Büyük patlamadaki tekilliğin yoğunluk matrisi ve dolaşıklık entropisi, büyük patlama sonucunda oluşan entropi ve yoğunluğa eşittir. Enerji yoktan yaratılmaz. Var olan enerji kaybolmaz.

Your assumptions are Axiomatic. do you have supporting documentation which can clarify your assumptions?

That is THE question… attempting to answer this could be soul destroying, given the

fundamental issues which will require to be resolved.

This is very much the high notes as far as an answer.

Calculating the density matrix for the singularity at the Big Bang, and determining the

entanglement entropy from it, involves advanced concepts in quantum cosmology and

quantum field theory in curved spacetime. Here's a high-level overview of how you

might approach this complex topic:

Density Matrix at the Big Bang Singularity

The Big Bang singularity is a point where classical theories of gravity (like General

Relativity) and standard cosmological models break down. To describe the state of the

universe at the Big Bang using a density matrix, you would likely need a theory of

quantum gravity, which remains elusive. However, here's a conceptual approach using

insights from general quantum field theory in curved spacetime:

• Quantum State of the Early Universe

Assume the early universe can be described by a quantum field theory on a curved

spacetime background. The metric of this spacetime needs to approximate the

conditions believed to exist near the Big Bang.

• State Description

The state of the universe can be modeled by a wavefunction or a state vector in a

Hilbert space that includes all fields and interactions. For simplicity, often a

vacuum state or a thermal state is assumed in the early universe models.

• Density Matrix Construction

The density matrix (ρ) for a system in state (|Ψ⟩) is given by (ρ = |Ψ⟩⟨Ψ|). For

mixed states, or states involving thermal fluctuations as one might expect near

the Big Bang, the density matrix would be a sum over these possible states

weighted by their probabilities.

Calculating the Entanglement Entropy

Entanglement entropy is a measure of quantum correlations between different parts of

a system. In the context of the early universe:

• Partition the System

Divide the system (the early universe) into two parts. This could be different regions

of space or different field degrees of freedom.

• Reduced Density Matrix

Trace out one part of the system from the density matrix to get the reduced

density matrix (ρ𝐴) for part A, where (ρ𝐴 = Tr𝐵 (ρ)).

• Entropy Calculation

The entanglement entropy \(S\) is then calculated using the von Neumann

entropy formula:

[𝑆 = −Tr(ρ𝐴 log ρ𝐴)]

This calculation involves computing the eigenvalues of (ρ𝐴) and summing over them.

Challenges and Considerations

• Singularity and Quantum Gravity

Near the singularity, spacetime curvature and quantum effects are extreme, and

classical descriptions fail. Quantum gravity theories such as Loop Quantum

Gravity or String Theory might provide a framework but are still under

development and often lack precise predictions for such extreme conditions.

• Computational Complexity

Calculating entanglement entropy, especially in a field theory context, can be

computationally intensive and analytically complex due to the infinite degrees of

freedom involved.

• Approximations and Models

Often, physicists use simpler models or approximations (like considering a

conformally coupled scalar field in a Robertson-Walker metric) to make

calculations tractable.

For practical calculations, advanced mathematical tools and numerical simulations

are often required, and these calculations are typically performed within the framework

of a specific theoretical model assuming certain conditions about the early universe.

way to GPT4 Rho ln Rho. i was already familiar, with some of these pieces, i am looking for refrences please and thank you.