The Weyl Anomaly in Interacting Quantum Field Theory on Curved Spacetimes

Abstract We define the notion of Weyl anomalies, measuring the violation of local scale invariance, in interacting quantum field theory on curved spacetimes in the framework of locally covariant field theory. We discuss some general properties of Weyl anomalies, such as their relation to the trace anomaly. We give a criterion for a theory to be conformal at the quantum level and show that even for a conformal theory Weyl transformations in general obtain quantum corrections. We study the trace anomaly in detail for the $$\phi ^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> theory, in particular determining it up to second order in the interaction. We also show that at third order in the interaction a potential $$\Box \phi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>□</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> term can be removed by finite renormalization.