituting what is known as the Supernova Relic Neutrino (SRN) background [1], could potentially be detected by large underground neutrino detectors, such as the Super - Kamiokande and Sudbury Neutrino Observatory (SNO) detectors.

The primary objective of these detectors is to capture traces of this elusive SRN background. The SN II Rate Evolution [2], coupled with the Metal Enrichment History [3], forms the basis of predicting the SRN Flux. By relating observations of Star Formation and metallicity enrichment, one establishes a robust framework for estimating the supernova rate and, consequently, the SRN flux.


Figure 1.   Representation of an Exploiting Supernova

The Photophysical Fingerprint of Neutrinos

The composition of neutrinos reaching Earth, originating from past supernovae, relies on several factors. First, it hinges on the differential flux of neutrinos per unit energy interval emitted by each supernova. Secondly, it is influenced by the distribution of supernova rates with respect to Redshift. Additionally, it's contingent upon a Friedmann-Robertson-Walker Cosmology, typically characterized by parameters such as the Hubble Parameter, H0 and the Matter Density Parameter, [math]\small{\Omega_{0}}[/math] (further information in here, Section 4).

The Spectrum (Figure 2) of neutrinos emitted from a supernova is characterized by a Fermi-Dirac Distribution with zero Chemical Potential, normalized to the total energy emitted by the supernova. For each Neutrino Species, [math]\small{\nu}[/math], the Neutrino Luminosity, L [math]\small{{_{\nu}}^{S}}[/math]([math]\small{\epsilon}[/math]) can be defined as:

L [math]\LARGE{{_{\nu}}^{S} (\epsilon) = E_{\nu} {120 \over 7{\pi}^4} {{\epsilon}^2 \over T_{\nu}^{4}} \Biggl[e^{\epsilon \over T_{\nu}} + 1 \Biggr]^{-1}}[/math]

L [math]\large{{_{\nu}}^{S} (\epsilon) = E_{\nu} {120 \over 7{\pi}^4} {{\epsilon}^2 \over T_{\nu}^{4}} \Biggl[e^{\epsilon \over T_{\nu}} + 1 \Biggr]^{-1}}[/math]

Equation 1.   Neutrinos Luminosity Equation

where the apex [math]\small{S}[/math] stands for "Spectrum"; [math]\small{\epsilon}[/math] is the Neutrino Species Energy for a singluar neutrino; [math]\small{E_{\nu}}[/math] and [math]\small{T_{\nu}}[/math] are, in the order, the Neutrino Species Total Energy and the Temperature Parameter derived by the neutrinosphere during the collapse, and both dependent on the Progenitor Mass of the supernova. However, obtaining the Initial Mass Function (IMF)-averaged neutrino flux is simplified because [math]\small{T_{\nu}}[/math] doesn't vary significantly with the progenitor mass.

Assuming that the Supernova Rate, [math]\small{N_{SN}}[/math]([math]\small{z}[/math]) follows the Metal Enrichment Rate, it can be expressed like below.

[math]\LARGE{N_{SN} (z) = {\dot{\rho}_{Z} (z) \over \langle M_{Z} \rangle}}[/math]

[math]\large{N_{SN} (z) = {\dot{\rho}_{Z} (z) \over \langle M_{Z} \rangle}}[/math]

Equation 2.   The form of Supernova Rate

[math]\Large{z}[/math] is the Redshift value; [math]\large{Z}[/math] is Atomic Number of a chemical element; [math]\large{\dot{\rho}_{Z} (z)}[/math] represents the Metal Enrichment Rate per unit comoving volume and [math]\large{\langle M_{Z} \rangle}[/math] denotes the Average Yield of Heavy Elements per supernova.

To track the metal enrichment rate, one can assume a constant supernova rate at higher redshifts ([math]\small{z > 1} [/math]) due to the limited knowledge of high-redshift evolution. This assumption is supported by various independent studies showing consistent evolutionary patterns.

Looking for Elusive Particles!

Detecting relic neutrinos from supernovae poses significant challenges, particularly across various energy ranges. SuperKamiokande, for instance, has an observable energy window estimated to span from 19 to 35 MeV. However, below 10 MeV, the contribution from neutrinos generated by reactors and those from Earth is expected to overshadow any relic neutrino signal. Beyond 10 MeV but still below the observable window, background sources include solar neutrinos, external radiation, and events induced by Cosmic - Ray Muons [4] within the detector.

Atmospheric neutrinos become the primary background above 19 MeV. As energy surpasses approximately 35 MeV, the rapidly diminishing flux of relic neutrinos becomes less significant compared to atmospheric neutrinos. The Efficiency [5] of this detector within the observable energy window is assumed to be 100%.

Given the detector's specifications, the predicted event rate at SuperKamiokande for SN relic neutrinos can be calculated, and it is suggested being a peaked distribution. The flux at the detector is assessed across various energy ranges, with considerations for background sources and detector capabilities.

Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group


Figure 2.   The Rainbow is an example of Electromagnetic Spectrum including Radiation in the Visible Energy Window

A Troublesome Research

One can present conservative upper bounds on the expected SRN event rates, indicating challenges in detecting these elusive particles. Despite advancements in detector technology, the Signal - to - Background Ratio remains a significant obstacle. Future research directions, including refining metal enrichment history models and exploring alternative detection strategies will be discussed.

