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

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