# String theory and background independence

I have read that string theory assumes strings live in spacetime defined by general relativity which make the theory background dependent (although general relativity is a background independent theory). Background independence dictates that spacetime emerge from more fundamental ingredient than spacetime. Quoting Brian Greene, “Then, the theories ingredients - be they strings, branes, loops, or something else discovered in the course of further research - coalesced to produce a familiar, large-scale spacetime” (Greene, The Fabric of the Cosmos, 2004: 491). My question: why couldn’t spacetime just be spacetime, a fundamental entity? If so, string theory, based on general relativity, would not be a “great unsolved problem” facing string theory.

I think you might be slightly misinterpreting what string theorists believe their theory says about quantum spacetime.

Start with a classical spacetime with a certain pseudo-Riemannian metric, and consider the theory of strings propagating on this spacetime. There are two very important results:

1. Unless $$R_{\mu \nu} = 0$$, the theory acquires a non-zero beta function under conformal transformations. Afaik this is widely believed to be unphysical, because strings theorists don't know how to make sense of the theory when it isn't invariant under conformal transformations. Therefore we conclude that string theories exist only on Ricci-flat spacetimes, that is, on spacetimes solving the vacuum EFE.
2. In the spectrum of the string, you will find spin-2 states which have been conjectured to model physical gravitons. They have all properties that one would expect gravitons of General Relativity to have, except that their ultraviolet completion is finite.

The question then becomes – why did we have to choose the classical background in the first place? After all, isn't quantum gravity supposed to model classical spacetime as a certain limit of the full theory?

An interesting observation is that if you try to build a "coherent" state from string theory gravitons, the setup can be re-interpreted exactly as if the string was in the vacuum state, but propagating in a different spacetime.

So it can be conjectured that there's a concealed duality between the excited states of the string/superstring propagating in one classical spacetime, and an unexcited string propagating in another spacetime. Thus, maybe string theory is background-independent after all, even though the perturbative formulation that we've found isn't manifestly background independent?

Though afaik there's no background independent nonperturbative formulations of string theory known (I am not considering any of the AdS/CFT stuff here because it probably isn't directly related to the perturbative string/superstring theories and still remains just a conjecture).

From my personal correspondence with people working in the field, I conclude that there's no consensus on this subject, despite some individuals' belief that there is consensus :) I personally know people working on topics related to string theory, who are absolutely convinced that background independence is a must-have property of the non-perturbative definition of string theory, whatever it is. But I also know at least one senior lecturer who is perfectly satisfied with a special-relativistic flat spacetime entering the definition of the theory.

• Thanks, this was what I thought might be the situation from my conceptual thinking (limited mathematical background). – Jim Johnson Sep 22 '18 at 22:24

The idea of presenting gravity as a standard quantum field of interacting gravitons is naive, because it does not explain how spacetime becomes curved. All other fields do not affect the background like pens writing on a sheet of paper. The sheet may be flat or curved, but writing on it doesn't change the curvature. GR defines the curvature and thus cannot be presented as a standard QFT.

Quantum gravity is not just about quantizing gravity. It is about building a theory on a background independent of spacetime. A theory built on this background can define spacetime as flat or curved as appropriate, like a projector projecting different images of reality. Then all other fields must be defined on the same background to be able to interact. So quantum gravity is about redefining both gravity and QFT to unite them on a new background.

• FWIW: There is a perfectly sensible interpretation of GR as a classical field on a flat spacetime: the metric $g$ is no different from the electromagnetic potential $A$, and the geodesic equation is nothing but a relativistic Newton equation where the Christoffel symbols represent a real force. This interpretation is equivalent to the standard one, and does not require a geometric perspective. In this setting, the "standard" QFT approach is not naïve anymore: it is the most natural approach to the problem (which just happens not to work for other reasons)... – AccidentalFourierTransform Sep 18 '18 at 2:10
• @AccidentalFourierTransform (1) How does this approach explain time dilation? (2) How would other quantum fields interact with this classical field? (I assume it would have to be quantized.) – safesphere Sep 18 '18 at 2:15
• Some good references are listed in researchgate.net/post/… see also physics.stackexchange.com/q/32544/84967 – AccidentalFourierTransform Sep 18 '18 at 2:21
• Safesphere, in a previous question about quantum fields, you said: “Other possibilities also have not been ruled out, such as that gravity may not have a quantum nature or have a nature that would change our understanding of “quantum” and what exactly we mean by it. No one yet knows if all fields are quantum.” Thus, in thinking about string theory, I wondered if background independence was a requirement. From what you said, I do not think it is, for example, if gravity does not have a quantum nature. Is this conclusion correct? thanks – Jim Johnson Sep 19 '18 at 1:02
• @JimJohnson It is not given that the String or M-theory is an adequate description of nature. Whatever the final theory is, it should have the current theories as special cases. In one extreme it should produce the QFT; in the other extreme it should produce the classical GR. It seems that to do so it should have space and time as functions depending of some other variables that we call the background. Everything seems to be based on symmetries. Not all symmetries have been revealed and neither have been all relationships among them or the reasons behind them. It should be done by 2050. – safesphere Sep 19 '18 at 2:39

If so, string theory, based on general relativity, would not be a “great unsolved problem” facing string theory.

The unsolved problem comes because the holy grail of physics at present is unification of all four forces, in one quantum mechanical theory. The weak the electromagnetic are unified , with the discovery of the Higgs validating the theory, and the strong has all the indications of unification. It is the quantization of gravity that is still elusive so cannot be harnessed except with speculations in one theory of everything.

At the moment it is only string theories that seem to be able to quantize gravity and also embed the standard model of physics in a unified way. The draw back is that there are a very large number of possible string theories, and research is ongoing for finding a model of our universe.

My question: why couldn’t spacetime just be spacetime, a fundamental entity?

The premise for present research is that the underlying level of nature is quantum mechanical. The aim is to show that classical ( and general relativity is a classical theory ) emerges from the basic underlying quantum level of reality. This can be mathematically shown for electromagnetism, and the hope is it will be true for gravity too. Note "hope", no definite theory yet. So spacetime has to be emergent in this theoretical outlook.

• Anna, thanks for your response; however, my question was not clear. The “great unsolved question” I was referring to was string theories lack of background independence, not unification. I was thinking that if spacetime were a fundamental entity, then string theory would be considered background independent. Could we have a quantum spacetime field that is fundamental, composed of no other entities? Is this a valid question? Thanks – Jim Johnson Sep 17 '18 at 20:50
• All mathematical models are valid . The ones chosen by physics have to fit the data and predict new verifiable manifestations so as to be validated for physics.In my answer above I am stating why string theory is attractive to theorists as a theory of everything: because it can include quantization of gravity naturally, it can be shown that general relativity emerges from some string theories . It is the goal of a quantized theory of everything , the problem to be solved as far as main stream physics goes, . en.wikipedia.org/wiki/Theory_of_everything . – anna v Sep 18 '18 at 2:55

You're asking a question that deserves an enormous amount and clarifications. I will just want to share some very illuminating references on the idea of "Background independence" in the context of string theory.

What is background independence and how important is it?

At least two philosophers understood background independence

Is space and time emergent? ER-EPR correspondence adds a voice