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In this study, we examined consequences of unconventional time development oftwo-dimensional conformal field theory induced by the $L_{1}$ and $L_{-1}$operators, employing the formalism previously developed in a study ofsine-square deformation. We discovered that the retainment of the Virasoroalgebra requires the presence of a cut-off near the fixed points. Theintroduction of a scale by the cut-off makes it possible to recapture theformula for entanglement entropy in a natural and straightforward manner.

Time development of conformal field theories associated with $L_{1}$ and $L_{-1}$ operators