415 lines
19 KiB
Text
415 lines
19 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import PhysLean.Meta.Basic
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import PhysLean.Meta.Remark.Properties
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import PhysLean.Meta.Notes.ToHTML
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import Mathlib.Lean.CoreM
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import PhysLean
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/-!
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# Extracting notes from Lean files
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-/
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open Lean System Meta PhysLean
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inductive NameStatus
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| complete : NameStatus
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| incomplete : NameStatus
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instance : ToString NameStatus where
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toString
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| NameStatus.complete => "complete"
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| NameStatus.incomplete => "incomplete"
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inductive NotePart
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| h1 : String → NotePart
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| h2 : String → NotePart
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| p : String → NotePart
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| name : Name → NameStatus → NotePart
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| warning : String → NotePart
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structure DeclInfo where
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line : Nat
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fileName : Name
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name : Name
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status : NameStatus
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declString : String
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docString : String
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isDef : Bool
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isThm : Bool
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def DeclInfo.ofName (n : Name) (ns : NameStatus): MetaM DeclInfo := do
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let line ← Name.lineNumber n
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let fileName ← Name.fileName n
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let declString ← Name.getDeclStringNoDoc n
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let docString ← Name.getDocString n
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let constInfo ← getConstInfo n
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let isDef := constInfo.isDef ∨ Lean.isStructure (← getEnv) n ∨ constInfo.isInductive
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let isThm := declString.startsWith "theorem" ∨ declString.startsWith "noncomputable theorem"
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pure {
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line := line,
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fileName := fileName,
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name := n,
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status := ns,
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declString := declString,
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docString := docString,
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isDef := isDef
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isThm := isThm}
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def DeclInfo.toYML (d : DeclInfo) : MetaM String := do
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let declStringIndent := d.declString.replace "\n" "\n "
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let docStringIndent := d.docString.replace "\n" "\n "
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let link := Name.toGitHubLink d.fileName d.line
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return s!"
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- type: name
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name: {d.name}
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line: {d.line}
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fileName: {d.fileName}
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status: \"{d.status}\"
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link: \"{link}\"
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isDef: {d.isDef}
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isThm: {d.isThm}
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docString: |
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{docStringIndent}
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declString: |
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{declStringIndent}"
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/-- In `(List String) × Nat × Nat` the first `Nat` is section number, the second `Nat`
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is subsection number, and the third `Nat` is defn. or lemma number.
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Definitions and lemmas etc are done by section not subsection. -/
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def NotePart.toYMLM : ((List String) × Nat × Nat × Nat) → NotePart →
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MetaM ((List String) × Nat × Nat × Nat)
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| x, NotePart.h1 s =>
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let newString := s!"
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- type: h1
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sectionNo: {x.2.1.succ}
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content: \"{s}\""
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return ⟨x.1 ++ [newString], ⟨Nat.succ x.2.1, 0, 0⟩⟩
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| x, NotePart.h2 s =>
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let newString := s!"
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- type: h2
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sectionNo: \"{x.2.1}.{x.2.2.1.succ}\"
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content: \"{s}\""
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return ⟨x.1 ++ [newString], ⟨x.2.1, Nat.succ x.2.2.1, x.2.2.2⟩⟩
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| x, NotePart.p s =>
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let newString := s!"
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- type: p
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content: \"{s}\""
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return ⟨x.1 ++ [newString], x.2⟩
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| x, NotePart.warning s =>
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let newString := s!"
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- type: warning
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content: \"{s}\""
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return ⟨x.1 ++ [newString], x.2⟩
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| x, NotePart.name n s => do
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match (← RemarkInfo.IsRemark n) with
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| true =>
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let remarkInfo ← RemarkInfo.getRemarkInfo n
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let content := remarkInfo.content
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let contentIndent := content.replace "\n" "\n "
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let shortName := remarkInfo.name.toString
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let newString := s!"
