Zero Theory: The Role of Vacuum Energy in Unifying All Phenomena through Energy Conversion

Zero Theory reinterprets the individual concepts of traditional physics (photons, gravity, dark energy, dark matter, gravitons, etc.) through the perspective that "everything can be explained by energy conversion." This approach, based on Einstein's famous energy-mass equivalence equation \(E=mc^2\), understands physical phenomena as a series of processes of energy conversion. Additionally, by applying Zero Theory to Einstein's equations, it delves deeper into how vacuum energy density influences the curvature of spacetime. This article elaborates on this Zero Theory perspective, focusing on the dynamic role of vacuum energy, forces arising from energy conversion, the transmission of photons and gravity, the reinterpretation of dark energy and dark matter, the understanding of quantum entanglement, and the redefinition of the law of energy conservation.

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1. Reinterpretation of Vacuum Energy

1.1 The Dynamic Role of Vacuum Energy as Energy Conversion

In traditional physics, the vacuum was often understood as a fluctuating field with zero-point energy. However, Zero Theory reinterprets the vacuum itself as the primary stage for energy conversion. This perspective posits that fluctuations in the vacuum's energy density are the foundation for all physical phenomena.

Equation:

The expectation value of zero-point energy is expressed as follows:

\[\langle 0 | H | 0 \rangle = \frac{1}{2} \sum_{k=0}^{k_{\text{max}}} \hbar \omega_k\]

This equation indicates that vacuum fluctuations cause finite energy variations. Here, the cutoff wave number \( k_{\text{max}} \) is set using the Planck length \( l_P \), preventing the divergence of vacuum energy to infinity.

\[k_{\text{max}} = \frac{1}{l_P}, \quad l_P = \sqrt{\frac{\hbar G}{c^3}}\]

This setting shows how energy conversion occurs at the smallest physical scale (Planck length), providing the foundation for Zero Theory.

Supplementary Equation:

The Planck energy \( E_P \) is defined as the energy scale corresponding to the Planck length:

\[E_P = \sqrt{\frac{\hbar c^5}{G}}\]

This definition is used to keep the vacuum energy fluctuations finite within a physically meaningful range.

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2. Extension of Einstein's Equation with Zero Theory

2.1 Application of Zero Theory to Einstein's Equation

Applying Zero Theory to Einstein's equation clarifies that vacuum energy density directly influences the curvature of spacetime. In traditional general relativity, the distribution of matter and energy determines the structure of spacetime, but Zero Theory adds the influence of vacuum energy to this relationship.

Extended Einstein's Equation with Zero Theory:

\[G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}^{\text{matter}} - \rho_{\text{vac}} g_{\mu\nu} \right)\]

This equation shows that in addition to the distribution of matter and energy, the vacuum energy density \(\rho_{\text{vac}}\) contributes to the curvature of spacetime. It emphasizes that vacuum energy is not merely a background energy but a dynamic element actively shaping the structure of spacetime.

2.2 Relationship between Dark Energy and the Cosmological Constant

In this extended equation, the dynamic changes in vacuum energy density play the role of dark energy, providing a deeper understanding beyond the traditional static cosmological constant \(\Lambda\). This naturally explains the accelerated expansion of the universe as a result of energy conversion.

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3. Forces Arising from Energy Conversion

3.1 Definition: Attractive Forces Generated by Distortion of Cosmic Medium through Energy Conversion

Attractive forces that arise from the distortion of the cosmic medium caused by energy conversion.

3.2 Concept: The Mechanism by which Energy Conversion Distorts the Medium (Spacetime), Generating Attractive Forces

The mechanism by which energy conversion distorts the medium (spacetime), resulting in the generation of attractive forces. This demonstrates that local changes in energy affect the entire medium.

3.3 Analogy between General Relativity and Zero Theory

3.3.1 Explanation of Gravity in General Relativity

In Einstein's general relativity, mass and energy curve spacetime, and this curvature exerts gravitational forces on other objects. The heavier an object, the more it curves the surrounding spacetime, causing other objects to move along these curves.

3.3.2 Analogy in Zero Theory

In Zero Theory, the universe is likened to a "medium" like a thread, with energy conversions "weaving" patterns into it. By weaving patterns into the thread, surrounding energy and matter are drawn in, creating gravitational-like forces. These patterns represent vibrations or conversions of energy within the medium, illustrating the mechanism by which attractive forces are generated.

3.4 Detailed Explanation of the Analogy

Vibrations of the Thread and Gravity:

In Zero Theory, energy conversion causes the medium (thread) to vibrate, creating "patterns" that manifest as gravitational-like phenomena. These vibrations influence the entire universe, explaining how energy conversions lead to the attraction between objects. Energy conversion alters the properties of the medium, resulting in gravitational forces.

Universality of the Medium:

In Zero Theory, the medium (thread) exists throughout the universe, and energy conversions within it distort the medium, generating gravity-like forces. This universal presence allows energy conversions to consistently influence the structure of spacetime, providing a unified explanation for gravitational phenomena and energy interactions.

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3.5 Understanding Gravity in Zero Theory

3.5.1 Origin of Attractive Forces:

In Zero Theory, attractive forces arise from the interaction between the cosmic medium and energy conversions. Here, the "cosmic medium" represents the state of energy conversion and fluctuations within the universe.

3.5.2 Macroscale Phenomena:

At cosmic scales, attraction is considered a result of the circulation and conversion of energy within Zero Theory. Attractive forces derive from distortions and energy distributions in the medium, understood as phenomena observable at macroscopic scales.

3.6 Examination of Theoretical Consistency

3.6.1 Consistency within Zero Theory

Energy Conversion and Attraction:

Zero Theory views the universe as a "medium," with energy conversions and fluctuations generating attractive forces. The explanation that attractive forces result from energy redistribution and conversion is consistent within the theory.

Fluctuations of the Medium:

In this theory, attractive forces emerge as interactions within the cosmic medium, meaning that fluctuations and energy variations directly generate these forces. This explains how attractive forces are produced as a consequence of energy transformations within the medium.

3.6.2 Consistency with Other Theories

Consistency with Relativity:

Gravity, explained by the curvature of spacetime in relativity, and gravity described in Zero Theory as vibrations or conversions in the cosmic medium, both contribute to explaining the same phenomenon (gravity) through different mechanisms.

Consistency with Quantum Mechanics:

If quantum entanglement and state changes are based on Zero Theory's energy conversion, they must align with established results in quantum mechanics. How Zero Theory integrates these phenomena within its framework is crucial for maintaining consistency with quantum theory.

3.6.3 Experimental Verification

Confirmation of Theoretical Consistency:

Alignment with experimental data is essential. Whether the forces and energy conversions predicted by Zero Theory match observations and experimental results is critical.

New Experimental Approaches:

To verify new predictions of Zero Theory, new experiments and observations are required. This includes precise measurements of energy conversions and the resulting forces within the cosmic medium.

3.7 Conclusion

The interpretation of attractive forces within Zero Theory conceptualizes the universe as a "medium," with energy conversions and fluctuations generating these forces. This approach explains how attractive forces emerge as a result of energy redistribution and conversion within the medium, providing a clear explanation of how these forces influence the motion and dynamics of matter.

