Merge branch 'master' into Workflows
This commit is contained in:
jstoobysmith
commit
7bb62fe9ad
23 changed files with 1002 additions and 488 deletions
|
|
@ -7,6 +7,17 @@ This assumes you have already installed Lean and setup HepLean.
|
|||
To add a result to HepLean:
|
||||
|
||||
- Create a new branch of HepLean.
|
||||
- Name the branch something which tells people what sort of things you are adding e.g. "StandardModel-HiggsPotential"
|
||||
- On pushing to a brach GitHub will run a number of continous integration checks on your code.
|
||||
- Once you are happy, and your branch has passed the continous integration checks, make a pull-request to the main branch of HepLean.
|
||||
- Name the branch something which tells people what sort of things you are adding e.g., "StandardModel-HiggsPotential"
|
||||
- On pushing to a branch GitHub will run a number of continuous integration checks on your code.
|
||||
- Once you are happy, and your branch has passed the continuous integration checks, make a pull-request to the main branch of HepLean.
|
||||
|
||||
## Naming commits
|
||||
|
||||
It is useful to prefix commits with one of the following.
|
||||
- `feat:` When you add one or more new lemma, definition, or theorems.
|
||||
- `refactor:` When you are improving the layout and structure of the code.
|
||||
- `fix:` When fixing a mistake in a definition or other Lean code.
|
||||
- `docs:` When adding a comment or updating documentation.
|
||||
- `style:` When adding e.g., white space, a semicolon etc. But does not change the overall
|
||||
structure of the code.
|
||||
- `chore:` Updating e.g., workflows.
|
||||
|
|
@ -56,7 +56,11 @@ import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic
|
|||
import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters
|
||||
import HepLean.SpaceTime.Basic
|
||||
import HepLean.SpaceTime.CliffordAlgebra
|
||||
import HepLean.SpaceTime.LorentzGroup
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
import HepLean.SpaceTime.LorentzGroup.Basic
|
||||
import HepLean.SpaceTime.LorentzGroup.Boosts
|
||||
import HepLean.SpaceTime.LorentzGroup.Orthochronous
|
||||
import HepLean.SpaceTime.LorentzGroup.Proper
|
||||
import HepLean.SpaceTime.Metric
|
||||
import HepLean.StandardModel.Basic
|
||||
import HepLean.StandardModel.HiggsBoson.Basic
|
||||
|
|
|
|||
|
|
@ -77,7 +77,7 @@ namespace ACCSystemLinear
|
|||
structure LinSols (χ : ACCSystemLinear) where
|
||||
/-- The underlying charge. -/
|
||||
val : χ.1.charges
|
||||
/-- The condition that the charge satifies the linear ACCs. -/
|
||||
/-- The condition that the charge satisfies the linear ACCs. -/
|
||||
linearSol : ∀ i : Fin χ.numberLinear, χ.linearACCs i val = 0
|
||||
|
||||
/-- Two solutions are equal if the underlying charges are equal. -/
|
||||
|
|
@ -172,7 +172,7 @@ namespace ACCSystemQuad
|
|||
|
||||
/-- The type of solutions to the linear and quadratic ACCs. -/
|
||||
structure QuadSols (χ : ACCSystemQuad) extends χ.LinSols where
|
||||
/-- The condition that the charge satifies the quadratic ACCs. -/
|
||||
/-- The condition that the charge satisfies the quadratic ACCs. -/
|
||||
quadSol : ∀ i : Fin χ.numberQuadratic, (χ.quadraticACCs i) val = 0
|
||||
|
||||
/-- Two `QuadSols` are equal if the underlying charges are equal. -/
|
||||
|
|
@ -225,7 +225,7 @@ namespace ACCSystem
|
|||
|
||||
/-- The type of solutions to the anomaly cancellation conditions. -/
|
||||
structure Sols (χ : ACCSystem) extends χ.QuadSols where
|
||||
/-- The condition that the charge satifies the cubic ACC. -/
|
||||
/-- The condition that the charge satisfies the cubic ACC. -/
|
||||
cubicSol : χ.cubicACC val = 0
|
||||
|
||||
/-- Two solutions are equal if the underlying charges are equal. -/
|
||||
|
|
|
|||
|
|
@ -206,7 +206,7 @@ lemma accSU2_ext {S T : MSSMCharges.charges}
|
|||
rw [hd, hu]
|
||||
rfl
|
||||
|
||||
/-- The anomaly cancelation condition for SU(3) anomaly. -/
|
||||
/-- The anomaly cancellation condition for SU(3) anomaly. -/
|
||||
@[simp]
|
||||
def accSU3 : MSSMCharges.charges →ₗ[ℚ] ℚ where
|
||||
toFun S := ∑ i, (2 * (Q S i) + (U S i) + (D S i))
|
||||
|
|
|
|||
|
|
@ -8,7 +8,7 @@ import Mathlib.Tactic.Polyrith
|
|||
/-!
|
||||
# Hypercharge in MSSM.
|
||||
|
||||
Relavent definitions for the MSSM hypercharge.
|
||||
Relevant definitions for the MSSM hypercharge.
|
||||
|
||||
-/
|
||||
|
||||
|
|
|
|||
|
|
@ -67,7 +67,7 @@ lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n.succ)).LinSols} (h : lineInCubic
|
|||
linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1) -
|
||||
(lineInCubic_expand h g f hS 1 2) / 6
|
||||
|
||||
/-- We say a `LinSol` satifies `lineInCubicPerm` if all its permutations satsify `lineInCubic`. -/
|
||||
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
|
||||
def lineInCubicPerm (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
|
||||
∀ (M : (FamilyPermutations (2 * n.succ)).group ),
|
||||
lineInCubic ((FamilyPermutations (2 * n.succ)).linSolRep M S)
|
||||
|
|
|
|||
|
|
@ -12,7 +12,7 @@ import Mathlib.RepresentationTheory.Basic
|
|||
/-!
|
||||
# Line in plane condition
|
||||
|
||||
We say a `LinSol` satifies the `line in plane` condition if for all distinct `i1`, `i2`, `i3` in
|
||||
We say a `LinSol` satisfies the `line in plane` condition if for all distinct `i1`, `i2`, `i3` in
|
||||
`Fin n`, we have
|
||||
`S i1 = S i2` or `S i1 = - S i2` or `2 S i3 + S i1 + S i2 = 0`.
|
||||
|
||||
|
|
@ -21,7 +21,7 @@ The main reference for this material is
|
|||
|
||||
- https://arxiv.org/pdf/1912.04804.pdf
|
||||
|
||||
We will show that `n ≥ 4` the `line in plane` condition on solutions inplies the
|
||||
We will show that `n ≥ 4` the `line in plane` condition on solutions implies the
|
||||
`constAbs` condition.
|
||||
|
||||
-/
|
||||
|
|
|
|||
|
|
@ -10,7 +10,7 @@ import Mathlib.Logic.Equiv.Fin
|
|||
/-!
|
||||
# Basis of `LinSols` in the odd case
|
||||
|
||||
We give a basis of `LinSols` in the odd case. This basis has the special propoerty
|
||||
We give a basis of `LinSols` in the odd case. This basis has the special property
|
||||
that splits into two planes on which every point is a solution to the ACCs.
|
||||
-/
|
||||
universe v u
|
||||
|
|
|
|||
|
|
@ -16,10 +16,10 @@ import Mathlib.RepresentationTheory.Basic
|
|||
# Line In Cubic Odd case
|
||||
|
||||
We say that a linear solution satisfies the `lineInCubic` property
|
||||
if the line through that point and through the two different planes formed by the baisis of
|
||||
if the line through that point and through the two different planes formed by the basis of
|
||||
`LinSols` lies in the cubic.
|
||||
|
||||
We show that for a solution all its permutations satsfiy this property,
|
||||
We show that for a solution all its permutations satisfy this property,
|
||||
then the charge must be zero.
|
||||
|
||||
The main reference for this file is:
|
||||
|
|
@ -34,7 +34,7 @@ open BigOperators
|
|||
variable {n : ℕ}
|
||||
open VectorLikeOddPlane
|
||||
|
||||
/-- A property on `LinSols`, statified if every point on the line between the two planes
|
||||
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
|
||||
in the basis through that point is in the cubic. -/
|
||||
def lineInCubic (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
|
||||
∀ (g f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
|
||||
|
|
@ -64,7 +64,7 @@ lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S
|
|||
|
||||
|
||||
|
||||
/-- We say a `LinSol` satifies `lineInCubicPerm` if all its permutations satsify `lineInCubic`. -/
|
||||
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
|
||||
def lineInCubicPerm (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
|
||||
∀ (M : (FamilyPermutations (2 * n + 1)).group ),
|
||||
lineInCubic ((FamilyPermutations (2 * n + 1)).linSolRep M S)
|
||||
|
|
|
|||
|
|
@ -13,7 +13,7 @@ import Mathlib.Logic.Equiv.Fin
|
|||
/-!
|
||||
# Anomaly cancellation conditions for n family SM.
|
||||
|
||||
We define the ACC system for the Standard Model with`n`-famlies and no RHN.
|
||||
We define the ACC system for the Standard Model with`n`-families and no RHN.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -21,7 +21,7 @@ universe v u
|
|||
open Nat
|
||||
open BigOperators
|
||||
|
||||
/-- Aassociate to each (including RHN) SM fermion a set of charges-/
|
||||
/-- Associate to each (including RHN) SM fermion a set of charges-/
|
||||
@[simps!]
|
||||
def SMCharges (n : ℕ) : ACCSystemCharges := ACCSystemChargesMk (5 * n)
|
||||
|
||||
|
|
|
|||
|
|
@ -7,7 +7,7 @@ import HepLean.AnomalyCancellation.SM.Basic
|
|||
/-!
|
||||
# Family maps for the Standard Model ACCs
|
||||
|
||||
We define the a series of maps between the charges for different numbers of famlies.
|
||||
We define the a series of maps between the charges for different numbers of families.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -84,7 +84,7 @@ other charges zero. -/
|
|||
def familyEmbedding (m n : ℕ) : (SMCharges m).charges →ₗ[ℚ] (SMCharges n).charges :=
|
||||
chargesMapOfSpeciesMap (speciesEmbed m n)
|
||||
|
||||
/-- For species, the embeddding of the `1`-family charges into the `n`-family charges in
|
||||
/-- For species, the embedding of the `1`-family charges into the `n`-family charges in
|
||||
a universal manor. -/
|
||||
@[simps!]
|
||||
def speciesFamilyUniversial (n : ℕ) :
|
||||
|
|
@ -98,7 +98,7 @@ def speciesFamilyUniversial (n : ℕ) :
|
|||
simp [HSMul.hSMul]
|
||||
--rw [show Rat.cast a = a from rfl]
|
||||
|
||||
/-- The embeddding of the `1`-family charges into the `n`-family charges in
|
||||
/-- The embedding of the `1`-family charges into the `n`-family charges in
|
||||
a universal manor. -/
|
||||
def familyUniversal ( n : ℕ) : (SMCharges 1).charges →ₗ[ℚ] (SMCharges n).charges :=
|
||||
chargesMapOfSpeciesMap (speciesFamilyUniversial n)
|
||||
|
|
|
|||
|
|
@ -12,7 +12,7 @@ import HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization
|
|||
The main result of this file is the conclusion of this paper:
|
||||
https://arxiv.org/abs/1907.00514
|
||||
|
||||
That eveery solution to the ACCs without gravity satifies for free the gravitational anomaly.
|
||||
That eveery solution to the ACCs without gravity satisfies for free the gravitational anomaly.
|
||||
-/
|
||||
|
||||
universe v u
|
||||
|
|
@ -67,7 +67,7 @@ lemma accGrav_Q_neq_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) ≠ 0
|
|||
rw [← hS']
|
||||
exact S'.grav_of_cubic hC FLTThree
|
||||
|
||||
/-- Any solution to the ACCs without gravity satifies the gravitational ACC. -/
|
||||
/-- Any solution to the ACCs without gravity satisfies the gravitational ACC. -/
|
||||
theorem accGravSatisfied {S : (SMNoGrav 1).Sols} (FLTThree : FermatLastTheoremWith ℚ 3) :
|
||||
accGrav S.val = 0 := by
|
||||
by_cases hQ : Q S.val (0 : Fin 1)= 0
|
||||
|
|
|
|||
|
|
@ -7,7 +7,7 @@ import HepLean.AnomalyCancellation.SMNu.Basic
|
|||
/-!
