Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Momentum space CFT correlators for Hamiltonian truncation

Nikhil Anand, Zuhair U. Khandker, Matthew T. Walters

TL;DR

This work develops a framework for Lorentzian momentum-space CFT correlators tailored to Hamiltonian/conformal truncation in real time. It provides explicit, numerically tractable formulas for two- and three-point functions in 1+1 and 2+1 dimensions, with the middle operator carrying zero spatial momentum, and expresses all results as finite sums of ${}_2F_1$ hypergeometric functions. The 3d implementation specializes to monomial tensor structures and yields concrete formulas for inner products and Hamiltonian matrix elements in a UV free CFT deformed by $c^2$ and $c^4$ operators, with cross-checks against Fock-space techniques. The AdS perspective is discussed as an alternative organizing principle, clarifying finiteness conditions and divergences, and the results serve as seed ingredients for upcoming numerical truncation studies of 3d $c^4$-theory. Altogether, the paper furnishes a practical, scalable toolkit for nonperturbative real-time QFT investigations via conformal truncation.

Abstract

We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infinite volume. Our momentum space formulas take the form of finite sums over ${}_2F_1$ hypergeometric functions, allowing for efficient numerical evaluation. As a concrete application, we work out matrix elements for 3d $φ^4$-theory, thus providing the seed ingredients for future truncation studies.

Momentum space CFT correlators for Hamiltonian truncation

TL;DR

This work develops a framework for Lorentzian momentum-space CFT correlators tailored to Hamiltonian/conformal truncation in real time. It provides explicit, numerically tractable formulas for two- and three-point functions in 1+1 and 2+1 dimensions, with the middle operator carrying zero spatial momentum, and expresses all results as finite sums of hypergeometric functions. The 3d implementation specializes to monomial tensor structures and yields concrete formulas for inner products and Hamiltonian matrix elements in a UV free CFT deformed by and operators, with cross-checks against Fock-space techniques. The AdS perspective is discussed as an alternative organizing principle, clarifying finiteness conditions and divergences, and the results serve as seed ingredients for upcoming numerical truncation studies of 3d -theory. Altogether, the paper furnishes a practical, scalable toolkit for nonperturbative real-time QFT investigations via conformal truncation.

Abstract

We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infinite volume. Our momentum space formulas take the form of finite sums over hypergeometric functions, allowing for efficient numerical evaluation. As a concrete application, we work out matrix elements for 3d -theory, thus providing the seed ingredients for future truncation studies.

Paper Structure

This paper contains 18 sections, 130 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. $i\epsilon$ prescription for Wightman functions
  3. 2d momentum space formulas
  4. Two-point functions
  5. Three-point functions
  6. 3d momentum space formulas
  7. Two-point functions
  8. Three-point functions
  9. AdS approach
  10. Application: 3d $\phi^4$-theory
  11. Inner product
  12. $\phi^2$ interaction
  13. $\phi^4$ interaction
  14. $n\rightarrow n$
  15. $n\rightarrow n+2$
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Integration contour for evaluating eq. \ref{['eq:IhPDef']}. The $i\epsilon$ prescription for this Wightman function places the branch point in the upper half of the complex plane, which ensures that the Fourier transform only has support for physical lightcone momentum $P>0$. The discontinuity along this branch cut gives the resulting momentum space expression in eq. \ref{['eq:IhP']}.
  • Figure 2: Witten diagram for the CFT Wightman function $\langle{\cal O}{\cal O}_R{\cal O}'\rangle$. The solid lines for the two external operators represent Wightman bulk-to-boundary propagators, while the dashed line for the middle operator represents a time-ordered propagator.