[Physics] energy in string theory


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?

Best Answer

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.