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4. Relationship between Zero Theory and Dark Energy, Dark Matter

4.1 Dark Energy: Cosmic Expansion as Energy Conversion

Concept:
Dark energy is the energy responsible for the accelerated expansion of the universe. It is uniformly distributed throughout the cosmos and affects space itself.

Relation to Zero Theory:

- Energy Vibrations:
Zero Theory explains changes in cosmic expansion and gravitational variations as energy vibrations and conversions within the cosmic medium. Dark energy's effects can potentially be understood as energy conversions within Zero Theory's cosmic medium.

- Relation to Gravity:
While dark energy works against gravitational effects, Zero Theory explains gravity as arising from energy vibrations and conversions within the medium. Dark energy's influence could also be explained through these energy conversions within the medium.

Equation:

The equation for energy exchange based on dark energy density fluctuations:

\[\rho_{\text{vac}}(t) = \rho_{\text{vac},0} \left( \frac{a_0}{a(t)} \right)^n\]

Here, \( a(t) \) is the scale factor of the universe, and \( n \) indicates the rate of expansion.

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4.2 Dark Matter: Energy Conversion of the Vacuum

Concept:
Dark matter is traditionally considered a hypothetical substance needed to account for the missing mass required to form and sustain galaxies. It interacts through gravity but is invisible and does not emit, absorb, or reflect light.

Relation to Zero Theory:

- Energy Vibrations:
Zero Theory explains the presence and behavior of matter and gravity through energy vibrations and conversions within the cosmic medium. Dark matter's gravitational effects can be understood as influences of energy conversions within Zero Theory's medium.

- Relation to Gravity:
Dark matter interacts gravitationally with ordinary matter, and in Zero Theory, gravity is generated by vibrations and conversions in the cosmic medium. Thus, dark matter's effects can be explained based on changes in the medium's energy.

Gravity Potential Equation:

\[\nabla^2 \Phi = 4\pi G \left( \rho_m + \delta \rho_{\text{vac}} \right)\]

Here, \(\delta \rho_{\text{vac}}\) represents fluctuations in vacuum energy density, understood not as a material entity but as a distortion in spacetime, characterized by the Planck length \( l_P \).

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5. Unified Explanation of Photons and Gravity

5.1 Reinterpretation of Wave-Particle Duality of Photons

In Zero Theory, photons are understood as transient phenomena resulting from the dynamic conversion of vacuum energy. The wave-like and particle-like nature of photons is reinterpreted as energy states arising from local fluctuations in vacuum energy density.

Photon Energy Equation:

\[E = h\nu\]

This equation shows that the energy of photons is determined by the conversion of vacuum energy, explaining how the propagation of light depends on the energy density of the vacuum.

5.2 Relationship between Gravity and Energy Conversion

In Zero Theory, gravity is also interpreted as a result of the energy conversion of the vacuum. Gravitational potential depends on fluctuations in vacuum energy density, which cause the curvature of spacetime.

\[\nabla^2 \Phi = 4\pi G \left( \rho_m + \delta \rho_{\text{vac}} \right)\]

Here, \(\delta \rho_{\text{vac}}\) represents fluctuations in vacuum energy density, clearly showing that gravity is not a material entity but a curvature of spacetime based on energy conversion.

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6. Redefinition of Gravitons: Understanding as Mediators of Energy Conversion

In Zero Theory, gravitons are reinterpreted as part of the energy conversion process itself, eliminating the need to assume gravitons as virtual particles. Gravity is seen as a phenomenon transmitted through fluctuations in vacuum energy, thus removing the necessity to posit gravitons as mediating particles.

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7. Reinterpretation of Quantum Entanglement and Vacuum Energy

7.1 The Role of Non-Local Correlation and Vacuum Energy

Quantum entanglement is interpreted as a phenomenon where energy states mutually influence each other through the vacuum energy field. This perspective explains why the correlation between entangled particles is maintained regardless of distance, from the viewpoint of energy conversion.

Correlation Function Equation:

\[\langle \psi_1 | \hat{O} | \psi_2 \rangle = \int e^{-i\omega t} \delta \rho_{\text{vac}}(x, t) dx\]

This equation indicates that fluctuations in vacuum energy play a role in maintaining the correlation between entangled particles.

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8. Unified Understanding through Vacuum Energy Conversion

8.1 Common Mechanism of Photons and Gravity

Photons and gravity are understood as phenomena transmitted through the conversion of vacuum energy. This common mechanism explains how energy state transformations are linked to various physical phenomena.

Equations:

\[E_{\text{photon}} = \hbar \omega, \quad G_{\mu\nu} = 8\pi G \rho_{\text{vac}} g_{\mu\nu}\]

This demonstrates that both photons and gravity rely on fluctuations in vacuum energy through a shared mechanism.

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9. Conclusion: Consistency of Zero Theory from the Perspective of Energy Conversion

Zero Theory aims to explain all physical phenomena through processes of energy conversion, providing a unified understanding of physical phenomena. By applying Zero Theory, the impact of vacuum energy density on the curvature of spacetime is clearly articulated in mathematical terms, enabling a unified understanding that maintains consistency with current physical theories while offering new interpretations. This approach returns to the foundations of Einstein's \(E=mc^2\), offering a simpler and more coherent theoretical system.

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Appendix: Importance of Theoretical Consistency and Experimental Verification

To ensure the consistency and reliability of Zero Theory, the following points are essential:

1. Detailed Mechanism Elucidation:
   Strengthening the physical basis of the theory by providing detailed explanations of energy conversion processes and mechanisms.
2. Examination of Consistency with Observational Data:
   Comparing the theory's predictions with observational data (e.g., cosmic microwave background radiation, galaxy rotation curves, gravitational lensing effects) to ensure alignment.
3. Theoretical Predictions and Experimental Verification:
   Enhancing the theory's practicality and reliability by outlining specific predictions and how they can be experimentally verified.
4. Clarification of Terminology:
   Defining and explaining unique terms and concepts within Zero Theory to facilitate reader understanding.
5. Theory's Limitations and Future Prospects:
   Discussing the current limitations of Zero Theory and outlining future research directions to address unresolved challenges.

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The above content provides a detailed explanation with mathematical equations necessary to comprehensively describe Zero Theory. By integrating Zero Theory with the new theory, a comprehensive physical model based on energy conversion is presented, offering a new perspective on existing physical theories. Future research and experimental verification are expected to further elucidate the validity and consistency of this theory.

Planck Length and Cutoff Wavenumber

Using the Planck length \(l_P\), we define the cutoff wavenumber \(k_{\text{max}}\). This represents a physical constraint to prevent the divergence of vacuum energy.

\[ k_{\text{max}} = \frac{1}{\sqrt{\frac{G \hbar}{c^3}}} \]

Explanation: The Planck length indicates the smallest scale at which quantum gravity effects become significant. This cutoff plays a role in preventing the divergence of zero-point energy, limiting it to a physically realizable range.