  1. ScienceDirect. "Discovery potential for supernova relic neutrinos with slow liquid scintillator detectors "https://www.sciencedirect.com/science/article/pii/S0370269317302629

  2. STScl. "The Evolution of the Cosmic Supernova Rates" https://www.stsci.edu/files/live/sites/www/files/home/jwst/about/history/design-reference-mission-drm/_documents/drm11.pdf

  3. Springer. "Metal enrichment history of the proto-galactic interstellar medium" https://link.springer.com/article/10.1023/A:1019527307631

  4. PhysicsOpenLab.org. "Cosmic Ray Muons & Muon Lifetime " https://physicsopenlab.org/2016/01/10/cosmic-muons-decay/

  5. KNS.org. "Detection Efficiency Calculation and Evaluation for Condenser Off-gas Radiation Monitoring System " https://www.kns.org/files/pre_paper/45/21S-123-%EA%B9%80%EC%9B%90%EA%B5%AC.pdf


How Can Teleportation involve Time Travels?

Albert Einstein's Theory of General Relativity permits the existence of Closed-Timelike Curves (CTCs) [1], which are paths within Space - Time that, if traversed, would enable a traveler to interact with their own Past self, whether that traveler be human or elemental particle. Kurt Gödel was among the first to highlight the possibility of CTCs, and subsequent research has proposed various Space - Time configurations accommodating these curves.

However, such scenarios of Time Travel inevitably introduce paradoxes, such as the infamous Grandfather Paradox [2], wherein the time traveler inadvertently alters the past in a way that prevents their own existence. This concept troubled even Einstein, who was close friends with Gödel. The reconciliation of CTCs with quantum mechanics poses a formidable challenge, tackled through various approaches, including Path-Integral Techniques [3].

The Removed “Memories” Approach...

Any theory aiming to unify quantum mechanics and gravity must address the complexities inherent in closed timelike curves, which introduce nonlinearities that challenge the Linearity [4] of conventional quantum mechanics. Deutsch proposed a resolution in his influential work, suggesting a Self-Consistency Condition (Equation 1) concerning the states within CTCs. This condition demands equivalence between measurements at the CTC's entrance and exit. However, this formulation necessitates the assumption of Factorization, implying invalidation of Future [5] "memories". However, Deutsch's theory has faced criticism for apparent inconsistencies.

[math]\LARGE{\rho_{CTC} = Tr_{A}[U(\rho_{A} \otimes \rho_{CTC})U^{\dagger}}][/math]

[math]\large{\rho_{CTC} = Tr_{A}[U(\rho_{A} \otimes \rho_{CTC})U^{\dagger}}][/math]

Equation 1.   Deutsch Self-Consistency Condition form

where [math]\small{\rho_{CTC}}[/math] is the density matrix (further information in here, Section 1) of the system state, [math]\small{A}[/math] inside the CTC; [math]\small{Tr_{A}}[/math] is the Trace of [math]\small{A}[/math]; [math]\small{U}[/math] is the Unitary Matrix; [math]\small{\rho_{A}}[/math] is the density matrix of [math]\small{A}[/math] and [math]\small{U^{\dagger}}[/math] is the transpose unitary matrix.

A New View: Overcoming the CTCs

In contrast, while acknowledging the strangeness of time travel quantum mechanics, P-CTCs, based on the Novikov Principle, appear to offer a less problematic framework. The concept of Probabilistic Closed Timelike Curves (P-CTCs) [6] was initially conceived to tackle the enigma posed by the integration of quantum mechanics into the framework of General Relativity, particularly concerning closed-timelike curves. However, its implications extend beyond this specific domain, offering insights into the potentiality of time travel in alternative scenarios.

N   [math]\LARGE{[\rho]\propto {Tr_{E}[U_{AE}] = C_{A}\rho C_{A}^{\dagger}}}[/math]

N  [math]\large{[\rho]\propto{Tr_{E}[U_{AE}] = C_{A}\rho C_{A}^{\dagger}}}[/math]

Equation 2.   P-CTC External System Time Evolution Equation

N [math]\small{[\rho]}[/math] is the Time Evolution of External System; [math]\small{\rho}[/math] is the density matrix of external system; [math]\small{Tr_{E}}[/math] is the trace of the Hilbert space (further information in here, Section 2), [math]\small{E}[/math] over the system into the CTC; [math]\small{C_{A}}[/math] is the partial trace of [math]\small{A}[/math]; [math]\small{U_{AE}}[/math] is the unitary matrix coupling the internal and external systems; [math]\small{{C_{A}}^{\dagger}}[/math] is the transpose partial trace of [math]\small{A}[/math].

Fundamentally, any quantum theory that permits non-linear processes like the Projection onto specific states, such as the Entangled States (further information in here, references) associated with P-CTCs, inherently allows for the prospect of time travel, even in the absence of Space - Time configurations supporting closed-timelike curves. The P-CTS mechanism translates mathematically to the time evolution of the external system being , with the absence of evolution enforced if certain conditions (Equation 2) are met. The paradigm of non-general relativistic P-CTCs can be instantiated through the generation and projection onto entangled pairs of particle-antiparticle. This method mirrors renowned Wheeler's Thought Experiment [7] of a telephone call through time.


