96 lines
4.1 KiB
Text
96 lines
4.1 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.FieldStatistics
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/-!
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# Super commutation coefficent.
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This is a complex number which is `-1` when commuting two fermionic operators and `1` otherwise.
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-/
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namespace Wick
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open FieldStatistic
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variable {𝓕 : Type} (q : 𝓕 → FieldStatistic)
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/-- Given two lists `la` and `lb` returns `-1` if they are both of grade `1` and
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`1` otherwise. This corresponds to the sign associated with the super commutator
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when commuting `la` and `lb` in the free algebra.
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In terms of physics it is `-1` if commuting two fermionic operators and `1` otherwise. -/
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def superCommuteCoef (la lb : List 𝓕) : ℂ :=
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if FieldStatistic.ofList q la = fermionic ∧
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FieldStatistic.ofList q lb = fermionic then - 1 else 1
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lemma superCommuteCoef_comm (la lb : List 𝓕) :
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superCommuteCoef q la lb = superCommuteCoef q lb la := by
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simp only [superCommuteCoef, Fin.isValue]
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congr 1
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exact Eq.propIntro (fun a => id (And.symm a)) fun a => id (And.symm a)
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/-- Given a list `l : List (Σ i, f i)` and a list `r : List I` returns `-1` if the
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grade of `l` is `1` and the grade of `r` is `1` and `1` otherwise. This corresponds
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to the sign associated with the super commutator when commuting
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the lift of `l` and `r` (by summing over fibers) in the
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free algebra over `Σ i, f i`.
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In terms of physics it is `-1` if commuting two fermionic operators and `1` otherwise. -/
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def superCommuteLiftCoef {f : 𝓕 → Type} (φs : List (Σ i, f i)) (φs' : List 𝓕) : ℂ :=
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(if FieldStatistic.ofList (fun i => q i.fst) φs = fermionic ∧
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FieldStatistic.ofList q φs' = fermionic then -1 else 1)
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lemma superCommuteLiftCoef_empty {f : 𝓕 → Type} (φs : List (Σ i, f i)) :
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superCommuteLiftCoef q φs [] = 1 := by
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simp [superCommuteLiftCoef]
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lemma superCommuteCoef_perm_snd (la lb lb' : List 𝓕)
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(h : lb.Perm lb') :
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superCommuteCoef q la lb = superCommuteCoef q la lb' := by
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rw [superCommuteCoef, superCommuteCoef, FieldStatistic.ofList_perm q h]
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lemma superCommuteCoef_mul_self (l lb : List 𝓕) :
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superCommuteCoef q l lb * superCommuteCoef q l lb = 1 := by
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simp only [superCommuteCoef, Fin.isValue, mul_ite, mul_neg, mul_one]
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have ha (a b : FieldStatistic) : (if a = fermionic ∧ b = fermionic then
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-if a = fermionic ∧ b = fermionic then -1 else 1
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else if a = fermionic ∧ b = fermionic then -1 else 1) = (1 : ℂ) := by
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fin_cases a <;> fin_cases b
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any_goals rfl
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simp
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exact ha (FieldStatistic.ofList q l) (FieldStatistic.ofList q lb)
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lemma superCommuteCoef_empty (la : List 𝓕) :
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superCommuteCoef q la [] = 1 := by
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simp only [superCommuteCoef, ofList_empty, reduceCtorEq, and_false, ↓reduceIte]
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lemma superCommuteCoef_append (la lb lc : List 𝓕) :
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superCommuteCoef q la (lb ++ lc) = superCommuteCoef q la lb * superCommuteCoef q la lc := by
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simp only [superCommuteCoef, Fin.isValue, ofList_append, ite_eq_right_iff, zero_ne_one, imp_false,
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mul_ite, mul_neg, mul_one]
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by_cases hla : ofList q la = fermionic
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· by_cases hlb : ofList q lb = fermionic
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· by_cases hlc : ofList q lc = fermionic
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· simp [hlc, hlb, hla]
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· have hc : ofList q lc = bosonic := by
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exact (neq_fermionic_iff_eq_bosonic (ofList q lc)).mp hlc
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simp [hc, hlb, hla]
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· have hb : ofList q lb = bosonic := by
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exact (neq_fermionic_iff_eq_bosonic (ofList q lb)).mp hlb
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by_cases hlc : ofList q lc = fermionic
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· simp [hlc, hb]
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· have hc : ofList q lc = bosonic := by
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exact (neq_fermionic_iff_eq_bosonic (ofList q lc)).mp hlc
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simp [hc, hb]
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· have ha : ofList q la = bosonic := by
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exact (neq_fermionic_iff_eq_bosonic (ofList q la)).mp hla
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simp [ha]
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lemma superCommuteCoef_cons (i : 𝓕) (la lb : List 𝓕) :
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superCommuteCoef q la (i :: lb) = superCommuteCoef q la [i] * superCommuteCoef q la lb := by
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trans superCommuteCoef q la ([i] ++ lb)
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simp only [List.singleton_append]
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rw [superCommuteCoef_append]
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end Wick
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