{"query": "Time-translation symmetry gives energy — the Hamiltonian", "count": 20, "results": [{"id": "card_noether_time_gives_energy", "title": "Time-translation symmetry gives energy — the Hamiltonian", "shelf": "codex", "surface": "secular", "snippet": "The laws do not change in time; Noether turns that one symmetry into conservation of energy - and energy is exactly the Hamiltonian. So the Lagrangian's symmetry produces the Hamiltonian's conserved q", "authority_tier": "engine_derived", "source": "Narrow Highway — Noether's theorem", "readable": false, "generated": false}, {"id": "card_floor_the_hamiltonian", "title": "The Hamiltonian — the law of the flow", "shelf": "codex", "surface": "secular", "snippet": "The energy operator and the generator of time evolution: it governs how the many potentials develop between the one source and the one end. Four pillars rest on the floors already seeded - the generat", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_n_041763374eca", "title": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "shelf": "science", "surface": "secular", "snippet": "The stochastic instantiation of probability-as-fluid. Fokker–Planck sets the current\nJ = μρ − D∇ρ (drift + diffusion); its steady state is the Boltzmann distribution ρ ∝ e^(−U/kT),\nwhich makes the cur", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-08", "readable": false, "generated": false}, {"id": "card_floor_noethers_theorem", "title": "Noether's theorem — symmetry is the source of conservation", "shelf": "codex", "surface": "secular", "snippet": "For every continuous symmetry of the action there is a conserved quantity (Emmy Noether, 1918). The seam between the Lagrangian and the Hamiltonian: the Lagrangian's time-symmetry PRODUCES the Hamilto", "authority_tier": "engine_derived", "source": "Narrow Highway — Noether's theorem", "readable": false, "generated": false}, {"id": "card_ham_noether", "title": "The conserved invariant — energy, the same end through the flow", "shelf": "codex", "surface": "secular", "snippet": "By Noether's theorem, time-translation symmetry gives one conserved quantity: the Hamiltonian, the energy. It is the invariant along every path the system can take. 'Many possibilities but the story e", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_floor_thermodynamics", "title": "Thermodynamics — the arrow and the one end", "shelf": "codex", "surface": "secular", "snippet": "The four laws. The first law is energy conservation (Noether's energy, now counting heat and work). The second law - entropy never decreases - is the one place physics is not time-reversible: the arro", "authority_tier": "engine_derived", "source": "Narrow Highway — thermodynamics", "readable": false, "generated": false}, {"id": "card_floor_relativity", "title": "Relativity — many frames, one invariant", "shelf": "codex", "surface": "secular", "snippet": "Special relativity: the laws and the speed of light are the same in every inertial frame; observers disagree on time and length but agree on the invariant interval - many frames, one invariant truth. ", "authority_tier": "engine_derived", "source": "Narrow Highway — relativity", "readable": false, "generated": false}, {"id": "card_ham_generator", "title": "The generator of time evolution — unitary, conserves the total", "shelf": "codex", "surface": "secular", "snippet": "H is Hermitian, so time evolution e^(-iHt/hbar) is unitary, and unitary evolution conserves total probability. The capstone's 'many potentials sum to one' is not just an instant - the Hamiltonian pres", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_floor_quantum_mechanics", "title": "Quantum mechanics — the quantum and its limits", "shelf": "codex", "surface": "secular", "snippet": "Not a floor beside the others - the framework they inhabit. Superposition and the Born rule are on the capstone, the Schrodinger equation on the Hamiltonian, the path integral on the Lagrangian. This ", "authority_tier": "engine_derived", "source": "Narrow Highway — quantum mechanics", "readable": false, "generated": false}, {"id": "card_thermo_statistical_mechanics", "title": "Statistical mechanics — many microstates, one macrostate", "shelf": "codex", "surface": "secular", "snippet": "The bridge from the reversible microscopic law to the irreversible macrostate: 10^23 particles, one temperature and one pressure. Boltzmann S = k_B ln W counts the microstates; the macrostate is what ", "authority_tier": "engine_derived", "source": "Narrow Highway — thermodynamics", "readable": false, "generated": false}, {"id": "card_c_c1d4f07c75cd", "title": "Fokker–Planck & Liouville — probability flow ↔ Fluid probability dynamics — one continuity ", "shelf": "connections", "surface": null, "snippet": "The stochastic instantiation of probability-as-fluid. ... In Hamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is incompressible.  — a concord the card itself states", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_floor_the_lagrangian", "title": "The Lagrangian — the Hamiltonian's dual, the path chosen", "shelf": "codex", "surface": "secular", "snippet": "L = T - V, and the action S = integral L dt. Its law is the principle of least action: of all paths, nature takes the one where the action is stationary. Four pillars rest on the floors already seeded", "authority_tier": "engine_derived", "source": "Narrow Highway — the Lagrangian", "readable": false, "generated": false}, {"id": "card_ham_no_cut", "title": "The no-cut bridge — one generator, classical to quantum", "shelf": "codex", "surface": "secular", "snippet": "The same generator - energy - runs classical mechanics (Hamilton's equations) and quantum mechanics (Schrodinger). The correspondence is continuous (Ehrenfest; Poisson bracket -> commutator). No scale", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_rel_mass_energy", "title": "Mass-energy — E = m c^2", "shelf": "codex", "surface": "secular", "snippet": "The electron's rest energy is m_e c^2 = 0.511 MeV (sealed). Mass and energy are the same thing, and because c^2 is enormous a tiny mass is a vast energy. Mass IS rest energy - it enters the Hamiltonia", "authority_tier": "engine_derived", "source": "Narrow Highway — relativity", "readable": false, "generated": false}, {"id": "card_question_measurement_gap", "title": "The measurement gap — what the Hamiltonian does not determine", "shelf": "codex", "surface": "secular", "snippet": "Unitary H evolution never selects a single outcome; it flows the whole superposition forward, deterministically and reversibly. What the Hamiltonian does not describe is the actualization - which one ", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_src_pron_hamiltonian", "title": "hamiltonian", "shelf": "pronunciation", "surface": "secular", "snippet": "hamiltonian: pronounced (ARPABET) HH AE1 M AH0 L T OW2 N Y AH0 N. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_src_book_37864", "title": "Know the Truth: A Critique on the Hamiltonian Theory of Limitation Including Some Strictures Upon the Theories of Rev. Henry L. Mansel and Mr. Herbert Spencer — Jesse Henry Jones", "shelf": "gutenberg", "surface": "secular", "snippet": "Know the Truth: A Critique on the Hamiltonian Theory of Limitation Including Some Strictures Upon the Theories of Rev. Henry L. Mansel and Mr. Herbert Spencer, by Jesse Henry Jones. Subjects: Theism; ", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_ham_eigenbasis", "title": "The energy eigenbasis — the stationary states", "shelf": "codex", "surface": "secular", "snippet": "H psi = E psi: the eigenvectors are the stationary states, the eigenvalues the allowed energies (hydrogen's ground state 13.6057 eV, the 1/n^2 ladder - sealed on the stick). Diagonalizing H is exactly", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_spine_the_hamiltonian", "title": "The Hamiltonian — a spine", "shelf": "spine", "surface": "secular", "snippet": "The pillars of the Hamiltonian.", "authority_tier": "engine_derived", "source": "Narrow Highway — the Hamiltonian", "readable": false, "generated": false}, {"id": "card_calc_energy_eigenvalues", "title": "Energy levels (stationary Schrodinger)", "shelf": "calculations", "surface": "secular", "snippet": "Energy levels (stationary Schrodinger) — physics. Formula: H psi = E psi. Canonical FORM: eigenvalue (A v = lambda v) — an atom's allowed energies are the eigenvalues of its Hamiltonian. Same form, di", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_noether_time_gives_energy"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}