Field transformations in functional integral, effective action and functional flow equations

Field transformations for the quantum effective action lead to different pictures of a given physical situation, as describing a given evolution of the universe by different geometries. Field transformations for functional flow equations can reveal important physical features, as the appearance of bound states. They also allow for technical simplifications. We make a critical assessment of the virtues and shortcomings of different versions of field transformations. Key issues are demonstrated by the quantum field theory for scalars with a field-dependent coefficient of the kinetic term. Our findings confirm the principle of field relativity for gravity and cosmology based on the quantum effective action.