Background Dependence, Background Independence, and Emergence

Background dependence, background independence, and emergence are not binary labels. They refer to different structures: fixed asymptotic data, a chosen perturbative vacuum, nondynamical geometric fields, and the way gauge-invariant observables reconstruct effective spacetime. A useful comparison states which structure is fixed and which can vary dynamically instead of equating “no metric in the action” with background independence.

Required background. The proposal comparison supplies the microscopic objects. Duality webs and parameter maps supplies changes of representation that need not change the physics.

Helpful background. Relational gauge-invariant observables supplies an operational observable language. The quantum-gravity handoff supplies the boundary of semiclassical geometry.

Four meanings that should be separated

A perturbative formulation is vacuum dependent when its fields and BRST operator are built from a chosen worldsheet CFT or classical solution. It is asymptotically conditioned when boundary topology, sources, or falloffs are fixed even though the interior metric fluctuates. It is background covariant when expansions about different representatives are related by field redefinitions or classical solutions. It is relationally geometric when distances and causal relations are inferred from correlations among dynamical observables rather than coordinates.

General relativity itself illustrates the distinction: the metric is dynamical, but a scattering problem can still fix asymptotic flatness and charges. Diffeomorphism invariance makes local coordinate values gauge dependent; it does not eliminate boundary conditions.

First application: compare three formulations

Closed string field theory starts with a conformal background, which fixes $Q$ and the string state space. A new classical solution can sometimes shift to a nearby background, and background-independence theorems relate compatible expansions perturbatively Sen and Zwiebach 1994. They do not show that every topology or asymptotic sector belongs to one convergent field space.

The IKKT action contains matrices rather than a spacetime metric, yet it fixes a target dimension, supersymmetry, matrix measure, signature or contour prescription, and large-$N$ limit Ishibashi et al. 1997. Anisotropic extent can emerge dynamically within those choices. Recovering a metric requires gauge-invariant spectral, causal, and interaction data, not just matrix eigenvalue plots.

AdS/CFT allows all interior bulk geometries compatible with a fixed boundary theory and boundary conditions. The radial coordinate and semiclassical bulk locality emerge in appropriate states and large-$N$ sectors Harlow 2018, while the asymptotic conformal structure remains defining data. This is interior background independence within a conditioned sector, not a sum over all asymptotics.

Adversarial control: remove the metric symbol

Rewrite a lattice or matrix action without an explicit $g_{\mu\nu}$ but keep a fixed causal ordering, dimension, topology, and boundary state. Declaring the result background independent ignores those nondynamical inputs. Conversely, expanding a covariant theory around one saddle does not prove fundamental background dependence if observables and transitions between saddles are contained in the same exact object.

An operational emergence claim should specify a family of states and a limit in which: a stable effective dimension appears; correlators admit approximately local propagation; causal response and equivalence-principle behavior are recovered; curvature is small compared with the microscopic cutoff; and deviations scale predictably with $1/N$, $\alpha'$, or another control parameter. Without these tests, “emergent spacetime” is an interpretation of collective variables.

The evidence ceiling varies by formulation. Background shifts in string field theory, matrix symmetry breaking, and holographic reconstruction each establish limited notions of emergence; none alone proves a universal sum over vacua. Numerical matrix evidence shows how to test one such recovery claim without building the conclusion into the observable.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Harlow, D. (2018), “TASI Lectures on the Emergence of Bulk Physics in AdS/CFT,” Proceedings of Science TASI2017, 002. arXiv:1802.01040.
• Ishibashi, N., Kawai, H., Kitazawa, Y., and Tsuchiya, A. (1997), “A Large-$N$ Reduced Model as Superstring,” Nuclear Physics B 498, 467–491. arXiv:hep-th/9612115.
• Sen, A., and Zwiebach, B. (1994), “Quantum Background Independence of Closed-String Field Theory,” Nuclear Physics B 423, 580–630. arXiv:hep-th/9311009.