The Glimm-Jaffe-Spencer expansion for the classical boundary conditions and coexistence of phases in the λφ24 Euclidean (quantum) field theory

The λφ₂⁴ Euclidean (quantum) field theory is studied in the multiphase region, and the following results are proven: (1) The “low temperature” expansion converges for Dirichlet (D), free (F), Neumann (N), and periodic (P), boundary conditions, and the even-point Schwinger functions for these boundary conditions have a mass gap; (2) $\text{〈}\text{o}\text{〉}{}^{\mathrm{b}}\text{=}\text{1}\text{2}\text{〈o〉}{}^{\mathrm{+}}\text{+}\text{1}\text{2}\text{〈o〉}{}^{\mathrm{?}}$, where b = D, F, N, P, and 〈o〉^± are the pure states of Glimm, Jaffe, and Spencer; (3) 〈o〉^±ξ = 〈o〉^± for all ξ > 0, where ξ is the buondary field; (4) alternative characterizations of the pure states 〈·〉^± are given.