feat: Add definition of Number of Loops
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3 changed files with 207 additions and 53 deletions
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@ -6,6 +6,11 @@ Authors: Joseph Tooby-Smith
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import HepLean.FeynmanDiagrams.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Algebra.Category.ModuleCat.Basic
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import Mathlib.LinearAlgebra.StdBasis
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import Mathlib.LinearAlgebra.Matrix.ToLin
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import Mathlib.Data.Matrix.Rank
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import Mathlib.Algebra.DirectSum.Module
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import Mathlib.LinearAlgebra.SesquilinearForm
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/-!
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# Momentum in Feynman diagrams
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@ -15,72 +20,176 @@ and constrain the momentums based conservation at each vertex and edge.
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The number of loops of a Feynman diagram is related to the dimension of the resulting
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vector space.
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## TODO
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- Prove lemmas that make the calcuation of the number of loops of a Feynman diagram easier.
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## Note
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This section is non-computable as we depend on the norm on `F.HalfEdgeMomenta`.
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-/
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namespace FeynmanDiagram
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open CategoryTheory
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open PreFeynmanRule
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variable {P : PreFeynmanRule} (F : FeynmanDiagram P)
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variable (d : ℕ)
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/-- The momentum space for a `d`-dimensional field theory for a single particle.
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TODO: Move this definition, and define it as a four-vector. -/
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def SingleMomentumSpace : Type := Fin d → ℝ
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instance : AddCommGroup (SingleMomentumSpace d) := Pi.addCommGroup
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instance : Module ℝ (SingleMomentumSpace d) := Pi.module _ _ _
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/-- The type which asociates to each half-edge a `d`-dimensional vector.
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This is to be interpreted as the momentum associated to that half-edge flowing from the
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corresponding `edge` to the corresponding `vertex`. So all momentums flow into vertices. -/
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def FullMomentumSpace : Type := F.𝓱𝓔 → Fin d → ℝ
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instance : AddCommGroup (F.FullMomentumSpace d) := Pi.addCommGroup
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instance : Module ℝ (F.FullMomentumSpace d) := Pi.module _ _ _
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/-- The linear map taking a half-edge to its momentum.
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(defined as flowing from the `edge` to the vertex.) -/
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def toHalfEdgeMomentum (i : F.𝓱𝓔) : F.FullMomentumSpace d →ₗ[ℝ] SingleMomentumSpace d where
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toFun x := x i
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map_add' x y := by rfl
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map_smul' c x := by rfl
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namespace Hom
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variable {F G : FeynmanDiagram P}
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variable (f : F ⟶ G)
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/-- The linear map induced by a morphism of Feynman diagrams. -/
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def toLinearMap : G.FullMomentumSpace d →ₗ[ℝ] F.FullMomentumSpace d where
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toFun x := x ∘ f.𝓱𝓔
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map_add' x y := by rfl
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map_smul' c x := by rfl
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end Hom
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/-- The contravariant functor from Feynman diagrams to Modules over `ℝ`. -/
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noncomputable def funcFullMomentumSpace : FeynmanDiagram P ⥤ (ModuleCat ℝ)ᵒᵖ where
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obj F := Opposite.op $ ModuleCat.of ℝ (F.FullMomentumSpace d)
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map f := Opposite.op $ Hom.toLinearMap d f
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variable {P : PreFeynmanRule} (F : FeynmanDiagram P) [IsFiniteDiagram F]
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/-!
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## Edge constraints
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There is a linear map from `F.FullMomentumSpace` to `F.EdgeMomentumSpace`, induced
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by the constraints at each edge.
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## Vector spaces associated with momenta in Feynman diagrams.
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We impose the constraint that we live in the kernal of this linear map.
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We define the vector space associated with momenta carried by half-edges,
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outflowing momenta of edges, and inflowing momenta of vertices.
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A similar result is true for the vertex constraints.
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We define the direct sum of the edge and vertex momentum spaces.
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-/
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/-- The type which assocaites to each half-edge a `1`-dimensional vector space.
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Corresponding to that spanned by its momentum. -/
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def HalfEdgeMomenta : Type := F.𝓱𝓔 → ℝ
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instance : AddCommGroup F.HalfEdgeMomenta := Pi.addCommGroup
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instance : Module ℝ F.HalfEdgeMomenta := Pi.module _ _ _
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/-- An auxillary function used to define the Euclidean inner product on `F.HalfEdgeMomenta`. -/
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def euclidInnerAux (x : F.HalfEdgeMomenta) : F.HalfEdgeMomenta →ₗ[ℝ] ℝ where
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toFun y := ∑ i, (x i) * (y i)
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map_add' z y :=
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show (∑ i, (x i) * (z i + y i)) = (∑ i, x i * z i) + ∑ i, x i * (y i) by
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simp only [mul_add, Finset.sum_add_distrib]
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map_smul' c y :=
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show (∑ i, x i * (c * y i)) = c * ∑ i, x i * y i by
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rw [Finset.mul_sum]
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refine Finset.sum_congr rfl (fun _ _ => by ring)
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lemma euclidInnerAux_symm (x y : F.HalfEdgeMomenta) :
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F.euclidInnerAux x y = F.euclidInnerAux y x := Finset.sum_congr rfl (fun _ _ => by ring)
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/-- The Euclidean inner product on `F.HalfEdgeMomenta`. -/
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def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ] ℝ where
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toFun x := F.euclidInnerAux x
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map_add' x y := by
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refine LinearMap.ext (fun z => ?_)
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simp only [euclidInnerAux_symm, map_add, LinearMap.add_apply]
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map_smul' c x := by
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refine LinearMap.ext (fun z => ?_)
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simp only [euclidInnerAux_symm, LinearMapClass.map_smul, smul_eq_mul, RingHom.id_apply,
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LinearMap.smul_apply]
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/-- The type which assocaites to each ege a `1`-dimensional vector space.
