Preserving Symmetry: Spontaneous Symmetry Breaking through Decoherence

Many-body quantum theory typically describes solids using translationally symmetric models, which theoretically preclude the localization of a center of mass. Conventional physics resolves this by introducing a semiclassical environment that breaks this symmetry. This paper investigates how such localization arises in finite quantum systems without invoking the thermodynamic limit, where the number of particles tends to infinity.

Instead of treating the environment as a classical background potential, the author models the environment as a quantum system coupled to the particles of interest. By analyzing the entanglement between the system and the environment, the study shows that the global wavefunction remains translationally symmetric while the system becomes localized relative to the environment. This framework is analogous to the Page-Wootters construction and quantum reference frames, where physical properties are defined relationally rather than against an absolute background.

• Decoherence-Induced Localization: The author proves that localization occurs within decoherent branches of the global wavefunction. From the perspective of a branch where the environment is at a specific location, the system appears to have a well-defined, pinned center of mass.
• Recovery of Semiclassical Models: The conventional semiclassical description of symmetry breaking is recovered as a limiting case when the environment's mass is significantly larger than the system's mass.
• Empirical Predictions: Unlike conventional models, this approach predicts that the ground-state width of a solid depends on the properties of the environment (such as its mass), offering a potential avenue for experimental verification.

This work provides a rigorous quantum-mechanical foundation for spontaneous symmetry breaking in finite systems. By removing the need for the thermodynamic limit, it resolves conceptual tensions in how localized macroscopic objects emerge from fundamentally symmetric quantum laws, aligning solid-state physics with modern decoherence theory and relational quantum mechanics.