74 lines
2.3 KiB
Text
74 lines
2.3 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Algebra.BigOperators.Group.Finset
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/-!
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# Creation and annihlation parts of fields
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-/
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/-- The type specifing whether an operator is a creation or annihilation operator. -/
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inductive CreateAnnihilate where
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| create : CreateAnnihilate
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| annihilate : CreateAnnihilate
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deriving Inhabited, BEq, DecidableEq
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namespace CreateAnnihilate
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/-- The type `CreateAnnihilate` is finite. -/
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instance : Fintype CreateAnnihilate where
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elems := {create, annihilate}
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complete := by
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intro c
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cases c
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· exact Finset.mem_insert_self create {annihilate}
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· refine Finset.insert_eq_self.mp ?_
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exact rfl
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lemma eq_create_or_annihilate (φ : CreateAnnihilate) : φ = create ∨ φ = annihilate := by
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cases φ <;> simp
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/-- The normal ordering on creation and annihlation operators.
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Under this relation, `normalOrder a b` is false only if `a` is annihlate and `b` is create. -/
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def normalOrder : CreateAnnihilate → CreateAnnihilate → Prop
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| create, _ => True
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| annihilate, annihilate => True
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| annihilate, create => False
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/-- The normal ordering on `CreateAnnihilate` is decidable. -/
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instance : (φ φ' : CreateAnnihilate) → Decidable (normalOrder φ φ')
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| create, create => isTrue True.intro
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| annihilate, annihilate => isTrue True.intro
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| create, annihilate => isTrue True.intro
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| annihilate, create => isFalse False.elim
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/-- Normal ordering is total. -/
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instance : IsTotal CreateAnnihilate normalOrder where
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total a b := by
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cases a <;> cases b <;> simp [normalOrder]
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/-- Normal ordering is transitive. -/
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instance : IsTrans CreateAnnihilate normalOrder where
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trans a b c := by
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cases a <;> cases b <;> cases c <;> simp [normalOrder]
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@[simp]
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lemma not_normalOrder_annihilate_iff_false (a : CreateAnnihilate) :
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(¬ normalOrder a annihilate) ↔ False := by
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cases a
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· simp [normalOrder]
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· simp [normalOrder]
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lemma sum_eq {M : Type} [AddCommMonoid M] (f : CreateAnnihilate → M) :
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∑ i, f i = f create + f annihilate := by
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change ∑ i in {create, annihilate}, f i = f create + f annihilate
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simp
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@[simp]
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lemma CreateAnnihilate_card_eq_two : Fintype.card CreateAnnihilate = 2 := by
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rfl
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end CreateAnnihilate
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