Spontaneous breakdown of symmetry in axiomatic theory

We prove two results; the first is that it is possible to prove Goldstone’s theorem in a Wightman theory in which there is a conserved current with a meaning for a sharp time and which generates a broken symmetry. The second is that for such theories there exist infinitely many positive linear functionals on the algebra of quasi-local operators, that are Lorentz invariant and lead to a theory with positive energy. That is, we may regard the vacuum functional as non-unique. These two results are illustrated in the theory of the free massless neutral scalar field.