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- type: remark
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name: \"{shortName}\"
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status : \"{s}\"
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link: \"{Name.toGitHubLink remarkInfo.fileName remarkInfo.line}\"
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content: |
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{contentIndent}"
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return ⟨x.1 ++ [newString], x.2⟩
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| false =>
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let newString ← (← DeclInfo.ofName n s).toYML
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let newString := newString ++ s!"\n declNo: \"{x.2.1}.{x.2.2.2.succ}\""
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return ⟨x.1 ++ [newString], ⟨x.2.1, x.2.2.1, Nat.succ x.2.2.2⟩⟩
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structure Note where
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title : String
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/-- The curators of the note are the people who put the note together.
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This may not conicide with the original authors of the material. -/
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curators : List String
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parts : List NotePart
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def Note.toYML : Note → MetaM String
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| ⟨title, curators, parts⟩ => do
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let parts ← parts.foldlM NotePart.toYMLM ([], ⟨0, 0⟩)
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return s!"
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title: \"{title}\"
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curators: {String.intercalate "," curators}
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parts:
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{String.intercalate "\n" parts.1}"
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def perturbationTheory : Note where
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title := "Proof of Wick's theorem"
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curators := ["Joseph Tooby-Smith"]
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parts := [
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.h1 "Introduction",
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.name `FieldSpecification.wicks_theorem_context .incomplete,
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.p "In this note we walk through the important parts of the proof of the three versions of
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Wick's theorem for field operators containing carrying both fermionic and bosonic statistics,
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as it appears in PhysLean. Not every lemma or definition is covered here.
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The aim is to give just enough that the story can be understood.",
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.p "
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Before proceeding with the steps in the proof, we review some basic terminology
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related to Lean and type theory. The most important notion is that of a type.
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We don't give any formal definition here, except to say that a type `T`, like a set, has
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elements `x` which we say are of type `T` and write `x : T`. Examples of types include,
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the type of natural numbers `ℕ`, the type of real numbers `ℝ`, the type of numbers
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`0, …, n-1` denoted `Fin n`. Given two types `T` and `S`, we can form the product type `T × S`,
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and the function type `T → S`.
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Types form the foundation of Lean and the theory behind them will be used both explicitly and
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implicitly throughout this note.",
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.h1 "Field operators",
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.h2 "Field statistics",
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.name ``FieldStatistic .complete,
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.name ``FieldStatistic.instCommGroup .complete,
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.name ``FieldStatistic.exchangeSign .complete,
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.h2 "Field specifications",
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.name ``FieldSpecification .complete,
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.h2 "Field operators",
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.name ``FieldSpecification.FieldOp .complete,
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.name ``FieldSpecification.fieldOpStatistic .complete,
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.name ``CreateAnnihilate .complete,
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.name ``FieldSpecification.CrAnFieldOp .complete,
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.name ``FieldSpecification.crAnFieldOpToCreateAnnihilate .complete,
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.name ``FieldSpecification.crAnStatistics .complete,
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.name `FieldSpecification.notation_remark .complete,
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.h2 "Field-operator free algebra",
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.name ``FieldSpecification.FieldOpFreeAlgebra .complete,
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.name `FieldSpecification.FieldOpFreeAlgebra.naming_convention .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.ofCrAnOpF .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.universality .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.ofCrAnListF .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.ofFieldOpF .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.ofFieldOpListF .complete,
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.name `FieldSpecification.FieldOpFreeAlgebra.notation_drop .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.fieldOpFreeAlgebraGrade .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.superCommuteF .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum .complete,
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.h2 "Field-operator algebra",
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.name ``FieldSpecification.FieldOpAlgebra .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ι .complete,
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.name ``FieldSpecification.FieldOpAlgebra.universality .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofCrAnOp .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofCrAnList .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofFieldOp .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofCrAnList .complete,
