Kara Farnsworth, Michael L. Graesser, Gabriel Herczeg
SciPost Phys. 13, 099 (2022) ·
published 24 October 2022

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Recently, we introduced the "NewmanPenrose map", a novel correspondence between a certain class of solutions of Einstein's equations and selfdual solutions of the vacuum Maxwell equations, which we showed was closely related to the classical double copy. Here, we give an alternative definition of this correspondence in terms of quantities that are defined naturally on twistor space, and a shearfree null geodesic congruence on Minkowski space whose twistorial character is articulated by the Kerr theorem. The advantage of this reformulation is that it is purely geometrical in nature, being manifestly invariant under both spacetime diffeomorphisms and projective transformations on twistor space. While the original formulation of the map may be more convenient for most explicit calculations, the twistorial formulation we present here may be of greater theoretical utility.
Timothy Cohen, Kara Farnsworth, Rachel Houtz, Markus A. Luty
SciPost Phys. 13, 011 (2022) ·
published 4 August 2022

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Hamiltonian truncation is a nonperturbative numerical method for calculating observables of a quantum field theory. The starting point for this method is to truncate the interacting Hamiltonian to a finitedimensional space of states spanned by the eigenvectors of the free Hamiltonian $H_0$ with eigenvalues below some energy cutoff $E_\text{max}$. In this work, we show how to treat Hamiltonian truncation systematically using effective field theory methodology. We define the finitedimensional effective Hamiltonian by integrating out the states above $E_\text{max}$. The effective Hamiltonian can be computed by matching a transition amplitude to the full theory, and gives corrections order by order as an expansion in powers of $1/E_\text{max}$. The effective Hamiltonian is nonlocal, with the nonlocality controlled in an expansion in powers of $H_0/E_\text{max}$. The effective Hamiltonian is also nonHermitian, and we discuss whether this is a necessary feature or an artifact of our definition. We apply our formalism to 2D $\lambda \phi^4$ theory, and compute the the leading $1/E_\text{max}^2$ corrections to the effective Hamiltonian. We show that these corrections nontrivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order $1/E_\text{max}^3$, as expected by our power counting. We also present the power counting for 3D $\lambda \phi^4$ theory and perform calculations that demonstrate the separation of scales in this theory.