Primary Fields

Definition

Primary fields are the fundamental building blocks in conformal field theory that correspond to the basic types of quantum states. These fields can be scalar, vector, or tensor fields and are essential for understanding how various symmetries operate in the context of two-dimensional quantum field theories. They also play a critical role in the representation of operators and in the classification of models within conformal field theory.

5 Must Know Facts For Your Next Test

1. Primary fields are classified by their scaling dimensions, which dictate how they transform under scale transformations.
2. In conformal field theory, primary fields are typically organized into representations of the Virasoro algebra.
3. There are primary fields associated with different types of particles, such as bosons and fermions, reflecting their intrinsic properties.
4. The correlation functions of primary fields contain vital information about the physical systems described by conformal field theories.
5. Primary fields can be combined to create composite fields, which may exhibit new scaling dimensions and transform according to different representations.

Review Questions

• How do primary fields transform under scaling transformations and what implications does this have for their classification?
  • Primary fields transform according to their scaling dimensions when subjected to scaling transformations. This transformation behavior is fundamental in classifying them within conformal field theory, as it determines their roles and interactions with other fields. The scaling dimensions also influence the correlation functions between primary fields, which provide insights into the physical properties of the quantum system being studied.
• Discuss the significance of the Virasoro algebra in relation to primary fields in conformal field theory.
  • The Virasoro algebra is significant because it encapsulates the infinite-dimensional symmetry structure that arises in two-dimensional conformal field theories. Primary fields are associated with specific representations of this algebra, allowing us to understand how these fields interact and evolve. This relationship not only aids in classifying the various primary fields but also helps to derive crucial physical results such as modular invariance in string theory.
• Evaluate the role of operator product expansion (OPE) in understanding the interactions of primary fields within conformal field theory.
  • The operator product expansion (OPE) plays a vital role by providing a systematic way to analyze how products of primary fields behave at short distances. Through OPE, we can express these products as sums over other operators, shedding light on how interactions occur between different primary fields. This framework helps reveal deeper structures within conformal field theories and allows for calculations that relate physical observables directly to primary field behavior.

Related terms

Conformal Symmetry: A symmetry that preserves angles but not necessarily distances, crucial for the behavior of primary fields in two-dimensional theories.

Operator Product Expansion (OPE): A method in conformal field theory that describes how products of local operators can be expressed as a sum of local operators, crucial for understanding interactions between primary fields.

Virasoro Algebra: An algebraic structure that arises in two-dimensional conformal field theory, associated with the symmetries of primary fields and important for their classification.