Modified Einstein Equation (Scalar Form)

This is a modified Einstein equation that dynamically takes into account the vacuum energy density.

\[ G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}^{\text{matter}} - \rho_{\text{vac}} \right) \]

Explanation: Here, the vacuum energy density \(\rho_{\text{vac}}\) directly affects the curvature of spacetime. In addition to the matter-energy term from conventional general relativity, the influence of vacuum energy is explicitly incorporated.

Dynamic Vacuum Energy Density

This equation expresses the time dependence of the vacuum energy density.

\[ \rho_{\text{vac}}(t) = \rho_{\text{vac},0} \left( \frac{a_0}{a(t)} \right)^n \]

Explanation: Here, \(\rho_{\text{vac},0}\) is the current vacuum energy density, \(a(t)\) is the scale factor of the universe, and \(n\) is a parameter of the model. This equation shows how the vacuum energy density evolves over time, reflecting the dynamics associated with the expansion of the universe.

Vacuum Energy Density Fluctuations as Dark Matter

This equation represents the role of dark matter as local fluctuations in the vacuum energy density.

\[ \rho_{\text{vac}}(t, \mathbf{x}) = \rho_{\text{vac},0} \left( \frac{a_0}{a(t)} \right)^n \left( 1 + \delta \rho_{\text{vac}} \right) \]

Explanation: \(\delta \rho_{\text{vac}}\) represents fluctuations in the vacuum energy density, which can cause distortions in spacetime that contribute to phenomena observed as dark matter.

Poisson Equation with Vacuum Energy Density Fluctuations

This equation describes how fluctuations in the vacuum energy density contribute to the formation of the gravitational potential.

\[ \nabla^2 \Phi = 4\pi G \left( \rho_m + \rho_{\text{vac},0} \left( \frac{a_0}{a(t)} \right)^n \left( 1 + \delta \rho_{\text{vac}} \right) \right) \]

Explanation: This equation shows that the gravitational potential \(\Phi\) depends on both the matter density \(\rho_m\) and the variations in vacuum energy density, clarifying how vacuum energy influences the creation of gravitational forces.

Equation for Quantum Entanglement via Vacuum Energy

This equation illustrates how quantum entanglement can be explained through fluctuations in vacuum energy density.

\[ \langle \psi_1 | \hat{O} | \psi_2 \rangle = \text{Piecewise}\left( \left( \frac{i \delta \rho_{\text{vac}} e^{-i\omega t}}{\omega}, \omega \neq 0 \right), \left( \delta \rho_{\text{vac}} t, \text{True} \right) \right) \]

Explanation: This formula suggests that fluctuations in vacuum energy help maintain non-local correlations between entangled particles, providing a way to understand the effects of quantum entanglement from an energy exchange perspective.

Energy Conservation Law for Matter

This equation shows energy exchange between matter and vacuum energy.

\[ \frac{d \rho_{\text{matter}}}{dt} + 3H \left( \rho_{\text{matter}} + \frac{p_{\text{matter}}}{c^2} \right) = -Q(t) \]

Explanation: The change in matter energy density depends on the energy exchange term \(Q(t)\), taking into account the conversion of energy between matter and vacuum energy.

Energy Conservation Law for Vacuum Energy

This equation describes the energy conservation law for the fluctuations in vacuum energy.

\[ \frac{d \rho_{\text{vac}}}{dt} + 3H \left( \rho_{\text{vac}} + \frac{p_{\text{vac}}}{c^2} \right) = Q(t) \]

Explanation: This equation shows that vacuum energy exchanges energy with matter through the term \(Q(t)\), describing the interaction between vacuum energy and matter energy.

Conclusion

These equations provide a framework based on the new theory to demonstrate how vacuum energy influences the evolution of the universe and various physical phenomena. They offer a unified explanation of these phenomena from the perspective of the dynamic role of vacuum energy and energy conversion. Let me know if you need further analysis or have any questions.

Mathematical Model of Temporal and Spatial Variations in Vacuum Energy Density

1. Introduction

To describe the temporal and spatial variations of vacuum energy density concretely, a framework combining quantum field theory and general relativity is necessary. Here, we construct a model that considers the influence of vacuum energy density on the gravitational field to explain large-scale structures and phenomena in the universe.

2. Definition of Vacuum Energy Density

The vacuum energy density \(\rho_{\text{vac}}\) is given as the expectation value of the zero-point energy of the field.

2.1 Calculation of Zero-Point Energy

Consider a scalar field \(\phi\), whose zero-point energy is calculated as follows:

\[\rho_{\text{vac}} = \langle 0 | T_{00} | 0 \rangle = \frac{1}{2} \int \frac{d^3k}{(2\pi)^3} \hbar \omega_k\]

where \(\omega_k = \sqrt{k^2 + m^2}\), with \(m\) being the mass and \(k\) the wave vector.

2.2 Introduction of a Cutoff

Since the integral diverges, a cutoff energy \(E_{\text{cutoff}}\) is introduced, setting the Planck energy \(E_P\) as the cutoff:

\[E_{\text{cutoff}} = E_P = \sqrt{\frac{\hbar c^5}{G}}\]

This ensures that the vacuum energy density takes a finite value.

2.3 Calculation of Vacuum Energy Density

Using the cutoff, the vacuum energy density is approximated as follows:

\[\rho_{\text{vac}} \approx \frac{1}{2} \int_{0}^{k_{\text{max}}} \frac{4\pi k^2 dk}{(2\pi)^3} \hbar \omega_k = \frac{\hbar}{8\pi^2} \int_{0}^{k_{\text{max}}} k^2 \sqrt{k^2 + m^2} \, dk\]

where \(k_{\text{max}} = \frac{E_{\text{cutoff}}}{\hbar c}\).

3. Modeling Temporal and Spatial Variations in Vacuum Energy Density

3.1 Introduction of Spatiotemporal Dependence

If vacuum energy density depends on time and space, it can be expressed as \(\rho_{\text{vac}} = \rho_{\text{vac}}(t, \mathbf{x})\). This implies that the expectation value or state of the field changes with time and space.

3.2 Introduction of Effective Action

Consider the effective action of the field \(S_{\text{eff}}\):

\[S_{\text{eff}} = \int d^4x \sqrt{-g} \left( \mathcal{L}_{\text{matter}} + \mathcal{L}_{\text{vac}} \right)\]

where \(\mathcal{L}_{\text{vac}}\) is the Lagrangian density corresponding to vacuum energy, given by:

\[\mathcal{L}_{\text{vac}} = -\rho_{\text{vac}}(t, \mathbf{x})\]

3.3 Influence on Einstein's Equation

Einstein's equation, including the vacuum energy density term, is expressed as:

\[G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}^{\text{matter}} + T_{\mu\nu}^{\text{vac}} \right)\]

where \(T_{\mu\nu}^{\text{vac}}\) is the energy-momentum tensor for vacuum energy, defined as:

\[T_{\mu\nu}^{\text{vac}} = -\rho_{\text{vac}}(t, \mathbf{x}) g_{\mu\nu}\]

3.4 Application to the Friedmann Equation

Assuming the universe is homogeneous and isotropic, we use the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is given by:

\[ds^2 = -c^2 dt^2 + a(t)^2 \left( \frac{dr^2}{1 - k r^2} + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 \right)\]

where \(a(t)\) is the scale factor and \(k\) is the curvature of space.

