The Lorentz invariant effective theory for the large momentum limit of a finite volume QFT

We develop methods for computing the effective action at infinite momentum for $1+1d$ QFTs at finite volume which do not rely on the theory having a Lagrangian description. We do this by taking the infinite momentum limit of equal-time quantization and integrating out all except for the chiral modes of the theory. Our main application of this method is to the Ising Field Theory (IFT), with an energy and magnetic deformation, where we compute the effective lightcone Hamiltonian numerically and check it against results from TCSA. Remarkably, in the low-temperature phase, the Lorentz invariant effective Hamiltonian at infinite momentum takes a very compact form and depends on the volume only through the finite volume vacuum expectation value of $\langle\sigma\rangle$, the spin operator.