In quantum field theory (QFT), anomalous dimensions play an important role in understanding the behavior of fields and particles. Anomalous dimensions are quantities that describe how the scaling properties of operators or fields change under renormalization.
Renormalization is a mathematical procedure used in QFT to remove or absorb divergences that arise when calculating physical quantities. It involves rescaling and redefining parameters and fields to ensure that the theory remains well-defined and meaningful.
Anomalous dimensions arise when a particular operator or field in a quantum field theory does not scale in the same way as its classical counterpart under renormalization. In other words, the scaling properties of the operator or field are modified by quantum corrections.
Mathematically, anomalous dimensions are expressed as the difference between the scaling dimension of an operator or field in the quantum theory and its classical scaling dimension. The scaling dimension determines how an operator or field scales under changes in length or energy scales. Anomalous dimensions can be positive, negative, or zero, depending on the specific theory and the operator under consideration.
Anomalous dimensions have important implications for the behavior of quantum field theories. They affect the running of coupling constants, which describe the strength of interactions, and can influence the behavior of particles and fields at different energy scales. Anomalous dimensions also appear in the context of the renormalization group equations, which describe how physical quantities change as the energy scale of a system is varied.
Overall, anomalous dimensions are a key concept in QFT, providing insights into the effects of quantum corrections and the behavior of fields and operators under renormalization. They are an essential tool for understanding and predicting the behavior of particles and interactions in the quantum realm.