Merge pull request #258 from HEPLean/PerturbationTheory

feat: Informal def of Feynman diagram

HepLean/PerturbationTheory/FeynmanDiagrams/Light.lean

@@ -3,6 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
+import HepLean.PerturbationTheory.Wick.Contract
 import HepLean.PerturbationTheory.Wick.Species
 /-!
 
@@ -13,5 +14,50 @@ This file currently contains a lighter implmentation of Feynman digrams than can
 
 The implmentation here is done in conjunction with Wicks species etc.
 
-This file is currently a stub.
 -/
+/-! TODO: Remove this namespace-/
+namespace LightFeynman
+
+informal_definition FeynmanDiagram where
+  math :≈ "
+    Let S be a WickSpecies
+    A Feynman diagram contains the following data:
+    - A type of vertices 𝓥 → S.𝓯 ⊕ S.𝓘.
+    - A type of edges ed : 𝓔 → S.𝓕.
+    - A type of half-edges 𝓱𝓔 with maps `e : 𝓱𝓔 → 𝓔`, `v : 𝓱𝓔 → 𝓥` and `f : 𝓱𝓔 → S.𝓯`
+    Subject to the following conditions:
+    - `𝓱𝓔` is a double cover of `𝓔` through `e`. That is,
+      for each edge `x : 𝓔` there exists an isomorphism between `i : Fin 2 → e⁻¹ 𝓱𝓔 {x}`.
+    - These isomorphisms must satisfy `⟦f(i(0))⟧ = ⟦f(i(1))⟧ = ed(e)` and `f(i(0)) = S.ξ (f(i(1)))`.
+    - For each vertex `ver : 𝓥` there exists an isomorphism between the object (roughly)
+      `(𝓘Fields v).2` and the pullback of `v` along `ver`."
+  deps :≈ [``Wick.Species]
+
+informal_definition _root_.Wick.Contract.toFeynmanDiagram where
+  math :≈ "The Feynman diagram constructed from a complete Wick contraction."
+  deps :≈ [``TwoComplexScalar.WickContract, ``FeynmanDiagram]
+
+informal_lemma _root_.Wick.Contract.toFeynmanDiagram_surj where
+  math :≈ "The map from Wick contractions to Feynman diagrams is surjective."
+  physics :≈ "Every Feynman digram corresponds to some Wick contraction."
+  deps :≈ [``TwoComplexScalar.WickContract, ``FeynmanDiagram]
+
+informal_definition FeynmanDiagram.toSimpleGraph where
+  math :≈ "The simple graph underlying a Feynman diagram."
+  deps :≈ [``FeynmanDiagram]
+
+informal_definition FeynmanDiagram.IsConnected where
+  math :≈ "A Feynman diagram is connected if its underlying simple graph is connected."
+  deps :≈ [``FeynmanDiagram]
+
+informal_definition _root_.Wick.Contract.toFeynmanDiagram_isConnected_iff where
+  math :≈ "The Feynman diagram corresponding to a Wick contraction is connected
+    if and only if the Wick contraction is connected."
+  deps :≈ [``TwoComplexScalar.WickContract.IsConnected, ``FeynmanDiagram.IsConnected]
+
+/-! TODO: Define an equivalence relation on Wick contracts related to the their underlying tensors
+  been equal after permutation. Show that two  Wick contractions are equal under this
+  equivalence relation if and only if they have the same Feynman diagram. First step
+  is to turn these statements into appropriate informal lemmas and definitions. -/
+
+end LightFeynman

HepLean/PerturbationTheory/Wick/Algebra.lean

@@ -124,7 +124,7 @@ informal_lemma timeOrder_pair where
 
 informal_definition WickMap where
   math :≈ "A linear map `vev` from the Wick algebra `A` to the underlying field such that
-    `vev(...ψd(t)) = 0` and `vev(ψc(t)...) = 0`."
+     `vev(...ψd(t)) = 0` and `vev(ψc(t)...) = 0`."
   physics :≈ "An abstraction of the notion of a vacuum expectation value, containing
     the necessary properties for lots of theorems to hold."
   deps :≈ [``WickAlgebra, ``WickMonomial]