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The relative entropy in two-dimensional Field Theory is studied for itsapplication as an irreversible quantity under the Renormalization Group,relying on a general monotonicity theorem for that quantity previouslyestablished. In the cylinder geometry, interpreted as finite-temperature fieldtheory, one can define from the relative entropy a monotonic quantity similarto Zamolodchikov's c function. On the other hand, the one-dimensional quantumthermodynamic entropy also leads to a monotonic quantity, with differentproperties. The relation of thermodynamic quantities with the complexcomponents of the stress tensor is also established and hence the entropic ctheorems are proposed as analogues of Zamolodchikov's c theorem for thecylinder geometry.Comment: 5 pages, Latex file, revtex, reorganized to best show the generality  of the results, version to appear in Phys. Rev. Let

Relative Entropy in 2D Quantum Field Theory, Finite-Size Corrections, and Irreversibility of the Renormalization Group