Borsboom and Dupont (2026) proposed a relational account of spontaneous symmetry breaking (SSB) in which a separating boundary condition, forcing the order parameter to vanish at the boundary of a finite system, creates superselection sectors labeled by the relational values the order parameter can take with respect to the environment; for infinite systems the boundary sits at infinity and the sectors appear intrinsic. The account was developed for lattice models and Abelian gauge theory, and is deliberately kinematic (equilibrium). The open problem, posed by the source itself, is to turn this partial picture into a complete physical account of SSB: supply (or rule out) a dynamical mechanism by which a symmetric initial state enters a single superselection sector, clarify whether the measurement process can itself be understood as boundary-condition imposition, extend the construction to non-Abelian gauge theories and gravity or identify obstructions, identify (or rule out) the physical origin of the symmetry-breaking perturbations that select a sector in superconductors, and determine whether boundary-condition-controlled experiments can observe or manipulate the boundary symmetries.

Background

Spontaneous symmetry breaking (SSB) is the phenomenon in which the equilibrium state of a many-body system is less symmetric than its dynamics: below the Curie temperature a ferromagnet magnetizes along one direction even though its Hamiltonian treats all directions equally. The rigorous theory of SSB is kinematic, i.e. it describes the equilibrium state space rather than time evolution, and is defined only for infinite systems via the thermodynamic limit, the limit of infinite system size. In that limit, symmetry-related pure phases become superselection sectors — subspaces between which no physically realizable operation interpolates, so that the symmetric mixture is unstable and the symmetry-broken states are dynamically isolated. In a finite quantum system the exact ground state is unique and symmetric; broken behavior emerges only because vanishingly small symmetry-breaking perturbations select one symmetry-related state when the thermodynamic limit and the perturbation-removal limit are taken in the wrong order, assisted by a 'thin spectrum' or 'tower of states' of collective excitations whose excitation energies vanish as the system size grows (Tasaki 2019). Canonical illustrations are the transverse-field Ising model of a ferromagnet (spin system with a discrete global bit-flip symmetry) and the BCS model of superconductivity, whose order parameter is the density of Cooper pairs — bound electron pairs of opposite momentum and spin condensing with a common quantum-mechanical phase. The conventional story thus leaves two gaps. First, SSB is defined only in an unphysical infinite limit, straining its application to finite systems such as a laboratory magnet. Second, the dynamics that carries a symmetric finite state into one broken sector is not supplied: exactly symmetric equations of motion cannot select a sector, an explanatory gap explicitly noted by Landsman (2013) and approached only obliquely by the 'flea' mechanism of exponentially small perturbations (Landsman and Reuvers 2013) and by decoherent-histories treatments of SSB in finite systems (Wallace 2018), where the decoherent-histories approach is a formulation of quantum theory in which coarse-grained alternative histories of a closed system acquire ordinary classical probabilities. For superconductors there is an additional puzzle: whereas a magnet is plausibly perturbed by stray magnetic moments in its environment, no comparably natural environmental source exists for a perturbation that couples to the Cooper-pair density and so breaks the global phase symmetry U(1)U(1); the dynamical emergence of a definite global phase remains without an agreed mechanism (Greiter 2005; van Wezel and van den Brink 2008). Borsboom and Dupont (2026) address the first gap with a relational, boundary-centered interpretation: a system exhibits SSB only relative to a reference environment that can induce perturbations across a boundary. A separating boundary condition — the order parameter, the quantity that distinguishes the competing broken-symmetry phases (the magnetization direction for a ferromagnet), required to vanish on the boundary — creates superselection sectors, labeled by the possible relational values of the order parameter with respect to the environment; for infinite systems the boundary is displaced to infinity and the sectors falsely appear intrinsic to the system, which is why the breaking looks 'spontaneous'. The authors develop this for short-range lattice models such as the transverse-field Ising model, for mean-field models, and for the Abelian Higgs model — the gauge theory of a charged scalar field coupled to electromagnetism, the field-theoretic prototype of gauge-symmetry breaking — where their construction connects to flux superselection sectors arising in Hamiltonian gauge theory on spatial regions with boundaries ('corners'; Riello and Schiavina 2024). They conclude that SSB is 'subtle' rather than spontaneous symmetry breaking. The account, however, is deliberately kinematic; the source paper closes by posing the open questions that constitute the present problem.

