Date:  2024-06-23

Article ID:  37301

Articles 00980 and 30983 studied the model of a single free scalar quantum field with a nonzero single-particle mass (m>0). When m=0, the model acquires a new symmetry called a shift symmetry. This article introduces two versions of the model with m=0: one in which the shift symmetry is spontaneously broken, and one in which observables are required to be invariant under the shift symmetry.