|
||||
# Family maps for the Standard Model for RHN ACCs
|
||||
|
||||
We define the a series of maps between the charges for different numbers of famlies.
|
||||
We define the a series of maps between the charges for different numbers of families.
|
||||
|
||||
-/
|
||||
|
||||
|
|
@ -89,7 +89,7 @@ other charges zero. -/
|
|||
def familyEmbedding (m n : ℕ) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
|
||||
chargesMapOfSpeciesMap (speciesEmbed m n)
|
||||
|
||||
/-- For species, the embeddding of the `1`-family charges into the `n`-family charges in
|
||||
/-- For species, the embedding of the `1`-family charges into the `n`-family charges in
|
||||
a universal manor. -/
|
||||
@[simps!]
|
||||
def speciesFamilyUniversial (n : ℕ) :
|
||||
|
|
@ -103,7 +103,7 @@ def speciesFamilyUniversial (n : ℕ) :
|
|||
simp [HSMul.hSMul]
|
||||
-- rw [show Rat.cast a = a from rfl]
|
||||
|
||||
/-- The embeddding of the `1`-family charges into the `n`-family charges in
|
||||
/-- The embedding of the `1`-family charges into the `n`-family charges in
|
||||
a universal manor. -/
|
||||
def familyUniversal (n : ℕ) : (SMνCharges 1).charges →ₗ[ℚ] (SMνCharges n).charges :=
|
||||
chargesMapOfSpeciesMap (speciesFamilyUniversial n)
|
||||
|
|
|
|||
|
|
@ -9,7 +9,7 @@ import Mathlib.Tactic.Polyrith
|
|||
/-!
|
||||
# The CKM Matrix
|
||||
|
||||
The definition of the type of CKM matries as unitary $3×3$-matries.
|
||||
The definition of the type of CKM matrices as unitary $3×3$-matrices.
|
||||
|
||||
An equivalence relation on CKM matrices is defined, where two matrices are equivalent if they are
|
||||
related by phase shifts.
|
||||
|
|
|
|||
|
|
@ -5,6 +5,8 @@ Authors: Joseph Tooby-Smith
|
|||
-/
|
||||
import Mathlib.Data.Complex.Exponential
|
||||
import Mathlib.Geometry.Manifold.VectorBundle.Basic
|
||||
import Mathlib.Analysis.InnerProductSpace.Adjoint
|
||||
|
||||
/-!
|
||||
# Space time
|
||||
|
||||
|
|
@ -36,6 +38,9 @@ open Matrix
|
|||
open Complex
|
||||
open ComplexConjugate
|
||||
|
||||
/-- The space part of spacetime. -/
|
||||
@[simp]
|
||||
def space (x : spaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
|
||||
|
||||
/-- The structure of a smooth manifold on spacetime. -/
|
||||
def asSmoothManifold : ModelWithCorners ℝ spaceTime spaceTime := 𝓘(ℝ, spaceTime)
|
||||
|
|
|
|||
249
HepLean/SpaceTime/FourVelocity.lean
Normal file
249
HepLean/SpaceTime/FourVelocity.lean
Normal file
|
|
@ -0,0 +1,249 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.Metric
|
||||
/-!
|
||||
# Four velocity
|
||||
|
||||
We define
|
||||
|
||||
- `PreFourVelocity` : as a space-time element with norm 1.
|
||||
- `FourVelocity` : as a space-time element with norm 1 and future pointing.
|
||||
|
||||
-/
|
||||
open spaceTime
|
||||
|
||||
/-- A `spaceTime` vector for which `ηLin x x = 1`. -/
|
||||
@[simp]
|
||||
def PreFourVelocity : Set spaceTime :=
|
||||
fun x => ηLin x x = 1
|
||||
|
||||
instance : TopologicalSpace PreFourVelocity := by
|
||||
exact instTopologicalSpaceSubtype
|
||||
|
||||
namespace PreFourVelocity
|
||||
lemma mem_PreFourVelocity_iff {x : spaceTime} : x ∈ PreFourVelocity ↔ ηLin x x = 1 := by
|
||||
rfl
|
||||
|
||||
/-- The negative of a `PreFourVelocity` as a `PreFourVelocity`. -/
|
||||
def neg (v : PreFourVelocity) : PreFourVelocity := ⟨-v, by
|
||||
rw [mem_PreFourVelocity_iff]
|
||||
simp only [map_neg, LinearMap.neg_apply, neg_neg]
|
||||
exact v.2 ⟩
|
||||
|
||||
lemma zero_sq (v : PreFourVelocity) : v.1 0 ^ 2 = 1 + ‖v.1.space‖ ^ 2 := by
|
||||
rw [time_elm_sq_of_ηLin, v.2]
|
||||
|
||||
lemma zero_abs_ge_one (v : PreFourVelocity) : 1 ≤ |v.1 0| := by
|
||||
have h1 := ηLin_leq_time_sq v.1
|
||||
rw [v.2] at h1
|
||||
exact (one_le_sq_iff_one_le_abs _).mp h1
|
||||
|
||||
|
||||
lemma zero_abs_gt_norm_space (v : PreFourVelocity) : ‖v.1.space‖ < |v.1 0| := by
|
||||
rw [(abs_norm _).symm, ← @sq_lt_sq, zero_sq]
|
||||
simp only [space, Fin.isValue, lt_add_iff_pos_left, zero_lt_one]
|
||||
|
||||
lemma zero_abs_ge_norm_space (v : PreFourVelocity) : ‖v.1.space‖ ≤ |v.1 0| :=
|
||||
le_of_lt (zero_abs_gt_norm_space v)
|
||||
|
||||
lemma zero_le_minus_one_or_ge_one (v : PreFourVelocity) : v.1 0 ≤ -1 ∨ 1 ≤ v.1 0 :=
|
||||
le_abs'.mp (zero_abs_ge_one v)
|
||||
|
||||
lemma zero_nonpos_iff (v : PreFourVelocity) : v.1 0 ≤ 0 ↔ v.1 0 ≤ - 1 := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
cases' zero_le_minus_one_or_ge_one v with h1 h1
|
||||
exact h1
|
||||
linarith
|
||||
· intro h
|
||||
linarith
|
||||
|
||||
lemma zero_nonneg_iff (v : PreFourVelocity) : 0 ≤ v.1 0 ↔ 1 ≤ v.1 0 := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
cases' zero_le_minus_one_or_ge_one v with h1 h1
|
||||
linarith
|
||||
exact h1
|
||||
· intro h
|
||||
linarith
|
||||
|
||||
/-- A `PreFourVelocity` is a `FourVelocity` if its time componenet is non-negative. -/
|
||||
def IsFourVelocity (v : PreFourVelocity) : Prop := 0 ≤ v.1 0
|
||||
|
||||
|
||||
lemma IsFourVelocity_abs_zero {v : PreFourVelocity} (hv : IsFourVelocity v) :
|
||||
|v.1 0| = v.1 0 := abs_of_nonneg hv
|
||||
|
||||
lemma not_IsFourVelocity_iff (v : PreFourVelocity) : ¬ IsFourVelocity v ↔ v.1 0 ≤ 0 := by
|
||||
rw [IsFourVelocity, not_le]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
exact le_of_lt h
|
||||
intro h
|
||||
have h1 := (zero_nonpos_iff v).mp h
|
||||
linarith
|
||||
|
||||
lemma not_IsFourVelocity_iff_neg {v : PreFourVelocity} : ¬ IsFourVelocity v ↔
|
||||
IsFourVelocity (neg v):= by
|
||||
rw [not_IsFourVelocity_iff, IsFourVelocity]
|
||||
simp only [Fin.isValue, neg]
|
||||
change _ ↔ 0 ≤ - v.1 0
|
||||
simp
|
||||
|
||||
lemma zero_abs_mul_sub_norm_space (v v' : PreFourVelocity) :
|
||||
0 ≤ |v.1 0| * |v'.1 0| - ‖v.1.space‖ * ‖v'.1.space‖ := by
|
||||
apply sub_nonneg.mpr
|
||||
apply mul_le_mul (zero_abs_ge_norm_space v) (zero_abs_ge_norm_space v') ?_ ?_
|
||||
exact norm_nonneg v'.1.space
|
||||
exact abs_nonneg (v.1 0)
|
||||
|
||||
|
||||
lemma euclid_norm_IsFourVelocity_IsFourVelocity {v v' : PreFourVelocity}
|
||||
(h : IsFourVelocity v) (h' : IsFourVelocity v') :
|
||||
0 ≤ (v.1 0) * (v'.1 0) + ⟪v.1.space, v'.1.space⟫_ℝ := by
|
||||
apply le_trans (zero_abs_mul_sub_norm_space v v')
|
||||
rw [IsFourVelocity_abs_zero h, IsFourVelocity_abs_zero h', sub_eq_add_neg]
|
||||
apply (add_le_add_iff_left _).mpr
|
||||
rw [@neg_le]
|
||||
apply le_trans (neg_le_abs _ : _ ≤ |⟪(v.1).space, (v'.1).space⟫_ℝ|) ?_
|
||||
exact abs_real_inner_le_norm (v.1).space (v'.1).space
|
||||
|
||||
lemma euclid_norm_not_IsFourVelocity_not_IsFourVelocity {v v' : PreFourVelocity}
|
||||
(h : ¬ IsFourVelocity v) (h' : ¬ IsFourVelocity v') :
|
||||
0 ≤ (v.1 0) * (v'.1 0) + ⟪v.1.space, v'.1.space⟫_ℝ := by
|
||||
have h1 := euclid_norm_IsFourVelocity_IsFourVelocity (not_IsFourVelocity_iff_neg.mp h)
|
||||
(not_IsFourVelocity_iff_neg.mp h')
|
||||
apply le_of_le_of_eq h1 ?_
|
||||
simp [neg, Fin.sum_univ_three, neg]
|
||||
repeat rw [show (- v.1) _ = - v.1 _ from rfl]
|
||||
repeat rw [show (- v'.1) _ = - v'.1 _ from rfl]
|
||||
ring
|
||||
|
||||
lemma euclid_norm_IsFourVelocity_not_IsFourVelocity {v v' : PreFourVelocity}
|
||||
(h : IsFourVelocity v) (h' : ¬ IsFourVelocity v') :
|
||||
(v.1 0) * (v'.1 0) + ⟪v.1.space, v'.1.space⟫_ℝ ≤ 0 := by
|
||||
rw [show (0 : ℝ) = - 0 by simp, le_neg]
|
||||
have h1 := euclid_norm_IsFourVelocity_IsFourVelocity h (not_IsFourVelocity_iff_neg.mp h')
|
||||
apply le_of_le_of_eq h1 ?_
|
||||
simp [neg, Fin.sum_univ_three, neg]
|
||||
repeat rw [show (- v'.1) _ = - v'.1 _ from rfl]
|
||||
ring
|
||||
|
||||
lemma euclid_norm_not_IsFourVelocity_IsFourVelocity {v v' : PreFourVelocity}
|
||||
(h : ¬ IsFourVelocity v) (h' : IsFourVelocity v') :
|
||||
(v.1 0) * (v'.1 0) + ⟪v.1.space, v'.1.space⟫_ℝ ≤ 0 := by
|
||||
rw [show (0 : ℝ) = - 0 by simp, le_neg]
|
||||
have h1 := euclid_norm_IsFourVelocity_IsFourVelocity h' (not_IsFourVelocity_iff_neg.mp h)
|
||||
apply le_of_le_of_eq h1 ?_
|
||||
simp [neg, Fin.sum_univ_three, neg]
|
||||
repeat rw [show (- v.1) _ = - v.1 _ from rfl]
|
||||
ring
|
||||
|
||||
end PreFourVelocity
|
||||
|
||||
/-- The set of `PreFourVelocity`'s which are four velocities. -/
|
||||
def FourVelocity : Set PreFourVelocity :=
|
||||
fun x => PreFourVelocity.IsFourVelocity x
|
||||
|
||||
instance : TopologicalSpace FourVelocity := instTopologicalSpaceSubtype
|
||||
|
||||
namespace FourVelocity
|
||||
open PreFourVelocity
|
||||
|
||||
lemma mem_FourVelocity_iff {x : PreFourVelocity} : x ∈ FourVelocity ↔ 0 ≤ x.1 0 := by
|
||||
rfl
|
||||
|
||||
lemma time_comp (x : FourVelocity) : x.1.1 0 = √(1 + ‖x.1.1.space‖ ^ 2) := by
|
||||
symm
|
||||
rw [Real.sqrt_eq_cases]
|
||||
refine Or.inl (And.intro ?_ x.2)
|
||||
rw [← PreFourVelocity.zero_sq x.1, sq]
|
||||
|
||||
/-- The `FourVelocity` which has all space components zero. -/
|
||||
def zero : FourVelocity := ⟨⟨![1, 0, 0, 0], by
|
||||
rw [mem_PreFourVelocity_iff, ηLin_expand]; simp⟩, by
|
||||
rw [mem_FourVelocity_iff]; simp⟩
|
||||
|
||||
|
||||
/-- A continuous path from the zero `FourVelocity` to any other. -/
|
||||
noncomputable def pathFromZero (u : FourVelocity) : Path zero u where
|
||||
toFun t := ⟨
|
||||
⟨![√(1 + t ^ 2 * ‖u.1.1.space‖ ^ 2), t * u.1.1 1, t * u.1.1 2, t * u.1.1 3],
|
||||
by
|
||||
rw [mem_PreFourVelocity_iff, ηLin_expand]
|
