Merge pull request #257 from HEPLean/PerturbationTheory

feat: Update species and Feynman diagram file

HepLean.lean

@@ -110,6 +110,7 @@ import HepLean.Meta.TransverseTactics
 import HepLean.PerturbationTheory.FeynmanDiagrams.Basic
 import HepLean.PerturbationTheory.FeynmanDiagrams.Instances.ComplexScalar
 import HepLean.PerturbationTheory.FeynmanDiagrams.Instances.Phi4
+import HepLean.PerturbationTheory.FeynmanDiagrams.Light
 import HepLean.PerturbationTheory.FeynmanDiagrams.Momentum
 import HepLean.PerturbationTheory.Wick.Algebra
 import HepLean.PerturbationTheory.Wick.Contract

HepLean/PerturbationTheory/FeynmanDiagrams/Light.lean

@@ -0,0 +1,17 @@
+/-
+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.Species
+/-!
+
+# Feynman diagrams
+
+This file currently contains a lighter implmentation of Feynman digrams than can be found in
+`HepLean.PerturbationTheory.FeynmanDiagrams.Basic`. Eventually this will superseed that file.
+
+The implmentation here is done in conjunction with Wicks species etc.
+
+This file is currently a stub.
+-/

HepLean/PerturbationTheory/Wick/Algebra.lean

@@ -11,8 +11,6 @@ import HepLean.PerturbationTheory.Wick.Species
 Currently this file is only for an example of Wick strings, correpsonding to a
 theory with two complex scalar fields. The concepts will however generalize.
 
-This file is currently a stub.
-
 We will formally define the operator ring, in terms of the fields present in the theory.
 
 ## Futher reading
@@ -21,28 +19,31 @@ We will formally define the operator ring, in terms of the fields present in the
 - Ryan Thorngren (https://physics.stackexchange.com/users/10336/ryan-thorngren), Fermions,
   different species and (anti-)commutation rules, URL (version: 2019-02-20) :
   https://physics.stackexchange.com/q/461929
+- Tong, https://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf
 -/
 
 namespace Wick
-open CategoryTheory
-open FeynmanDiagram
-open PreFeynmanRule
 
 informal_definition WickAlgebra where
   math :≈ "
-    Modifications of this may be needed, in particular
-    need to add asympotic states.
-
+    Modifications of this may be needed.
     A structure with the following data:
     - A ℤ₂-graded algebra A.
     - A map from `ψ : 𝓔 × SpaceTime → A` where 𝓔 are field colors.
     - A map `ψc : 𝓔 × SpaceTime → A`.
     - A map `ψd : 𝓔 × SpaceTime → A`.
     Subject to the conditions:
     - The sum of `ψc` and `ψd` is `ψ`.
+    - All maps land on homogeneous elements.
     - Two fields super-commute if there colors are not dual to each other.
     - The super-commutator of two fields is always in the
-      center of the algebra. "
+      center of the algebra.
+    Asympotic states:
+    - `φc : 𝓔 × SpaceTime → A`. The creation asympotic state (incoming).
+    - `φd : 𝓔 × SpaceTime → A`. The destruction asympotic state (outgoing).
+    Subject to the conditions:
+    ...
+      "
   physics :≈ "This is defined to be an
     abstraction of the notion of an operator algebra."
   ref :≈ "https://physics.stackexchange.com/questions/24157/"
@@ -84,6 +85,17 @@ informal_definition normalOrder where
 
 end WickMonomial
 
+informal_definition asymptoicContract where
+  math :≈ "Given two `i j : 𝓔 × SpaceTime`, the super-commutator [φd(i), ψ(j)]."
+  ref :≈ "See e.g. http://www.dylanjtemples.com:82/solutions/QFT_Solution_I-6.pdf"
+
+informal_definition contractAsymptotic where
+  math :≈ "Given two `i j : 𝓔 × SpaceTime`, the super-commutator [ψ(i), φc(j)]."
+
+informal_definition asymptoicContractAsymptotic where
+  math :≈ "Given two `i j : 𝓔 × SpaceTime`, the super-commutator
+    [φd(i), φc(j)]."
+
 informal_definition contraction where
   math :≈ "Given two `i j : 𝓔 × SpaceTime`, the element of WickAlgebra
     defined by subtracting the normal ordering of `ψ i ψ j` from the time-ordering of
@@ -112,7 +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]

HepLean/PerturbationTheory/Wick/Contract.lean

@@ -4,6 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.PerturbationTheory.Wick.String
+import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Data.Fintype.Sum
+import Mathlib.Logic.Equiv.Fin
 /-!
 
