Conformal anomaly

Definition

A conformal anomaly refers to the breakdown of conformal invariance in a quantum field theory, particularly in the context of two-dimensional conformal field theories. It indicates that, despite the classical theory being conformally invariant, quantum corrections can introduce a dependence on the background geometry, leading to a violation of this invariance.

5 Must Know Facts For Your Next Test

1. Conformal anomalies arise due to quantum effects that break the classical symmetry present in a theory, showing that quantum field theories can behave differently than their classical counterparts.
2. In two-dimensional conformal field theories, the presence of a conformal anomaly is often tied to the central charge of the theory, which quantifies how many degrees of freedom are present.
3. The study of conformal anomalies has deep implications in string theory, particularly in how these anomalies affect the consistency of models.
4. Conformal anomalies can influence correlation functions and play a critical role in determining physical observables in quantum field theories.
5. Mathematically, conformal anomalies can be detected by examining the divergence of the trace of the energy-momentum tensor and identifying contributions that depend on curvature.

Review Questions

• How does a conformal anomaly affect the symmetry properties of a quantum field theory?
  • A conformal anomaly signifies that although a quantum field theory might exhibit conformal invariance at the classical level, quantum corrections can introduce dependencies on background geometry. This means that certain physical predictions may change based on how we scale or transform our coordinates. Consequently, while classical equations maintain symmetry under these transformations, the presence of an anomaly shows that this symmetry is not preserved when moving to the quantum regime.
• Discuss the significance of Weyl anomalies in relation to conformal anomalies within curved spacetime.
  • Weyl anomalies are a specific instance of conformal anomalies that emerge when analyzing theories in curved spacetime. They indicate how the energy-momentum tensor's trace fails to vanish in curved settings due to quantum effects. This is significant because it illustrates how classical geometrical properties can be modified by quantum mechanics, affecting not only theoretical formulations but also practical applications in areas such as string theory and cosmology.
• Evaluate how conformal anomalies can impact physical predictions and implications for string theory models.
  • Conformal anomalies play a critical role in shaping physical predictions by influencing correlation functions and other observables. In string theory models, they help ensure consistency conditions are met; for instance, if anomalies are present, they must be canceled to avoid inconsistencies in the theory. This requirement leads to constraints on model building and has led to significant insights into the underlying symmetries and structures within string theory, ultimately guiding our understanding of fundamental physics.

Related terms

Conformal invariance: A property of a field theory where the equations of motion remain unchanged under conformal transformations, which include scaling and angle-preserving transformations.

Weyl anomaly: A specific type of conformal anomaly that arises when considering the trace of the energy-momentum tensor in curved spacetime, indicating a deviation from scale invariance.

Two-dimensional conformal field theory: A quantum field theory defined on two-dimensional spacetime that possesses conformal symmetry, often used as a framework for studying critical phenomena and string theory.