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Facts agreed on by most Physicists -

GR: One can't apply Noether's theorem to argue there is a conserved energy. QFT: One can apply Noether's theorem to argue there is a conserved energy. String Theory: A mathematically consistent quantum theory of gravity.

Conclusion -

If one can apply Noether's theorem in String Theory to argue there is a conserved energy, String Theory is not compatible with GR. If one can't, it is not compatible with QFT.

Questions -

Is the conclusion wrong? What is wrong with it? Is there a definition of energy in String Theory? If yes, what is the definition?

Shelly Adhikari
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    You can apply Noether's theorem in GR. I think what you meant to say is that most of the space-times are not invariant to time translations and so you can't derive that energy is conserved. Is this your problem, that energy might not be conserved in string theory? Because conservation of energy is no big deal. If it is conserved, great, we can use that. If not, well, that's life. But we can still do physics. – Marek Mar 19 '11 at 18:15
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    At any rate, all those questions - about the existence of a conserved quantity and its possible derivation from Noether's theorem if one exists - proceed identically in general relativity and in string theory. It doesn't really matter whether the theory is classical or quantum - and indeed, string theory is the only consistent extension of "quantum field theory" that incorporates general relativity, but this fact is irrelevant for the existence of Noether's conserved quantities, too. So it just makes no sense to talk about as advanced a thing as string theory in this simple context. – Luboš Motl Mar 19 '11 at 18:18
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    Page 148 "Superstring Theory : Introduction" by Green, Schwarz and Witten - "Upon coupling this theory to a curved world sheet, energy momentum conservation breaks down unless c = d." One of the reasons, we got to a 26D theory was that energy momentum conservation was imposed. I am thoroughly confused about the definition/role of energy in String Theory. Thanks in advance for any clarification. – Shelly Adhikari Mar 19 '11 at 20:23
  • Unless I am misremembering (don't have the book here), this statement refers to the two dimensional energy momentum tensor on the worldsheet, whose conservation is one of the consistency conditions of perturbative string theory. This is all separate from the question of spacetime energy which I addressed in my answer. –  Mar 19 '11 at 22:08

2 Answers2

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In any theory which includes General Relativity, there is no locally conserved energy. The reason is that energy creates a gravitational field which has energy itself, so "gravity gravitates". There is a local quantity (the energy-momentum tensor) which is covariantly conserved, and there are global quantities (like the ADM mass) which expressed the total energy of the system and are conserved. But there are no currents which give locally conserved quantities.

(One more technical way to express that is that the spacetime transformations corresponding to local energy and momentum conservation are now no longer global symmetries but are gauge redundancies).

In field theory (classical or quantum) without gravity spacetime translations can be a global symmetry (if spacetime is flat), and correspondingly energy is locally conserved. Once you couple the theory weakly to gravity (even if gravity is only a background with no dynamics of its own), it is then only approximately conserved. When gravitational effects are large, there is no approximately conserved quantity correpsonding to energy.

As Lubos said in comments, nothing really new happens with respect to this question in string theory. In the most general situation there is no locally conserved quantity, and when the theory reduces to QFT on spacetime which is approximately flat and when gravitational effects are small, then there is an approximately conserved quantity. String theory is compatible with both QFT and GR, which means it reproduces their results approximately in the appropriate limit. But of course away from those limits it has its own features that are generally different from either one of those theories. For that specific question it is much closer in spirit to GR.

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    There are prefectly fine stress-energy pseudotensors, which give a local picture of energy flows, and should not be dismissed because their transformation properties are not intuitive and shift stuff around nonlocally. – Ron Maimon Oct 08 '11 at 21:25
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Within GR, there is a conserved stress energy pseudo-tensor. It is called a pseudo-tensor because it is not a tensorial quantity, it's transformation properties allow you to make the stress-energy of the gravitational field vanish at any point. This quantity can be derived by using Noether's prescription on the Einstein-Hilbert action, and it was proposed by Einstein as the correct stress-energy of a gravitational field. It has been controversial, because of its non-tensorial property, but it is the correct thing. If you integrate to find the total charge, you find sensible total masses in asymptotically flat space time, but the objects are defined relative to boundaries at infinity.

String theory has exactly the same type of stress-energy definitions. These are on the worldsheet, and not in space-time, and so are asymptotic S-matrix energy and momentum. The global/local issues in string theory exactly match the global/local issues in General Relativity, and your argument is actually one that supports an S-matrix/string description.

Ron Maimon
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