emergence of time from unitary equivalence

We discuss the concept of unitary equivalence ˆh ∼ ˆu† ˆhmodˆu betweenthe modular hamiltonian ˆhmod and the subsystem hamiltonian ˆh in the context ofrealizing the emergence of time through a unitary operator ˆu . this concept suggests aduality between the modular flow and time evolution. additionally, requiring unitaryequivalence implies a connection between the ”modular chaos bound” and the ”chaosbound”. furthermore, we demonstrate this duality using quantum chaos diagnosticquantities in the thermofield double state of a fermionic system. quantum chaos diagnosticquantities are mathematical measures that characterize chaotic behavior inquantum systems. by examining these quantities in the thermofield double state, weillustrate the duality between them and the modular hamiltonian. we show a specificduality between correlators, the spectral form factor, and the loschmidt echo withthe modular hamiltonian. the spectral form factor is a quantity that provides informationabout the energy spectrum of a quantum system, while the loschmidt echocharacterizes the sensitivity of a system’s modular time evolution to perturbations.finally, we demonstrate that a different entanglement spectrum does not impose thesame constraint on the subsystem hamiltonian. the entanglement spectrum is relatedto entanglement entropy and provides information about the eigenvalues of the reduceddensity matrix associated with a subsystem. we discuss complex concepts related tothe interplay between quantum chaos, time emergence, and the relationship betweenmodular and subsystem hamiltonians. these ideas are part of ongoing research inquantum information theory and related fields.