v 1 2 M ar 1 99 7 Assumption of nonvanishingness of vacuum expectation of the scalar field for spontaneous symmetry breaking is superfluous

For spontaneous breaking of global or gauge symmetry, it is superfluous to assume that the vacuum expectation value of the scalar field manifesting the symmetry is nonvanishing. PACS Numbers: 11.30.Qc Electronic address: [email protected] 1 A crucial ingredient of the theories of spontaneous symmetry breaking is the assumption that the vaccum expectation value of the scalar field manifesting the the symmetry, φ, is nonzero [1][2][3][4][5][6]. This is contradictory with the basic fact that the Hilbert space is spanned by states with definite number of particles and/or antiparticles in each mode. Though it is widely accepted as a property of vacuum, here we point out that this assumption is superfluous. In fact, φ 6= 0 is only a sufficient but not a necessary condition for the required results, and can be simply replaced by φ†φ 6= 0, which can be automatically satisfied in the ground state. The similar situation for superfluidity and superconductivity has been discussed elsewhere [7]. We follow the notations and line of development in [1], and start with Goldstone theorem. Under a continuous symmetry which transforms a set of Hermitian scalar fields φn(x) as φn(x) → φn(x) + iǫ ∑ m tnmφm(x), (1) where tnm is the matrix corresponding to the symmetry transformation, the invariance of the action and measure, and thus the effective potential leads to