Merge branch 'master' into SpaceTime/LorentzGroup
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jstoobysmith
commit
7107ba8b80
16 changed files with 36 additions and 34 deletions
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name: continuous integration
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@ -16,7 +16,7 @@ It defines a module structure on the charges, and the solutions to the linear AC
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## TODO
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- Derive ACC systems from gauge algebras and fermionic representations.
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- Relate ACCSystems to algebraic varities.
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- Relate ACCSystems to algebraic varieties.
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-/
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@ -86,7 +86,7 @@ lemma LinSols.ext {χ : ACCSystemLinear} {S T : χ.LinSols} (h : S.val = T.val)
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cases' S
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simp_all only
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/-- An instance providng the operations and properties for `LinSols` to form an
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/-- An instance providing the operations and properties for `LinSols` to form an
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addative commutative monoid. -/
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@[simps!]
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instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
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@ -121,7 +121,7 @@ instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
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apply LinSols.ext
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exact χ.chargesAddCommMonoid.nsmul_succ _ _
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/-- An instance providng the operations and properties for `LinSols` to form an
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/-- An instance providing the operations and properties for `LinSols` to form an
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module over `ℚ`. -/
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@[simps!]
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instance linSolsModule (χ : ACCSystemLinear) : Module ℚ χ.LinSols where
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@ -149,7 +149,7 @@ instance linSolsModule (χ : ACCSystemLinear) : Module ℚ χ.LinSols where
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exact χ.chargesModule.add_smul _ _ _
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/-- An instance providing the operations and properties for `LinSols` to form an
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an addative community. -/
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an additive community. -/
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instance linSolsAddCommGroup (χ : ACCSystemLinear) : AddCommGroup χ.LinSols :=
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Module.addCommMonoidToAddCommGroup ℚ
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@ -271,7 +271,7 @@ structure Hom (χ η : ACCSystem) where
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charges : χ.charges →ₗ[ℚ] η.charges
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/-- The map between solutions. -/
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anomalyFree : χ.Sols → η.Sols
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/-- The condition that the map commutes with the relevent inclusions. -/
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/-- The condition that the map commutes with the relevant inclusions. -/
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commute : charges ∘ χ.solsIncl = η.solsIncl ∘ anomalyFree
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/-- The definition of composition between two ACCSystems. -/
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@ -13,7 +13,7 @@ under which the anomaly equations are invariant.
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From this we define
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- The representation acting on the vector space of solutions to the linear ACCs.
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- The group action acting on solutions to the linera + quadratic equations.
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- The group action acting on solutions to the linear + quadratic equations.
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- The group action acting on solutions to the anomaly cancellation conditions.
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-/
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@ -34,7 +34,7 @@ def MSSMSpecies : ACCSystemCharges := ACCSystemChargesMk 3
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namespace MSSMCharges
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/-- An equivalence between `MSSMCharges.charges` and the space of maps
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`(Fin 18 ⊕ Fin 2 → ℚ)`. The first 18 factors corresponds to the SM fermions, whils the last two
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`(Fin 18 ⊕ Fin 2 → ℚ)`. The first 18 factors corresponds to the SM fermions, while the last two
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are the higgsions. -/
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@[simps!]
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def toSMPlusH : MSSMCharges.charges ≃ (Fin 18 ⊕ Fin 2 → ℚ) :=
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@ -173,7 +173,7 @@ lemma accGrav_ext {S T : MSSMCharges.charges}
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rw [hd, hu]
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rfl
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/-- The anomaly cancelation condition for SU(2) anomaly. -/
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/-- The anomaly cancellation condition for SU(2) anomaly. -/
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@[simp]
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def accSU2 : MSSMCharges.charges →ₗ[ℚ] ℚ where
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toFun S := ∑ i, (3 * Q S i + L S i) + Hd S + Hu S
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@ -10,7 +10,7 @@ import Mathlib.Algebra.BigOperators.Fin
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/-!
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# Pure U(1) ACC system.
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We define the anomaly cancellation conditions for a pure U(1) gague theory with `n` fermions.
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We define the anomaly cancellation conditions for a pure U(1) gauge theory with `n` fermions.