||||
simp only [space, Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,
|
||||
Matrix.cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, Matrix.tail_cons,
|
||||
Matrix.cons_val_three]
|
||||
rw [Real.mul_self_sqrt, ← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp only [Fin.isValue, Matrix.cons_val_zero, RCLike.inner_apply, conj_trivial,
|
||||
Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_two, Nat.succ_eq_add_one,
|
||||
Nat.reduceAdd, Matrix.tail_cons]
|
||||
ring
|
||||
refine Right.add_nonneg (zero_le_one' ℝ) $
|
||||
mul_nonneg (sq_nonneg _) (sq_nonneg _) ⟩,
|
||||
by
|
||||
rw [mem_FourVelocity_iff]
|
||||
exact Real.sqrt_nonneg _⟩
|
||||
continuous_toFun := by
|
||||
refine Continuous.subtype_mk ?_ _
|
||||
refine Continuous.subtype_mk ?_ _
|
||||
apply (continuous_pi_iff).mpr
|
||||
intro i
|
||||
fin_cases i
|
||||
<;> continuity
|
||||
source' := by
|
||||
simp only [Set.Icc.coe_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, space,
|
||||
Fin.isValue, zero_mul, add_zero, Real.sqrt_one]
|
||||
rfl
|
||||
target' := by
|
||||
simp only [Set.Icc.coe_one, one_pow, space, Fin.isValue, one_mul]
|
||||
refine SetCoe.ext ?_
|
||||
refine SetCoe.ext ?_
|
||||
funext i
|
||||
fin_cases i
|
||||
simp only [Fin.isValue, Fin.zero_eta, Matrix.cons_val_zero]
|
||||
exact (time_comp _).symm
|
||||
all_goals rfl
|
||||
|
||||
lemma isPathConnected_FourVelocity : IsPathConnected (@Set.univ FourVelocity) := by
|
||||
use zero
|
||||
apply And.intro trivial ?_
|
||||
intro y _
|
||||
use pathFromZero y
|
||||
simp only [Set.mem_univ, implies_true]
|
||||
|
||||
lemma η_pos (u v : FourVelocity) : 0 < ηLin u v := by
|
||||
refine lt_of_lt_of_le ?_ (ηLin_ge_sub_norm u v)
|
||||
apply sub_pos.mpr
|
||||
refine mul_lt_mul_of_nonneg_of_pos ?_ ?_ ?_ ?_
|
||||
simpa [IsFourVelocity_abs_zero u.2] using zero_abs_gt_norm_space u.1
|
||||
simpa [IsFourVelocity_abs_zero v.2] using zero_abs_ge_norm_space v.1
|
||||
exact norm_nonneg u.1.1.space
|
||||
have h2 := (mem_FourVelocity_iff).mp v.2
|
||||
rw [zero_nonneg_iff] at h2
|
||||
linarith
|
||||
|
||||
lemma one_plus_ne_zero (u v : FourVelocity) : 1 + ηLin u v ≠ 0 := by
|
||||
linarith [η_pos u v]
|
||||
|
||||
lemma η_continuous (u : spaceTime) : Continuous (fun (a : FourVelocity) => ηLin u a) := by
|
||||
simp only [ηLin_expand]
|
||||
refine Continuous.add ?_ ?_
|
||||
refine Continuous.add ?_ ?_
|
||||
refine Continuous.add ?_ ?_
|
||||
refine Continuous.comp' (continuous_mul_left _) ?_
|
||||
apply (continuous_pi_iff).mp
|
||||
exact Isometry.continuous fun x1 => congrFun rfl
|
||||
all_goals apply Continuous.neg
|
||||
all_goals apply Continuous.comp' (continuous_mul_left _)
|
||||
all_goals apply (continuous_pi_iff).mp
|
||||
all_goals exact Isometry.continuous fun x1 => congrFun rfl
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
end FourVelocity
|
||||
|
|
@ -1,449 +0,0 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.Metric
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
/-!
|
||||
# The Lorentz Group
|
||||
|
||||
We define the Lorentz group.
|
||||
|
||||
## TODO
|
||||
|
||||
- Show that the Lorentz is a Lie group.
|
||||
- Prove that the restricted Lorentz group is equivalent to the connected component of the
|
||||
identity.
|
||||
- Define the continuous maps from `ℝ³` to `restrictedLorentzGroup` defining boosts.
|
||||
|
||||
## References
|
||||
|
||||
- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf
|
||||
- A proof that the restricted Lorentz group is connected can be found at:
|
||||
https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
|
||||
The proof involves the introduction of boosts.
|
||||
|
||||
-/
|
||||
|
||||
noncomputable section
|
||||
|
||||
namespace spaceTime
|
||||
|
||||
open Manifold
|
||||
open Matrix
|
||||
open Complex
|
||||
open ComplexConjugate
|
||||
|
||||
/-- The Lorentz group as a subgroup of the general linear group over the reals. -/
|
||||
def lorentzGroup : Subgroup (GeneralLinearGroup (Fin 4) ℝ) where
|
||||
carrier := {Λ | ∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y}
|
||||
mul_mem' {a b} := by
|
||||
intros ha hb x y
|
||||
simp only [Units.val_mul, mulVec_mulVec]
|
||||
rw [← mulVec_mulVec, ← mulVec_mulVec, ha, hb]
|
||||
one_mem' := by
|
||||
intros x y
|
||||
simp
|
||||
inv_mem' {a} := by
|
||||
intros ha x y
|
||||
simp only [coe_units_inv, ← ha ((a.1⁻¹) *ᵥ x) ((a.1⁻¹) *ᵥ y), mulVec_mulVec]
|
||||
have hx : (a.1 * (a.1)⁻¹) = 1 := by
|
||||
simp only [@Units.mul_eq_one_iff_inv_eq, coe_units_inv]
|
||||
simp [hx]
|
||||
|
||||
/-- The Lorentz group is a topological group with the subset topology. -/
|
||||
instance : TopologicalGroup lorentzGroup :=
|
||||
Subgroup.instTopologicalGroupSubtypeMem lorentzGroup
|
||||
|
||||
lemma mem_lorentzGroup_iff (Λ : GeneralLinearGroup (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ ∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y := by
|
||||
rfl
|
||||
|
||||
lemma mem_lorentzGroup_iff' (Λ : GeneralLinearGroup (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ ∀ (x y : spaceTime), ηLin (x) ((η * Λ.1ᵀ * η * Λ.1) *ᵥ y) = ηLin x y := by
|
||||
rw [mem_lorentzGroup_iff]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
intro x y
|
||||
have h1 := h x y
|
||||
rw [ηLin_mulVec_left, mulVec_mulVec] at h1
|
||||
exact h1
|
||||
intro h
|
||||
intro x y
|
||||
rw [ηLin_mulVec_left, mulVec_mulVec]
|
||||
exact h x y
|
||||
|
||||
lemma mem_lorentzGroup_iff'' (Λ : GL (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ η * Λ.1ᵀ * η * Λ.1 = 1 := by
|
||||
rw [mem_lorentzGroup_iff', ηLin_matrix_eq_identity_iff (η * Λ.1ᵀ * η * Λ.1)]
|
||||
apply Iff.intro
|
||||
· simp_all only [ηLin_apply_apply, implies_true, iff_true, one_mulVec]
|
||||
· simp_all only [ηLin_apply_apply, mulVec_mulVec, implies_true]
|
||||
|
||||
lemma mem_lorentzGroup_iff''' (Λ : GL (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ η * Λ.1ᵀ * η = Λ⁻¹ := by
|
||||
rw [mem_lorentzGroup_iff'']
|
||||
exact Units.mul_eq_one_iff_eq_inv
|
||||
|
||||
|
||||
lemma mem_lorentzGroup_iff_transpose (Λ : GL (Fin 4) ℝ) :
|
||||
Λ ∈ lorentzGroup ↔ ⟨transpose Λ, (transpose Λ)⁻¹, by
|
||||
rw [← transpose_nonsing_inv, ← transpose_mul, (coe_units_inv Λ).symm]
|
||||
rw [show ((Λ)⁻¹).1 * Λ.1 = (1 : Matrix (Fin 4) (Fin 4) ℝ) by rw [@Units.inv_mul_eq_one]]
|
||||
rw [@transpose_eq_one], by
|
||||
rw [← transpose_nonsing_inv, ← transpose_mul, (coe_units_inv Λ).symm]
|
||||
rw [show Λ.1 * ((Λ)⁻¹).1 = (1 : Matrix (Fin 4) (Fin 4) ℝ) by rw [@Units.mul_inv_eq_one]]
|
||||
rw [@transpose_eq_one]⟩ ∈ lorentzGroup := by
|
||||
rw [mem_lorentzGroup_iff''', mem_lorentzGroup_iff''']
|
||||
simp only [coe_units_inv, transpose_transpose, Units.inv_mk]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
have h1 := congrArg transpose h
|
||||
rw [transpose_mul, transpose_mul, transpose_transpose, η_transpose, ← mul_assoc] at h1
|
||||
rw [h1]
|
||||
exact transpose_nonsing_inv _
|
||||
intro h
|
||||
have h1 := congrArg transpose h
|
||||
rw [transpose_mul, transpose_mul, η_transpose, ← mul_assoc] at h1
|
||||
rw [h1, transpose_nonsing_inv, transpose_transpose]
|
||||
|
||||
/-- The transpose of an matrix in the Lorentz group is an element of the Lorentz group. -/
|
||||
def lorentzGroupTranspose (Λ : lorentzGroup) : lorentzGroup :=
|
||||
⟨⟨transpose Λ.1, (transpose Λ.1)⁻¹, by
|
||||
rw [← transpose_nonsing_inv, ← transpose_mul, (coe_units_inv Λ.1).symm]
|
||||
rw [show ((Λ.1)⁻¹).1 * Λ.1 = (1 : Matrix (Fin 4) (Fin 4) ℝ) by rw [@Units.inv_mul_eq_one]]
|
||||
rw [@transpose_eq_one], by
|
||||
rw [← transpose_nonsing_inv, ← transpose_mul, (coe_units_inv Λ.1).symm]
|
||||
rw [show Λ.1.1 * ((Λ.1)⁻¹).1 = (1 : Matrix (Fin 4) (Fin 4) ℝ) by rw [@Units.mul_inv_eq_one]]
|
||||
rw [@transpose_eq_one]⟩,
|
||||
(mem_lorentzGroup_iff_transpose Λ.1).mp Λ.2 ⟩
|
||||
|
||||
|
||||
/-- A continous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` for which the Lorentz
|
||||
group is the kernal. -/
|
||||
def lorentzMap (Λ : GL (Fin 4) ℝ) : Matrix (Fin 4) (Fin 4) ℝ := η * Λ.1ᵀ * η * Λ.1
|
||||
|
||||
lemma lorentzMap_continuous : Continuous lorentzMap := by
|
||||
apply Continuous.mul _ Units.continuous_val
|
||||
apply Continuous.mul _ continuous_const
|
||||
exact Continuous.mul continuous_const (Continuous.matrix_transpose (Units.continuous_val))
|
||||
|
||||
lemma lorentzGroup_kernal : lorentzGroup = lorentzMap ⁻¹' {1} := by
|
||||
ext Λ
|
||||
erw [mem_lorentzGroup_iff'' Λ]
|
||||
rfl
|
||||
|
||||
/-- The Lorentz Group is a closed subset of `GL (Fin 4) ℝ`. -/
|
||||
theorem lorentzGroup_isClosed : IsClosed (lorentzGroup : Set (GL (Fin 4) ℝ)) := by
|
||||
rw [lorentzGroup_kernal]
|
||||
exact continuous_iff_isClosed.mp lorentzMap_continuous {1} isClosed_singleton
|
||||
|
||||
namespace lorentzGroup
|
||||
|
||||
|
||||
lemma det_eq_one_or_neg_one (Λ : lorentzGroup) : Λ.1.1.det = 1 ∨ Λ.1.1.det = -1 := by
|
||||
simpa [← sq, det_one, det_mul, det_mul, det_mul, det_transpose, det_η] using
|
||||
(congrArg det ((mem_lorentzGroup_iff'' Λ.1).mp Λ.2))
|
||||
|
||||
local notation "ℤ₂" => Multiplicative (ZMod 2)
|
||||
|
||||
/-- A group homomorphism from `lorentzGroup` to `ℤ₂` taking a matrix to
|
||||
0 if it's determinant is 1 and 1 if its determinant is -1.-/
|
||||
@[simps!]
