Abstract
General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics. This convergence is guaranteed to hold in the sense of a distribution, meaning that it holds for correlation functions smeared by smooth test functions. The conformal blocks for this OPE are conceptually extremely simple: they are products of 3-point functions. We construct the conformal blocks in 2-dimensional conformal field theory and show that the OPE in fact converges pointwise to an ordinary function in a specific kinematic region. Using microcausality, we also formulate a bootstrap equation directly in terms of momentum space Wightman functions.
Highlights
Unless the time differences in the exponents are all positive, the time evolution operators are ill-defined because the Hamiltonian H is not bounded above
General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics
In this paper we have studied the operator product expansion (OPE) of conformal field theory (CFT) in momentum space, focusing on two spacetime dimensions
Summary
The basic objects of our study are the momentum-space Wightman correlation functions defined in eq (1.1). (Our conventions are given in appendix A.) The OPE for these correlation functions comes from the Hilbert space completeness relation eq (1.4), where the sum is over all primary operators ψ. (Our conventions are given in appendix A.) The OPE for these correlation functions comes from the Hilbert space completeness relation eq (1.4), where the sum is over all primary operators ψ. Descendant operators are multiples of primary operators in momentum space, so no sum over descendants is required to define the conformal blocks. For a 4-point function of primary operators, this gives the OPE. The momentum space conformal blocks have a conceptually simple structure, but in practice the 3-point functions are difficult to compute in general d, especially for operators with spin. A detailed discussion on computing 3-point functions with the second approach in general spacetime dimension d was given recently in [5]. We will focus on 2D CFT, where we can perform the Fourier transforms for arbitrary operators, including spin
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