Figure 1.   A typical Macroscopic Example of Quantum Tunneling: a Ball (Subatomic Particle) which overcomes a Wall (Potential Energy Barrier)

Although the process of projection is inherently nonlinear, defying deterministic implementation within conventional quantum mechanics, it can be executed in a probabilistic manner. Consequently, experimental validation of P-CTCs is achievable through Quantum Teleportation experiments, where outcomes corresponding to the desired entangled-state output are selectively post-processed. Should it transpire that the linearity of quantum mechanics is merely an approximation, and projection onto specific states indeed manifests, such occurrences could potentially be witnessed at the singularities (further information in here) of black holes.

In such a scenario, even in the absence of general relativistic closed-timelike curves, the realization of time travel might still be feasible. The theoretical framework of P-CTCs elucidates that quantum time travel can be conceived as a form of retrograde Quantum Tunneling (Figure 1), permitting temporal traversal devoid of a classical trajectory from future to past. P-CTCs rely on Destructive Interference (Figure 2) to prevent self-contradictory events, emphasizing a different self-consistency condition from Deutsch's approach.

Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

Two Different Perspectives

Illustrating the link between P-CTCs and teleportation provides further insights, showcasing their behavior through Qubits. This demonstration underscores the compatibility of P-CTCs with Higher-Dimensional Systems and the extension to infinite-dimensional scenarios. Presently, no definitive conclusion favors either approach (Deutsch or P-CTCs), given their respective foundations and consistency with different theoretical frameworks. The aspiration in elaborating on the theory of P-CTCs is that it may furnish valuable insights for formulating a Quantum Theory of Gravity. By shedding light on one of the most enigmatic ramifications of general relativity—the prospect of time travel—this theory may contribute significantly to our understanding of Gravity at the quantum level.


Figure 2.   A classic Experiment of Waves Interference: Ripples in the Water

  1. arXiv. "Can we travel to the past? Irreversible physics along closed timelike curves "https://arxiv.org/pdf/1912.04702.pdf

  2. ResearchGate. "Grandfather paradox from a new perspective" https://www.researchgate.net/publication/361446083_Grandfather_paradox_from_a_new_perspective#fullTextFileContent

  3. Galileo.phys. "Path Integrals in Quantum Mechanics" https://galileo.phys.virginia.edu/classes/751.mf1i.fall02/PathIntegrals.htm

  4. ScienceDirect. "Consistency and linearity in quantum theory" https://pdf.sciencedirectassets.com/271541/1-s2.0-S0375960100X01843/1-s2.0-S0375960198002898/main.pdf?X

  5. LinkedIn. "Time Travel is Real: Unraveling the Wonders of Traveling to the Future" https://www.linkedin.com/pulse/time-travel-real-unraveling-wonders-traveling-future-manjunath-m-r/

  6. American Physical Society. "Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency" https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.106.040403

  7. Horizon IIT. "The Delayed Choice Quantum Eraser – does the future affect the past?" https://horizoniitm.github.io/dcqe/


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.

Figure 1.   write

write (Figure 3) write,

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.

Figure 2.   write write

Figure 3.   write

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/

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 Inflationary Theory(ies). The divergence of tensors in the Einstein Equations near the singularity suggests that classical General Relativity may not be applicable under such extreme conditions. Quantum Gravitational Effects likely become 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]) Operators is modified. This modification yields a generalized uncertainty relation, leading to novel consequences in the Statistical Mechanics of free particle systems.

[math]\LARGE{\Delta{x}\Delta{p} \geq {{\hbar} \over {2}}}[/math]

[math]\Large{\Delta{x}\Delta{p} \geq {{\hbar} \over {2}}}[/math]

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:

[math]\LARGE{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2})}[/math]

[math]\Large{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2})}[/math]

Equation 2.   Kempf - Mangano - Mann Commutation relation

Another example is Maggiore's generalization [4], as follows.

[math]\LARGE{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2} + {m^2{c^2}})}[/math]

[math]\Large{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2} + {m^2{c^2}})}[/math]

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,

[math]\LARGE{\Delta{x}\Delta{p} \geq {1 + \lambda^2{\Delta{p}^2}}}[/math]

[math]\Large{\Delta{x}\Delta{p} \geq {1 + \lambda^2{\Delta{p}^2}}}[/math]

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 particles 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 Function1 for a non-relativistic Ideal Gas2 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 Maxwell - Boltzmann Statistics. Consequently, the grand canonical partition function takes the following form

[math]\LARGE{ln{Z} = \beta{P}{V} = \int{\frac{e^{-{\beta{E}}{d^3{x}}{d^3{p}}}} {h^3(1 + {\lambda^2}{p^2})^3}}}[/math]

[math]\Large{ln{Z} = \beta{P}{V} = \int{{e^{-{\beta{E}}{d^3{x}}{d^3{p}}}} \over {h^3(1 + {\lambda^2}{p^2})^3}}}[/math]

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

  • [math]\Large{Z}[/math]         is the Partition Function(1);

  • [math]\Large{\beta}[/math]          is the Thermodynamic Beta(2);

  • [math]\Large{P}[/math]         is the Pressure(2) of the system;

  • [math]\Large{V}[/math]        is the Volume(2) of the system;

  • [math]\Large{E}[/math]        is the Total Energy(2) of the system;

  • [math]\Large{x}[/math]         is the Position of the particle ([math]\Large{d^3}[/math] is the total differential cube);

  • [math]\Large{p}[/math]         is the Linear Momentum of the particle;

  • [math]\Large{\lambda}[/math]        is the Lenght Parameter of the uncertainty principle;

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].

