Operator-state correpondence in CFT

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SUMMARY

The operator-state correspondence is a fundamental concept in conformal field theory (CFT) that establishes a mapping between states and operators. This correspondence holds true specifically for CFTs due to their conformal invariance, which allows for a consistent mapping from the cylinder to the complex plane. General two-dimensional quantum field theories (QFTs) do not maintain this property, as they can be altered by the mapping. The discussion also raises questions about the validity of this correspondence in boundary CFTs, which remains unresolved.

PREREQUISITES
  • Understanding of conformal field theory (CFT)
  • Familiarity with two-dimensional quantum field theories (QFT)
  • Knowledge of the mapping from the cylinder to the complex plane
  • Basic concepts of boundary conditions in quantum field theories
NEXT STEPS
  • Research the implications of conformal invariance in CFTs
  • Study the mapping techniques between different geometries in quantum field theories
  • Explore the concept of boundary CFTs and their properties
  • Examine David Tong's script on string theory, particularly chapter 4, for deeper insights into CFT
USEFUL FOR

This discussion is beneficial for theoretical physicists, particularly those specializing in string theory and conformal field theories, as well as graduate students seeking to deepen their understanding of operator-state correspondence.

physicus
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I am having trouble to understand a conceptionally important points of the operator-state corredondence in CFT. I am using David Tong's script on string theory, chapter 4, to learn CFT. My questions are the following:
1. Why is the state-operator map only true for conformal field theories. If I consider a general 2D QFT on the cylinder, why can't I use the same procedure to map it to the complex plane and then identify states in the far past to operator insertions at the origin of the complex plane? Is this connected to the fact that the map from the cylinder to the complex plane is conformal and, therefore, conformally invariant theories are not altered by the mapping but general QFTs are?
2. Is the correspondence only true for CFTs on the cylinder or is it also valid for boundary CFTs?

Thank you very much for your help.

physicus
 
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It's been a while since I thought about CFT, but from what I remember you're right about point 1. Afraid I can't help you with point 2. because I don't know what a "boundary CFT" is.
 

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