|
||||
def detRep : lorentzGroup →* ℤ₂ where
|
||||
toFun Λ := if Λ.1.1.det = 1 then (Additive.toMul 0) else Additive.toMul (1 : ZMod 2)
|
||||
map_one' := by
|
||||
simp
|
||||
map_mul' := by
|
||||
intro Λ1 Λ2
|
||||
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
|
||||
mul_ite, mul_one, ite_mul, one_mul]
|
||||
cases' (det_eq_one_or_neg_one Λ1) with h1 h1
|
||||
<;> cases' (det_eq_one_or_neg_one Λ2) with h2 h2
|
||||
<;> simp [h1, h2]
|
||||
rfl
|
||||
|
||||
/-- A Lorentz Matrix is proper if its determinant is 1. -/
|
||||
def IsProper (Λ : lorentzGroup) : Prop := Λ.1.1.det = 1
|
||||
|
||||
instance : DecidablePred IsProper := by
|
||||
intro Λ
|
||||
apply Real.decidableEq
|
||||
|
||||
lemma IsProper_iff (Λ : lorentzGroup) :
|
||||
IsProper Λ ↔ detRep Λ = Additive.toMul 0 := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
erw [detRep_apply, if_pos h]
|
||||
simp only [toMul_zero]
|
||||
intro h
|
||||
by_cases h1 : IsProper Λ
|
||||
exact h1
|
||||
erw [detRep_apply, if_neg h1] at h
|
||||
contradiction
|
||||
|
||||
section Orthochronous
|
||||
|
||||
/-- The space-like part of the first row of of a Lorentz matrix. -/
|
||||
@[simp]
|
||||
def fstRowToEuclid (Λ : lorentzGroup) : EuclideanSpace ℝ (Fin 3) := fun i => Λ.1 0 i.succ
|
||||
|
||||
/-- The space-like part of the first column of of a Lorentz matrix. -/
|
||||
@[simp]
|
||||
def fstColToEuclid (Λ : lorentzGroup) : EuclideanSpace ℝ (Fin 3) := fun i => Λ.1 i.succ 0
|
||||
|
||||
lemma fst_col_normalized (Λ : lorentzGroup) :
|
||||
(Λ.1 0 0) ^ 2 - (Λ.1 1 0) ^ 2 - (Λ.1 2 0) ^ 2 - (Λ.1 3 0) ^ 2 = 1 := by
|
||||
have h00 := congrFun (congrFun ((mem_lorentzGroup_iff'' Λ).mp Λ.2) 0) 0
|
||||
simp only [Fin.isValue, mul_apply, transpose_apply, Fin.sum_univ_four, ne_eq, zero_ne_one,
|
||||
not_false_eq_true, η_off_diagonal, zero_mul, add_zero, Fin.reduceEq, one_ne_zero, mul_zero,
|
||||
zero_add, one_apply_eq] at h00
|
||||
simp only [η, Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
|
||||
one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul, cons_val_two, Nat.succ_eq_add_one,
|
||||
Nat.reduceAdd, tail_cons, cons_val_three, head_fin_const] at h00
|
||||
rw [← h00]
|
||||
ring
|
||||
|
||||
lemma fst_row_normalized (Λ : lorentzGroup) :
|
||||
(Λ.1 0 0) ^ 2 - (Λ.1 0 1) ^ 2 - (Λ.1 0 2) ^ 2 - (Λ.1 0 3) ^ 2 = 1 := by
|
||||
simpa using fst_col_normalized (lorentzGroupTranspose Λ)
|
||||
|
||||
|
||||
lemma zero_zero_sq_col (Λ : lorentzGroup) :
|
||||
(Λ.1 0 0) ^ 2 = 1 + (Λ.1 1 0) ^ 2 + (Λ.1 2 0) ^ 2 + (Λ.1 3 0) ^ 2 := by
|
||||
rw [← fst_col_normalized Λ]
|
||||
ring
|
||||
|
||||
lemma zero_zero_sq_row (Λ : lorentzGroup) :
|
||||
(Λ.1 0 0) ^ 2 = 1 + (Λ.1 0 1) ^ 2 + (Λ.1 0 2) ^ 2 + (Λ.1 0 3) ^ 2 := by
|
||||
rw [← fst_row_normalized Λ]
|
||||
ring
|
||||
|
||||
lemma zero_zero_sq_col' (Λ : lorentzGroup) :
|
||||
(Λ.1 0 0) ^ 2 = 1 + ‖fstColToEuclid Λ‖ ^ 2 := by
|
||||
rw [zero_zero_sq_col, ← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp only [Fin.isValue, fstColToEuclid, Fin.succ_zero_eq_one, RCLike.inner_apply, conj_trivial,
|
||||
Fin.succ_one_eq_two]
|
||||
rw [show (Fin.succ 2) = 3 by rfl]
|
||||
ring_nf
|
||||
|
||||
lemma zero_zero_sq_row' (Λ : lorentzGroup) :
|
||||
(Λ.1 0 0) ^ 2 = 1 + ‖fstRowToEuclid Λ‖ ^ 2 := by
|
||||
rw [zero_zero_sq_row, ← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp only [Fin.isValue, fstRowToEuclid, Fin.succ_zero_eq_one, RCLike.inner_apply, conj_trivial,
|
||||
Fin.succ_one_eq_two]
|
||||
rw [show (Fin.succ 2) = 3 by rfl]
|
||||
ring_nf
|
||||
|
||||
lemma norm_col_leq_abs_zero_zero (Λ : lorentzGroup) : ‖fstColToEuclid Λ‖ ≤ |Λ.1 0 0| := by
|
||||
rw [(abs_norm (fstColToEuclid Λ)).symm, ← @sq_le_sq, zero_zero_sq_col']
|
||||
simp only [le_add_iff_nonneg_left, zero_le_one]
|
||||
|
||||
lemma norm_row_leq_abs_zero_zero (Λ : lorentzGroup) : ‖fstRowToEuclid Λ‖ ≤ |Λ.1 0 0| := by
|
||||
rw [(abs_norm (fstRowToEuclid Λ)).symm, ← @sq_le_sq, zero_zero_sq_row']
|
||||
simp only [le_add_iff_nonneg_left, zero_le_one]
|
||||
|
||||
|
||||
|
||||
lemma zero_zero_sq_ge_one (Λ : lorentzGroup) : 1 ≤ (Λ.1 0 0) ^ 2 := by
|
||||
rw [zero_zero_sq_col Λ]
|
||||
refine le_add_of_le_of_nonneg ?_ (sq_nonneg _)
|
||||
refine le_add_of_le_of_nonneg ?_ (sq_nonneg _)
|
||||
refine le_add_of_le_of_nonneg (le_refl 1) (sq_nonneg _)
|
||||
|
||||
lemma abs_zero_zero_ge_one (Λ : lorentzGroup) : 1 ≤ |Λ.1 0 0| :=
|
||||
(one_le_sq_iff_one_le_abs _).mp (zero_zero_sq_ge_one Λ)
|
||||
|
||||
lemma zero_zero_le_minus_one_or_ge_one (Λ : lorentzGroup) : (Λ.1 0 0) ≤ - 1 ∨ 1 ≤ (Λ.1 0 0) :=
|
||||
le_abs'.mp (abs_zero_zero_ge_one Λ)
|
||||
|
||||
lemma zero_zero_nonpos_iff (Λ : lorentzGroup) : (Λ.1 0 0) ≤ 0 ↔ (Λ.1 0 0) ≤ - 1 := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
cases' zero_zero_le_minus_one_or_ge_one Λ with h1 h1
|
||||
exact h1
|
||||
linarith
|
||||
· intro h
|
||||
linarith
|
||||
|
||||
lemma zero_zero_nonneg_iff (Λ : lorentzGroup) : 0 ≤ (Λ.1 0 0) ↔ 1 ≤ (Λ.1 0 0) := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
cases' zero_zero_le_minus_one_or_ge_one Λ with h1 h1
|
||||
linarith
|
||||
exact h1
|
||||
· intro h
|
||||
linarith
|
||||
|
||||
/-- An element of the lorentz group is orthochronous if its time-time element is non-negative. -/
|
||||
@[simp]
|
||||
def IsOrthochronous (Λ : lorentzGroup) : Prop := 0 ≤ Λ.1 0 0
|
||||
|
||||
instance : Decidable (IsOrthochronous Λ) :=
|
||||
Real.decidableLE 0 (Λ.1 0 0)
|
||||
|
||||
lemma not_IsOrthochronous_iff (Λ : lorentzGroup) : ¬ IsOrthochronous Λ ↔ Λ.1 0 0 ≤ 0 := by
|
||||
rw [IsOrthochronous, not_le]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
exact le_of_lt h
|
||||
intro h
|
||||
have h1 := (zero_zero_nonpos_iff Λ).mp h
|
||||
linarith
|
||||
|
||||
|
||||
lemma IsOrthochronous_abs_zero_zero {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
|
||||
|Λ.1 0 0| = Λ.1 0 0 :=
|
||||
abs_eq_self.mpr h
|
||||
|
||||
lemma not_IsOrthochronous_abs_zero_zero {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
|
||||
Λ.1 0 0 = - |Λ.1 0 0| := by
|
||||
refine neg_eq_iff_eq_neg.mp (abs_eq_neg_self.mpr ?_).symm
|
||||
simp only [IsOrthochronous, Fin.isValue, not_le] at h
|
||||
exact le_of_lt h
|
||||
|
||||
lemma schwartz_identity_fst_row_col (Λ Λ' : lorentzGroup) :
|
||||
‖⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ‖ ≤ ‖fstRowToEuclid Λ‖ * ‖fstColToEuclid Λ'‖ :=
|
||||
norm_inner_le_norm (fstRowToEuclid Λ) (fstColToEuclid Λ')
|
||||
|
||||
lemma zero_zero_mul (Λ Λ' : lorentzGroup) :
|
||||
(Λ * Λ').1 0 0 = (Λ.1 0 0) * (Λ'.1 0 0) + ⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ := by
|
||||
rw [@Subgroup.coe_mul, @GeneralLinearGroup.coe_mul, mul_apply, Fin.sum_univ_four]
|
||||
rw [@PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp
|
||||
rw [show (Fin.succ 2) = 3 by rfl]
|
||||
ring_nf
|
||||
example (a b c : ℝ ) (h : a - b ≤ a + c) : - b ≤ c := by
|
||||
exact (add_le_add_iff_left a).mp h
|
||||
|
||||
lemma zero_zero_mul_sub_row_col (Λ Λ' : lorentzGroup) :