                                                a)     [math]\LARGE{{\dot{a} \over a^2} = {1 \over 3}{\rho}}[/math]

                                                b)     [math]\LARGE{{\ddot{a} \over a} = - {1 \over 6}(\rho + 3P)}[/math]

       a)     [math]\large{{\dot{a} \over a^2} = {1 \over 3}{\rho}}[/math]

       b)     [math]\large{{\ddot{a} \over a} = - {1 \over 6}(\rho + 3P)}[/math]

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.

[math]\LARGE{a^3({1 - {3 \over 2}{x^2}}) = \mathbf{Constant}}[/math]

[math]\Large{a^3({1 - {3 \over 2}{x^2}}) = constant}[/math]

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.

craiyon_185234_Growing_Virtual_Universe (1)

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.

  1. American Scientist. "Is String Theory Even Wrong?"https://www.americanscientist.org/article/is-string-theory-even-wrong

  2. Nature. "Experimental simulation of loop quantum gravity on a photonic chip" https://www.nature.com/articles/s41534-023-00702-y

  3. 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

  4. 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

  5. IOPscience. "Critical dynamics in the early universe" https://iopscience.iop.org/article/10.1088/0264-9381/10/S/009

  6. Scientific American. "Origin of the Universe" https://www.jstor.org/stable/26001524

  7. 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


The Implications of Quantum Entanglement on Space-Time: A Focus on the Time Direction

About the "entangled" Microscopic and Macroscopic Realities

Through a detailed investigations on both the Conformal Field Theory (CFT) and Gravity (Space - Time) sides, it’s possible to find the profound connections between initial correlations in quantum systems and the geometric structure of Dual Space-Times. Recent developments in Anti - de - Sitter (AdS)/CFT correspondence [1] have uncovered intriguing links between Quantum Information Theory and Gravity, specifically focusing on the structure of Quantum Entanglement [2] (Figure 1) in conformal field theories (CFTs) and its impact on the dual spacetime. The entanglement structure of quantum subsystems is argued to be a key determinant of classically connected spacetimes.

Let's start with the Quantum Entanglement!

In the standard approach, we commence by examining two independent conformal field theories (CFTs) on the sphere [math]\normalsize{S^d}[/math] (x time) (Figure 2). These CFTs correspond to subsystems, Left (L) and Right (R), with their Hilbert Spaces decomposed as:

[math]\LARGE{{H_{LR}} = {H_{L}} \otimes {H_{R}}}[/math]

[math]\Large{H_{LR} = H_{L} \otimes H_{R}}[/math]

Equation 1.   Decomposed Hilbert Space of LR Entangled State

[math]\large{H_{LR}}[/math] is the Hilbert Space for LR Entangled State; [math]\large{H_{L}}[/math] is the Hilbert Space for L Subsystem and [math]\large{H_{R}}[/math] is the Hilbert Space for R Subsystem ([math]\large{\otimes}[/math] is the Product Operator).

Initially uncorrelated, the joint state is a product state, [math]\normalsize{\rho_{LR}}[/math]

[math]\LARGE{\rho_{LR} = \rho_{L} \otimes \rho_{R} = |\Psi_{\beta}\Psi_{\beta}|} [/math]

[math]\Large{\rho_{LR} = \rho_{L} \otimes \rho_{R} = |\Psi_{\beta}\Psi_{\beta}|} [/math]

Equation 2.   LR Entangled State Density Matrix

where [math]\normalsize{\rho_{L}}[/math] and [math]\normalsize{\rho_{R}}[/math] are the Density Matrix(ces) for the left and right subsystems, representing Thermal States [3]; [math]\normalsize{\Psi_{\beta}}[/math] is the LR Entangled State Wavefunction. For initially entangled states, such as the thermofield double state, the joint state [math]\normalsize{\rho_{LR}}[/math] involves entangled pure states for subsystems L and R.

Due particelle interagenti con un fascio di energia

Figure 1.   An abstract illustration of the two-Particles Quantum Entanglement

In the AdS/CFT framework, this uncorrelated state corresponds to disconnected AdS spacetimes. We quantify correlations using mutual information as follows.

[math]\LARGE{{I({\rho_{LR}})} = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR}})}}[/math]

[math]\large{{I({\rho_{LR}})} = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR}})}}[/math]

Equation 3.   The Quantum Information equation for the LR State

[math]\large{S(\rho_{LR})}[/math], [math]\large{S(\rho_{L})}[/math] and [math]\large{S(\rho_{R})}[/math] are the LR State, L and R Subsystems Entropies, respectively.