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.name `FieldSpecification.FieldOpAlgebra.notation_drop .complete,
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.name ``FieldSpecification.FieldOpAlgebra.anPart .complete,
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.name ``FieldSpecification.FieldOpAlgebra.crPart .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofFieldOp_eq_crPart_add_anPart .complete,
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.name ``FieldSpecification.FieldOpAlgebra.fieldOpAlgebraGrade .complete,
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.name ``FieldSpecification.FieldOpAlgebra.superCommute .complete,
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.h1 "Time ordering",
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.name ``FieldSpecification.crAnTimeOrderRel .complete,
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.name ``FieldSpecification.crAnTimeOrderList .complete,
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.name ``FieldSpecification.crAnTimeOrderSign .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.timeOrderF .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeOrder .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeOrder_eq_maxTimeField_mul_finset .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeOrder_timeOrder_mid .complete,
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.h1 "Normal ordering",
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.name ``FieldSpecification.normalOrderRel .complete,
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.name ``FieldSpecification.normalOrderList .complete,
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.name ``FieldSpecification.normalOrderSign .complete,
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.name ``FieldSpecification.normalOrderSign_eraseIdx .complete,
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.name ``FieldSpecification.FieldOpFreeAlgebra.normalOrderF .complete,
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.name ``FieldSpecification.FieldOpAlgebra.normalOrder .complete,
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.name ``FieldSpecification.FieldOpAlgebra.normalOrder_superCommute_eq_zero .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum .complete,
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.name ``FieldSpecification.FieldOpAlgebra.ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum .complete,
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.h1 "Wick Contractions",
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.h2 "Definition",
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.name ``WickContraction .complete,
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.name ``WickContraction.mem_three .complete,
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.name ``WickContraction.mem_four .complete,
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.name `WickContraction.contraction_notation .complete,
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.name ``WickContraction.GradingCompliant .complete,
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.h2 "Aside: Cardinality",
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.name ``WickContraction.card_eq_cardFun .complete,
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.h2 "Uncontracted elements",
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.name ``WickContraction.uncontracted .complete,
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.name ``WickContraction.uncontractedListGet .complete,
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.h2 "Constructors",
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.name ``WickContraction.insertAndContract .complete,
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.name ``WickContraction.insertLift_sum .complete,
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.name ``WickContraction.join .complete,
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.h2 "Sign",
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.name ``WickContraction.sign .complete,
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.name ``WickContraction.join_sign .complete,
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.name ``WickContraction.sign_insert_none .complete,
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.name ``WickContraction.sign_insert_none_zero .complete,
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.name ``WickContraction.sign_insert_some_of_not_lt .complete,
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.name ``WickContraction.sign_insert_some_of_lt .complete,
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.name ``WickContraction.sign_insert_some_zero .complete,
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.h2 "Normal order",
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.name ``FieldSpecification.FieldOpAlgebra.normalOrder_uncontracted_none .complete,
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.name ``FieldSpecification.FieldOpAlgebra.normalOrder_uncontracted_some .complete,
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.h1 "Static Wick's theorem",
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.h2 "Static contractions",
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.name ``WickContraction.staticContract .complete,
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.name ``WickContraction.staticContract_insert_none .complete,
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.name ``WickContraction.staticContract_insert_some .complete,
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.h2 "Static Wick terms",
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.name ``WickContraction.staticWickTerm .complete,
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.name ``WickContraction.staticWickTerm_empty_nil .complete,
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.name ``WickContraction.staticWickTerm_insert_zero_none .complete,
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.name ``WickContraction.staticWickTerm_insert_zero_some .complete,
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.name ``WickContraction.mul_staticWickTerm_eq_sum .complete,
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.h2 "The Static Wick's theorem",
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.name ``FieldSpecification.FieldOpAlgebra.static_wick_theorem .complete,
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.h1 "Wick's theorem",
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.h2 "Time contractions",
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.name ``FieldSpecification.FieldOpAlgebra.timeContract .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeContract_of_timeOrderRel .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeContract_of_not_timeOrderRel_expand .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeContract_mem_center .complete,
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.name ``WickContraction.timeContract .complete,
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.name ``WickContraction.timeContract_insert_none .complete,
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.name ``WickContraction.timeContract_insert_some_of_not_lt .complete,
