Deducing counter-terms for canonically quantised GR

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In summary: This will give me the counter-terms needed to renormalize the theory.In summary, the conversation discusses the action of a massive scalar field minimally coupled to gravity and the one loop contribution to graviton to graviton scattering with a single scalar loop. In order to eliminate a divergence, the speaker plans to add counter-terms to the action, which can be determined by looking at interactions that reproduce the amplitude at tree-level. These counter-terms may include terms like ##R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## and ##\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}##.
  • #1
GogoJS
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Consider the action of a massive scalar field minimally coupled to gravity, that is,

$$S = \int d^4x \, \sqrt{-g} \, \left( 2\kappa^{-1} R + \partial_\mu \phi \partial^\mu \phi - m^2 \phi^2\right)$$

The theory I consider is canonically quantised gravity, with ##g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}## and with the corresponding Feynman rules I evaluated the one loop contribution to graviton to graviton scattering with a single scalar loop. I found the amplitude was of the form,

$$\mathcal M \sim -\frac{\kappa^2}{32\pi^2\epsilon} \left[ \eta_{\mu\nu}\eta_{\lambda\sigma} f_1(p) + (\eta_{\mu\lambda}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\lambda})f_2(p)\right] + \mathrm{finite \, \, (\epsilon \to 0)}$$

in dimensional regularisation, where the functions ##f_1(p)## and ##f_2(p)## are polynomials in the external momentum ##p## and scalar mass ##m##. To eliminate the divergence I would like to add the required counter-terms to the action, as one does in effective field theory. However, I am not sure how to deduce which terms to add.

Those allowed by symmetries would include terms like ##R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## but I'm a little iffy on how I can relate these to the amplitude I've computed to absorb the divergence. In addition, I am unsure as to where or not we would need terms like ##\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## involving the scalar. How can we reason what diagrams would require counter-terms coupling the scalar to curvature scalars?
 
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  • #2
GogoJS said:
Consider the action of a massive scalar field minimally coupled to gravity, that is,

$$S = \int d^4x \, \sqrt{-g} \, \left( 2\kappa^{-1} R + \partial_\mu \phi \partial^\mu \phi - m^2 \phi^2\right)$$

The theory I consider is canonically quantised gravity, with ##g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}## and with the corresponding Feynman rules I evaluated the one loop contribution to graviton to graviton scattering with a single scalar loop. I found the amplitude was of the form,

$$\mathcal M \sim -\frac{\kappa^2}{32\pi^2\epsilon} \left[ \eta_{\mu\nu}\eta_{\lambda\sigma} f_1(p) + (\eta_{\mu\lambda}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\lambda})f_2(p)\right] + \mathrm{finite \, \, (\epsilon \to 0)}$$

in dimensional regularisation, where the functions ##f_1(p)## and ##f_2(p)## are polynomials in the external momentum ##p## and scalar mass ##m##. To eliminate the divergence I would like to add the required counter-terms to the action, as one does in effective field theory. However, I am not sure how to deduce which terms to add.

Those allowed by symmetries would include terms like ##R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## but I'm a little iffy on how I can relate these to the amplitude I've computed to absorb the divergence. In addition, I am unsure as to where or not we would need terms like ##\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## involving the scalar. How can we reason what diagrams would require counter-terms coupling the scalar to curvature scalars?
You are looking for an interaction which, at tree level, will give an amplitude be proportional to the divergent one you wrote above. Since your contribution comes from a diagram with the scalar in the loop and only gravitons in the external legs you should find that you will need a counter term that contains only the Riemann tensor. Of course at some point you will need terms like ##\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## but these will arise to cure divergences of loop diagrams with a scalar in an external leg.
 
  • #3
nrqed said:
You are looking for an interaction which, at tree level, will give an amplitude be proportional to the divergent one you wrote above. Since your contribution comes from a diagram with the scalar in the loop and only gravitons in the external legs you should find that you will need a counter term that contains only the Riemann tensor. Of course at some point you will need terms like ##\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## but these will arise to cure divergences of loop diagrams with a scalar in an external leg.

Thanks, I see you what you mean. I will expand ##R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## and other curvature scalars in powers of ##h_{\mu\nu}##, and then find the terms that are required at tree-level for the amplitude to be reproduced.
 

Related to Deducing counter-terms for canonically quantised GR

1. What is meant by "Deducing counter-terms for canonically quantised GR"?

Deducing counter-terms for canonically quantised GR refers to the process of finding mathematical expressions that can be added to the equations of general relativity (GR) in order to remove divergences that arise in the quantum field theory calculations of GR. These counter-terms are necessary to make the theory predictive and physically meaningful.

2. Why is it important to deduce counter-terms for canonically quantised GR?

It is important to deduce counter-terms for canonically quantised GR because without them, the theory would be plagued with divergences and would not be able to make accurate predictions about the behavior of physical systems. These counter-terms help to make the theory consistent and reliable.

3. How are counter-terms deduced for canonically quantised GR?

Counter-terms for canonically quantised GR are typically deduced using a combination of analytical and computational techniques. The process involves identifying the divergent terms in the quantum field theory calculations and then finding mathematical expressions that can cancel out these divergences. This can be a complex and iterative process, and often requires significant mathematical expertise.

4. What challenges are involved in deducing counter-terms for canonically quantised GR?

One of the main challenges in deducing counter-terms for canonically quantised GR is that the theory is non-renormalizable, meaning that infinite terms cannot be completely cancelled out. This makes it difficult to find a finite number of counter-terms that can fully remove all divergences. Additionally, the calculations involved can be highly complex and require advanced mathematical techniques.

5. Are there any current research developments in the field of deducing counter-terms for canonically quantised GR?

Yes, there are ongoing research efforts to improve our understanding of how to deduce counter-terms for canonically quantised GR. This includes exploring new mathematical techniques and approaches, as well as studying the implications of these counter-terms for the behavior of the theory at different energy scales. Additionally, there are ongoing efforts to apply these techniques to other theories, such as supersymmetric theories, which also require the use of counter-terms.

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