Conformal bootstrap in momentum space at finite volume

In this paper, we Fourier transform the Wightman function concerning energy and angular momentum on the SD-1 spatial slice in radial quantization in D = 2, 3 dimensions. In each case, we use the conformal Ward Identities to solve systematically for the Fourier components. We then use these Fourier components to build conformal blocks for the four-point function in momentum space, giving a finite-volume version of the momentum-space conformal blocks. We check that this construction is consistent with the known result in infinite volume. Our construction may help to find bootstrap equations that can give nontrivial constraints that do not appear in analysis in infinite volume. We show some examples of bootstrap equations and their nontriviality.

Automated summary

In this paper, we systematically solve for the Fourier components of the Wightman function concerning energy and angular momentum on the SD-1 spatial slice in radial quantization in D = 2, 3 dimensions using conformal Ward Identities. These Fourier components are then used to construct conformal blocks for the four-point function in momentum space, resulting in a finite-volume version. The construction is consistent with the known result in infinite volume and may provide nontrivial constraints through bootstrap equations.