In a low - Entropy (Figure 3) environment,

[math]\boxed{\LARGE{{I({\rho_{LR}})}} = 0} \hspace{0.5cm} \LARGE{\Rightarrow}[/math]     [math]\LARGE{0 = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR})}}}[/math];

[math]\LARGE{{S({\rho_{LR}})} = {S({\rho_{L}})} + {S{(\rho_{R})}}}[/math]

[math]\boxed{\large{{I({\rho_{LR}})} = 0}}[/math]


[math]\large{0 = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR})}}}[/math];

[math]\large{{S({\rho_{LR}})} = {S({\rho_{L}})} + {S{(\rho_{R})}}}[/math]

Equation 4.   Low-Entropy Conditions Quantum Information

According to the Second Law of Thermodinamics (Equation 5), the Entropy of the composite system must increase as individual entropies evolve, giving:

[math]\LARGE{{\Delta{S}({\rho_{L}})} + {\Delta{S}({\rho_{R}})} \geq 0}[/math]

[math]\Large{{\Delta{S}({\rho_{L}})} + {\Delta{S}({\rho_{R}})} \geq 0}[/math]

Equation 5.   The Second Law of Thermodynamics for the LR State

[math]\large{\Delta{S}(\rho)}[/math] is the Entropy Variation for each subsystem and state involved.

In the absence of initial correlations, the dual Space - Time is composed of disconnected AdS regions, while initial entanglement leads to classical connectivity. The degree of entanglement is shown to dynamically influence the connectivity of the dual spacetime. Disentangling Degrees of Freedom decreases mutual information and Entropy.

Traveling the Space-Time aboard the Thermodynamic Arrow of Time

Building upon recent debates on the Thermodynamic Arrow of Time [4], it has been established a connection between the initial conditions of quantum correlations and the emergence of a preferred direction for the arrow of time. If there are no initial correlations, the arrow of time is directed toward increasing Entropy. However, in contrast to the uncorrelated case, initial correlations alter the entropy evolution. The thermodynamic arrow can now reverse, allowing for both orientations.


Figure 2.   A 3D Representation of a [math]\small{S^d}[/math] Sphere

And on Gravity … Side?

Furthermore, the concepts of Space - Time Sidedness [5] and Time - Orientability have to be discussed. Initial entanglement in the composite quantum system is argued to lead to a time-unoriented, one-sided Space - Time, while decreasing entanglement results in a time-oriented, two-sided Space - Time. In the latter condition, the dual spacetime features disconnected components with opposing time orientations, reflecting the reversed arrows of time in the individual CFTs.

The Fluctuations between Entanglement States

The effects of varying the degree of entanglement between the dual CFTs affect the Space - Time. High correlations are associated with a connected one-sided spacetime, while disentangling the degrees of freedom leads to a disconnected two-sided Space - Time. The maximal entanglement is interpreted as building a connection between the two sides of Space - Time.


Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

Just a Multi-Effect Dynamics

As shown, the insights into the relationship between quantum entanglement, Space - Time sidedness, and the thermodynamic arrow of time, within the AdS/CFT correspondence framework, highlight the crucial play of dynamic between initial correlations and the geometric dual structure, in understanding the emergence and orientation of the thermodynamic arrow of time.

  1. nLab.org. "AdS-CFT correspondence in nLab"https://ncatlab.org/nlab/show/AdS-CFT+correspondence

  2. IOPscience. "Quantum Entanglement and Its Application in Quantum Communication" https://iopscience.iop.org/article/10.1088/1742-6596/1827/1/012120

  3. Astronomy & Astrophysics. "The thermal state of molecular clouds in the Galactic center: evidence for non-photon-driven heating" https://www.aanda.org/articles/aa/full_html/2013/02/aa20096-12/aa20096-12.html

  4. Forbes. "No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time" https://www.forbes.com/sites/startswithabang/2019/11/22/no-thermodynamics-does-not-explain-our-perceived-arrow-of-time/?sh=4694b68c3109

  5. vXra.org. "The Placement of Two-sided Time in Physics" https://vixra.org/pdf/1906.0353v2.pdf


Is it Really Possible to Overcome the Wave-Particle Duality?

Undoubtedly, at the dawn of Quantum Theory birth there is the comparison between the David Bohm's Deterministic Interpretation, with hidden variables, and the Schrödinger and Born Probabilistic Paradigm, which introduces the Wavefunction, 𝜓 as a representation of a physical system's state (Figure 1).

Towards a Unified Quantum Mechanics Theory

In the last years, a Unified Quantum Mechanics framework has been proposed, consisting of a comprehensive analysis of the WaveParticle Dynamics of quantum particles. The conventional wavefunction, [math]\Large{\psi}[/math] (Figure 2) is redefined in terms of real scalar [math]\Large{{R}_0}[/math] and vector [math]\Large{\vec{R}}[/math] functions, shedding light on the intrinsic nature of quantum particles. The resulting expression unveils a dual composition of massless and massive fields within the particle, with the scalar [math]\Large{{R}_0}[/math] and vector [math]\Large{\vec{R}}[/math] functions satisfying the Quantum Telegraph Equation (Equation 1) [1] and the phase, [math]\Large{S}[/math] representing a massless field.

The interaction between these three functions is investigated, revealing intriguing connections. Notably, the product
[math]\Large{S \cdot {R_0}}[/math] satisfies the telegraph equation, representing the potential energy of the particle field, while [math]\Large{S \cdot {\vec{R}}}[/math] signifies its momentum. This article explores scenarios where the particle exhibits fluid-like (wave) behavior, characterized by a conserved Energy–Momentum Tensor, [math]\Large{{\sigma_{ij}}^m}[/math] (Equation 2).