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-/
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universe v u
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@ -24,7 +24,7 @@ variable {n : ℕ}
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/-- The condition for two rationals to have the same square (equivalent to same abs). -/
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def constAbsProp : ℚ × ℚ → Prop := fun s => s.1^2 = s.2^2
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/-- The condition on a charge assigment `S` to have constant absolute value among charges. -/
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/-- The condition on a charge assignment `S` to have constant absolute value among charges. -/
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@[simp]
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def constAbs (S : (PureU1 n).charges) : Prop := ∀ i j, (S i) ^ 2 = (S j) ^ 2
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@ -137,7 +137,7 @@ lemma boundary_accGrav'' (k : Fin n) (hk : boundary S k) :
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rw [boundary_castSucc hS hk, boundary_succ hS hk]
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ring
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/-- We say a `S ∈ charges` has a boundry if there exists a `k ∈ Fin n` which is a boundary. -/
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/-- We say a `S ∈ charges` has a boundary if there exists a `k ∈ Fin n` which is a boundary. -/
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@[simp]
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def hasBoundary (S : (PureU1 n.succ).charges) : Prop :=
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∃ (k : Fin n), boundary S k
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@ -10,7 +10,7 @@ import Mathlib.Logic.Equiv.Fin
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/-!
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# Basis of `LinSols` in the even case
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We give a basis of `LinSols` in the even case. This basis has the special propoerty
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We give a basis of `LinSols` in the even case. This basis has the special property
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that splits into two planes on which every point is a solution to the ACCs.
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-/
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universe v u
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@ -15,10 +15,10 @@ import Mathlib.Tactic.Polyrith
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# Line In Cubic Even case
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We say that a linear solution satisfies the `lineInCubic` property
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if the line through that point and through the two different planes formed by the baisis of
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if the line through that point and through the two different planes formed by the basis of
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`LinSols` lies in the cubic.
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We show that for a solution all its permutations satsfiy this property, then there exists
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We show that for a solution all its permutations satisfy this property, then there exists
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a permutation for which it lies in the plane spanned by the first part of the basis.
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The main reference for this file is:
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@ -34,7 +34,7 @@ open BigOperators
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variable {n : ℕ}
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open VectorLikeEvenPlane
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/-- A property on `LinSols`, statified if every point on the line between the two planes
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/-- A property on `LinSols`, satisfied if every point on the line between the two planes
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in the basis through that point is in the cubic. -/
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def lineInCubic (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
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∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
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@ -16,7 +16,7 @@ import Mathlib.Tactic.Polyrith
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Given maps `g : Fin n.succ → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
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equations. We show that every solution can be got in this way, up to permutation, unless it, up to
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permutaiton, lives in the plane spanned by the firt part of the basis vector.
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permutation, lives in the plane spanned by the first part of the basis vector.
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The main reference is:
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@ -32,7 +32,7 @@ open BigOperators
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variable {n : ℕ}
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open VectorLikeEvenPlane
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/-- Given coefficents `g` of a point in `P` and `f` of a point in `P!`, and a rational, we get a
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/-- Given coefficients `g` of a point in `P` and `f` of a point in `P!`, and a rational, we get a
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rational `a ∈ ℚ`, we get a
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point in `(PureU1 (2 * n.succ)).AnomalyFreeLinear`, which we will later show extends to an anomaly
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free point. -/
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@ -7,7 +7,7 @@ import HepLean.AnomalyCancellation.SM.Basic
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/-!
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# Anomaly Cancellation in the Standard Model without Gravity
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This file defines the system of anaomaly equations for the SM without RHN, and
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This file defines the system of anomaly equations for the SM without RHN, and
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without the gravitational ACC.
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-/
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@ -12,7 +12,7 @@ import HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization
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The main result of this file is the conclusion of this paper:
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https://arxiv.org/abs/1907.00514
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That eveery solution to the ACCs without gravity satisfies for free the gravitational anomaly.
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That every solution to the ACCs without gravity satisfies for free the gravitational anomaly.
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-/
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universe v u
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@ -46,7 +46,7 @@ lemma ext {S T : linearParameters} (hQ : S.Q' = T.Q') (hY : S.Y = T.Y) (hE : S.E
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cases' S
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simp_all only
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/-- The map from the linear paramaters to elements of `(SMNoGrav 1).charges`. -/
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/-- The map from the linear parameters to elements of `(SMNoGrav 1).charges`. -/
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@[simp]
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def asCharges (S : linearParameters) : (SMNoGrav 1).charges := fun i =>
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match i with
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@ -11,8 +11,8 @@ This file defines the Gamma matrices.
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## TODO
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- Prove that the algebra generated by the gamma matrices is ismorphic to the
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Clifford algebra assocaited with spacetime.
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- Prove that the algebra generated by the gamma matrices is isomorphic to the
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Clifford algebra associated with spacetime.
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- Include relations for gamma matrices.