Problem Statement

Complete the kinematic account of spontaneous symmetry breaking as superselection by boundary conditions into a full physical account, by resolving the interlocking open questions posed by Borsboom and Dupont (2026). (1) Dynamical sector selection: provide, within the boundary-superselection framework, a dynamical mechanism whereby a symmetric initial state of a finite quantum system interacting with its environment through the boundary evolves into a single symmetry-broken superselection sector on timescales compatible with observation, or prove that no such mechanism exists even when the environment coupling through the boundary is included, so that sector selection must be traced to a physical asymmetry of the environment. (2) Relation to the measurement problem (why a quantum measurement yields a single definite outcome rather than a superposition): determine whether measurement can be rigorously understood as the imposition of boundary conditions that collapses a symmetric superposition into a broken-symmetry state, or show that the analogy is merely heuristic. (3) Generality: extend the superselection-by-boundary-conditions construction from Abelian gauge theories and lattice models to non-Abelian gauge theories (gauge theories, such as Yang–Mills theory, whose symmetry transformations do not commute) and gravity, where the boundary structure is more complex, or identify the obstructions. (4) Superconductor sector selection: identify a physically realistic environmental source of the symmetry-breaking perturbation that couples to the Cooper-pair density and breaks the global U(1)U(1) phase symmetry, analogous to the stray moments that select a magnetization direction in ferromagnets, or show that no such environmental source exists. (5) Experimental testability: determine whether experiments with precise control over boundary conditions can directly observe or manipulate the 'boundary symmetries' — the symmetry structures associated with the boundary in this framework — and their breaking, in a way distinguishable from ordinary explicit symmetry breaking. Each of these five questions stands on its own: a definite determination of any single one of them (a mechanism or a no-go proof, a determination of the measurement analogy, an extension or an identified obstruction, an identified physical source or a proof that no such source exists, or an experimental determination) resolves that question independently of the others, and the five are not required to be settled jointly, although settling all of them is what would complete the account. Together they serve one objective: establishing whether SSB is completely explained as environment-induced superselection through boundaries, rather than as an intrinsic spontaneous phenomenon defined only in the thermodynamic limit.

References

  1. Silvester G. A. Borsboom and José P. Dupont, Symmetry Breaking through Superselection by Boundary Conditions (2026), arXiv:2606.15272, https://arxiv.org/abs/2606.15272
  2. Silvester G. A. Borsboom and Sebastian De Haro, Global Gauge Symmetry Breaking in the Abelian Higgs Mechanism, Philosophy of Physics 4(1):6 (2026; preprint posted 2025), arXiv:2504.17483, doi:10.31389/pop.248, https://arxiv.org/abs/2504.17483
  3. N. P. Landsman, Spontaneous Symmetry Breaking in Quantum Systems: Emergence or Reduction?, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44(4):379–394 (2013), doi:10.1016/j.shpsb.2013.07.003
  4. N. P. Landsman and R. Reuvers, A Flea on Schrödinger's Cat, Foundations of Physics 43(3):373–407 (2013), doi:10.1007/s10701-013-9700-1
  5. M. Greiter, Is electromagnetic gauge invariance spontaneously violated in superconductors?, Annals of Physics 319(1):217–249 (2005), doi:10.1016/j.aop.2005.03.008
  6. J. van Wezel and J. van den Brink, Spontaneous symmetry breaking and decoherence in superconductors, Physical Review B 77:064523 (2008), doi:10.1103/PhysRevB.77.064523
  7. David Wallace, Spontaneous Symmetry Breaking in Finite Quantum Systems: a decoherent-histories approach (2018), arXiv:1808.09547, https://arxiv.org/abs/1808.09547
  8. A. Riello and M. Schiavina, Hamiltonian gauge theory with corners: constraint reduction and flux superselection, Advances in Theoretical and Mathematical Physics 28(4):1241–1424 (2024), doi:10.4310/ATMP.241029014101
  9. H. Tasaki, Long-Range Order, 'Tower' of States, and Symmetry Breaking in Lattice Quantum Systems, Journal of Statistical Physics 174(4):735–761 (2019), doi:10.1007/s10955-018-2193-8