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.name ``WickContraction.timeContract_insert_some_of_lt .complete,
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.name ``WickContraction.join_sign_timeContract .complete,
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.h2 "Wick terms",
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.name ``WickContraction.wickTerm .complete,
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.name ``WickContraction.wickTerm_empty_nil .complete,
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.name ``WickContraction.wickTerm_insert_none .complete,
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.name ``WickContraction.wickTerm_insert_some .complete,
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.name ``WickContraction.mul_wickTerm_eq_sum .complete,
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.h2 "Wick's theorem",
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.name ``FieldSpecification.wicks_theorem .complete,
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.h1 "Normal-ordered Wick's theorem",
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.name ``WickContraction.EqTimeOnly.timeOrder_staticContract_of_not_mem .complete,
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.name ``WickContraction.EqTimeOnly.staticContract_eq_timeContract_of_eqTimeOnly .complete,
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.name ``WickContraction.EqTimeOnly.timeOrder_timeContract_mul_of_eqTimeOnly_left .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeOrder_ofFieldOpList_eqTimeOnly .complete,
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.name ``FieldSpecification.FieldOpAlgebra.normalOrder_timeOrder_ofFieldOpList_eq_eqTimeOnly_empty .complete,
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.name ``FieldSpecification.FieldOpAlgebra.timeOrder_haveEqTime_split .complete,
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.name ``FieldSpecification.FieldOpAlgebra.wicks_theorem_normal_order .complete
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]
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def harmonicOscillator : Note where
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title := "The Quantum Harmonic Oscillator"
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curators := ["Joseph Tooby-Smith"]
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parts := [
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.h1 "Introduction",
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.p "The quantum harmonic oscillator is a fundamental example of a one-dimensional
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quantum mechanical system. It is often one of the first models encountered by undergraduate
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students studying quantum mechanics.",
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.p " This note presents the formalization of the quantum harmonic oscillator in the theorem prover
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Lean 4, as part of the larger project PhysLean.
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What this means is that every definition and theorem in this note has been formally checked
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for mathematical correctness for by a computer. There are a number of
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motivations for doing this which are discussed <a href = 'https://heplean.com'>here</a>.",
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.p "Note that we do not give every definition and theorem which is part of
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the formalization.
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Our goal is give key aspects, in such a way that we hope this note will be useful
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to newcomers to Lean or those simply intrested in learning about the
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quantum harmonic oscillator.",
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.h1 "Hilbert Space",
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.name ``QuantumMechanics.OneDimension.HilbertSpace .complete,
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.name ``QuantumMechanics.OneDimension.HilbertSpace.MemHS .complete,
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.name ``QuantumMechanics.OneDimension.HilbertSpace.memHS_iff .complete,
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.name ``QuantumMechanics.OneDimension.HilbertSpace.mk .complete,
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.name ``QuantumMechanics.OneDimension.HilbertSpace.parity .complete,
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.name ``QuantumMechanics.OneDimension.HilbertSpace.memHS_of_parity .complete,
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.h1 "The Schrodinger Operator",
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.ξ .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.schrodingerOperator .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.schrodingerOperator_parity .complete,
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.h1 "The eigenfunctions of the Schrodinger Operator",
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.name ``PhysLean.physHermite .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction .complete,
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.h2 "Properties of the eigenfunctions",
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_integrable .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_conj .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_aeStronglyMeasurable .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_square_integrable .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_memHS .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_differentiableAt .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_orthonormal .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_parity .complete,
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.h1 "The time-independent Schrodinger Equation",
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.schrodingerOperator_eigenfunction
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.complete,
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.h1 "Completeness",
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.orthogonal_power_of_mem_orthogonal .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.orthogonal_exp_of_mem_orthogonal .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.fourierIntegral_zero_of_mem_orthogonal .complete,
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.name ``QuantumMechanics.OneDimension.HarmonicOscillator.eigenfunction_completeness .complete,
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]
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def higgsPotential : Note where
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title := "The Higgs potential"
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curators := ["Joseph Tooby-Smith"]
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parts := [
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.h1 "Introduction",
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.p "The Higgs potential is a key part of the Standard Model of particle physics.
