AFF, Fubini Vacuum, and Modular FlowA Modern Interpretation

This work revisits the classical results of de Alfaro–Fubini–Furlan (AFF) and Sergio
Fubini (1976) from the modern perspectives of quantum information theory and emergent
spacetime. While conventional quantum field theory assumes that time, scale, and geometry
are fixed structures encoded in the Lagrangian, Fubini and AFF proposed a radically different
viewpoint: these structures may instead emerge from the choice of generator or vacuum state.
We first analyze the AFF construction based on the sl(2, R) algebra and show how the selection of a time-evolution generator G modifies the spectral properties of the system, turning
a continuous spectrum into a discrete one without explicitly breaking conformal invariance.
We then examine Fubini’s “vacuum selection” mechanism, in which the vacuum itself acquires spacetime structure and induces position-dependent expectation values h0|φ(x)|0i,
anticipating the modern notion of state-dependent geometry familiar from AdS/CFT.
Finally, we connect these ideas to modular flow in algebraic quantum field theory. Using
a finite-dimensional matrix model, we demonstrate that vacuum selection effectively determines the modular Hamiltonian, illustrating the Tomita–Takesaki principle that the state
fixes the dynamics. We argue that classical time may emerge as a stable, coarse-grained
modular flow shaped by decoherence.
Overall, this work highlights a paradigm shift from the traditional view of time as a
fundamental background structure to a picture in which time emerges from state-dependent
modular dynamics.