Schwinger-Dyson equations for Quantum Gravity

[mhill]

using the Schwinger Dyson equations that gives us a differential expression for the functional Z[J] so Z[0] is just the path integral over 4-dimensional spaces .then for Einstein equation (no matter) they read (system of 10 functional equations)

$$R _{a,b}( -i \frac{ \delta Z (J)}{\delta J})+ J(x)Z(J)=0$$

then let's suppose we had a super-powerfull computer so we could solve these S-D equations Numerically could it be a solution to the problem of QG ?? , in fact could someone say me what methods are used to solve these kind of equations with functional derivatives ?? .. if possible in the perturbative and Non-perturbative expansions, thanks

 

[lbrits]

Well the functional derivative expansion maps directly into an expansion onto Feynman diagrams. But the expression you have written will probably blow up due in an uncontrollable way due to non-renormalizability. Also, $$R_{ab}$$ is a function of position on that manifold, but position is only meaningful for a particular metric. If the metric itself is fluctuating, what exactly is the meaning of your equation then? I don't think this is the way to go about quantum gravity. What is Z[0]?

 

[denian]

The Schwinger-Dyson equations for Quantum Gravity are a set of functional equations that relate the quantum field theory to the underlying gravitational theory. These equations provide a powerful framework for studying the dynamics of quantum gravity and have been extensively studied by physicists.

The first equation mentioned in the content, the functional expression for Z[J], is known as the generating functional. It is a mathematical tool used to calculate expectation values of observables in quantum field theory. The Schwinger-Dyson equations then relate this functional to the Einstein equations, which describe the behavior of gravity.

Solving these equations numerically is a challenging task, and it is currently an area of active research in theoretical physics. One approach is to use perturbative methods, where the equations are solved order by order in a small parameter. This is useful for studying weak gravitational interactions, but it becomes increasingly difficult to use for strong gravitational interactions.

Non-perturbative methods are also used to solve the Schwinger-Dyson equations. These methods involve finding exact solutions without relying on approximations or expansions. However, these methods are often more difficult to implement and require advanced mathematical techniques.

In summary, solving the Schwinger-Dyson equations for Quantum Gravity is a complex and ongoing research topic. While a super-powerful computer could potentially help with solving these equations numerically, it is not a guaranteed solution to the problem of quantum gravity. A combination of perturbative and non-perturbative methods is necessary to fully understand the dynamics of quantum gravity.