[math]\LARGE{ - {\tau\hbar}{{\partial^2\psi} \over {\partial t^2}} + {i\hbar} {{\partial\psi} \over {\partial t}} = - {{\hbar^2} \over 2m} \Delta\psi}[/math]

[math]\Large{ - {\tau\hbar}{{\partial^2\psi} \over {\partial t^2}} + {i\hbar} {{\partial\psi} \over {\partial t}} = - {{\hbar^2} \over 2m} \Delta\psi}[/math]

Equation 1.   The mathematical general form of the Quantum Telegraph Equation

  • [math]\Large{\tau}[/math]          is a Time-dimension Parameter;

  • [math]\Large{\hbar}[/math]          is the Planck Constant;

  • [math]\Large{\psi}[/math]         is the three-dimensional Electron Wavefunction;

  • [math]\Large{t}[/math]           is the Time coordinate;

  • [math]\Large{i}[/math]           is the Imaginary Number;

  • [math]\Large{m}[/math]       is the Electron Mass;

  • [math]\Large{\Delta{\psi}}[/math]   is the Electron Wavefunction three-dimensional Espansion;

  • [math]\Large{v_p}[/math]       is the Phase Velocity;

  • [math]\Large{p^{\infty}}[/math]    is the Phase at high-frequency ([math]\Large{\omega}[/math]) limit;

Furthermore, Bohmian Mechanics can be incorporated into the Dirac Equation, resulting in a relativistic Hamilton – Jacobi Equation and a Continuity Equation. The new formalism eliminates the conventional quantum potential energy, providing a fresh perspective on Quantum Mechanics. Application of the Bohmian Method to Maxwell's Equations demonstrates the Particle–Wave Duality [2] of the electromagnetic field. It can also be established that when the electromagnetic field behaves as matter, it resembles a massless Dirac particle, supporting the de Broglie Hypothesis [3]. 

[math]\LARGE{{{\sigma_{ij}}^{m}} = { - {\vec{R}_i}{\vec{R}_j} + {\delta_{ij}}{\Biggr({{\vec{R}^2} \over {2}} - {{\vec{R}_0}^2 \over {2{c_{0}}^2}}\Biggl)}}}[/math]

[math]\large{{{\sigma_{ij}}^{m}} = { - {\vec{R}_i}{\vec{R}_j} + {\delta_{ij}}{\Biggr({{\vec{R}^2} \over {2}} - {{\vec{R}_0}^2 \over {2{c_{0}}^2}}\Biggl)}}}[/math]

Equation 2.   The Energy-Momentum Tensor mathematical expression

where [math]\vec{R}_i[/math] and [math]\vec{R}_j[/math] are the [math]R[/math] vector components along the [math]i[/math] and [math]j[/math] arbitrary directions, respectively; while [math]\delta_{ij}[/math] is the Delta Function, which modulates the tensor magnitude.

What about a New Potential Energy?

In the so-called Schrödinger-Bohm Mechanics [4], the motion of a non-relativistic quantum particle in a Potential Energy, [math]\Large{V}[/math] is described using the Schrödinger Equation. The wavefunction is expressed as:

[math]\LARGE{\psi = {Re^{{iS} \over {\hbar}}}}[/math]

[math]\Large{\psi = {Re^{{iS} \over {\hbar}}}}[/math]

Equation 3.   One of the Electron Wavefunction scalar forms

where [math]\Large{R}[/math] and [math]\Large{S}[/math] are real functions. The resulting equations are analyzed, revealing the dual nature of the particle as both wave and matter. A new potential energy dependent on the phase [math]\Large{S}[/math] is introduced, termed the Phase (Spin) Potential Energy.

A hybrid Theory of Vector and Scalar Wavefunctions

The Vector Quantum Mechanics represents a free quantum particle with mass, [math]\Large{m}[/math] using scalar and vector wavefunctions [math]\Large{\psi{_0}}[/math] and [math]\Large{\vec{\psi}}[/math], respectively. The quantum telegraph equation governs their evolution, illustrating the fluid-like motion [5] of the particle.


Figure 1.   A representative 3-Dimensional Electron Wavefunction Energy Distribution graph

From Bohmian Vector Quantum Mechanics to Relativistic Dirac Equation

The combination of Bohmian Vector Quantum Mechanics to vector quantum mechanics yields to a relativistic Dirac Hamiltonian for massless and massive particles. The Hamiltonian equality implies a dual description of the particle, with all information propagating at the speed of light. The mass of the particle is uniquely determined by its wavefunctions, showcasing the interconnectedness of the fields.

Application to Maxwell’s Equations

The Bohmian approach is further extended to Maxwell's equations, revealing the particle–wave duality of the Electromagnetic Field. The electromagnetic field is expressed as a wave and particle, with the phase [math]\Large{S}[/math] governing the particle dynamics and [math]\Large{\vec{E}_0}[/math] and [math]\Large{\vec{B}_0}[/math] (Electric Field and Magnetic Field in vacuum, respectively, interacting with particle) representing wave aspects. The Photon, when behaving as matter, resembles a massless Dirac particle, supporting the de Broglie hypothesis.


Figure 2.   A fountain plan of Electron Wavefunctions Interference

A Delicious Chance to Connect the Matter and Wave features of Particles

The Unified Quantum Mechanics framework highlights the inherent duality and interconnectedness of massless and massive fields within quantum particles. Particularly, the elimination of the quantum potential energy in this new formalism opens avenues for further exploration in quantum theory.

