# ADM Formalism & Wheeler-deWitt Equation: What's the Relation?

• jesusrods
In summary, ADM formalism is a mathematical framework used in general relativity to describe the dynamics of spacetime, while the Wheeler-deWitt equation is a key equation in quantum gravity derived from ADM formalism. The Wheeler-deWitt equation addresses the problem of time in quantum gravity by describing the universe in a timeless manner. However, it is difficult to solve and has implications for our understanding of the universe, but also has criticisms such as not considering observers or probabilities and lacking integration with other areas of physics.
jesusrods
Hello! I am interested in knowing the existing relation between the ADM Formalism and the Wheeler-deWitt Equation.

From the articles and lectures I was read, I understand that ADM formulation is the Hamiltonian formulation of General Relativity at classical level. Then, the Hamiltonian is quantized converting the metric and the momentums in the correct operators, and the Wheeler-deWitt equation is then Hψ=0 with H the quantized Hamiltonian.

I am right or wrong? I would be grateful if you can give me an answer.

PS: If I have some mistakes, please excuse me, english is not my born language and I am learning it. Thanks.

Hello,

Thank you for your interest in the relationship between the ADM Formalism and the Wheeler-deWitt Equation. You are correct in your understanding that the ADM Formalism is a classical formulation of General Relativity, while the Wheeler-deWitt Equation is a quantum version of the theory.

To elaborate further, the ADM Formalism breaks down the spacetime metric into its spatial and temporal components, allowing for a Hamiltonian formulation of General Relativity. This Hamiltonian is then quantized, as you mentioned, resulting in the Wheeler-deWitt Equation. This equation, as you stated, is of the form Hψ=0, where H is the quantized Hamiltonian and ψ is the wave function of the universe.

In essence, the ADM Formalism provides a classical framework for understanding the dynamics of spacetime, while the Wheeler-deWitt Equation introduces a quantum aspect to the theory. Both are important in understanding the behavior of the universe at different scales.

I hope this helps answer your question. If you have any further inquiries, please feel free to ask. And don't worry about any mistakes in your English, it is clear that you have a strong understanding of the topic. Keep up the good work!

## 1. What is ADM formalism and how is it related to the Wheeler-deWitt equation?

ADM formalism, also known as the Arnowitt-Deser-Misner formalism, is a mathematical framework used to describe the dynamics of a spacetime in general relativity. It breaks down the spacetime into a set of spatial hypersurfaces and time, allowing for the formulation of the equations of motion. The Wheeler-deWitt equation is a key equation in quantum gravity that attempts to merge general relativity and quantum mechanics. It is derived using the ADM formalism and describes the evolution of the wave function of the universe.

## 2. How does the Wheeler-deWitt equation address the problem of time in quantum gravity?

In classical physics, time is considered to be a universal and absolute concept. However, in quantum mechanics, time is treated as a parameter and is not included as an operator. This creates a fundamental conflict between the two theories when trying to describe the universe. The Wheeler-deWitt equation attempts to resolve this issue by eliminating the concept of time altogether and describing the universe in a timeless manner.

## 3. Can the Wheeler-deWitt equation be solved?

The Wheeler-deWitt equation is a highly complex and non-linear partial differential equation, making it extremely difficult to solve. Currently, there is no exact solution for the equation, but there have been attempts to find approximate solutions using various techniques such as perturbation theory and numerical methods.

## 4. What are the implications of the Wheeler-deWitt equation for our understanding of the universe?

The Wheeler-deWitt equation is a key component of theories of quantum gravity, which aim to unify general relativity and quantum mechanics. By addressing the problem of time and describing the universe in a timeless manner, it provides a potential framework for understanding the nature of the universe at its most fundamental level. It also has implications for the concept of causality and the nature of space and time.

## 5. Are there any criticisms of the Wheeler-deWitt equation?

One major criticism of the Wheeler-deWitt equation is that it does not take into account the role of observers or measurements, which are essential in quantum mechanics. It also does not incorporate the concept of probabilities, which is a fundamental aspect of quantum mechanics. Additionally, the equation has not yet been successfully integrated with other areas of physics, such as particle physics, and has not been experimentally tested.

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