Area operator and fixed area states in conformal field theories

Abstract

The fixed area states are previously discussed in the quantum error-correction codes interpretation of $\mathrm{AdS}/\mathrm{CFT}$. The dual bulk geometry is constructed by gravitational path integrals. In this paper we show the fixed area states correspondence in conformal field theories (CFTs), which are associated with the spectrum decomposition of reduced density matrix ${\ensuremath{\rho}}_{A}$ for a subsystem $A$. For two-dimensional CFTs we directly build the bulk metric, which is consistent with the expected geometry of the fixed area states. For arbitrary pure state $|\ensuremath{\psi}⟩$ with a geometric dual in the bulk we also find the consistency by using the gravity dual of R\'enyi entropy. We also obtain the parameters relation between the bulk geometry and boundary state. The pure state $|\ensuremath{\psi}⟩$ can be expanded as a superposition of the fixed area states. Motivated by this, we propose an area operator ${\stackrel{^}{A}}^{\ensuremath{\psi}}$. The fixed area state is the eigenstate of ${\stackrel{^}{A}}^{\ensuremath{\psi}}$, the associated eigenvalue is related to the R\'enyi entropy of subsystem $A$ in this state. The Ryu-Takayanagi formula can be expressed as the expectation value $⟨\ensuremath{\psi}|{\stackrel{^}{A}}^{\ensuremath{\psi}}|\ensuremath{\psi}⟩$ divided by $4G$, where $G$ is the Newton constant. We further show the fluctuation of the area operator in the geometric state $|\ensuremath{\psi}⟩$ is suppressed in the semiclassical limit $G\ensuremath{\rightarrow}0$.