  1. LinkedIn. "Classical and Quantum Wave Equations: A Journey into the Core Concepts"https://www.linkedin.com/pulse/classical-quantum-wave-equations-journey-core-concepts-dadhich/

  2. ScienceDirect.com. "Complementarity, wave-particle duality, and domains of applicability" https://www.sciencedirect.com/science/article/pii/S1355219817301028

  3. ScienceReady.com. "de Broglie's Matter Wave Duality and Experimental Evidence" https://scienceready.com.au/pages/matter-wave-duality

  4. IOPscience. "Is the de Broglie-Bohm interpretation of quantum mechanics really plausible?" https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012060/pdf

  5. Springer.com. "Visualization of hydrodynamic pilot-wave phenomena" https://link.springer.com/article/10.1007/s12650-016-0383-5


The Effect of Dark Matter and Dark Energy on Gravitational Time Advancement

Nowadays, the effects of Dark Matter (Figure 1) and Dark Energy on Gravitational Time Advancement, on local and large distance scales, are quite known. Some research consider various models of dark matter and dark energy, including the Cosmological Constant, Λ and compares their impact on gravitational phenomena with the standard Schwarzschild Geometry (Equation 1).

Beyond the Standard Theories...

Postulating the acceleration of the Universe's expansion, the latest advancements led to the inclusion of dark energy, characterized by a negative pressure, into the Energy - Momentum Tensor, Tik of the Universe. Dark matter, a non-luminous component dominating galaxies, is supported by observations such as rotation curve surveys, cosmic microwave background (Figure 2) measurements [1], and baryon acoustic oscillations [2]. 

[math]\Large{ds^2 = - \Biggl( - 1 - {2\mu \over r}\Biggr) dt^2 + \Biggl(1 - {2\mu \over r}\Biggr)dr^2 + {r^2 \over d\Omega^2}}[/math]

[math]\small{ds^2 = - \Biggl( - 1 - {2\mu \over r}\Biggr) dt^2 + \Biggl(1 - {2\mu \over r}\Biggr)dr^2 + {r^2 \over d\Omega^2}}[/math]

Equation 1.   The Schwarzschild Geometry equation


[math]\Large{\mu = {{G \cdot M} \over {{c_0}^2}}}[/math]


[math]\large{\mu = {{G \cdot M} \over {{c_0}^2}}}[/math]

  • [math]\Large{\textit{s}}[/math]        is a spherical Surface;

  • [math]\Large{\mu}[/math]       is the Schwarzschild Radius;

  • [math]\Large{G}[/math]      is the Gravitational Constant;

  • [math]\Large{M}[/math]    is the body Mass;

  • [math]\Large{c_{0}}[/math]     is the Light Speed in Vacuum;

  • [math]\Large{\textit{r}}[/math]        is the Radial Coordinate;

  • [math]\Large{\textit{t}}[/math]         is the Time Coordinate;

  • [math]\Large{\Omega}[/math]      is a Point on the two sphere [math]\Large{{\textit{S}} \hspace{0.1cm} ^2}[/math];

Dark Energy and Dark Matter Models

The first and important candidate for dark energy is the cosmological constant, characterized by a constant Energy Density with an Equation-Of-State Parameter, w  equal to -1. However, its theoretical problem lies in its size, significantly lower than the expected Vacuum Energy Density. Various alternative theories, including Quintessence [3]  and Chaplygin Gas Models, (CG) [4] have been proposed to address this issue. Modifications to General Relativity, such as scalar tensor theories and [math]\Large{f(R)}[/math] gravity models, have also been suggested to explain late-time accelerated expansion with no reference to dark energy.


Figure 1.   A figurative representation of Cosmos objects (white lights) surrounded
by Dark Matter lattice (red and blue structure)

Similarly, dark matter candidates include WIMPs [5], Axions, and sterile Neutrinos. Theories like MOND propose modifications to Newtonian Dynamics, while Conformal Gravitational Theory [6], based on Weyl Symmetry, aims to explain flat rotation curves of Galaxies without the need for dark matter.

What is the Gravitational Time Advancement?

Gravitational Time Advancement occurs when an observer is in a stronger gravitational field compared to the photon's trajectory. A study finds that dark energy leads to a gravitational time delay, reducing the gravitational time advancement effect. In contrast, the conformal theory suggests a large time advancement effect, particularly at distances beyond approximately 30 kpc (Kiloparsec).


Figure 2.   An instance of small ground station satellite antenna to pick up
the Milky Way Microwave Background

Experimental Verification and Future Missions

This article emphasizes the challenging nature of experimentally verifying the gravitational time advancement effect. BEACON [7] and GRACE-FO [8] missions are expected to provide a higher accuracy in probing the Earth's gravitational field, offering opportunities to measure gravitational time advancement effects at large distances.

Gravitational Time Advancement as a Proof of Dark Matter and Dark Energy

However, analytical expressions are derived, revealing that dark energy induces a gravitational time delay, while the Schwarzschild Metric results in both time delay and time advancement effects. The findings suggest that measuring gravitational time advancement at large distances could serve as a means to prove the dark matter and certain dark energy models. Also parameters, on upper bound conditions, will be affected by this gravitational phenomenon.