The Friedmann equation is then modified as follows:

\[\left( \frac{\dot{a}}{a} \right)^2 + \frac{k c^2}{a^2} = \frac{8\pi G}{3} \left( \rho_{\text{matter}} + \rho_{\text{vac}}(t) \right)\]

3.5 Modeling the Temporal Variation of Vacuum Energy Density

To model the time variation of vacuum energy density, we assume a functional form as follows:

\[\rho_{\text{vac}}(t) = \rho_{\text{vac},0} \cdot f(a(t))\]

where \(\rho_{\text{vac},0}\) is the current vacuum energy density and \(f(a(t))\) is a function that depends on the scale factor \(a(t)\).

3.5.1 Specific Model Example

For example, let \(f(a)\) be expressed as:

\[f(a) = a^{-n}\]

where \(n\) is a model parameter. For \(n=0\), \(\rho_{\text{vac}}\) remains constant, corresponding to the cosmological constant model.

3.5.2 Introduction of the Equation of State Parameter

The equation of state for vacuum energy is given by:

\[p_{\text{vac}} = w_{\text{vac}} \rho_{\text{vac}} c^2\]

Typically, \(w_{\text{vac}} = -1\), but if it varies with time, \(w_{\text{vac}}\) depends on both time and the scale factor \(a(t)\).

3.6 Application of the Energy Conservation Law

The energy conservation law (continuity equation) is expressed as:

\[\dot{\rho} + 3H \left( \rho + \frac{p}{c^2} \right) = 0\]

where \(\rho\) and \(p\) are the total energy density and pressure, respectively.

If there is energy exchange between vacuum energy and matter, the continuity equation is modified as follows:

\[\dot{\rho}_{\text{matter}} + 3H \left( \rho_{\text{matter}} + \frac{p_{\text{matter}}}{c^2} \right) = -Q(t)\]

\[\dot{\rho}_{\text{vac}} + 3H \left( \rho_{\text{vac}} + \frac{p_{\text{vac}}}{c^2} \right) = Q(t)\]

where \(Q(t)\) represents the energy exchange term between matter and vacuum energy.

4. Modeling Spatial Variations in Vacuum Energy Density

4.1 Introduction of Spatial Fluctuations

If the vacuum energy density varies spatially, it can be expressed as \(\rho_{\text{vac}} = \rho_{\text{vac}}(t, \mathbf{x})\). To model this, we introduce the density fluctuation \(\delta_{\text{vac}}(\mathbf{x}, t)\):

\[\rho_{\text{vac}}(t, \mathbf{x}) = \rho_{\text{vac}}(t) [1 + \delta_{\text{vac}}(\mathbf{x}, t)]\]

4.2 Derivation of Perturbation Equations

Using the gravitational potential perturbation \(\Phi(\mathbf{x}, t)\), we write the perturbed Poisson equation as:

\[\nabla^2 \Phi = 4\pi G \rho_{\text{total}}(t) \delta_{\text{total}}(\mathbf{x}, t)\]

where \(\delta_{\text{total}} = \delta_{\text{matter}} + \delta_{\text{vac}}\).

4.3 Growth Equation of Fluctuations

The growth of density fluctuations is described by the following differential equation:

\[\ddot{\delta}_{\text{matter}} + 2H \dot{\delta}_{\text{matter}} - 4\pi G \rho_{\text{total}} \delta_{\text{total}} = 0\]

We analyze the influence of vacuum energy fluctuations on the growth of density fluctuations.

5. Microscopic Model of Vacuum Energy Density

5.1 Quantum Field Theoretical Approach

In quantum field theory, the temporal and spatial variations of vacuum energy density are described by changes in the state of the field and its interactions.

5.2 Introduction of Effective Potential

The effective potential \(V_{\text{eff}}(\phi)\) of a scalar field \(\phi\) is used to model the energy density:

\[\rho_{\text{vac}}(t, \mathbf{x}) = V_{\text{eff}}(\phi(t, \mathbf{x}))\]

The dynamics of the field are described by the Klein-Gordon equation:

\[\Box \phi - \frac{dV_{\text{eff}}}{d\phi} = 0\]

5.3 Interactions and Symmetry Breaking

Field interactions and symmetry breaking may induce variations in vacuum energy density, such as changes due to phase transitions.

6. Comparison with Observations and Constraints

6.1 Accelerated Expansion of the Universe

It is observationally known that the universe is undergoing accelerated expansion. We examine whether the time variation of \(\rho_{\text{vac}}(t)\) in the model can reproduce this acceleration.

6.2 Fluctuations in the Cosmic Microwave Background (CMB)

We analyze the impact of spatial fluctuations in vacuum energy on the anisotropy of the CMB and compare it with observational data.

6.3 Formation of Large-Scale Structures

We calculate the growth rate of density fluctuations and assess whether it is consistent with the formation process of galaxies and galaxy clusters.

7. Conclusion

By concretely modeling the temporal and spatial variations in vacuum energy density, we can quantitatively assess the role of vacuum energy in the evolution and structure formation of the universe. By adjusting the model parameters and checking consistency with observational data, we can validate the theory's plausibility.

1. Introduction of Zero-Point Energy and Setting of Cutoff Wavenumber

The new theory introduces the expectation value of zero-point energy and sets the cutoff wavenumber to ensure the finiteness of vacuum energy.

The expectation value of zero-point energy is expressed as follows:

\[ \langle 0 | H | 0 \rangle = \frac{1}{2} \sum_{k=0}^{k_{\text{max}}} \hbar \omega_k \]

Here, the cutoff wavenumber \( k_{\text{max}} \) is set using the Planck length \( l_P \):

\[ k_{\text{max}} = \frac{1}{l_P}, \quad l_P = \sqrt{\frac{\hbar G}{c^3}} \]

This equation prevents the divergence of vacuum energy density, providing the foundation for the new theory.

2. Extension of Einstein Equation by Zero Theory

Applying the zero theory to the Einstein equation clarifies the impact of vacuum energy density on the curvature of spacetime.

The modified Einstein equation is expressed as follows:

\[ G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}^{\text{matter}} - \rho_{\text{vac}} g_{\mu\nu} \right) \]

This equation demonstrates that vacuum energy is not merely a background energy but an active element that shapes the structure of spacetime dynamically.

3. Reinterpretation of Photon Energy and Gravitational Potential

In the new theory, the wave-particle duality of photons is determined by the local fluctuations in vacuum energy density.

The energy of a photon is expressed as:

\[ E = h\nu \]

The equation indicating that gravity transmission depends on the fluctuations of vacuum energy density is as follows:

\[ \nabla^2 \Phi = 4\pi G \left( \rho_m + \delta \rho_{\text{vac}} \right) \]

This equation leads to the understanding of gravity not as a material entity but as a distortion of spacetime based on energy conversion.

4. Reinterpretation of Dark Energy and Dark Matter

Dark energy is explained as an energy conversion resulting from temporal fluctuations in vacuum energy.

