# Can Schwinger-Dyson Equations Compute Green Functions in Quantum Gravity?

In summary, the conversation discussed the possibility of computing Green functions, specifically in regards to Quantum Gravity, through an exact, non-perturbative approach using the action S and the Einstein-Hilbert Lagrangian. However, the conversation concluded that this is not feasible due to the non-renormalizable nature of the quantum theory and the non-linearity of the classical field equations.
A question about them (i have looked it up at wikipedia) Can they produce (a solution to them) a way to compute "Green functions" (and hence the propagator) in an "exact" (Non-perturbative approach) way?? .. for example the S-D equation read:

$$\frac{\delta S}{\delta \phi(x)}[-i \frac{\delta}{\delta J}]Z[J]+J(x)Z[J]=0$$ (1)

Then if we put the action S to be $$S[\phi]=\int d^{4}xL_{E-H}$$

where L is the Einstein-Hilbert Lagrangian..a solution to (1) if exist would be a form to compute the Green-function for the "Quantum Gravity"?:shy:

In principle, yes, you can put the HE Lagrangian in the path integral. However, it leads you nowhere, as the quantum theory is not renormalizable. For any interacting theory the path integral is not exactly computable and so neither the Green functions. This comes from the fact that the classical Lagrangian (or even Hamiltonian, if you know how to obtain them) field equations are not linear, besides being a system of coupled PDE-s.

Daniel.

The Schwinger-Dyson equations are a set of functional equations that govern the dynamics of quantum field theories. They play a central role in non-perturbative approaches to quantum field theory, as they provide a way to compute Green functions and propagators in an exact manner.

The equation you have cited (1) is a general form of the Schwinger-Dyson equation, which applies to any quantum field theory. However, in order to use it to compute Green functions, one needs to have an explicit expression for the action S[\phi]. This is where the challenge lies in applying the Schwinger-Dyson equations to quantum gravity.

The Einstein-Hilbert Lagrangian, which you have mentioned as a possible choice for the action, is the classical action for general relativity. In order to use it in the Schwinger-Dyson equation, one would need to find a way to quantize the theory, i.e. to express it in terms of quantum fields. This is still an open problem in theoretical physics and there is currently no agreed upon method for quantizing general relativity.

Therefore, while the Schwinger-Dyson equations provide a powerful tool for computing Green functions in non-perturbative approaches, their application to quantum gravity remains a challenging and ongoing area of research. Further developments in this field may provide insights into the nature of quantum gravity and potentially lead to a better understanding of the fundamental laws of our universe.

## What are Schwinger-Dyson equations?

Schwinger-Dyson equations are a set of integral equations used in quantum field theory to describe the dynamics of a quantum system. They are derived from the principles of quantum mechanics and special relativity.

## What is the significance of Schwinger-Dyson equations?

Schwinger-Dyson equations play a crucial role in understanding the properties and behavior of quantum systems, particularly in the study of gauge theories and quantum electrodynamics. They also provide a framework for calculating the behavior of particles and interactions in the quantum field theory.

## How are Schwinger-Dyson equations derived?

Schwinger-Dyson equations are derived by using the path integral formalism in quantum field theory. This involves summing over all possible paths of particles in a given interaction and taking into account the contributions of virtual particles.

## What are the limitations of Schwinger-Dyson equations?

Schwinger-Dyson equations are perturbative in nature and are only valid for weak interactions. This means that they cannot accurately describe strong interactions between particles. Additionally, they are only applicable to systems in equilibrium, so they cannot be used to study non-equilibrium phenomena.

## How are Schwinger-Dyson equations used in practical applications?

Schwinger-Dyson equations are used in a wide range of applications, including the study of quantum chromodynamics, the calculation of Feynman diagrams, and the development of effective field theories. They are also used in the development of quantum computing and in the study of condensed matter systems.

• Quantum Physics
Replies
13
Views
979
• Quantum Physics
Replies
11
Views
1K
• Quantum Physics
Replies
33
Views
3K
• Quantum Physics
Replies
2
Views
1K
• Advanced Physics Homework Help
Replies
1
Views
998
• Quantum Physics
Replies
5
Views
2K
• Classical Physics
Replies
1
Views
508
• Quantum Physics
Replies
5
Views
915
• Special and General Relativity
Replies
7
Views
766
• Quantum Physics
Replies
2
Views
1K