 # Wick Contract
@@ -18,9 +21,6 @@ theory with two complex scalar fields. The concepts will however generalize.
 -/
 
 namespace TwoComplexScalar
-open CategoryTheory
-open FeynmanDiagram
-open PreFeynmanRule
 
 /-- A Wick contraction for a Wick string is a series of pairs `i` and `j` of indices
   to be contracted, subject to ordering and subject to the condition that they can
@@ -256,8 +256,8 @@ lemma mem_snoc' {ni : ℕ} {i : Fin ni → 𝓔} {n : ℕ} {c : Fin n → 𝓔}
     (hilej : i < j) → (hb1 : ∀ r, b1 r < i) → (hb2i : ∀ r, b2 r ≠ i) → (hb2j : ∀ r, b2 r ≠ j) →
     (hb1' : Fin.snoc b1 i = b1' ∘ Fin.cast hk') →
     (hb2' : Fin.snoc b2 j = b2' ∘ Fin.cast hk') →
-    ∃ (w' : WickContract str b1 b2), w = castMaps hk' hb1' hb2' (
-      contr i j h hilej hb1 hb2i hb2j w') := fun
+    ∃ (w' : WickContract str b1 b2), w = castMaps hk' hb1' hb2'
+      (contr i j h hilej hb1 hb2i hb2j w') := fun
   | string => fun hk' => by
     simp at hk'
   | contr i' j' h' hilej' hb1' hb2i' hb2j' w' => by

HepLean/PerturbationTheory/Wick/PositionSpace.lean

@@ -21,5 +21,4 @@ The following reference provides a good resource for Wick contractions of extern
 
 namespace Wick
 
-
 end Wick

HepLean/PerturbationTheory/Wick/Species.lean

@@ -3,7 +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.FeynmanDiagrams.Basic
+import Mathlib.Logic.Function.Basic
 import HepLean.Meta.Informal
 /-!
 
@@ -23,17 +23,34 @@ namespace Wick
   calculate the corresponding Feynman diagrams.
 
   WARNING: This definition is not yet complete.
-   -/
+  -/
 structure Species where
-  /-- The color of Field operators which appear in a theory. -/
-  𝓕 : Type
+  /-- The color of Field operators which appear in a theory.
+    One may wish to call these `half-edges`, however we restrict this terminology
+    to Feynman diagrams. -/
+  𝓯 : Type
   /-- The map taking a field operator to its dual operator. -/
-  ξ : 𝓕 → 𝓕
+  ξ : 𝓯 → 𝓯
   /-- The condition that `ξ` is an involution. -/
   ξ_involutive : Function.Involutive ξ
-  /-- The color of vertices which appear in a theory. -/
-  𝓥 : Type
-  /-- The edges each vertex corresponds to. -/
-  𝓥Fields : 𝓥 → Σ n, Fin n → 𝓕
+  /-- The color of interaction terms which appear in a theory.
+    One may wish to call these `vertices`, however we restrict this terminology
+    to Feynman diagrams. -/
+  𝓘 : Type
+  /-- The fields associated to each interaction term. -/
+  𝓘Fields : 𝓘 → Σ n, Fin n → 𝓯
+
+namespace Species
+
+variable (S : Species)
+
+informal_definition 𝓕 where
+  math :≈ "The orbits of the involution `ξ`.
+    May have to define a multiplicative action of ℤ₂ on `𝓯`, and
+    take the orbits of this."
+  physics :≈ "The different types of fields present in a theory."
+  deps :≈ [``Species]
+
+end Species
 
 end Wick

HepLean/PerturbationTheory/Wick/String.lean

@@ -3,7 +3,8 @@ 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.Species
+import HepLean.Meta.Informal
+import Mathlib.Data.Fin.Tuple.Basic
 /-!
 # Wick strings
 
@@ -19,9 +20,6 @@ term in the ring of operators. This has yet to be implemented.
 -/
 
 namespace TwoComplexScalar
-open CategoryTheory
-open FeynmanDiagram
-open PreFeynmanRule
 
 /-- The colors of edges which one can associate with a vertex for a theory
   with two complex scalar fields. -/

HepLean/PerturbationTheory/Wick/Theorem.lean

@@ -13,9 +13,6 @@ Wick's theorem is related to a result in probability theory called Isserlis' the
 -/
 
 namespace Wick
-open CategoryTheory
-open FeynmanDiagram
-open PreFeynmanRule
 
 informal_lemma wicks_theorem where
   math :≈ "Wick's theorem for fields which are not normally ordered."