The equation representing the effect of dark energy is as follows:

\[ \rho_{\text{vac}}(t) = \rho_{\text{vac},0} \left( \frac{a_0}{a(t)} \right)^n \]

Additionally, the equation indicating that the effect of dark matter arises from fluctuations in vacuum energy is as follows:

\[ \nabla^2 \Phi = 4\pi G \left( \rho_m + \delta \rho_{\text{vac}} \right) \]

This equation explains that dark matter is a phenomenon caused by fluctuations in vacuum energy.

5. Relationship between Quantum Entanglement and Vacuum Energy

Quantum entanglement is interpreted as a phenomenon where energy states mutually influence each other through the vacuum energy field.

The equation representing the correlation between entangled particles is as follows:

\[ \langle \psi_1 | \hat{O} | \psi_2 \rangle = \int e^{-i\omega t} \delta \rho_{\text{vac}}(x, t) dx \]

This equation shows that the fluctuations in vacuum energy play a role in maintaining the non-local correlations between entangled particles.

6. Common Mechanism of Photons and Gravity

Photons and gravity are understood as phenomena transmitted through the conversion of vacuum energy in a unified manner.

The equations representing the energy relationship between photons and gravity are as follows:

\[ E_{\text{photon}} = \hbar \omega, \quad G_{\mu\nu} = 8\pi G \rho_{\text{vac}} g_{\mu\nu} \]

These equations demonstrate that both photons and gravity rely on a common mechanism, depending on the fluctuations in vacuum energy.

Conclusion

These modifications and additions to the equations more clearly demonstrate that the new theory provides a unified explanation of the dynamic role of vacuum energy while maintaining consistency with general relativity and quantum electrodynamics.

Establishing the Consistency of the New Theory with General Relativity and Quantum Electrodynamics

1. Introduction

The new theory reinterprets vacuum energy as a dynamic medium of energy conversion, aiming to provide a unified explanation for the transmission of photons, gravity, dark energy, dark matter, quantum entanglement, and more. This section details how this theory aligns with general relativity and quantum electrodynamics, highlighting the differences and similarities with existing theories.

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2. Consistency with General Relativity

2.1 Fundamental Principles of General Relativity

- Equivalence Principle: The equality of inertial and gravitational mass.

- Curvature of Spacetime: Energy-momentum determines the curvature of spacetime.

- Einstein's Equation:

\[ G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]

where \( G_{\mu\nu} \) is the Einstein tensor and \( T_{\mu\nu} \) is the energy-momentum tensor.

2.2 The Role of Vacuum Energy in the New Theory

- Contribution to the Energy-Momentum Tensor:

In the new theory, the vacuum energy density \( \rho_{\text{vac}} \) contributes to the energy-momentum tensor.

\[ T_{\mu\nu}^{\text{total}} = T_{\mu\nu}^{\text{matter}} + T_{\mu\nu}^{\text{vac}} \]

\[ T_{\mu\nu}^{\text{vac}} = -\rho_{\text{vac}} g_{\mu\nu} \]

- Modified Einstein's Equation:

\[ G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}^{\text{matter}} - \rho_{\text{vac}} g_{\mu\nu} \right) \]

- Relation to the Cosmological Constant:

The vacuum energy density is related to the cosmological constant \( \Lambda \):

\[ \Lambda = \frac{8\pi G}{c^4} \rho_{\text{vac}} \]

2.3 Introduction of Dynamic Vacuum Energy

- Temporal and Spatial Changes in Vacuum Energy Density:

The new theory allows for the vacuum energy density to vary with time and space.

- Relationship to the Conservation of Energy:

In general relativity, the local conservation of energy-momentum holds:

\[ \nabla^\mu T_{\mu\nu} = 0 \]

If vacuum energy varies with time and space, an energy exchange term must be introduced to satisfy this conservation law.

- Impact on the Dynamics of Spacetime:

Dynamic vacuum energy influences the expansion or contraction of spacetime, offering new insights into the evolution of the universe.

2.4 Conclusion

The new theory integrates vacuum energy density into the energy-momentum tensor and modifies Einstein's equation, maintaining consistency with general relativity. By allowing for dynamic variations in vacuum energy, it holds potential for explaining phenomena like the accelerated expansion of the universe.

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3. Consistency with Quantum Electrodynamics

3.1 Fundamental Principles of Quantum Electrodynamics

- Field Quantization: Treating electromagnetic fields as quantized, with interactions mediated by photons.

- Dirac Equation: Describing the behavior of fermions such as electrons.

- Perturbative Interaction Treatment: Calculations using Feynman diagrams.

3.2 Vacuum Energy and Quantum Field Theory

- Zero-Point Energy:

The quantization of fields implies the existence of energy (zero-point energy) even in a vacuum.

- Casimir Effect:

An example of the physical effects of zero-point energy, highlighting the importance of vacuum in physics.

3.3 Photon Transmission in the New Theory

- Transmission of Photons through Vacuum Energy:

The wave-particle duality of photons is explained through dynamic changes in vacuum energy.

- Reinterpretation of Creation and Annihilation Operators:

The creation and annihilation of photons are expressed as local fluctuations in vacuum energy density.

\[ a_k^\dagger |0\rangle = |k\rangle, \quad a_k |k\rangle = |0\rangle \]

3.4 Reevaluation of Interactions

- Field Interactions:

The interaction between electromagnetic and matter fields is reinterpreted as a process mediated by vacuum energy.

- Need for New Perturbative Calculations:

Perturbative calculations considering the dynamic role of vacuum energy become necessary.

3.5 Conclusion

The new theory incorporates the dynamic role of vacuum energy into the framework of quantum electrodynamics, reinterpreting photon transmission and field interactions. This approach maintains consistency with quantum electrodynamics while offering new physical insights.

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4. Clarification of Differences and Similarities with Existing Theories

4.1 Similarities

- Recognition of the Importance of Vacuum:

Existing theories also acknowledge the critical role of vacuum through zero-point energy and the Casimir effect.

- Contribution to the Energy-Momentum Tensor:

In general relativity, vacuum energy contributes to the energy-momentum tensor as the cosmological constant.

- Field Quantization:

Quantum electrodynamics describes the properties of photons and vacuum through field quantization.

4.2 Differences

- Dynamic Role of Vacuum Energy:

The new theory asserts that vacuum energy dynamically varies with time and space, actively influencing physical phenomena, unlike existing theories that usually assume vacuum energy to be constant.

- Reinterpretation of Dark Energy and Dark Matter:

Dark energy and dark matter are explained as effects of vacuum energy fluctuations or density changes, whereas existing theories treat them as unknown energies or substances.

- Interpretation of Quantum Entanglement:

The new theory considers vacuum energy as mediating non-local correlations in quantum entanglement, unlike existing theories where vacuum energy's role is not explicitly addressed.

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5. Clarifying the Positioning of the Theory

5.1 Features of the New Theory

- Unified Framework:

Aiming to explain photons, gravity, dark energy, dark matter, and quantum entanglement through the central role of vacuum energy.

- Dynamic Vacuum Energy:

Considering vacuum energy as a dynamic quantity that varies with time and space.

- Extension of Existing Theories:

Expanding or reinterpreting general relativity and quantum electrodynamics to explain new physical phenomena.

5.2 Positioning

- Approach to a New Unified Theory:

Positioned as part of an effort to unify gravity and quantum mechanics.

