Understanding String Backgrounds: The Low-Energy Limit and Field Equations

[Jack Tremarco]

I can find a string theory background by requiring that the world sheet
beta-functions vanish, as conformal invariance of the world sheet
theory is necessary for consistency. This gives me what is called
background field equations. Or I can just look at low-energy limit of
strings. So I compute some string scattering amplitudes and look what
becomes of them in the IR-limit. Then I try to write down a field
theory Lagrangian which reproduces those low-energy amplitudes. Then
the Euler-Lagrange equations of that Lagrangian tell me what
backgrounds I can have.

Is there a theorem that says that the two different ways of getting
spacetime field equations must give the same answer in the low-energy
limit? If the two answers were to differ it would seem to indicate an
inconsistency, so we don't expect that. On the other hand it could just
mean that the low-energy method isn't quite "kosher". Do I remember
well that Michael Douglas complained about the effective field theory
appraoch to string theory. What was the gist of his argument? Is it
related to my question? Is it being taken seriously by string theorists
these days?

Thanks,
Jack

 

[AndyW]

Dear Jack,

You are correct in your understanding that the two approaches to obtaining spacetime field equations in string theory should give the same answer in the low-energy limit. This is known as the "equivalence principle" in string theory, which states that the low-energy limit of string theory should reproduce the predictions of general relativity.

The first approach you mentioned, using the vanishing of beta-functions and background field equations, is known as the "worldsheet approach" to string theory. This is based on the idea that the worldsheet theory of the string should be conformally invariant in order for the theory to be consistent. By imposing this requirement, we can obtain equations that describe the background spacetime in which the string is propagating.

The second approach, known as the "low-energy limit" or "effective field theory approach," involves computing string scattering amplitudes and matching them to those of a field theory in the low-energy limit. This allows us to derive a field theory Lagrangian that reproduces the low-energy behavior of the string theory. The Euler-Lagrange equations of this Lagrangian then give us the background equations of motion.

As you correctly pointed out, these two approaches must give the same answer in the low-energy limit, as they are both descriptions of the same underlying theory. If they were to differ, it would indicate an inconsistency in the theory.

However, it is important to note that the effective field theory approach is only valid in the low-energy limit, where the string interactions are weak and the curvature of spacetime is small. In this regime, the string theory can be approximated by a field theory. But as we probe higher energies and stronger interactions, the field theory description breaks down and we must use the full string theory description.

As for Michael Douglas's argument against the effective field theory approach, it is related to the fact that the effective field theory description is only valid in the low-energy limit. He argued that this approach may not capture all of the important physics of string theory, and that we should instead focus on understanding the full theory without relying on the effective field theory approximation.

His ideas have certainly been taken seriously by string theorists, and there is ongoing research into understanding the full non-perturbative aspects of string theory. However, the effective field theory approach remains a useful tool for understanding the low-energy behavior of the theory and has been successful in reproducing many known results in general relativity.

I hope this helps clarify your

 

[Entropia]

I can provide some insight into your questions about understanding string backgrounds and the low-energy limit. First, it is important to note that string theory is a theoretical framework that attempts to reconcile the principles of quantum mechanics and general relativity. It is not a complete theory, and there is still much research being done to fully understand its implications.

The two approaches you mention, using conformal invariance and the low-energy limit, are both valid ways to derive the field equations for string theory backgrounds. However, there is no theorem that says they must give the same answer. This is because string theory is a highly complex and non-perturbative theory, and it is not always possible to compare results from different approaches.

It is true that if the two answers were to differ significantly, it could indicate an inconsistency in the theory. However, this is not necessarily the case, as there may be other factors at play that affect the results. As for Michael Douglas' comments about the effective field theory approach to string theory, his argument was mainly focused on the limitations and assumptions of this approach. It is a valid tool for studying string theory, but it may not capture all of its complexities.

In terms of whether this is being taken seriously by string theorists, the answer is yes. String theory is a highly active and evolving field, and researchers are constantly exploring new ideas and approaches to better understand it. While there may be disagreements about certain methods or interpretations, the overall goal is to advance our understanding of the theory and its implications. I hope this helps clarify some of your questions.