{ "$schema": "https://qft.org/data/schemas/qft-many-body-comparison-1.0.0.schema.json", "schema_version": "1.0.0", "artifact_id": "qft.artifact.many-body-quantum-matter.nonrelativistic-field-theory-claim-validity-ledger", "table_id": "nonrelativistic-field-theory-claim-validity-ledger", "canonical_source": { "route": "/many-body-quantum-matter/nonrelativistic-field-theory/", "heading": "A chapter-wide convention and validity table", "anchor": "a-chapter-wide-convention-and-validity-table" }, "caption": "Construction, control, and failure tests in nonrelativistic field theory.", "common_scope": { "markdown": "", "math": [], "links": [] }, "qualifications": { "markdown": "", "math": [], "links": [] }, "columns": [ { "key": "construction_or_claim", "label": "Construction or claim" }, { "key": "data_that_must_be_fixed", "label": "Data that must be fixed" }, { "key": "invariant_or_observable_check", "label": "Invariant or observable check" }, { "key": "controlled_domain", "label": "Controlled domain" }, { "key": "characteristic_failure", "label": "Characteristic failure" } ], "rows": [ { "key": "galilean-field", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--galilean-field", "cells": { "construction_or_claim": { "markdown": "Galilean field", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Species, mass charge, field normalization, dimension", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Boosted dispersion changes by $\\mathbf p\\mapsto\\mathbf p+m\\mathbf v$", "math": [ { "kind": "inlineMath", "tex": "\\mathbf p\\mapsto\\mathbf p+m\\mathbf v" } ], "links": [] }, "controlled_domain": { "markdown": "Momenta below the nonrelativistic breakdown scale", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Mixing fields with different mass charges as one irreducible field", "math": [], "links": [] } } }, { "key": "second-quantization", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--second-quantization", "cells": { "construction_or_claim": { "markdown": "Second quantization", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Box or continuum normalization; bosonic or fermionic brackets", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Number eigenvalues and equal-time delta function", "math": [], "links": [] }, "controlled_domain": { "markdown": "Chosen one-particle basis is complete in the retained sector", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Losing a volume or lattice-spacing factor", "math": [], "links": [] } } }, { "key": "coherent-state-integral", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--coherent-state-integral", "cells": { "construction_or_claim": { "markdown": "Coherent-state integral", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Normal ordering, time step, endpoints, thermal boundary condition", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Exact single-mode partition function", "math": [], "links": [] }, "controlled_domain": { "markdown": "Continuum notation taken only after the discrete kernel is fixed", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Replacing the normal symbol by a naive classical Hamiltonian", "math": [], "links": [] } } }, { "key": "continuum-reduction", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--continuum-reduction", "cells": { "construction_or_claim": { "markdown": "Continuum reduction", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Retained modes, matching observables, cutoff, symmetries", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Agreement with microscopic dispersion and amplitudes through the claimed order", "math": [], "links": [] }, "controlled_domain": { "markdown": "External scales small compared with the band or interaction cutoff", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Double counting modes or silently restoring a broken symmetry", "math": [], "links": [] } } }, { "key": "hubbard-stratonovich-rewriting", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--hubbard-stratonovich-rewriting", "cells": { "construction_or_claim": { "markdown": "Hubbard–Stratonovich rewriting", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Channel, sign, integration contour, normalization", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Integrating out the auxiliary field reproduces the original interaction", "math": [], "links": [] }, "controlled_domain": { "markdown": "Identity is exact; a saddle needs a separate small parameter", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Treating a convenient channel or saddle as unique", "math": [], "links": [] } } }, { "key": "power-counted-action", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--power-counted-action", "cells": { "construction_or_claim": { "markdown": "Power-counted action", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Fixed point, dynamical exponent, operator basis, breakdown scale", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Cutoff stability after refitting the required couplings", "math": [], "links": [] }, "controlled_domain": { "markdown": "Expansion parameters are demonstrably small or promoted terms are resummed", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Using vacuum counting at a Fermi surface without translation", "math": [], "links": [] } } }, { "key": "current-and-response", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--current-and-response", "cells": { "construction_or_claim": { "markdown": "Current and response", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Source sign, charge, contact terms, limit order", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Continuity equation and Ward identity", "math": [], "links": [] }, "controlled_domain": { "markdown": "Regulator and approximation preserve the stated symmetry", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Omitting diamagnetic terms or interchanging $\\omega\\to0$ and $\\mathbf q\\to0$", "math": [ { "kind": "inlineMath", "tex": "\\omega\\to0" }, { "kind": "inlineMath", "tex": "\\mathbf q\\to0" } ], "links": [] } } }, { "key": "goldstone-count", "row_id": "nonrelativistic-field-theory-claim-validity-ledger--goldstone-count", "cells": { "construction_or_claim": { "markdown": "Goldstone count", "math": [], "links": [] }, "data_that_must_be_fixed": { "markdown": "Broken charges, thermodynamic state, commutator densities", "math": [], "links": [] }, "invariant_or_observable_check": { "markdown": "Rank of $\\rho_{ab}=-i\\langle[Q_a,Q_b]\\rangle/V$ and low-momentum poles", "math": [ { "kind": "inlineMath", "tex": "\\rho_{ab}=-i\\langle[Q_a,Q_b]\\rangle/V" } ], "links": [] }, "controlled_domain": { "markdown": "Internal symmetries with the stated locality and limit hypotheses", "math": [], "links": [] }, "characteristic_failure": { "markdown": "Applying the internal-symmetry formula unchanged to spacetime or gauged symmetries", "math": [], "links": [] } } } ], "references": [ { "id": "ref-4568b81986303a12", "markdown": "Watanabe, Haruki. “Counting Rules of Nambu–Goldstone Modes.” *Annual Review of Condensed Matter Physics* **11** (2020): 169–187. [DOI](https://doi.org/10.1146/annurev-conmatphys-031119-050644). [Open PDF](https://arxiv.org/pdf/1904.00569).", "urls": [ "https://doi.org/10.1146/annurev-conmatphys-031119-050644", "https://arxiv.org/pdf/1904.00569" ] } ], "reference_scope": "The reference catalog reproduces the canonical chapter bibliography. Cell links are associated with references only when their resolved URL matches; uncited chapter references are not asserted as claim-specific support." }