fix: Typo

HepLean/PerturbationTheory/FeynmanDiagrams/Momentum.lean

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+/-
+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.Data.Real.Basic
+import Mathlib.Algebra.DirectSum.Module
+import Mathlib.LinearAlgebra.SesquilinearForm
+import Mathlib.LinearAlgebra.Dimension.Finrank
+/-!
+# Momentum in Feynman diagrams
+
+The aim of this file is to associate with each half-edge of a Feynman diagram a momentum,
+and constrain the momentums based conservation at each vertex and edge.
+
+The number of loops of a Feynman diagram is related to the dimension of the resulting
+vector space.
+
+## TODO
+
+- Prove lemmas that make the calculation of the number of loops of a Feynman diagram easier.
+
+## Note
+
+This section is non-computable as we depend on the norm on `F.HalfEdgeMomenta`.
+-/
+
+namespace FeynmanDiagram
+
+open CategoryTheory
+open PreFeynmanRule
+
+variable {P : PreFeynmanRule} (F : FeynmanDiagram P) [IsFiniteDiagram F]
+
+/-!
+
+## Vector spaces associated with momenta in Feynman diagrams.
+
+We define the vector space associated with momenta carried by half-edges,
+outflowing momenta of edges, and inflowing momenta of vertices.
+
+We define the direct sum of the edge and vertex momentum spaces.
+
+-/
+
+/-- The type which assocaites to each half-edge a `1`-dimensional vector space.
+  Corresponding to that spanned by its momentum. -/
+def HalfEdgeMomenta : Type := F.𝓱𝓔 → ℝ
+
+/-- The half momenta carries the structure of an addative commutative group. -/
+instance : AddCommGroup F.HalfEdgeMomenta := Pi.addCommGroup
+
+/-- The half momenta carries the structure of a module over `ℝ`. Defined via its target. -/
+instance : Module ℝ F.HalfEdgeMomenta := Pi.module _ _ _
+
+/-- An auxiliary function used to define the Euclidean inner product on `F.HalfEdgeMomenta`. -/
+def euclidInnerAux (x : F.HalfEdgeMomenta) : F.HalfEdgeMomenta →ₗ[ℝ] ℝ where
+  toFun y := ∑ i, (x i) * (y i)
+  map_add' z y :=
+    show (∑ i, (x i) * (z i + y i)) = (∑ i, x i * z i) + ∑ i, x i * (y i) by
+      simp only [mul_add, Finset.sum_add_distrib]
+  map_smul' c y :=
+    show (∑ i, x i * (c * y i)) = c * ∑ i, x i * y i by
+      rw [Finset.mul_sum]
+      refine Finset.sum_congr rfl (fun _ _ => by ring)
+
+lemma euclidInnerAux_symm (x y : F.HalfEdgeMomenta) :
+    F.euclidInnerAux x y = F.euclidInnerAux y x := Finset.sum_congr rfl (fun _ _ => by ring)
+
+/-- The Euclidean inner product on `F.HalfEdgeMomenta`. -/
+def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ] ℝ where
+  toFun x := F.euclidInnerAux x
+  map_add' x y := by
+    refine LinearMap.ext (fun z => ?_)
+    simp only [euclidInnerAux_symm, map_add, LinearMap.add_apply]
+  map_smul' c x := by
+    refine LinearMap.ext (fun z => ?_)
+    simp only [euclidInnerAux_symm, LinearMapClass.map_smul, smul_eq_mul, RingHom.id_apply,
+      LinearMap.smul_apply]
+
+/-- The type which associates to each edge a `1`-dimensional vector space.
+  Corresponding to that spanned by its total outflowing momentum. -/
+def EdgeMomenta : Type := F.𝓔 → ℝ
+
+/-- The edge momenta form an addative commuative group. -/
+instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
+
+/-- The edge momenta form a module over `ℝ`. -/
+instance : Module ℝ F.EdgeMomenta := Pi.module _ _ _
+
+/-- The type which assocaites to each ege a `1`-dimensional vector space.