- Challenge to Unresolved Problems:

Aiming to provide a deeper understanding of dark energy, dark matter, and quantum entanglement.

- Potential Paradigm Shift in Physics:

Reevaluating the role of vacuum energy could deepen our fundamental understanding of physics.

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6. Future Challenges and Outlook

6.1 Refinement of the Theory

- Strengthening Mathematical Support:

Developing detailed calculations and predictive models is necessary.

- Clarification of Interaction Mechanisms:

Elucidating the mechanism of interactions mediated by vacuum energy.

6.2 Experimental Verification

- Validation with Observational Data:

Comparing the model's predictions with observed data on the expansion rate of the universe and the formation of large-scale structures.

- Proposing New Experiments:

Developing experimental techniques to detect the dynamic variations in vacuum energy.

6.3 Comparison with Other Theories

- Relationship with Existing Unified Theories:

Evaluating the relationship with theories like string theory and loop quantum gravity.

- Clarification of Physical Predictions:

Providing specific predictions to distinguish this theory from others.

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7. Conclusion

The new theory, by focusing on the dynamic role of vacuum energy, offers new interpretations of unresolved problems in physics while maintaining consistency with general relativity and quantum electrodynamics. Clarifying its similarities and differences with existing theories, the theory's positioning aims to contribute to the advancement of physics.

New Theory: The Role of Vacuum Energy in Unifying All Phenomena through Energy Conversion

The new theory reinterprets the individual concepts of traditional physics (photons, gravity, dark energy, dark matter, gravitons, etc.) through the perspective that "everything can be explained by energy conversion." This approach, based on Einstein's famous energy-mass equivalence equation \(E=mc^2\), understands physical phenomena as a series of processes of energy conversion. Additionally, by applying Zero Theory to Einstein's equations, it delves deeper into how vacuum energy density influences the curvature of spacetime. This article elaborates on this new perspective, focusing on the dynamic role of vacuum energy, forces arising from energy conversion, the transmission of photons and gravity, the reinterpretation of dark energy and dark matter, the understanding of quantum entanglement, and the redefinition of the law of energy conservation.

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1. Reinterpretation of Vacuum Energy

1.1 The Dynamic Role of Vacuum Energy as Energy Conversion

In traditional physics, the vacuum is often considered an empty space without energy. However, the new theory interprets the vacuum as a field with zero-point energy fluctuations, viewing the vacuum itself as the primary stage for energy conversion. This perspective posits that fluctuations in the vacuum's energy density are the foundation for all physical phenomena.

Equation:

The expectation value of zero-point energy is expressed as follows:

\[\langle 0 | H | 0 \rangle = \frac{1}{2} \sum_{k=0}^{k_{\text{max}}} \hbar \omega_k\]

This equation indicates that vacuum fluctuations cause finite energy variations. Here, the cutoff wave number \( k_{\text{max}} \) is set using the Planck length \( l_P \), preventing the divergence of vacuum energy to infinity.

\[k_{\text{max}} = \frac{1}{l_P}, \quad l_P = \sqrt{\frac{\hbar G}{c^3}}\]

This setting shows how energy conversion occurs at the smallest physical scale (Planck length), providing the foundation for the new theory.

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2. Extension of Einstein's Equation with Zero Theory

2.1 Application of Zero Theory to Einstein's Equation

Applying Zero Theory to Einstein's equation clarifies that vacuum energy density directly influences the curvature of spacetime. In traditional general relativity, the distribution of matter and energy determines the structure of spacetime, but Zero Theory adds the influence of vacuum energy to this relationship.

Extended Einstein's Equation with Zero Theory:

\[G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}^{\text{matter}} - \rho_{\text{vac}} g_{\mu\nu} \right)\]

This equation shows that in addition to the distribution of matter and energy, the vacuum energy density \(\rho_{\text{vac}}\) contributes to the curvature of spacetime. It emphasizes that vacuum energy is not merely a background energy but a dynamic element actively shaping the structure of spacetime.

2.2 Relationship between Dark Energy and the Cosmological Constant

In this extended equation, the dynamic changes in vacuum energy density play the role of dark energy, providing a deeper understanding beyond the traditional static cosmological constant \(\Lambda\). This naturally explains the accelerated expansion of the universe as a result of energy conversion.

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3. Forces Arising from Energy Conversion

3.1 Definition: Attractive Forces Generated by Distortion of Cosmic Medium through Energy Conversion

Attractive forces that arise from the distortion of the cosmic medium caused by energy conversion.

3.2 Concept: The Mechanism by which Energy Conversion Distorts the Medium (Spacetime), Generating Attractive Forces

Energy conversion distorts the medium (spacetime), resulting in the generation of attractive forces. This mechanism demonstrates that local changes in energy affect the entire medium. The mechanisms of these two forces are based on different energy conversions and interactions within the framework of Zero Theory, making them essential elements for explaining the dynamics of the universe.

3.3 Analogy between General Relativity and Zero Theory

3.3.1 Explanation of Gravity in General Relativity

Curvature of Space:

In Einstein's general relativity, mass and energy curve spacetime, and this curvature exerts gravitational forces on other objects. The heavier an object, the more it curves the surrounding spacetime, causing other objects to move along these curves.

3.3.2 Analogy in Zero Theory

Thread and Pattern Analogy:

In Zero Theory, the universe is likened to a "medium" like a thread, with energy conversions "weaving" patterns into it. By weaving patterns into the thread, the surrounding energy and matter are drawn in, creating gravitational-like forces. These patterns represent the vibrations or conversions of energy within the medium, illustrating the mechanism by which attractive forces are generated.

3.4 Detailed Explanation of the Analogy

Vibrations of the Thread and Gravity:

In Zero Theory, energy conversion causes the medium (thread) to vibrate, creating "patterns" that manifest as gravitational-like phenomena. These vibrations influence the entire universe, explaining how energy conversions lead to the attraction between objects. This perspective views energy conversion as altering the properties of the medium, resulting in the emergence of gravitational forces.

Universality of the Medium:

In Zero Theory, the medium (thread) exists throughout the universe, and energy conversions within this medium distort it, generating forces similar to gravity. This universal presence allows energy conversions to influence the structure of spacetime consistently, providing a unified explanation for gravitational phenomena and energy interactions.

3.5 Understanding Gravity in Zero Theory

3.5.1 Origin of Attractive Forces:

In Zero Theory, attractive forces arise from the interaction between the cosmic medium and energy conversions. Here, the "cosmic medium" represents the state of energy conversion and fluctuations within the universe.

3.5.2 Macroscale Phenomena:

At cosmic scales, the attraction is considered a result of the circulation and conversion of energy within Zero Theory. Attractive forces are derived from the distortions and energy distributions in the medium, understood as phenomena observable at macroscopic scales.

3.6 Examination of Theoretical Consistency

3.6.1 Consistency within the Framework of Zero Theory

Energy Conversion and Attraction:

Zero Theory views the universe as a "medium," with energy conversions and fluctuations generating attractive forces. The explanation that attractive forces are a result of energy redistribution and conversion is consistent within the theory.

