Generalized boost transformations in finite volumes and application to Hamiltonian methods

The extraction of physical observables from lattice energy spectra using finite volume quantization conditions is one of the key methods in the study of hadron physics. Due to the expensive and limited lattice configurations, there is significant interest in extracting as much information as possible from these datasets. One effective approach involves varying the total momentum within the finite volume, which allows for the identification of additional finite volume energy levels in moving systems. However, the finite volume quantization conditions applicable to moving systems require careful consideration of momentum transformations between different reference frames, and there exists several different method to perform the three-momentum transformation. This work systematically presents a general scheme for three-momentum transformations in a finite volume, and this scheme is able to generate two existed transformations in literature. In addition, we propose a new transformation method that circumvents reliance on the total energy during the transformation process, which is crucial for employing the Hamiltonian Effective Field Theory (HEFT) approach to extract scattering amplitudes. At last, we also demonstrate the consistence between our method with others through numerical comparisons, employing a phenomenological $\pi\pi$ scattering example.