{"id": "card_question_yang_mills", "kind": "reference", "title": "Yang-Mills existence and the mass gap", "body": "Yang-Mills existence and the mass gap. A quantum Yang-Mills theory exists on four-dimensional Euclidean space for every compact simple gauge group, and has a mass gap greater than zero. Its chart is the tick stick stick_yang_mills_existence_and_mass_gap: what is sealed, what is cited and what stays open are read live at /stick?id=stick_yang_mills_existence_and_mass_gap. Open on the continuum; the gap is proven on the lattice at strong coupling (Osterwalder-Seiler 1978).", "source": {"label": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "url": "https://www.claymath.org/millennium-problems/", "ref": "stick_yang_mills_existence_and_mass_gap", "domain": "mathematics", "authority_tier": "reference"}, "shelf": "millennium", "box": "question", "bands": ["millennium", "open question", "clay", "yang mills", "gauge theory", "quantum field theory", "physics"], "subject": "Yang-Mills existence and the mass gap", "connections": [{"to_card_id": "card_spine_millennium_sources", "relationship": "member_of", "evidence": "one of the seven questions the Millennium shelf is about"}, {"to_card_id": "card_floor_millennium", "relationship": "open_end_of", "evidence": "an open question hanging off the floor of what is proven"}, {"to_card_id": "card_joint_gue_statistics", "relationship": "connects_at", "evidence": "Zeta zeros and the lattice Dirac operator share GUE statistics. montgomery 1973; odlyzko 1987; verbaarschot 1994; berry keating 1999. Sealed: the Riemann stick's spacing witness (seal deefbb6d)"}, {"to_card_id": "card_joint_renormalization_group", "relationship": "connects_at", "evidence": "The renormalization group, and the continuum limit as the question. wilson 1974; forster nelson stephen 1977. Sealed: the running coupling (Yang-Mills stick) and the Kolmogorov scale (Navier-Stokes stick)"}, {"to_card_id": "card_joint_sign_problem", "relationship": "connects_at", "evidence": "The lattice sign problem is NP-hard. troyer wiese 2005. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_floor_logarithm", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_src_mill_gross_wilczek_1973", "relationship": "connects_at", "evidence": "where the logarithm enters: the coupling runs as one over the logarithm of the scale - sealed on the stick (alpha_s at 10 GeV)"}, {"to_card_id": "card_src_mill_jaffe_witten_2000", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_src_mill_osterwalder_seiler_1978", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_src_mill_morningstar_peardon_1999", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_floor_yang_mills", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chart_ym_measure", "relationship": "charted_by", "evidence": "the stick reads the gap as Delta = E1 - E0, a measured spectral difference - the quantity the existence-and-mass-gap question asks to be positive"}, {"to_card_id": "card_chart_grassmannian", "relationship": "charted_by", "evidence": "the stick seals the amplituhedron's dimensions - the positive geometry that computes Yang-Mills scattering amplitudes, admissibility found, never generated"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "millennium.question", "facets": {"subject": ["clay", "gauge theory", "millennium", "open question", "physics", "quantum field theory", "yang mills"]}, "presentation": {"glyph": "?", "kind_label": "A question", "by": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "authority": "reference", "posted": "", "standing": "", "link": {"url": "https://www.claymath.org/millennium-problems/", "host": "claymath.org", "provider": "", "titled": "Yang-Mills existence and the mass gap", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "relationship": "on the shelf of", "why": "one of the seven questions the Millennium shelf is about", "href": "/card/card_spine_millennium_sources", "resolved": true}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "open end of", "why": "an open question hanging off the floor of what is proven", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_joint_gue_statistics", "title": "Zeta zeros and the lattice Dirac operator share GUE statistics", "relationship": "connects at", "why": "Zeta zeros and the lattice Dirac operator share GUE statistics. montgomery 1973; odlyzko 1987; verbaarschot 1994; berry keating 1999. 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