Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model

It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin-Kasteleyn-Ginibre property in this model and the Ghirlanda-Guerra identities in artificial models in a path integral representation based on the Lie-Trotter-Suzuki formula enable us to extend Chatterjee's proof for the random field Ising model to the quantum model.