  1. Space.com. "What is the cosmic microwave background?"https://www.space.com/33892-cosmic-microwave-background.html

  2. Nasa (.gov). "Baryon Acoustic Oscillations - Roman Space Telescope" https://roman.gsfc.nasa.gov/BAO.html

  3. Cerncourier.com. "The quintessence of cosmology" https://cerncourier.com/a/the-quintessence-of-cosmology/

  4. Academic Accelerator. "Chaplygin Gas: Most Up-to-Date Encyclopedia, News & Reviews" https://academic-accelerator.com/encyclopedia/chaplygin-gas

  5. Arstechnica. "No WIMPS! Heavy particles don’t explain gravitational lensing oddities" https://arstechnica.com/science/2023/04/gravitational-lensing-may-point-to-lighter-dark-matter-candidate/

  6. Physics.wustl. "Quantum Conformal Gravity" https://physics.wustl.edu/events/quantum-conformal-gravity

  7. ResearchGate. "A Search for New Physics with the BEACON Mission" https://www.researchgate.net/publication/1732985_A_Search_for_New_Physics_with_the_BEACON_Mission

  8. Forbes. "NASA's GRACE-FO Mission Will Study How Earth's Climate Is Evolving" https://www.forbes.com/sites/jesseshanahan/2018/05/22/nasas-grace-fo-mission-will-study-how-earths-climate-is-evolving/


Many types of Fuels for Space-Time Travels


Figure 1.  A Spaceship refueling with liquid Hydrogen (LH2) in deep Space

Overview of different Fuel types for Space-Time Travel

You'll need different fuels for different things. LH2 is really too bulky for most air-breathing aircraft, but it's a waste to operate nuclear rockets on anything else, and spaceplanes almost require it for cooling and combustion speed (Figure 1). Methane is a much better launch vehicle fuel due to its density, which makes tanks far smaller and improves thrust density. And any decently developed industrial base will allow production of heavier hydrocarbons, for example by the Fischer-Tropsch process. And then there's various room-temperature chemical fuels and ion thruster propellants [1].

That’s good for planet Earth, especially when compared with rocket launches that rely on a popular alternative: Kerosene-based propellant. In the case of SpaceX, a single Falcon 9 flight emits about 336 tons of Carbon Dioxide—the equivalent of a car traveling around the World 70 times—according to John Cumbers, a former NASA synthetic biologist and CEO of SynBioBeta [2].

Advantages and Disadvantages of each Fuel Type

Whilst liquid fuels present disadvantages such as the potential for hazardous spills or leaks, one of the biggest issues discovered with such fuels is the relatively complex design, with an increased likelihood of things going wrong. If the liquid substance is cryogenic the fuel cannot be stored for long, and so the foundations for cryogenic storage facilities must be set up at the launch site. This is an area where Skyrora stands out from market competitors, with the propellants of our Skyrora XL vehicle designed to be stored for a longer launch window which is crucial for UK launches where weather conditions make go-for-launch difficult [3].

: Thirty-percent better fuel economy than an equivalent gasoline vehicle, widely available, lower cost premium than for hybrid vehicles, engines deliver lots of torque for a given displacement, and any Diesel car can run on a blend of renewable Biodiesel fuel. With effort and investment, older diesel engines can be converted to run on pure waste vegetable oil (Figure 2).

Cons: Traditionally more engine noise and vibration. Additional emissions equipement drives up vehicle prices, which along with currently higher cost of Diesel fuel takes a big bite out of any savings. Most clean diesels require refills of Urea solution. Manufacturers won't warranty Biodiesel blends of more than five-percent of Biodiesel [4].


Figure 2.  A Solar Sail in a Space-Time Travel

Current research and development in Alternative Fuel Sources for Space-Time Travel

Picking up fuel along the way — the Ramjet approach — will lose efficiency as the Space craft's speed increases relative to the planetary reference (Figure 3). This happens because the fuel must be accelerated to the spaceship's velocity before its energy can be extracted, and that will cut the fuel efficiency dramatically [5].

Whilst reusability of rockets benefits science, exploration and human spaceflight – one of the greatest drivers for stakeholders in the global launch segment is the scale and demand for in-orbit assets by industry and economy, fuelled in tandem by the plummeting costs and size of satellites (e.g. CubeSats), instrumentation, and even ride-sharing platform services [6].


Figure 3.  A futuristic Warp Drive spaceship slicing through the Galaxy with neon blue lights

  1. Headed For Space. "Using Liquid Methane As Rocket Fuel – Advantages & Drawbacks" https://headedforspace.com/using-liquid-methane-as-rocket-fuel/

  2. Fortune.com. "Space travel is heating up—and so are rocket fuel emissions. These companies are developing cleaner alternatives to protect earth first." https://fortune.com/2022/12/05/space-travel-is-heating-up-and-so-are-rocket-fuel-emissions-these-companies-are-developing-cleaner-alternatives-to-protect-earth-first/

  3. Skyrora. "Rocket fuel: is it rocket science?." https://www.skyrora.com/rocket-fuel-is-it-rocket-science/

  4. Consumer Reports. "The Pros and Cons on Alternative Fuels." https://www.consumerreports.org/cro/2011/05/pros-and-cons-a-reality-check-on-alternative-fuels/index.htm

  5. Wikipedia. "Spacetravel under constant acceleration" https://en.wikipedia.org/wiki/Space_travel_under_constant_acceleration

  6. Spaceaustralia. "Renewable Rocket Fuels – Going Green and Into Space" https://spaceaustralia.com/feature/renewable-rocket-fuels-going-green-and-space

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