|
||||
0 ≤ |Λ.1 0 0| * |Λ'.1 0 0| - ‖fstRowToEuclid Λ‖ * ‖fstColToEuclid Λ'‖ :=
|
||||
sub_nonneg.mpr (mul_le_mul (norm_row_leq_abs_zero_zero Λ) (norm_col_leq_abs_zero_zero Λ')
|
||||
(norm_nonneg (fstColToEuclid Λ')) (abs_nonneg (Λ.1 0 0)))
|
||||
|
||||
lemma orthchro_mul_orthchro {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
|
||||
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
|
||||
apply le_trans (zero_zero_mul_sub_row_col Λ Λ')
|
||||
have h1 := zero_zero_mul Λ Λ'
|
||||
rw [← IsOrthochronous_abs_zero_zero h, ← IsOrthochronous_abs_zero_zero h'] at h1
|
||||
refine le_trans
|
||||
(?_ : _ ≤ |Λ.1 0 0| * |Λ'.1 0 0| + (- |⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ|)) ?_
|
||||
· erw [add_le_add_iff_left]
|
||||
simp only [neg_mul, neg_neg, neg_le]
|
||||
exact schwartz_identity_fst_row_col Λ Λ'
|
||||
· rw [h1, add_le_add_iff_left]
|
||||
exact neg_abs_le ⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ
|
||||
|
||||
lemma not_orthchro_mul_not_orthchro {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
|
||||
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
|
||||
apply le_trans (zero_zero_mul_sub_row_col Λ Λ')
|
||||
refine le_trans
|
||||
(?_ : _ ≤ |Λ.1 0 0| * |Λ'.1 0 0| + (- |⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ|)) ?_
|
||||
· erw [add_le_add_iff_left]
|
||||
simp only [neg_mul, neg_neg, neg_le]
|
||||
exact schwartz_identity_fst_row_col Λ Λ'
|
||||
· rw [zero_zero_mul Λ Λ', not_IsOrthochronous_abs_zero_zero h,
|
||||
not_IsOrthochronous_abs_zero_zero h']
|
||||
simp only [Fin.isValue, abs_neg, _root_.abs_abs, conj_trivial, mul_neg, neg_mul, neg_neg,
|
||||
add_le_add_iff_left]
|
||||
exact neg_abs_le ⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ
|
||||
|
||||
lemma orthchro_mul_not_orthchro {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
|
||||
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
|
||||
rw [not_IsOrthochronous_iff]
|
||||
refine le_trans
|
||||
(?_ : _ ≤ - (|Λ.1 0 0| * |Λ'.1 0 0| + (- |⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ|))) ?_
|
||||
· rw [zero_zero_mul Λ Λ', ← IsOrthochronous_abs_zero_zero h,
|
||||
not_IsOrthochronous_abs_zero_zero h']
|
||||
simp only [Fin.isValue, mul_neg, _root_.abs_abs, abs_neg, neg_add_rev, neg_neg,
|
||||
le_add_neg_iff_add_le, neg_add_cancel_comm]
|
||||
exact le_abs_self ⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ
|
||||
· apply le_trans ?_ (neg_nonpos_of_nonneg (zero_zero_mul_sub_row_col Λ Λ'))
|
||||
simp only [Fin.isValue, neg_add_rev, neg_neg, neg_sub, add_neg_le_iff_le_add, sub_add_cancel]
|
||||
exact schwartz_identity_fst_row_col _ _
|
||||
|
||||
lemma not_orthchro_mul_orthchro {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
|
||||
(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
|
||||
rw [not_IsOrthochronous_iff]
|
||||
refine le_trans
|
||||
(?_ : _ ≤ - (|Λ.1 0 0| * |Λ'.1 0 0| + (- |⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ|))) ?_
|
||||
· rw [zero_zero_mul Λ Λ', ← IsOrthochronous_abs_zero_zero h',
|
||||
not_IsOrthochronous_abs_zero_zero h]
|
||||
simp only [Fin.isValue, neg_mul, PiLp.inner_apply, fstRowToEuclid, fstColToEuclid,
|
||||
RCLike.inner_apply, conj_trivial, abs_neg, _root_.abs_abs, neg_add_rev, neg_neg,
|
||||
le_add_neg_iff_add_le, neg_add_cancel_comm]
|
||||
exact le_abs_self ⟪fstRowToEuclid Λ, fstColToEuclid Λ'⟫_ℝ
|
||||
· apply le_trans ?_ (neg_nonpos_of_nonneg (zero_zero_mul_sub_row_col Λ Λ'))
|
||||
simp only [Fin.isValue, neg_add_rev, neg_neg, neg_sub, add_neg_le_iff_le_add, sub_add_cancel]
|
||||
exact schwartz_identity_fst_row_col _ _
|
||||
|
||||
/-- The homomorphism from `lorentzGroup` to `ℤ₂` whose kernal consists of precisely the
|
||||
orthochronous elements. -/
|
||||
@[simps!]
|
||||
def orthochronousRep : lorentzGroup →* ℤ₂ where
|
||||
toFun Λ := if IsOrthochronous Λ then (Additive.toMul 0) else Additive.toMul (1 : ZMod 2)
|
||||
map_one' := by
|
||||
simp
|
||||
map_mul' := by
|
||||
intro Λ1 Λ2
|
||||
simp only [toMul_zero, mul_ite, mul_one, ite_mul, one_mul]
|
||||
cases' (em (IsOrthochronous Λ1)) with h h
|
||||
<;> cases' (em (IsOrthochronous Λ2)) with h' h'
|
||||
rw [if_pos (orthchro_mul_orthchro h h'), if_pos h, if_pos h']
|
||||
rw [if_neg (orthchro_mul_not_orthchro h h'), if_neg h', if_pos h]
|
||||
rw [if_neg (not_orthchro_mul_orthchro h h'), if_pos h', if_neg h]
|
||||
rw [if_pos (not_orthchro_mul_not_orthchro h h'), if_neg h', if_neg h]
|
||||
rfl
|
||||
|
||||
lemma IsOrthochronous_iff (Λ : lorentzGroup) :
|
||||
IsOrthochronous Λ ↔ orthochronousRep Λ = Additive.toMul 0 := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
erw [orthochronousRep_apply, if_pos h]
|
||||
simp only [toMul_zero]
|
||||
intro h
|
||||
by_cases h1 : IsOrthochronous Λ
|
||||
exact h1
|
||||
erw [orthochronousRep_apply, if_neg h1] at h
|
||||
contradiction
|
||||
|
||||
|
||||
|
||||
end Orthochronous
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
open lorentzGroup in
|
||||
/-- The restricted lorentz group consists of those elements of the `lorentzGroup` which are
|
||||
proper and orthochronous. -/
|
||||
def restrictedLorentzGroup : Subgroup (lorentzGroup) where
|
||||
carrier := {Λ | IsProper Λ ∧ IsOrthochronous Λ}
|
||||
mul_mem' {Λa Λb} := by
|
||||
intro ha hb
|
||||
apply And.intro
|
||||
rw [IsProper_iff, detRep.map_mul, (IsProper_iff Λa).mp ha.1, (IsProper_iff Λb).mp hb.1]
|
||||
simp only [toMul_zero, mul_one]
|
||||
rw [IsOrthochronous_iff, orthochronousRep.map_mul, (IsOrthochronous_iff Λa).mp ha.2,
|
||||
(IsOrthochronous_iff Λb).mp hb.2]
|
||||
simp
|
||||
one_mem' := by
|
||||
simp only [IsOrthochronous, Fin.isValue, Set.mem_setOf_eq, OneMemClass.coe_one, Units.val_one,
|
||||
one_apply_eq, zero_le_one, and_true]
|
||||
rw [IsProper_iff, detRep.map_one]
|
||||
simp
|
||||
inv_mem' {Λ} := by
|
||||
intro h
|
||||
apply And.intro
|
||||
rw [IsProper_iff, detRep.map_inv, (IsProper_iff Λ).mp h.1]
|
||||
simp only [toMul_zero, inv_one]
|
||||
rw [IsOrthochronous_iff, orthochronousRep.map_inv, (IsOrthochronous_iff Λ).mp h.2]
|
||||
simp
|
||||
|
||||
/-- The restricted lorentz group is a topological group. -/
|
||||
instance : TopologicalGroup restrictedLorentzGroup :=
|
||||
Subgroup.instTopologicalGroupSubtypeMem restrictedLorentzGroup
|
||||
|
||||
end spaceTime
|
||||
|
||||
end
|
||||
151
HepLean/SpaceTime/LorentzGroup/Basic.lean
Normal file
151
HepLean/SpaceTime/LorentzGroup/Basic.lean
Normal file
|
|
@ -0,0 +1,151 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.Metric
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
/-!
|
||||
# The Lorentz Group
|
||||
|
||||
We define the Lorentz group.
|
||||
|
||||
## TODO
|
||||
|
||||
- Show that the Lorentz is a Lie group.
|
||||
- Prove that the restricted Lorentz group is equivalent to the connected component of the
|
||||
identity.
|
||||
- Define the continuous maps from `ℝ³` to `restrictedLorentzGroup` defining boosts.