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Corresponding to that spanned by its total outflowing momentum. -/
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def EdgeMomenta : Type := F.𝓔 → ℝ
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instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
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instance : Module ℝ F.EdgeMomenta := Pi.module _ _ _
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/-- The type which assocaites to each ege a `1`-dimensional vector space.
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Corresponding to that spanned by its total inflowing momentum. -/
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def VertexMomenta : Type := F.𝓥 → ℝ
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instance : AddCommGroup F.VertexMomenta := Pi.addCommGroup
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instance : Module ℝ F.VertexMomenta := Pi.module _ _ _
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/-- The map from `Fin 2` to `Type` landing on `EdgeMomenta` and `VertexMomenta`. -/
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def EdgeVertexMomentaMap : Fin 2 → Type := fun i =>
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match i with
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| 0 => F.EdgeMomenta
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| 1 => F.VertexMomenta
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instance (i : Fin 2) : AddCommGroup (EdgeVertexMomentaMap F i) :=
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match i with
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| 0 => instAddCommGroupEdgeMomenta F
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| 1 => instAddCommGroupVertexMomenta F
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instance (i : Fin 2) : Module ℝ (EdgeVertexMomentaMap F i) :=
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match i with
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| 0 => instModuleRealEdgeMomenta F
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| 1 => instModuleRealVertexMomenta F
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/-- The direct sum of `EdgeMomenta` and `VertexMomenta`.-/
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def EdgeVertexMomenta : Type := DirectSum (Fin 2) (EdgeVertexMomentaMap F)
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instance : AddCommGroup F.EdgeVertexMomenta := DirectSum.instAddCommGroup _
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instance : Module ℝ F.EdgeVertexMomenta := DirectSum.instModule
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/-!
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## Linear maps between the vector spaces.
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We define various maps into `F.HalfEdgeMomenta`.
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In particular, we define a map from `F.EdgeVertexMomenta` to `F.HalfEdgeMomenta`. This
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map represents the space orthogonal (with respect to the standard Euclidean inner product)
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to the allowed momenta of half-edges (up-to an offset determined by the
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external momenta).
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The number of loops of a diagram is defined as the number of half-edges minus
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the rank of this matrix.
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-/
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/-- The linear map from `F.EdgeMomenta` to `F.HalfEdgeMomenta` induced by
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the map `F.𝓱𝓔To𝓔.hom`. -/
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def edgeToHalfEdgeMomenta : F.EdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta where
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toFun x := x ∘ F.𝓱𝓔To𝓔.hom
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map_add' _ _ := by rfl
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map_smul' _ _ := by rfl
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/-- The linear map from `F.VertexMomenta` to `F.HalfEdgeMomenta` induced by
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the map `F.𝓱𝓔To𝓥.hom`. -/
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def vertexToHalfEdgeMomenta : F.VertexMomenta →ₗ[ℝ] F.HalfEdgeMomenta where
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toFun x := x ∘ F.𝓱𝓔To𝓥.hom
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map_add' _ _ := rfl
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map_smul' _ _ := rfl
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/-- The linear map from `F.EdgeVertexMomenta` to `F.HalfEdgeMomenta` induced by
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`F.edgeToHalfEdgeMomenta` and `F.vertexToHalfEdgeMomenta`. -/
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def edgeVertexToHalfEdgeMomenta : F.EdgeVertexMomenta →ₗ[ℝ] F.HalfEdgeMomenta :=
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DirectSum.toModule ℝ (Fin 2) F.HalfEdgeMomenta
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(fun i => match i with | 0 => F.edgeToHalfEdgeMomenta | 1 => F.vertexToHalfEdgeMomenta)
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/-!
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## Submodules
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We define submodules of `F.HalfEdgeMomenta` which correspond to
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the orthogonal space to allowed momenta (up-to an offset), and the space of
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allowed momenta.
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-/
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/-- The submodule of `F.HalfEdgeMomenta` corresponding to the range of
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`F.edgeVertexToHalfEdgeMomenta`. -/
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def orthogHalfEdgeMomenta : Submodule ℝ F.HalfEdgeMomenta :=
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LinearMap.range F.edgeVertexToHalfEdgeMomenta
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/-- The submodule of `F.HalfEdgeMomenta` corresponding to the allowed momenta. -/
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def allowedHalfEdgeMomenta : Submodule ℝ F.HalfEdgeMomenta :=
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Submodule.orthogonalBilin F.orthogHalfEdgeMomenta F.euclidInner
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/-!
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## Number of loops
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We define the number of loops of a Feynman diagram as the dimension of the
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allowed space of half-edge momenta.
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-/
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/-- The number of loops of a Feynman diagram. Defined as the dimension
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of the space of allowed Half-loop momenta. -/
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noncomputable def numberOfLoops : ℕ := FiniteDimensional.finrank ℝ F.allowedHalfEdgeMomenta
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end FeynmanDiagram
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