Abstract
Hamiltonian Truncation Methods are a useful numerical tool to study strongly coupled QFTs. In this work we present a new method to compute the exact corrections, at any order, in the Hamiltonian Truncation approach presented by Rychkov et al. in Refs. [1-3]. The method is general but as an example we calculate the exact $g^2$ and some of the $g^3$ contributions for the $\phi^4$ theory in two dimensions. The coefficients of the local expansion calculated in Ref. [1] are shown to be given by phase space integrals. In addition we find new approximations to speed up the numerical calculations and implement them to compute the lowest energy levels at strong coupling. A simple diagrammatic representation of the corrections and various tests are also introduced.
Highlights
The Hamiltonian truncation method consists in truncating the Hamiltonian H into a large finite matrix (HT )ij and diagonalizing it numerically
In ref. [1] the φ4 theory in two dimensions was studied at strong coupling using the Hamiltonian truncation method just presented in the Fock basis
14In the φ4 theory the strong coupling can be estimated to be g 1, see eqs. (5.39) and (5.40). 15For the φ2 perturbation studied in section 4 we find that the error in the computed eigenvalues can be decreased by increasing ET even without introducing EW
Summary
Equivalent to renormalize to zero the UV divergences from closed loops with propagators starting and ending on the same vertex To study these theories using the Hamiltonian truncation method we begin by defining them on the cylinder R × S1 where the circle corresponds to the space direction which we take to have a length Lm 1, and R is the time. The Hamiltonian H can be diagonalized by sectors with given quantum numbers associated with operators that commute with H. These are the total momentum P , the spatial parity P : x → −x and the field parity Z2 : φ(x) → −φ(x), which act on the H0-eigenstates as P | Ei =.
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