|
||||
|
||||
## References
|
||||
|
||||
- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf
|
||||
|
||||
-/
|
||||
|
||||
|
||||
noncomputable section
|
||||
|
||||
namespace spaceTime
|
||||
|
||||
open Manifold
|
||||
open Matrix
|
||||
open Complex
|
||||
open ComplexConjugate
|
||||
|
||||
/-- We say a matrix `Λ` preserves `ηLin` if for all `x` and `y`,
|
||||
`ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y`. -/
|
||||
def PreservesηLin (Λ : Matrix (Fin 4) (Fin 4) ℝ) : Prop :=
|
||||
∀ (x y : spaceTime), ηLin (Λ *ᵥ x) (Λ *ᵥ y) = ηLin x y
|
||||
|
||||
namespace PreservesηLin
|
||||
|
||||
variable (Λ : Matrix (Fin 4) (Fin 4) ℝ)
|
||||
|
||||
lemma iff_on_right : PreservesηLin Λ ↔
|
||||
∀ (x y : spaceTime), ηLin x ((η * Λᵀ * η * Λ) *ᵥ y) = ηLin x y := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
intro x y
|
||||
have h1 := h x y
|
||||
rw [ηLin_mulVec_left, mulVec_mulVec] at h1
|
||||
exact h1
|
||||
intro h
|
||||
intro x y
|
||||
rw [ηLin_mulVec_left, mulVec_mulVec]
|
||||
exact h x y
|
||||
|
||||
lemma iff_matrix : PreservesηLin Λ ↔ η * Λᵀ * η * Λ = 1 := by
|
||||
rw [iff_on_right, ηLin_matrix_eq_identity_iff (η * Λᵀ * η * Λ)]
|
||||
apply Iff.intro
|
||||
· simp_all [ηLin, implies_true, iff_true, one_mulVec]
|
||||
· simp_all only [ηLin, LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply,
|
||||
mulVec_mulVec, implies_true]
|
||||
|
||||
lemma iff_matrix' : PreservesηLin Λ ↔ Λ * (η * Λᵀ * η) = 1 := by
|
||||
rw [iff_matrix]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
exact mul_eq_one_comm.mp h
|
||||
intro h
|
||||
exact mul_eq_one_comm.mp h
|
||||
|
||||
lemma iff_transpose : PreservesηLin Λ ↔ PreservesηLin Λᵀ := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
have h1 := congrArg transpose ((iff_matrix Λ).mp h)
|
||||
rw [transpose_mul, transpose_mul, transpose_mul, η_transpose,
|
||||
← mul_assoc, transpose_one] at h1
|
||||
rw [iff_matrix' Λ.transpose, ← h1]
|
||||
repeat rw [← mul_assoc]
|
||||
intro h
|
||||
have h1 := congrArg transpose ((iff_matrix Λ.transpose).mp h)
|
||||
rw [transpose_mul, transpose_mul, transpose_mul, η_transpose,
|
||||
← mul_assoc, transpose_one, transpose_transpose] at h1
|
||||
rw [iff_matrix', ← h1]
|
||||
repeat rw [← mul_assoc]
|
||||
|
||||
/-- The lift of a matrix which preserves `ηLin` to an invertible matrix. -/
|
||||
def liftGL {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) : GL (Fin 4) ℝ :=
|
||||
⟨Λ, η * Λᵀ * η , (iff_matrix' Λ).mp h , (iff_matrix Λ).mp h⟩
|
||||
|
||||
end PreservesηLin
|
||||
|
||||
/-- The Lorentz group as a subgroup of the general linear group over the reals. -/
|
||||
def lorentzGroup : Subgroup (GL (Fin 4) ℝ) where
|
||||
carrier := {Λ | PreservesηLin Λ}
|
||||
mul_mem' {a b} := by
|
||||
intros ha hb x y
|
||||
simp only [Units.val_mul, mulVec_mulVec]
|
||||
rw [← mulVec_mulVec, ← mulVec_mulVec, ha, hb]
|
||||
one_mem' := by
|
||||
intros x y
|
||||
simp
|
||||
inv_mem' {a} := by
|
||||
intros ha x y
|
||||
simp only [coe_units_inv, ← ha ((a.1⁻¹) *ᵥ x) ((a.1⁻¹) *ᵥ y), mulVec_mulVec]
|
||||
have hx : (a.1 * (a.1)⁻¹) = 1 := by
|
||||
simp only [@Units.mul_eq_one_iff_inv_eq, coe_units_inv]
|
||||
simp [hx]
|
||||
|
||||
/-- The Lorentz group is a topological group with the subset topology. -/
|
||||
instance : TopologicalGroup lorentzGroup :=
|
||||
Subgroup.instTopologicalGroupSubtypeMem lorentzGroup
|
||||
|
||||
/-- The lift of a matrix perserving `ηLin` to a Lorentz Group element. -/
|
||||
def PreservesηLin.liftLor {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) :
|
||||
lorentzGroup := ⟨liftGL h, h⟩
|
||||
|
||||
namespace lorentzGroup
|
||||
|
||||
lemma mem_iff (Λ : GL (Fin 4) ℝ): Λ ∈ lorentzGroup ↔ PreservesηLin Λ := by
|
||||
rfl
|
||||
|
||||
/-- The transpose of an matrix in the Lorentz group is an element of the Lorentz group. -/
|
||||
def transpose (Λ : lorentzGroup) : lorentzGroup :=
|
||||
PreservesηLin.liftLor ((PreservesηLin.iff_transpose Λ.1).mp Λ.2)
|
||||
|
||||
|
||||
/-- The continuous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` whose kernal is
|
||||
the Lorentz group. -/
|
||||
def kernalMap : C(GL (Fin 4) ℝ, Matrix (Fin 4) (Fin 4) ℝ) where
|
||||
toFun Λ := η * Λ.1ᵀ * η * Λ.1
|
||||
continuous_toFun := by
|
||||
apply Continuous.mul _ Units.continuous_val
|
||||
apply Continuous.mul _ continuous_const
|
||||
exact Continuous.mul continuous_const (Continuous.matrix_transpose (Units.continuous_val))
|
||||
|
||||
lemma kernalMap_kernal_eq_lorentzGroup : lorentzGroup = kernalMap ⁻¹' {1} := by
|
||||
ext Λ
|
||||
erw [mem_iff Λ, PreservesηLin.iff_matrix]
|
||||
rfl
|
||||
|
||||
/-- The Lorentz Group is a closed subset of `GL (Fin 4) ℝ`. -/
|
||||
theorem isClosed_of_GL4 : IsClosed (lorentzGroup : Set (GL (Fin 4) ℝ)) := by
|
||||
rw [kernalMap_kernal_eq_lorentzGroup]
|
||||
exact continuous_iff_isClosed.mp kernalMap.2 {1} isClosed_singleton
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
end spaceTime
|
||||
173
HepLean/SpaceTime/LorentzGroup/Boosts.lean
Normal file
173
HepLean/SpaceTime/LorentzGroup/Boosts.lean
Normal file
|
|
@ -0,0 +1,173 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.LorentzGroup.Orthochronous
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
import Mathlib.Topology.Constructions
|
||||
/-!
|
||||
# Boosts
|
||||
|
||||
This file defines those Lorentz which are boosts.
|
||||
|
||||
We first define generalised boosts, which are restricted lorentz transforamations taking
|
||||
a four velocity `u` to a four velcoity `v`.
|
||||
|
||||
A boost is the speical case of a generalised boost when `u = stdBasis 0`.
|
||||
|
||||
## TODO
|
||||
|
||||
- Show that generalised boosts are in the restircted Lorentz group.
|
||||
- Define boosts.
|
||||
|
||||
## References
|
||||
|
||||
- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
|
||||
https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
|
||||
|
||||
-/
|
||||
noncomputable section
|
||||
namespace spaceTime
|
||||
|
||||
namespace lorentzGroup
|
||||
|
||||
open FourVelocity
|
||||
|
||||
/-- An auxillary linear map used in the definition of a genearlised boost. -/
|
||||
def genBoostAux₁ (u v : FourVelocity) : spaceTime →ₗ[ℝ] spaceTime where
|
||||
toFun x := (2 * ηLin x u) • v.1.1
|
||||
map_add' x y := by
|
||||
simp only [map_add, LinearMap.add_apply]
|
||||
rw [mul_add, add_smul]
|
||||
map_smul' c x := by
|
||||
simp only [LinearMapClass.map_smul, LinearMap.smul_apply, smul_eq_mul,
|
||||
RingHom.id_apply]
|
||||
rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
|
||||
|
||||
/-- An auxillary linear map used in the definition of a genearlised boost. -/
|
||||
def genBoostAux₂ (u v : FourVelocity) : spaceTime →ₗ[ℝ] spaceTime where
|
||||
toFun x := - (ηLin x (u + v) / (1 + ηLin u v)) • (u + v)
|
||||
map_add' x y := by
|
||||
simp only
|
||||
rw [ηLin.map_add, div_eq_mul_one_div]
|
||||
rw [show (ηLin x + ηLin y) (↑u + ↑v) = ηLin x (u+v) + ηLin y (u+v) from rfl]
|
||||
rw [add_mul, neg_add, add_smul, ← div_eq_mul_one_div, ← div_eq_mul_one_div]
|
||||
map_smul' c x := by
|
||||
simp only
|
||||
rw [ηLin.map_smul]
|
||||
rw [show (c • ηLin x) (↑u + ↑v) = c * ηLin x (u+v) from rfl]
|
||||
rw [mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
|
||||
rfl
|
||||
|
||||
/-- An genearlised boost. This is a Lorentz transformation which takes the four velocity `u`
|
||||
to `v`. -/
|
||||
def genBoost (u v : FourVelocity) : spaceTime →ₗ[ℝ] spaceTime :=
|
||||
LinearMap.id + genBoostAux₁ u v + genBoostAux₂ u v
|
||||
|
||||
namespace genBoost
|
||||
|
||||
lemma self (u : FourVelocity) : genBoost u u = LinearMap.id := by
|
||||
ext x
|
||||
simp [genBoost]
|
||||
rw [add_assoc]
|
||||
rw [@add_right_eq_self]
|
||||
rw [@add_eq_zero_iff_eq_neg]
|
||||
rw [genBoostAux₁, genBoostAux₂]
|
||||
simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]
|
||||
rw [← add_smul]
|
||||
apply congr
|
||||
apply congrArg
|
||||
repeat rw [u.1.2]
|
||||
field_simp
|
||||
ring
|
||||
rfl
|
||||
|
||||
/-- A generalised boost as a matrix. -/
|
||||
def toMatrix (u v : FourVelocity) : Matrix (Fin 4) (Fin 4) ℝ :=
|
||||
LinearMap.toMatrix stdBasis stdBasis (genBoost u v)
|
||||
|
||||
lemma toMatrix_mulVec (u v : FourVelocity) (x : spaceTime) :
|
||||
(toMatrix u v).mulVec x = genBoost u v x :=
|
||||
LinearMap.toMatrix_mulVec_repr stdBasis stdBasis (genBoost u v) x
|
||||
|
||||
lemma toMatrix_apply (u v : FourVelocity) (i j : Fin 4) :
|
||||
(toMatrix u v) i j =
|
||||
η i i * (ηLin (stdBasis i) (stdBasis j) + 2 * ηLin (stdBasis j) u * ηLin (stdBasis i) v -
|
||||
ηLin (stdBasis i) (u + v) * ηLin (stdBasis j) (u + v) / (1 + ηLin u v)) := by
|
||||
rw [ηLin_matrix_stdBasis' j i (toMatrix u v), toMatrix_mulVec]
|
||||
simp only [genBoost, genBoostAux₁, genBoostAux₂, map_add, smul_add, neg_smul, LinearMap.add_apply,
|
||||
LinearMap.id_apply, LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, smul_eq_mul,
|
||||
map_neg, mul_eq_mul_left_iff]
|
||||
apply Or.inl
|
||||
ring
|
||||
|
||||
lemma toMatrix_continuous (u : FourVelocity) : Continuous (toMatrix u) := by
|
||||
refine continuous_matrix ?_
|
||||
intro i j
|
||||
simp only [toMatrix_apply]
|
||||
refine Continuous.comp' (continuous_mul_left (η i i)) ?hf
|
||||
refine Continuous.sub ?_ ?_
|
||||
refine Continuous.comp' (continuous_add_left ((ηLin (stdBasis i)) (stdBasis j))) ?_
|
||||
refine Continuous.comp' (continuous_mul_left (2 * (ηLin (stdBasis j)) ↑u)) ?_
|
||||
exact η_continuous _
|
||||
refine Continuous.mul ?_ ?_
|
||||
refine Continuous.mul ?_ ?_
|
||||
simp only [(ηLin _).map_add]
|
||||
refine Continuous.comp' ?_ ?_
|
||||
exact continuous_add_left ((ηLin (stdBasis i)) ↑u)
|
||||
exact η_continuous _
|
||||
simp only [(ηLin _).map_add]
|
||||
refine Continuous.comp' ?_ ?_
|
||||
exact continuous_add_left _
|
||||
exact η_continuous _
|
||||
refine Continuous.inv₀ ?_ ?_
|
||||
refine Continuous.comp' (continuous_add_left 1) ?_
|
||||
exact η_continuous _
|
||||
exact fun x => one_plus_ne_zero u x
|
||||
|
||||
|
||||
lemma toMatrix_PreservesηLin (u v : FourVelocity) : PreservesηLin (toMatrix u v) := by
|
||||
intro x y
|
||||
rw [toMatrix_mulVec, toMatrix_mulVec, genBoost, genBoostAux₁, genBoostAux₂]
|
||||
have h1 : (1 + (ηLin ↑u) ↑v) ≠ 0 := one_plus_ne_zero u v
|
||||
simp only [map_add, smul_add, neg_smul, LinearMap.add_apply, LinearMap.id_coe,
|
||||
id_eq, LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, map_neg, LinearMap.smul_apply,
|
||||
smul_eq_mul, LinearMap.neg_apply]
|
||||
field_simp
|
||||
rw [u.1.2, v.1.2, ηLin_symm v u, ηLin_symm u y, ηLin_symm v y]
|
||||
ring
|
||||
|
||||
/-- A generalised boost as an element of the Lorentz Group. -/
|
||||
def toLorentz (u v : FourVelocity) : lorentzGroup :=
|
||||
PreservesηLin.liftLor (toMatrix_PreservesηLin u v)
|
||||
|
||||
lemma toLorentz_continuous (u : FourVelocity) : Continuous (toLorentz u) := by
|
||||
refine Continuous.subtype_mk ?_ fun x => (PreservesηLin.liftLor (toMatrix_PreservesηLin u x)).2
|
||||
refine Units.continuous_iff.mpr (And.intro ?_ ?_)
|
||||
exact toMatrix_continuous u
|
||||
change Continuous fun x => (η * (toMatrix u x).transpose * η)
|
||||
refine Continuous.matrix_mul ?_ continuous_const
|
||||
refine Continuous.matrix_mul continuous_const ?_
|
||||
exact Continuous.matrix_transpose (toMatrix_continuous u)
|
||||
|
||||
|
||||
lemma toLorentz_joined_to_1 (u v : FourVelocity) : Joined 1 (toLorentz u v) := by
|
||||
obtain ⟨f, _⟩ := isPathConnected_FourVelocity.joinedIn u trivial v trivial
|
||||
use ContinuousMap.comp ⟨toLorentz u, toLorentz_continuous u⟩ f
|
||||
· simp only [ContinuousMap.toFun_eq_coe, ContinuousMap.comp_apply, ContinuousMap.coe_coe,
|
||||
Path.source, ContinuousMap.coe_mk]
|
||||
rw [@Subtype.ext_iff, toLorentz, PreservesηLin.liftLor]
|
||||
refine Units.val_eq_one.mp ?_
|
||||
simp [PreservesηLin.liftGL, toMatrix, self u]
|
||||
· simp
|
||||
|
||||
|
||||
end genBoost
|
||||
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
|
||||
end spaceTime
|
||||
end
|
||||
199
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
Normal file
199
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
Normal file
|
|
@ -0,0 +1,199 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
import HepLean.SpaceTime.LorentzGroup.Proper
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
/-!
