16
Background Independence
Abstract
Spacetime background independence is a fundamental property of any candidate
quantum gravity theory. In this chapter, we outline the proof of background
independence for the closed SFT by proving that the equations of motion of
two backgrounds related by a marginal deformation are equivalent after a field
redefinition.
16.1 The Concept of Background Independence
Background independence means that the formalism does not depend on the
background—if any—used to write the theory. A dependence in the background
would imply that there is a distinguished background among all possibilities, which
seems in tension with the dynamics of spacetime and the superposition principle
from quantum mechanics. Moreover, one would expect a fundamental theory to
tell which backgrounds are consistent and that they could be derived instead of
postulated. Background independence allows spacetime to emerge as a consequence
of the dynamics of the theory and of its defining fundamental laws.
Background independence can be manifest or not. In the second case, one needs
to fix a background to define the theory, but the dynamics on different backgrounds
are physically equivalent. 1 This implies that two theories with different backgrounds
can be related, for example, by a field redefinition.
While fields other than the metric can also be expanded around a background, no
difficulty is expected in this case. Indeed, the topic of background independence is
particularly sensible only for the metric because it provides the frame for all other
1 This does not mean that the physics in all backgrounds are identical, but that the laws are. Hence,
a computation made in one specific background can be translated into another background.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_16
327
Background Independence
Abstract
Spacetime background independence is a fundamental property of any candidate
quantum gravity theory. In this chapter, we outline the proof of background
independence for the closed SFT by proving that the equations of motion of
two backgrounds related by a marginal deformation are equivalent after a field
redefinition.
16.1 The Concept of Background Independence
Background independence means that the formalism does not depend on the
background—if any—used to write the theory. A dependence in the background
would imply that there is a distinguished background among all possibilities, which
seems in tension with the dynamics of spacetime and the superposition principle
from quantum mechanics. Moreover, one would expect a fundamental theory to
tell which backgrounds are consistent and that they could be derived instead of
postulated. Background independence allows spacetime to emerge as a consequence
of the dynamics of the theory and of its defining fundamental laws.
Background independence can be manifest or not. In the second case, one needs
to fix a background to define the theory, but the dynamics on different backgrounds
are physically equivalent. 1 This implies that two theories with different backgrounds
can be related, for example, by a field redefinition.
While fields other than the metric can also be expanded around a background, no
difficulty is expected in this case. Indeed, the topic of background independence is
particularly sensible only for the metric because it provides the frame for all other
1 This does not mean that the physics in all backgrounds are identical, but that the laws are. Hence,
a computation made in one specific background can be translated into another background.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_16
327