Fluctuations of the Medium:

In this theory, attractive forces emerge as interactions within the cosmic medium, meaning that fluctuations and energy variations within the medium directly generate these forces. This explains how attractive forces are produced as a consequence of energy transformations within the medium.

3.6.2 Consistency with Other Theories

Consistency with Relativity:

Gravity, explained by the curvature of spacetime in relativity, and gravity described in Zero Theory as vibrations or conversions in the cosmic medium, contribute to explaining the same phenomenon (gravity) through different mechanisms. While the mechanisms differ, both aim to describe the gravitational force's effects.

Consistency with Quantum Mechanics:

If quantum entanglement and state changes based on Zero Theory's energy conversion, they must align with established results in quantum mechanics. How Zero Theory integrates these phenomena within its framework is crucial for maintaining consistency with quantum theory.

3.6.3 Experimental Verification

Confirmation of Theoretical Consistency:

Alignment with experimental data is essential. Whether the forces and energy conversions predicted by Zero Theory match observations and experimental results is critical.

New Experimental Approaches:

To verify the new predictions of Zero Theory, new experiments and observations are required. This includes precise measurements of energy conversions and the resulting forces within the cosmic medium.

3.7 Conclusion

The interpretation of attractive forces within Zero Theory conceptualizes the universe as a "medium," with energy conversions and fluctuations generating these forces. This approach explains how attractive forces emerge as a result of energy redistribution and conversion within the medium, providing a clear explanation of how these forces influence the motion and dynamics of matter.

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4. Relationship between Zero Theory and Dark Energy, Dark Matter

4.1 Dark Energy: Cosmic Expansion as Energy Conversion

Concept:
Dark energy is the energy responsible for the accelerated expansion of the universe. It is uniformly distributed throughout the cosmos and affects space itself.

Relation to Zero Theory:

- Energy Vibrations:
In Zero Theory, changes in cosmic expansion and gravitational variations are explained by the vibrations and conversions of energy within the cosmic medium. The effects of dark energy can potentially be understood as energy conversions within Zero Theory's cosmic medium.

- Relation to Gravity:
While dark energy works against the effects of gravity, Zero Theory explains gravity as arising from energy vibrations and conversions within the medium. The influence of dark energy could also be explained through these energy conversions within the medium.

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4.2 Dark Matter

Concept:
Dark matter is traditionally considered a hypothetical substance that accounts for the missing mass required to form and sustain galaxies. It interacts through gravity but is invisible and does not emit, absorb, or reflect light.

Relation to Zero Theory:

- Energy Vibrations:
In Zero Theory, the presence and behavior of matter and gravity are explained by energy vibrations and conversions within the cosmic medium. The gravitational effects of dark matter can be understood as the influence of energy conversions within Zero Theory's medium.

- Relation to Gravity:
Dark matter interacts gravitationally with ordinary matter, and in Zero Theory, gravity is generated by vibrations and conversions in the cosmic medium. Thus, the effects of dark matter can be explained based on the changes in the medium's energy.

\[\nabla^2 \Phi = 4\pi G \left( \rho_m + \delta \rho_{\text{vac}} \right)\]

Here, \(\delta \rho_{\text{vac}}\) represents fluctuations in vacuum energy density, understood not as a material entity but as a distortion in spacetime, characterized by the Planck length \( l_P \).

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5. Redefinition of Gravitons: Understanding as Mediators of Energy Conversion

Gravitons are traditionally considered quantum particles that mediate gravity in quantum mechanics. However, the new theory reinterprets them as part of the energy conversion process itself. Gravity is seen as a phenomenon transmitted through fluctuations in vacuum energy, eliminating the need to assume the existence of gravitons as virtual particles.

In this new interpretation, vacuum energy itself acts as the medium of space, with distortions in space manifesting as increases in density, observed as gravity. Thus, the conversion of vacuum energy directly shapes the structure of spacetime and generates gravitational phenomena.

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6. Reinterpretation of Quantum Entanglement and Vacuum Energy

6.1 The Role of Non-Local Correlation and Vacuum Energy

Quantum entanglement is interpreted as a phenomenon where energy states mutually influence each other through the vacuum energy field. This perspective explains why the correlation between entangled particles is maintained regardless of distance, from the viewpoint of energy conversion.

\[\langle \psi_1 | \hat{O} | \psi_2 \rangle = \int e^{-i\omega t} \delta \rho_{\text{vac}}(x, t) dx\]

This equation indicates that fluctuations in vacuum energy play a role in maintaining the correlation between entangled particles.

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7. Unified Understanding through Vacuum Energy Conversion

7.1 Common Mechanism of Photons and Gravity

Photons and gravity are understood as phenomena transmitted through the conversion of vacuum energy. This common mechanism explains how energy state transformations are linked to various physical phenomena.

\[E_{\text{photon}} = \hbar \omega, \quad G_{\mu\nu} = 8\pi G \rho_{\text{vac}} g_{\mu\nu}\]

This shows that both photons and gravity rely on the fluctuations in vacuum energy through a shared mechanism.

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8. Conclusion: Consistency of the New Theory from the Perspective of Energy Conversion

The new theory aims to explain all physical phenomena through processes of energy conversion, providing a consistent framework for understanding photons, gravity, dark energy, dark matter, and quantum entanglement. By applying Zero Theory, the impact of vacuum energy density on the curvature of spacetime is clearly articulated in mathematical terms, enabling a unified understanding that maintains consistency with current physical theories while offering new interpretations. This approach returns to the foundations of Einstein's \(E=mc^2\), offering a simpler and more coherent theoretical system.

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Appendix: Importance of Theoretical Consistency and Experimental Verification

To ensure the consistency and reliability of the new theory, the following points are essential:

1. Detailed Mechanism Elucidation:
   Strengthening the physical basis of the theory by providing detailed explanations of energy conversion processes and mechanisms.
2. Examination of Consistency with Observational Data:
   Comparing the theory's predictions with observational data (e.g., cosmic microwave background radiation, galaxy rotation curves, gravitational lensing effects) to ensure alignment.
3. Theoretical Predictions and Experimental Verification:
   Enhancing the theory's practicality and reliability by outlining specific predictions and how they can be experimentally verified.
4. Clarification of Terminology:
   Defining and explaining unique terms and concepts within the new theory to facilitate reader understanding.
5. Theory's Limitations and Future Prospects:
   Discussing the current limitations of the new theory and outlining future research directions to address unresolved challenges.

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The above content provides a detailed explanation with mathematical equations necessary to comprehensively describe the new theory. By integrating Zero Theory with the new theory, a comprehensive physical model based on energy conversion is presented, offering a new perspective on existing physical theories. Future research and experimental verification are expected to further elucidate the validity and consistency of this theory.

The new theory reinterprets the vacuum not as a mere empty background but as a dynamic energy conversion medium that actively participates in the transmission of photons and gravity. Vacuum energy forms the foundation of physical phenomena, opening the way to a unified explanation of photon and gravitational propagation, dark energy and dark matter phenomena, quantum entanglement, and more. This paper focuses on the role of vacuum energy, explaining the perspective of the new theory, equations, comparisons with existing theories, and detailed descriptions of each concept.

