Quantum Decoherence & Gravity from Quantum Entropy

Quantum Decoherence, Entropy, and Gravity from Entropic Principles: From Density Matrices to Modified Einstein Equations

What you'll learn
- Learn how decoherence is related to the transition from the quantum to the classical world
- Define the density matrix and its connection with decoherence, as well as the Von Neumann equation
- Write down the density matrix for general mixed states
- Learn how Quantum Entropy naturally arises from the density matrix, and how it is related to the definition of entropy in statistical mechanics
- Write down the Wigner function, and learn how it is an extension of classical probabilities
- Understand the Moyal equation, which is the dynamical equation satisfied by the Wigner function, and study its classical limit
- Understand a recent research paper written by a mathematical physicist: Ginestra Bianconi. This paper will be the key to investigating gravity, quantum entropy.
- Define eigenvalues and the logarithm of tensors
- Learn the connection between Lagrangian densities and the entropic action in Ginestra Bianconi's article
- Learn the generalization of the Klein Gordon equation from the quantum entropic action
- Understand Topological fields, metric between vectors and bivectors
- Define and learn the intuition behind the Codifferential and Dirac operator (differential forms will be used. There is also an appendix dedicated to them)
- Learn the the G-field and the Θ-Field, which are used in Ginestra Bianconi's article to derive modified Einstein field equations
- Derive the the cosmological constant from modified gravity
- Take the variation of the entropic action to derived modified Einstein field equations
- Understand how to caclulate the entropy of a black hole

Requirements
- Differential forms (these will be necessary for the second part of the course. However, there is a full appendix dedicated to them)
- The formalism of quantum physics (operators, observables, bra-ket notation, Hermiticity, etc). However, there is a preliminary section which serves as a refresher on the formalism of quantum mechanics. It might be consulted by those students who need to review a few concepts.
- Classical field theory
- Linear algebra
- Classical General Relativity (the mathematics of tensors)

Description
In this course we explore the connection between entropy, quantum mechanics, and gravity.

In this advanced theoretical physics course, we examine the fundamental role ofquantum decoherencein the transition from quantum to classical behavior, and we intrpduce the concept ofquantum entropy(this will be the first part of the course). After that, we take a step forward, investigating howgravity itself may emerge from entropic principles.

Starting from thedensity matrix formalism, we develop a clear understanding of decoherence and how it explains the classical appearance of a fundamentally quantum world. We also analyze the important concept ofWigner function, which serves as a tool for connecting quantum dynamics with classical phase space. Then, we rigorously definequantum entropy, using the Von Neumann formulation.

In the second half of the course, we apply these tools to modern research topics. We exploretopological metrics,codifferential operators, and thevariation of entropic actions. Special emphasis is placed on a recent and influential work byGinestra Bianconi, which derives modified Einstein field equations using entropy as a fundamental physical quantity.

This course integrates insights fromquantum physics,general relativity,field theory,differential forms, andinformation theory, making it suitable for physicists, mathematicians, and engineers interested in the cutting-edge theoretical landscape.

What You’ll Learn

How to describe decoherence using thedensity matrixandvon Neumann equation

The role of theWigner functionin bridging classical and quantum dynamics

The concept and computation ofquantum entropy

How entropy can lead toentropic actionsfor matter and gauge fields

The structure and variation oftopological and geometrical actions

A detailed walkthrough ofGinestra Bianconi’s paper “Gravity from Entropy”

Derivation ofmodified Einstein equationsfrom entropic considerations

The emergence of acosmological constantfrom an entropic action

How to calculate the (quantum) entropy of a blackhole (by analyzing another article written by Ginestra Bianconi)

Who Is This Course For?

Physicists and mathematicians interested inquantum gravityorfoundations of quantum theory

Researchers or students intheoretical physics,mathematical physics, orcomplex systems

Anyone curious about howinformation and entropymay be fundamental to space, time, and gravity

Who this course is for:
- Advanced Physics Students: Learners pursuing upper-level undergraduate or graduate studies in physics who want to deepen their understanding of quantum decoherence, entropy, and foundational quantum theory.
- Researchers and Academics: Individuals involved in theoretical or mathematical physics who are interested in modern approaches to quantum-classical transition and the role of entropy in gravitational frameworks.
- Curious Enthusiasts in Quantum Foundations: Passionate learners with a strong grasp of quantum mechanics who wish to explore deeper ideas such as the Wigner function, density matrices, and the emergence of classicali behavior from quantum behavior
- Those Exploring Entropic Gravity and Modified Theories: Physicists or mathematically inclined learners interested in cutting-edge theories connecting quantum information, entropy, and gravity — including Ginestra Bianconi’s work on deriving gravity from entropic principles.
- In summary, this course is intended for advanced physics students, researchers, and enthusiasts with a solid background in quantum mechanics and mathematical physics. It is ideal for those interested in the foundational aspects of quantum theory, the transition to classical behavior via decoherence, and the emerging connections between entropy and gravitational dynamics. The course will also benefit learners curious about modern theoretical frameworks, such as entropic gravity and modifications to Einstein’s field equations.
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