# Hamiltonian Weak Gravitational Field - Learn Free Particle Theory

• Andre' Quanta
In summary, the Lagrangian for the weak gravitational field satisfies the D'Alembert equation and the Hamiltonian is the same as the one for the free Klein-Gordon and free electromagnetic fields.
Andre' Quanta
In weak field regime i know that it is possible to quantize the gravitational field obtaining a quantum theory of free particles, called gravitons, which is very similar to the one for the electtromagnetic field.
Do you know some book in wiich i can study this theory?
In anycase what is the expression for the Hamiltonian of this theory in terms of creations and destructions operators?

I'd suggest starting with Donoghue's lecture notes http://arxiv.org/abs/gr-qc/9512024. The first part of DeWitt's http://arxiv.org/abs/0711.2445, as well as the original papers by Feynman and DeWitt might be useful at some point too, but I think that the effective field theory point of view is important and those older references predate that whole perspective, so might not be the best pedagogy anymore.

Andre' Quanta
I was interested in the Hamiltonian operator for the theory, is it the same as the one of the free electromagnetic field?

Andre' Quanta said:
I was interested in the Hamiltonian operator for the theory, is it the same as the one of the free electromagnetic field?

The field operator in linearized gravity has spin 2 so it cannot be the same. I don't know off hand any place to find an explict expression for the Hamiltonian. It's best if you learn what the Lagrangian is, then you can see if it's easy to make a Legendre transformation to write a Hamiltonian. The Lagrangian is more useful for covariant QFT, though you might have some particular reason to study the Hamiltonian.

The weak gravitational field, the usual h mu nu, satisfies the simple D' Alemebert equation, so the lagrangian is the one of Klein-Gordon replacing the scalar field with the tensorial one in our theory: this will be valid also for the Hamiltonian so i would say the Hamiltonian that i am looking for is simply the same as the one used for the free Klein-Gordon field and the free electromagnetic field, am i wrong?
ps: there would be difference in some costant for the dimensionality, but we can neglect it, i am only looking for a formal expression

Maybe you want to use the Wheeler-DeWitt and linearize. Otherwise, you can start with the linearized Lagrangian and you'll have to apply constraints for the gauge conditions etc.

## 1. What is a Hamiltonian weak gravitational field?

A Hamiltonian weak gravitational field refers to a specific type of weak gravitational field, which is a region in space where the gravitational force is relatively weak compared to other forces. In this type of field, the equations describing the motion of particles can be simplified using Hamiltonian mechanics.

## 2. What is the free particle theory?

The free particle theory is a theoretical framework that describes the behavior of particles moving in a weak gravitational field, where the gravitational force is considered negligible. It is based on the principles of classical mechanics and uses Hamiltonian mechanics to simplify the equations of motion for particles in such a field.

## 3. How does the Hamiltonian weak gravitational field affect the motion of particles?

In a Hamiltonian weak gravitational field, the equations of motion for particles can be simplified using Hamiltonian mechanics, making it easier to predict and analyze the motion of particles. This type of field also allows for the study of the motion of particles under the influence of other forces, such as electromagnetic forces, in addition to the weak gravitational force.

## 4. What are the applications of studying Hamiltonian weak gravitational fields and free particle theory?

Studying Hamiltonian weak gravitational fields and free particle theory is important for understanding the behavior of particles in weak gravitational fields, such as those found in our solar system. It has applications in various fields, including astrophysics, cosmology, and space exploration. Additionally, it can help in the development of new technologies, such as spacecraft propulsion systems.

## 5. Are there any current research areas related to Hamiltonian weak gravitational fields and free particle theory?

Yes, there are ongoing research efforts to further understand and improve our knowledge of Hamiltonian weak gravitational fields and free particle theory. Some current research areas include studying the effects of strong gravitational fields on the motion of particles, investigating the connection between quantum mechanics and gravity, and developing new techniques for analyzing and predicting the behavior of particles in these fields.

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