1. Reinterpretation of Vacuum Energy

1.1 The Dynamic Role of Vacuum Energy

In conventional physics, the vacuum is often treated as an empty space without energy, but in quantum mechanics and quantum field theory, the vacuum is recognized as a fluctuating field with zero-point energy.

Equation:

The expectation value of zero-point energy is expressed as follows:

\[ \langle 0 | H | 0 \rangle = \frac{1}{2} \sum_{k=0}^{k_{\text{max}}} \hbar \omega_k \]

Explanation:

This equation shows that vacuum fluctuations cause finite energy variations. The cutoff wavenumber \( k_{\text{max}} \) is introduced to prevent the divergence of infinite energy.

Setting an Appropriate Cutoff Standard:

The cutoff wavenumber \( k_{\text{max}} \) is set using the Planck length \( l_P \) as follows:

\[ k_{\text{max}} = \frac{1}{l_P} \]

The Planck length \( l_P \) is expressed by the following equation:

\[ l_P = \sqrt{\frac{\hbar G}{c^3}} \]

This allows energy calculations to converge to finite values that are physically meaningful, preventing vacuum energy from diverging to infinity. The Planck length provides the minimum length scale, indicating the region where quantum gravity effects become prominent.

In the new theory, this vacuum energy is reinterpreted as a dynamic medium for energy conversion, and the vacuum itself is seen as an entity actively involved in physical phenomena.

2. The Role of the Vacuum in Photon Transmission

2.1 The Wave Nature of Photons and Vacuum Energy

The wave nature of light refers to its behavior as an electromagnetic wave, described by Maxwell's equations. In existing theories, the wave nature of light propagates through the interaction of electric and magnetic fields, with energy spreading as a wave. It is well known that the energy of a photon depends on its wavelength and frequency, expressed by the following equation:

Equation:

\[ E = h\nu \]

In the new theory, this wave nature of photons is interpreted as a process of energy conversion via vacuum energy. It is shown that dynamic changes in vacuum energy affect the wave nature of photons, and the propagation of light depends on the vacuum energy density \(\rho_{\text{vac}}\).

2.2 The Particle Nature of Photons and Energy Localization

The particle nature of photons refers to their discrete nature as particles carrying energy and momentum. Photons behave as localized particles when they collide with detectors. When position is observed, the wave function collapses, and photons are observed as particles.

Equation in Existing Theory:

\[ E = \hbar \omega, \quad p = \hbar k \]

Equation in the New Theory:

Using creation and annihilation operators, the localization of energy is shown:

\[ a_k^\dagger |0\rangle = |k\rangle, \quad a_k |k\rangle = |0\rangle \]

Explanation:

Through the creation and annihilation operators, the mechanism by which photons appear locally from the vacuum state is demonstrated. The particle nature of photons depends on the local concentration of vacuum energy. The Planck length plays an important role as the scale of this local concentration, with vacuum energy fluctuations leading to the localization of photon energy.

3. The Role of the Vacuum in Gravity Transmission

3.1 Gravitational Potential and Vacuum Energy Density

The gravitational potential is influenced not only by the mass density of matter but also by the vacuum energy density. The new theory understands gravity transmission as a dynamic fluctuation in vacuum energy.

Equation:

\[ \nabla^2 \Phi(\mathbf{x}) = 4\pi G \left[ \rho_m(\mathbf{x}) + \rho_{\text{vac}}(\mathbf{x}) \right] \]

Explanation:

This equation shows that vacuum energy density \(\rho_{\text{vac}}\) contributes to the gravitational potential, interpreting gravity as part of energy conversion through the vacuum.

4. Reinterpretation of Dark Energy, Dark Matter, and Gravitons

4.1 Dark Energy

As the universe expands, the vacuum energy density changes, resulting in fluctuations in the vacuum. These fluctuations serve as the energy source, and in the current theory, this is referred to as "dark energy." In other words, the density change due to expansion generates this energy, which is interpreted as the phenomenon that seems to accelerate the expansion of the universe.

4.2 Dark Matter and Gravitons

Dark matter is reinterpreted as the vacuum itself. Local fluctuations in vacuum energy density cause spacetime distortions, which are observed as gravity. These distortions in spacetime due to changes in vacuum density act as a force that attracts matter, understood as the effect of dark matter. In current theory, this transmission of gravity is explained by a hypothetical energy particle called the "graviton." However, in the new theory, it is believed that fluctuations in vacuum density themselves cause spacetime distortion, resulting in gravity and the attraction of matter. In other words, dark matter is not a material entity but is explained by the distortion of spacetime based on changes in vacuum density, i.e., gravitational effects. The scale of these spacetime distortions is also characterized by the Planck length \( l_P \).

5. The Role of Vacuum Energy in Quantum Entanglement

5.1 A New Interpretation of Quantum Entanglement

Quantum entanglement is a phenomenon where energy states influence each other through the vacuum. Entangled particles maintain their correlation through the vacuum energy field, regardless of distance. Fluctuations in vacuum energy preserve the correlation between entangled particles, manifesting as phase differences when observed. These fluctuations in vacuum energy explain the non-local correlations observed in quantum entanglement.

6. Unified Understanding of Physical Phenomena Through the Vacuum

6.1 Common Mechanisms of Photon and Gravity Transmission

The propagation of photons and the transmission of gravity are both understood as results of energy conversion via vacuum energy.

• Photon propagation: Energy localization and transmission through the dynamic conversion of vacuum energy.
• Gravity transmission: Changes in spacetime curvature due to fluctuations in vacuum energy density. Fluctuations on the Planck scale influence the transmission of gravity.

6.2 Unified Understanding of Dark Energy, Dark Matter, and Quantum Entanglement

• Dark Energy: Fluctuations in the vacuum caused by changes in vacuum energy density due to the expansion of the universe. This is referred to as "dark energy."
• Dark Matter: Changes in vacuum density cause spacetime distortions, resulting in gravity and the attraction of matter. In current theory, this transmission of gravity is explained by hypothetical energy particles called "gravitons." The scale of spacetime distortions caused by vacuum fluctuations is explained based on the Planck length.
• Quantum Entanglement: Fluctuations in vacuum energy maintain the correlation between entangled particles, transmitting the influence of energy states through the vacuum, explaining the non-local correlation.

7. Redefinition of the Law of Energy Conservation

The new theory redefines the law of energy conservation by considering vacuum energy. Fluctuations in vacuum energy provide a unified framework for the conservation of both wave and particle energy.

Equation:

\[ \frac{\partial}{\partial t} (u_{\text{field}} + \rho_{\text{vac}}) + \nabla \cdot (\mathbf{S} + \mathbf{S}_{\text{vac}}) = 0 \]

Explanation:

This equation shows that the energy of the electromagnetic field \( u_{\text{field}} \) and the vacuum energy density \( \rho_{\text{vac}} \) are conserved in a unified manner, emphasizing the active role of vacuum energy in physical phenomena.

Conclusion

Through energy conversion and transmission via the vacuum, physical phenomena such as photon and gravitational propagation, dark energy, dark matter, and quantum entanglement can be understood in a unified way. The new theory provides a fresh perspective that sees vacuum energy as dynamically involved in physical phenomena, offering a path for further developments in physics while maintaining consistency with existing theories.