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-/
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@ -113,7 +113,7 @@ def lorentzGroup : Subgroup (GL (Fin 4) ℝ) where
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instance : TopologicalGroup lorentzGroup :=
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Subgroup.instTopologicalGroupSubtypeMem lorentzGroup
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/-- The lift of a matrix perserving `ηLin` to a Lorentz Group element. -/
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/-- The lift of a matrix preserving `ηLin` to a Lorentz Group element. -/
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def PreservesηLin.liftLor {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) :
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lorentzGroup := ⟨liftGL h, h⟩
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@ -127,24 +127,24 @@ def transpose (Λ : lorentzGroup) : lorentzGroup :=
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PreservesηLin.liftLor ((PreservesηLin.iff_transpose Λ.1).mp Λ.2)
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/-- The continuous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` whose kernal is
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/-- The continuous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` whose kernel is
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the Lorentz group. -/
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def kernalMap : C(GL (Fin 4) ℝ, Matrix (Fin 4) (Fin 4) ℝ) where
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def kernelMap : C(GL (Fin 4) ℝ, Matrix (Fin 4) (Fin 4) ℝ) where
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toFun Λ := η * Λ.1ᵀ * η * Λ.1
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continuous_toFun := by
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apply Continuous.mul _ Units.continuous_val
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apply Continuous.mul _ continuous_const
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exact Continuous.mul continuous_const (Continuous.matrix_transpose (Units.continuous_val))
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lemma kernalMap_kernal_eq_lorentzGroup : lorentzGroup = kernalMap ⁻¹' {1} := by
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lemma kernelMap_kernel_eq_lorentzGroup : lorentzGroup = kernelMap ⁻¹' {1} := by
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ext Λ
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erw [mem_iff Λ, PreservesηLin.iff_matrix]
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rfl
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/-- The Lorentz Group is a closed subset of `GL (Fin 4) ℝ`. -/
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theorem isClosed_of_GL4 : IsClosed (lorentzGroup : Set (GL (Fin 4) ℝ)) := by
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rw [kernalMap_kernal_eq_lorentzGroup]
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exact continuous_iff_isClosed.mp kernalMap.2 {1} isClosed_singleton
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rw [kernelMap_kernel_eq_lorentzGroup]
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exact continuous_iff_isClosed.mp kernelMap.2 {1} isClosed_singleton
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end lorentzGroup
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@ -42,7 +42,7 @@ abbrev higgsVec := EuclideanSpace ℂ (Fin 2)
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section higgsVec
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/-- The continous linear map from the vector space `higgsVec` to `(Fin 2 → ℂ)` acheived by
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/-- The continuous linear map from the vector space `higgsVec` to `(Fin 2 → ℂ)` achieved by
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casting vectors. -/
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def higgsVecToFin2ℂ : higgsVec →L[ℝ] (Fin 2 → ℂ) where
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toFun x := x
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@ -69,7 +69,7 @@ noncomputable def higgsRepUnitary : gaugeGroup →* unitaryGroup (Fin 2) ℂ whe
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map_one' := by
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simp only [Prod.snd_one, _root_.map_one, Prod.fst_one, mul_one]
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/-- An orthonomral basis of higgsVec. -/
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/-- An orthonormal basis of higgsVec. -/
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noncomputable def orthonormBasis : OrthonormalBasis (Fin 2) ℂ higgsVec :=
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EuclideanSpace.basisFun (Fin 2) ℂ
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@ -306,8 +306,8 @@ lemma IsMinOn_potential_iff_of_μSq_nonpos {μSq : ℝ} (hμSq : μSq ≤ 0) :
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exact potential_eq_bound_IsMinOn_of_μSq_nonpos hLam hμSq φ h
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end potentialProp
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/-- Given a Higgs vector, a rotation matrix which puts the fst component of the
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vector to zero, and the snd componenet to a real -/
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/-- Given a Higgs vector, a rotation matrix which puts the first component of the
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vector to zero, and the second component to a real -/
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def rotateMatrix (φ : higgsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
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![![φ 1 /‖φ‖ , - φ 0 /‖φ‖], ![conj (φ 0) / ‖φ‖ , conj (φ 1) / ‖φ‖] ]
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@ -353,8 +353,8 @@ lemma rotateMatrix_specialUnitary {φ : higgsVec} (hφ : φ ≠ 0) :
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(rotateMatrix φ) ∈ specialUnitaryGroup (Fin 2) ℂ :=
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mem_specialUnitaryGroup_iff.mpr ⟨rotateMatrix_unitary hφ, rotateMatrix_det hφ⟩
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/-- Given a Higgs vector, an element of the gauge group which puts the fst component of the
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vector to zero, and the snd componenet to a real -/
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/-- Given a Higgs vector, an element of the gauge group which puts the first component of the
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vector to zero, and the second component to a real -/
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def rotateGuageGroup {φ : higgsVec} (hφ : φ ≠ 0) : gaugeGroup :=
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⟨1, ⟨(rotateMatrix φ), rotateMatrix_specialUnitary hφ⟩, 1⟩
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