+  Corresponding to that spanned by its total inflowing momentum. -/
+def VertexMomenta : Type := F.𝓥 → ℝ
+
+/-- The vertex momenta carries the structure of an addative commuative group. -/
+instance : AddCommGroup F.VertexMomenta := Pi.addCommGroup
+
+/-- The vertex momenta carries the structure of a module over `ℝ`. -/
+instance : Module ℝ F.VertexMomenta := Pi.module _ _ _
+
+/-- The map from `Fin 2` to `Type` landing on `EdgeMomenta` and `VertexMomenta`. -/
+def EdgeVertexMomentaMap : Fin 2 → Type := fun i =>
+  match i with
+  | 0 => F.EdgeMomenta
+  | 1 => F.VertexMomenta
+
+/-- The target of the map `EdgeVertexMomentaMap` is either the type of edge momenta
+  or vertex momenta and thus carries the structure of an addative commuative group. -/
+instance (i : Fin 2) : AddCommGroup (EdgeVertexMomentaMap F i) :=
+  match i with
+  | 0 => instAddCommGroupEdgeMomenta F
+  | 1 => instAddCommGroupVertexMomenta F
+
+/-- The target of the map `EdgeVertexMomentaMap` is either the type of edge momenta
+  or vertex momenta and thus carries the structure of a module over `ℝ`. -/
+instance (i : Fin 2) : Module ℝ (EdgeVertexMomentaMap F i) :=
+  match i with
+  | 0 => instModuleRealEdgeMomenta F
+  | 1 => instModuleRealVertexMomenta F
+
+/-- The direct sum of `EdgeMomenta` and `VertexMomenta`. -/
+def EdgeVertexMomenta : Type := DirectSum (Fin 2) (EdgeVertexMomentaMap F)
+
+/-- The structure of a addative commutative group on `EdgeVertexMomenta` for a
+  Feynman diagram `F`. -/
+instance : AddCommGroup F.EdgeVertexMomenta := DirectSum.instAddCommGroup _
+
+/-- The structure of a module over `ℝ` on `EdgeVertexMomenta` for a Feynman diagram `F`. -/
+instance : Module ℝ F.EdgeVertexMomenta := DirectSum.instModule
+
+/-!
+
+## Linear maps between the vector spaces.
+
+We define various maps into `F.HalfEdgeMomenta`.
+
+In particular, we define a map from `F.EdgeVertexMomenta` to `F.HalfEdgeMomenta`. This
+map represents the space orthogonal (with respect to the standard Euclidean inner product)
+to the allowed momenta of half-edges (up-to an offset determined by the
+external momenta).
+
+The number of loops of a diagram is defined as the number of half-edges minus
+the rank of this matrix.
+
+-/
+
+/-- The linear map from `F.EdgeMomenta` to `F.HalfEdgeMomenta` induced by
+  the map `F.𝓱𝓔To𝓔.hom`. -/
+def edgeToHalfEdgeMomenta : F.EdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta where
+  toFun x := x ∘ F.𝓱𝓔To𝓔.hom
+  map_add' _ _ := by rfl
+  map_smul' _ _ := by rfl
+
+/-- The linear map from `F.VertexMomenta` to `F.HalfEdgeMomenta` induced by
+  the map `F.𝓱𝓔To𝓥.hom`. -/
+def vertexToHalfEdgeMomenta : F.VertexMomenta →ₗ[ℝ] F.HalfEdgeMomenta where
+  toFun x := x ∘ F.𝓱𝓔To𝓥.hom
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+/-- The linear map from `F.EdgeVertexMomenta` to `F.HalfEdgeMomenta` induced by
+  `F.edgeToHalfEdgeMomenta` and `F.vertexToHalfEdgeMomenta`. -/
+def edgeVertexToHalfEdgeMomenta : F.EdgeVertexMomenta →ₗ[ℝ] F.HalfEdgeMomenta :=
+  DirectSum.toModule ℝ (Fin 2) F.HalfEdgeMomenta
+    (fun i => match i with | 0 => F.edgeToHalfEdgeMomenta | 1 => F.vertexToHalfEdgeMomenta)
+
+/-!
+
+## Submodules
+
+We define submodules of `F.HalfEdgeMomenta` which correspond to
+the orthogonal space to allowed momenta (up-to an offset), and the space of
+allowed momenta.
+
+-/
+
+/-- The submodule of `F.HalfEdgeMomenta` corresponding to the range of
+  `F.edgeVertexToHalfEdgeMomenta`. -/
+def orthogHalfEdgeMomenta : Submodule ℝ F.HalfEdgeMomenta :=
+  LinearMap.range F.edgeVertexToHalfEdgeMomenta
+
+/-- The submodule of `F.HalfEdgeMomenta` corresponding to the allowed momenta. -/
+def allowedHalfEdgeMomenta : Submodule ℝ F.HalfEdgeMomenta :=
+  Submodule.orthogonalBilin F.orthogHalfEdgeMomenta F.euclidInner
+
+/-!
+
+## Number of loops
+
+We define the number of loops of a Feynman diagram as the dimension of the
+allowed space of half-edge momenta.
+
+-/
+
+/-- The number of loops of a Feynman diagram. Defined as the dimension
+  of the space of allowed Half-loop momenta. -/
+noncomputable def numberOfLoops : ℕ := Module.finrank ℝ F.allowedHalfEdgeMomenta
+
+/-!
+
+## Lemmas regarding `numberOfLoops`
+
+We now give a series of lemmas which be used to help calculate the number of loops
+for specific Feynman diagrams.
+
+### TODO
+
+- Complete this section.
+
+-/
+
+/-!
+
+## Category theory
+
+### TODO
+
+- Complete this section.
+
+-/
+
+end FeynmanDiagram