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It is a scalar potential which is used to give mass to the elementary particles.
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The Higgs potential is a polynomial of degree four in the Higgs field.",
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.h1 "The Higgs field",
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.name ``StandardModel.HiggsVec .complete,
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.name ``StandardModel.HiggsBundle .complete,
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.name ``StandardModel.HiggsField .complete,
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.h1 "The Higgs potential",
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.name ``StandardModel.HiggsField.Potential .complete,
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.name ``StandardModel.HiggsField.normSq .complete,
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.name ``StandardModel.HiggsField.Potential.toFun .complete,
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.h2 "Properties of the Higgs potential",
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.name ``StandardModel.HiggsField.Potential.neg_𝓵_quadDiscrim_zero_bound .complete,
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.name ``StandardModel.HiggsField.Potential.pos_𝓵_quadDiscrim_zero_bound .complete,
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.name ``StandardModel.HiggsField.Potential.neg_𝓵_sol_exists_iff .complete,
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.name ``StandardModel.HiggsField.Potential.pos_𝓵_sol_exists_iff .complete,
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.h2 "Boundedness of the Higgs potential",
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.name ``StandardModel.HiggsField.Potential.IsBounded .complete,
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.name ``StandardModel.HiggsField.Potential.isBounded_𝓵_nonneg .complete,
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.name ``StandardModel.HiggsField.Potential.isBounded_of_𝓵_pos .complete,
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.h2 "Maximum and minimum of the Higgs potential",
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.name ``StandardModel.HiggsField.Potential.isMaxOn_iff_field_of_𝓵_neg .complete,
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.name ``StandardModel.HiggsField.Potential.isMinOn_iff_field_of_𝓵_pos .complete,
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]
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def tensors : Note where
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title := "Tensors and index notation 🚧"
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curators := ["Joseph Tooby-Smith"]
|
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parts := [
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.warning "This note is a work in progress. (5th March 2025)",
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||
.h1 "Introduction",
|
||
.p "This note is related to: https://arxiv.org/pdf/2411.07667, and concerns the
|
||
implmentation of tensors and index notation into PhysLean, and
|
||
its mathematical structure.",
|
||
.p "This note is not intended to be a first-introduction to tensors and index notation.",
|
||
.h1 "Tensor Species",
|
||
.name ``TensorSpecies .incomplete,
|
||
.h2 "Example: Complex Lorentz tensors",
|
||
.h1 "Tensor trees",
|
||
.name ``TensorTree .incomplete,
|
||
.name ``TensorTree.tensor .incomplete,
|
||
.h2 "Node identities",
|
||
.h1 "Elaboration",
|
||
.h1 "Example use: Pauli matrices"
|
||
]
|
||
|
||
def notesToMake : List (Note × String) := [
|
||
(perturbationTheory, "perturbationTheory"),
|
||
(harmonicOscillator, "harmonicOscillator"),
|
||
(higgsPotential, "higgsPotential"),
|
||
(tensors, "tensors")]
|
||
|
||
def makeYML (nt : Note × String) : IO UInt32 := do
|
||
let n := nt.1
|
||
let s := nt.2
|
||
let ymlString ← CoreM.withImportModules #[`PhysLean] (n.toYML).run'
|
||
let fileOut : System.FilePath := {toString := s!"./docs/_data/{s}.yml"}
|
||
IO.FS.writeFile fileOut ymlString
|
||
IO.println (s!"YML file made for {n.title}.")
|
||
pure 0
|
||
|
||
unsafe def main (_ : List String) : IO UInt32 := do
|
||
initSearchPath (← findSysroot)
|
||
let _ ← notesToMake.mapM makeYML
|
||
pure 0
|