|
||||
# The Orthochronous Lorentz Group
|
||||
|
||||
We define the give a series of lemmas related to the orthochronous property of lorentz
|
||||
matrices.
|
||||
|
||||
## TODO
|
||||
|
||||
- Prove topological properties.
|
||||
|
||||
-/
|
||||
|
||||
|
||||
noncomputable section
|
||||
|
||||
namespace spaceTime
|
||||
|
||||
open Manifold
|
||||
open Matrix
|
||||
open Complex
|
||||
open ComplexConjugate
|
||||
|
||||
namespace lorentzGroup
|
||||
open PreFourVelocity
|
||||
|
||||
/-- The first column of a lorentz matrix as a `PreFourVelocity`. -/
|
||||
@[simp]
|
||||
def fstCol (Λ : lorentzGroup) : PreFourVelocity := ⟨Λ.1 *ᵥ stdBasis 0, by
|
||||
rw [mem_PreFourVelocity_iff, ηLin_expand]
|
||||
simp only [Fin.isValue, stdBasis_mulVec]
|
||||
have h00 := congrFun (congrFun ((PreservesηLin.iff_matrix Λ.1).mp ((mem_iff Λ.1).mp Λ.2)) 0) 0
|
||||
simp only [Fin.isValue, mul_apply, transpose_apply, Fin.sum_univ_four, ne_eq, zero_ne_one,
|
||||
not_false_eq_true, η_off_diagonal, zero_mul, add_zero, Fin.reduceEq, one_ne_zero, mul_zero,
|
||||
zero_add, one_apply_eq] at h00
|
||||
simp only [η, Fin.isValue, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
|
||||
vecCons_const, one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul, cons_val_two,
|
||||
Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, cons_val_three, head_fin_const] at h00
|
||||
rw [← h00]
|
||||
ring⟩
|
||||
|
||||
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
|
||||
def IsOrthochronous (Λ : lorentzGroup) : Prop := 0 ≤ Λ.1 0 0
|
||||
|
||||
lemma IsOrthochronous_iff_transpose (Λ : lorentzGroup) :
|
||||
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by
|
||||
simp only [IsOrthochronous, Fin.isValue, transpose, PreservesηLin.liftLor, PreservesηLin.liftGL,
|
||||
transpose_transpose, transpose_apply]
|
||||
|
||||
lemma IsOrthochronous_iff_fstCol_IsFourVelocity (Λ : lorentzGroup) :
|
||||
IsOrthochronous Λ ↔ IsFourVelocity (fstCol Λ) := by
|
||||
simp [IsOrthochronous, IsFourVelocity]
|
||||
rw [stdBasis_mulVec]
|
||||
|
||||
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
|
||||
def mapZeroZeroComp : C(lorentzGroup, ℝ) := ⟨fun Λ => Λ.1 0 0, by
|
||||
refine Continuous.matrix_elem ?_ 0 0
|
||||
refine Continuous.comp' Units.continuous_val continuous_subtype_val⟩
|
||||
|
||||
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
|
||||
def stepFunction : ℝ → ℝ := fun t =>
|
||||
if t ≤ -1 then -1 else
|
||||
if 1 ≤ t then 1 else t
|
||||
|
||||
lemma stepFunction_continuous : Continuous stepFunction := by
|
||||
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
|
||||
<;> intro a ha
|
||||
rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
|
||||
rw [ha]
|
||||
simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
|
||||
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
|
||||
rw [if_neg h1]
|
||||
rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
|
||||
simp [ha]
|
||||
|
||||
/-- The continuous map from `lorentzGroup` to `ℝ` wh
|
||||
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
|
||||
def orthchroMapReal : C(lorentzGroup, ℝ) := ContinuousMap.comp
|
||||
⟨stepFunction, stepFunction_continuous⟩ mapZeroZeroComp
|
||||
|
||||
lemma orthchroMapReal_on_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
|
||||
orthchroMapReal Λ = 1 := by
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
|
||||
simp only [IsFourVelocity] at h
|
||||
rw [zero_nonneg_iff] at h
|
||||
simp [stdBasis_mulVec] at h
|
||||
have h1 : ¬ Λ.1 0 0 ≤ (-1 : ℝ) := by linarith
|
||||
change stepFunction (Λ.1 0 0) = 1
|
||||
rw [stepFunction, if_neg h1, if_pos h]
|
||||
|
||||
|
||||
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
|
||||
orthchroMapReal Λ = - 1 := by
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
|
||||
rw [not_IsFourVelocity_iff, zero_nonpos_iff] at h
|
||||
simp only [fstCol, Fin.isValue, stdBasis_mulVec] at h
|
||||
change stepFunction (Λ.1 0 0) = - 1
|
||||
rw [stepFunction, if_pos h]
|
||||
|
||||
lemma orthchroMapReal_minus_one_or_one (Λ : lorentzGroup) :
|
||||
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
|
||||
by_cases h : IsOrthochronous Λ
|
||||
apply Or.inr $ orthchroMapReal_on_IsOrthochronous h
|
||||
apply Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
|
||||
|
||||
local notation "ℤ₂" => Multiplicative (ZMod 2)
|
||||
|
||||
/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
|
||||
def orthchroMap : C(lorentzGroup, ℤ₂) :=
|
||||
ContinuousMap.comp coeForℤ₂ {
|
||||
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
|
||||
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
|
||||
|
||||
lemma orthchroMap_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
|
||||
orthchroMap Λ = 1 := by
|
||||
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
|
||||
|
||||
lemma orthchroMap_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
|
||||
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
|
||||
simp [orthchroMap, orthchroMapReal_on_not_IsOrthochronous h]
|
||||
rfl
|
||||
|
||||
lemma zero_zero_mul (Λ Λ' : lorentzGroup) :
|
||||
(Λ * Λ').1 0 0 = (fstCol (transpose Λ)).1 0 * (fstCol Λ').1 0 +
|
||||
⟪(fstCol (transpose Λ)).1.space, (fstCol Λ').1.space⟫_ℝ := by
|
||||
rw [@Subgroup.coe_mul, @GeneralLinearGroup.coe_mul, mul_apply, Fin.sum_univ_four]
|
||||
rw [@PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp [transpose, stdBasis_mulVec, PreservesηLin.liftLor, PreservesηLin.liftGL]
|
||||
ring
|
||||
|
||||
lemma mul_othchron_of_othchron_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
|
||||
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous, zero_zero_mul]
|
||||
exact euclid_norm_IsFourVelocity_IsFourVelocity h h'
|
||||
|
||||
lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
|
||||
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous, zero_zero_mul]
|
||||
exact euclid_norm_not_IsFourVelocity_not_IsFourVelocity h h'
|
||||
|
||||
lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
|
||||
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity, not_IsFourVelocity_iff]
|
||||
simp [stdBasis_mulVec]
|
||||
change (Λ * Λ').1 0 0 ≤ _
|
||||
rw [zero_zero_mul]
|
||||
exact euclid_norm_IsFourVelocity_not_IsFourVelocity h h'
|
||||
|
||||
lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
|
||||
(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity, not_IsFourVelocity_iff]
|
||||
simp [stdBasis_mulVec]
|
||||
change (Λ * Λ').1 0 0 ≤ _
|
||||
rw [zero_zero_mul]
|
||||
exact euclid_norm_not_IsFourVelocity_IsFourVelocity h h'
|
||||
|
||||
/-- The representation from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
|
||||
def orthchroRep : lorentzGroup →* ℤ₂ where
|
||||
toFun := orthchroMap
|
||||
map_one' := by
|
||||
have h1 : IsOrthochronous 1 := by simp [IsOrthochronous]
|
||||
rw [orthchroMap_IsOrthochronous h1]
|
||||
map_mul' Λ Λ' := by
|
||||
simp only
|
||||
by_cases h : IsOrthochronous Λ
|
||||
<;> by_cases h' : IsOrthochronous Λ'
|
||||
rw [orthchroMap_IsOrthochronous h, orthchroMap_IsOrthochronous h',
|
||||
orthchroMap_IsOrthochronous (mul_othchron_of_othchron_othchron h h')]
|
||||
simp only [mul_one]
|
||||
rw [orthchroMap_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
|
||||
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_othchron_not_othchron h h')]
|
||||
simp only [Nat.reduceAdd, one_mul]
|
||||
rw [orthchroMap_not_IsOrthochronous h, orthchroMap_IsOrthochronous h',
|
||||
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_not_othchron_othchron h h')]
|
||||
simp only [Nat.reduceAdd, mul_one]
|
||||
rw [orthchroMap_not_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
|
||||
orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
|
||||
simp only [Nat.reduceAdd]
|
||||
rfl
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
end spaceTime
|
||||
end
|
||||
123
HepLean/SpaceTime/LorentzGroup/Proper.lean
Normal file
123
HepLean/SpaceTime/LorentzGroup/Proper.lean
Normal file
|
|
@ -0,0 +1,123 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.LorentzGroup.Basic
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
/-!
|
||||
# The Proper Lorentz Group
|
||||
|
||||
We define the give a series of lemmas related to the determinant of the lorentz group.
|
||||
|
||||
-/
|
||||
|
||||
|
||||
noncomputable section
|
||||
|
||||
namespace spaceTime
|
||||
|
||||
open Manifold
|
||||
open Matrix
|
||||
open Complex
|
||||
open ComplexConjugate
|
||||
|
||||
namespace lorentzGroup
|
||||
|
||||
/-- The determinant of a member of the lorentz group is `1` or `-1`. -/
|
||||
lemma det_eq_one_or_neg_one (Λ : lorentzGroup) : Λ.1.1.det = 1 ∨ Λ.1.1.det = -1 := by
|
||||
simpa [← sq, det_one, det_mul, det_mul, det_mul, det_transpose, det_η] using
|
||||
(congrArg det ((PreservesηLin.iff_matrix' Λ.1).mp ((mem_iff Λ.1).mp Λ.2)))
|
||||
|
||||
local notation "ℤ₂" => Multiplicative (ZMod 2)
|
||||
|
||||
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
|
||||
|
||||
instance : DiscreteTopology ℤ₂ := by
|
||||
exact forall_open_iff_discrete.mp fun _ => trivial
|
||||
|
||||
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
|
||||
|
||||
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
|
||||
@[simps!]
