A theory of a free massless fusion field composed of free massless spinor fields are investigated in the framework of the indefinite-metric quantum field theory. It is found that the free theory already interesting features characteristic of two-dimensional massless models. On the basis of finite-volume considerations, a massless ``scalar'' fusion theory is constructed consistently in the infinite-volume limit. Two different sets of charge operators are found to exist. Subsidiary conditions in terms of the charges define the positive-definite physical Hilbert space Hphys. However the conditions are not unique and lead us to different structures of the Hilbert space. Some cases have the unique vacuum in Hphys, whereas other cases have vacuum degeneracy, i.e., the vacua characterized in terms of one or two parameters (θ-vacua). Which realization of the symmetries generated by the charges shows up depends on the vacuum chosen. Various cases are possible. Poincaré transformations are discussed. The fusion fields are not Lorentz scalars in a strict sense, but they behave as scalars in Hphys.