The formulation of spontaneous breakdown in the path-integral method

We examine the problem of formulating spontaneous breakdown of symmetries in the path-integral framework; since different solutions of the field equations correspond to different boundary conditions, the possibility of different boundary conditions must be incorporated in the generating functional itself. We propose a modification of the generating functional that provides the required flexibility that can lead to different solutions. We apply our formalism to a model of scalar complex fields with invariance under phase transformations of the first kind and derive Ward-Takahashi identities and the Goldstone theorem.