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

[Karlisbad]

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:

 

[dextercioby]

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.

 

[denian]

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.