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Steven Weinberg's 1967 lepton model combined the weak interaction and electromagnetic interaction in a gauge-theory framework. Spontaneous symmetry breaking gives different masses to the weak vector bosons while leaving the photon massless. This joins interactions with very different apparent ranges under a common structure, with their differences arising from the vacuum and couplings. The paper also identifies renormalizability as a crucial consistency question, rather than claiming that all higher-order calculations were already established.^[ref]

A physicist.^[ref]

At zero field and with spin inversion symmetry in its boundary conditions, a finite Ising model pairs every configuration with its spin-reversed configuration. Their equal Boltzmann weights force the expected value of its spin magnetization $m=N^{-1}{\sum }_i{s}_i$ to vanish. Below a model's ordering critical temperature, an arbitrarily small positive field can instead select the positive spontaneous magnetization if size is taken to infinity first:The distinction is the order of limits and the stated symmetric boundary conditions; plus boundary conditions already break spin inversion symmetry. A zero-field symmetric mixture of the two ordered states can retain zero mean spin magnetization even in the thermodynamic limit. Also, $⟨|m|⟩>0$ in a finite system is not by itself proof of spontaneous symmetry breaking.^[ref]

The thermodynamic limit describes a sequence of larger systems, not a single enormous specimen: particle number $N$ and volume $V$ tend to infinity with number density $n=N/V$ fixed. Hold the chosen intensive and extensive thermodynamic quantities such as temperature $T$ and an applied magnetic field fixed along that sequence. In the canonical ensemble the controls are $T,N,V$; in the grand canonical ensemble one instead fixes $T$ and chemical potential, and the limiting number density is a result, possibly with a jump at phase coexistence. These are different specifications of a limiting experiment.

The useful limit is usually an intensive and extensive thermodynamic quantities normalization of the Helmholtz free energy, not a finite limit of the total extensive quantity itself:Here $Z_{N,V}$ is the canonical partition function, $k_B$ is the Boltzmann constant, $f$ is free-energy density, and $a$ is Helmholtz free energy per particle. Thus $F_{N,V}=Vf+o(V)$ when the limit exists; finite systems need not be exactly extensive quantities because of the surface correction to thermodynamic free energy.

Specify the interactions and the growing containers too. For a conventional fluid, use a fixed stable pair potential with a tempered pair potential tail, and containers whose boundary layer of any fixed thickness occupies a vanishing fraction of the volume; uniformly enlarged cubes work, a tube of fixed cross-section does not give the same bulk limit. For a finite-range Ising model, keep the lattice spacing and bond coupling fixed while enlarging all intended spatial directions. Ordinary bounded wall interactions or periodic boundary conditions can then give the same bulk free-energy density, although boundary conditions can select different spontaneous symmetry breaking states. This is a set of hypotheses, not a theorem that every interaction has a boundary-independent limit.^[ref]

The Curie–Weiss model below deliberately uses a different limiting prescription: its all-to-all coupling is divided by $N$. Without that factor its interaction energy would grow as $N^2$, rather than as an extensive quantity. The model has no spatial wall or correlation length to compare with a container; it is an exactly summable illustration of finite-size rounding, not a quantitative model of the fluid experiment below.

Tom Kibble related spontaneous symmetry breaking in the early universe to the formation of topological defects. Different regions can choose different states on a vacuum manifold. Its homotopy groups determine whether boundaries, loops or surrounding surfaces can be continuously unwound, giving possible domain walls, cosmic strings or monopoles. His 1976 paper also examines their subsequent cosmological effects. The connection makes the global shape of a space of field states relevant to large-scale physical structures.^[ref]

For a stable relativistic quantum field theory in Minkowski space, a vacuum is a lowest-energy state for the chosen time evolution; a physical vacuum is normally also invariant under spacetime translations. It need not have zero fluctuations or vanish under every field operator. A free field's Fock vacuum is annihilated by its annihilation operators, while an interacting vacuum need not equal the free-theory Fock state. Multiple degenerate vacua permit spontaneous symmetry breaking. In curved spacetime the choice of time evolution and positive-frequency modes introduces additional vacuum ambiguity in a nonstationary spacetime.

Yoichiro Nambu transferred the idea of dynamical symmetry breaking from superconductivity into particle. His model with Giovanni Jona-Lasinio treats particle mass as a self-consistent interaction effect, while collective excitations reflect the underlying chiral symmetry breaking. This makes the physical vacuum and its excitations part of the explanation of particle properties. Their simplified four-fermion theory is a model of the mechanism, rather than the modern fundamental theory of the strong interaction.^[ref]

A physicist working on spontaneous symmetry breaking.^[ref]