Spontaneous symmetry breaking is studied in the ultralocal limit of a scalar quantum field theory, that when [Formula: see text] (or infrared limit). In this approximation theory formally two-dimensional and its Euclidean solutions are instantons. For BPST-like with text], map between two dimensions self-dual Yang–Mills carefully discussed.

Spontaneous symmetry breaking is determined by quantum mechanical measurement, and is classified to two types according to whether Hamiltonian commutes the relevant observable, the average of which is the order parameter. A supplement regarding phases is given to the expansion postulate and validate ensemble description of many-particle systems. Interference between many-particle states, macros...