Momentum space CFT correlators for Hamiltonian 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 2 F 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.

Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data

“…Note that, for simplicity, in these expressions we have suppressed the i prescription needed to ensure Wightman ordering of the correlators, step functions enforcing positivity of lightcone momenta, and any resulting overall phases. For a detailed derivation of these formulas, including such additional subtleties, see [27]. At an operational level, though, (3.57) is all we need.…”

“…Explicit examples so that the theory of ∂ψ is a Generalized Free Theory (GFT). 27 Correlators of a GFT are indeed conformally covariant, with the scaling dimensions of fields set by their two-point functions.…”

“…35 We must also include an i prescription in this analytic continuation to define a particular operator ordering for the resulting Lorentzian correlator, which will affect the Fourier transform to momentum space. These subtleties have already been accounted for in evaluating the Fourier transform formulas (3.57), and are discussed in detail in [27].…”

“…The most successful variational methods in QFT have been Hamiltonian truncation methods, where the variational parameters are the coefficients of states in a basis of a well-chosen subspace of the full Hilbert space. 1 This review is meant to serve as an introduction and user's guide for a specific kind of Hamiltonian truncation known as Lightcone Conformal Truncation (LCT) [20][21][22][23][24][25][26][27][28] that works in lightcone quantization for QFTs that have UV CFT fixed points. As we will describe in detail, lightcone quantization circumvents a number of issues that are typically challenging for Hamiltonian truncation methods in QFT.…”

Exploring Hamiltonian Truncation in $\bf{d=2+1}$

“…Meanwhile ref. [18] laid down formulas to efficiently compute the matrix elements of φ 4 3 in the Conformal Truncation approach. Finally, other recent work has been done in [19].…”