|
||||
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
|
||||
toFun x := if x = ⟨1, by simp⟩ then (Additive.toMul 0) else (Additive.toMul (1 : ZMod 2))
|
||||
continuous_toFun := by
|
||||
haveI : DiscreteTopology ({-1, 1} : Set ℝ) := discrete_of_t1_of_finite
|
||||
exact continuous_of_discreteTopology
|
||||
|
||||
/-- The continuous map taking a lorentz matrix to its determinant. -/
|
||||
def detContinuous : C(lorentzGroup, ℤ₂) :=
|
||||
ContinuousMap.comp coeForℤ₂ {
|
||||
toFun := fun Λ => ⟨Λ.1.1.det, Or.symm (lorentzGroup.det_eq_one_or_neg_one _)⟩,
|
||||
continuous_toFun := by
|
||||
refine Continuous.subtype_mk ?_ _
|
||||
exact Continuous.matrix_det $
|
||||
Continuous.comp' Units.continuous_val continuous_subtype_val}
|
||||
|
||||
lemma detContinuous_eq_iff_det_eq (Λ Λ' : lorentzGroup) :
|
||||
detContinuous Λ = detContinuous Λ' ↔ Λ.1.1.det = Λ'.1.1.det := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
simp [detContinuous] at h
|
||||
cases' det_eq_one_or_neg_one Λ with h1 h1
|
||||
<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
|
||||
<;> simp_all [h1, h2, h]
|
||||
rw [← toMul_zero, @Equiv.apply_eq_iff_eq] at h
|
||||
change (0 : Fin 2) = (1 : Fin 2) at h
|
||||
simp only [Fin.isValue, zero_ne_one] at h
|
||||
nth_rewrite 2 [← toMul_zero] at h
|
||||
rw [@Equiv.apply_eq_iff_eq] at h
|
||||
change (1 : Fin 2) = (0 : Fin 2) at h
|
||||
simp [Fin.isValue, zero_ne_one] at h
|
||||
intro h
|
||||
simp [detContinuous, h]
|
||||
|
||||
|
||||
/-- The representation taking a lorentz matrix to its determinant. -/
|
||||
@[simps!]
|
||||
def detRep : lorentzGroup →* ℤ₂ where
|
||||
toFun Λ := detContinuous Λ
|
||||
map_one' := by
|
||||
simp [detContinuous]
|
||||
map_mul' := by
|
||||
intro Λ1 Λ2
|
||||
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
|
||||
mul_ite, mul_one, ite_mul, one_mul]
|
||||
cases' (det_eq_one_or_neg_one Λ1) with h1 h1
|
||||
<;> cases' (det_eq_one_or_neg_one Λ2) with h2 h2
|
||||
<;> simp [h1, h2, detContinuous]
|
||||
rfl
|
||||
|
||||
lemma detRep_continuous : Continuous detRep := detContinuous.2
|
||||
|
||||
lemma det_on_connected_component {Λ Λ' : lorentzGroup} (h : Λ' ∈ connectedComponent Λ) :
|
||||
Λ.1.1.det = Λ'.1.1.det := by
|
||||
obtain ⟨s, hs, hΛ'⟩ := h
|
||||
let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
|
||||
haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
|
||||
simpa [f, detContinuous_eq_iff_det_eq] using
|
||||
(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
|
||||
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
|
||||
|
||||
lemma det_of_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') : Λ.1.1.det = Λ'.1.1.det :=
|
||||
det_on_connected_component $ pathComponent_subset_component _ h
|
||||
|
||||
/-- A Lorentz Matrix is proper if its determinant is 1. -/
|
||||
@[simp]
|
||||
def IsProper (Λ : lorentzGroup) : Prop := Λ.1.1.det = 1
|
||||
|
||||
instance : DecidablePred IsProper := by
|
||||
intro Λ
|
||||
apply Real.decidableEq
|
||||
|
||||
lemma IsProper_iff (Λ : lorentzGroup) : IsProper Λ ↔ detRep Λ = 1 := by
|
||||
rw [show 1 = detRep 1 by simp]
|
||||
rw [detRep_apply, detRep_apply, detContinuous_eq_iff_det_eq]
|
||||
simp only [IsProper, OneMemClass.coe_one, Units.val_one, det_one]
|
||||
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
end spaceTime
|
||||
end
|
||||
|
|
@ -24,7 +24,7 @@ open ComplexConjugate
|
|||
|
||||
/-- The metric as a `4×4` real matrix. -/
|
||||
def η : Matrix (Fin 4) (Fin 4) ℝ :=
|
||||
![![1, 0, 0, 0], ![0, -1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]
|
||||
!![1, 0, 0, 0; 0, -1, 0, 0; 0, 0, -1, 0; 0, 0, 0, -1]
|
||||
|
||||
lemma η_off_diagonal {μ ν : Fin 4} (h : μ ≠ ν) : η μ ν = 0 := by
|
||||
fin_cases μ <;>
|
||||
|
|
@ -46,8 +46,8 @@ lemma η_transpose : η.transpose = η := by
|
|||
rw [transpose_apply, η_symmetric]
|
||||
|
||||
lemma det_η : η.det = - 1 := by
|
||||
simp only [η, det_succ_row_zero, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val',
|
||||
empty_val', cons_val_fin_one, cons_val_zero, submatrix_apply, Fin.succ_zero_eq_one,
|
||||
simp only [η, det_succ_row_zero, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, of_apply,
|
||||
cons_val', empty_val', cons_val_fin_one, cons_val_zero, submatrix_apply, Fin.succ_zero_eq_one,
|
||||
cons_val_one, head_cons, submatrix_submatrix, Function.comp_apply, Fin.succ_one_eq_two,
|
||||
cons_val_two, tail_cons, det_unique, Fin.default_eq_zero, cons_val_succ, head_fin_const,
|
||||
Fin.sum_univ_succ, Fin.val_zero, pow_zero, one_mul, Fin.zero_succAbove, Finset.univ_unique,
|
||||
|
|
@ -67,6 +67,9 @@ lemma η_sq : η * η = 1 := by
|
|||
vecHead, vecTail, Function.comp_apply]
|
||||
|
||||
|
||||
lemma η_diag_mul_self (μ : Fin 4) : η μ μ * η μ μ = 1 := by
|
||||
fin_cases μ
|
||||
<;> simp [η]
|
||||
|
||||
lemma η_mulVec (x : spaceTime) : η *ᵥ x = ![x 0, -x 1, -x 2, -x 3] := by
|
||||
rw [explicit x]
|
||||
|
|
@ -92,7 +95,6 @@ def linearMapForSpaceTime (x : spaceTime) : spaceTime →ₗ[ℝ] ℝ where
|
|||
rfl
|
||||
|
||||
/-- The metric as a bilinear map from `spaceTime` to `ℝ`. -/
|
||||
@[simps!]
|
||||
def ηLin : LinearMap.BilinForm ℝ spaceTime where
|
||||
toFun x := linearMapForSpaceTime x
|
||||
map_add' x y := by
|
||||
|
|
@ -116,6 +118,39 @@ lemma ηLin_expand (x y : spaceTime) : ηLin x y = x 0 * y 0 - x 1 * y 1 - x 2 *
|
|||
cons_val_zero, cons_val_one, head_cons, mul_neg, cons_val_two, tail_cons, cons_val_three]
|
||||
ring
|
||||
|
||||
lemma ηLin_expand_self (x : spaceTime) : ηLin x x = x 0 ^ 2 - ‖x.space‖ ^ 2 := by
|
||||
rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three, ηLin_expand]
|
||||
simp only [Fin.isValue, space, cons_val_zero, RCLike.inner_apply, conj_trivial, cons_val_one,
|
||||
head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons]
|
||||
ring
|
||||
|
||||
lemma time_elm_sq_of_ηLin (x : spaceTime) : x 0 ^ 2 = ηLin x x + ‖x.space‖ ^ 2 := by
|
||||
rw [ηLin_expand_self]
|
||||
ring
|
||||
|
||||
lemma ηLin_leq_time_sq (x : spaceTime) : ηLin x x ≤ x 0 ^ 2 := by
|
||||
rw [time_elm_sq_of_ηLin]
|
||||
apply (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖x.space‖
|
||||
|
||||
lemma ηLin_space_inner_product (x y : spaceTime) :
|
||||
ηLin x y = x 0 * y 0 - ⟪x.space, y.space⟫_ℝ := by
|
||||
rw [ηLin_expand, @PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp only [Fin.isValue, space, cons_val_zero, RCLike.inner_apply, conj_trivial, cons_val_one,
|
||||
head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons]
|
||||
ring
|
||||
|
||||
lemma ηLin_ge_abs_inner_product (x y : spaceTime) :
|
||||
x 0 * y 0 - ‖⟪x.space, y.space⟫_ℝ‖ ≤ (ηLin x y) := by
|
||||
rw [ηLin_space_inner_product, sub_le_sub_iff_left]
|
||||
exact Real.le_norm_self ⟪x.space, y.space⟫_ℝ
|
||||
|
||||
lemma ηLin_ge_sub_norm (x y : spaceTime) :
|
||||
x 0 * y 0 - ‖x.space‖ * ‖y.space‖ ≤ (ηLin x y) := by
|
||||
apply le_trans ?_ (ηLin_ge_abs_inner_product x y)
|
||||
rw [sub_le_sub_iff_left]
|
||||
exact norm_inner_le_norm x.space y.space
|
||||
|
||||
|
||||
lemma ηLin_symm (x y : spaceTime) : ηLin x y = ηLin y x := by
|
||||
rw [ηLin_expand, ηLin_expand]
|
||||
ring
|
||||
|
|
@ -138,7 +173,7 @@ lemma ηLin_η_stdBasis (μ ν : Fin 4) : ηLin (stdBasis μ) (stdBasis ν) = η
|
|||
|
||||
lemma ηLin_mulVec_left (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
ηLin (Λ *ᵥ x) y = ηLin x ((η * Λᵀ * η) *ᵥ y) := by
|
||||
simp only [ηLin_apply_apply, mulVec_mulVec]
|
||||
simp only [ηLin, LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply, mulVec_mulVec]
|
||||
rw [(vecMul_transpose Λ x).symm, ← dotProduct_mulVec, mulVec_mulVec]
|
||||
rw [← mul_assoc, ← mul_assoc, η_sq, one_mul]
|
||||
|
||||
|
|
@ -151,12 +186,16 @@ lemma ηLin_matrix_stdBasis (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
|||
ηLin (stdBasis ν) (Λ *ᵥ stdBasis μ) = η ν ν * Λ ν μ := by
|
||||
rw [ηLin_stdBasis_apply, stdBasis_mulVec]
|
||||
|
||||
lemma ηLin_matrix_stdBasis' (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
Λ ν μ = η ν ν * ηLin (stdBasis ν) (Λ *ᵥ stdBasis μ) := by
|
||||
rw [ηLin_matrix_stdBasis, ← mul_assoc, η_diag_mul_self, one_mul]
|
||||
|
||||
lemma ηLin_matrix_eq_identity_iff (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
|
||||
Λ = 1 ↔ ∀ (x y : spaceTime), ηLin x y = ηLin x (Λ *ᵥ y) := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
subst h
|
||||
simp only [ηLin_apply_apply, one_mulVec, implies_true]
|
||||
simp only [ηLin, one_mulVec, implies_true]
|
||||
intro h
|
||||
funext μ ν
|
||||
have h1 := h (stdBasis μ) (stdBasis ν)
|
||||
|
|
|
|||
19
README.md
19
README.md
|
|
@ -1,16 +1,24 @@
|
|||
# High Energy Physics in Lean
|
||||
|
||||
[](https://heplean.github.io/HepLean/)
|
||||
[](https://github.com/HEPLean/HepLean/pulls)
|
||||
[](https://leanprover.zulipchat.com)
|
||||
|
||||
A project to digitalize high energy physics.
|
||||
|
||||
## Aims of this project
|
||||
|
||||
- Use Lean to create a exhaustive database of definitions, theorems, proofs and calculations in high energy physics.
|
||||
- Make a libary that is easy to use by the high energy physics community.
|
||||
- Use Lean to create an exhaustive database of definitions, theorems, proofs, and calculations in high energy physics.
|
||||
- Make a library that is easy to use by the high energy physics community.
|
||||
- Keep the database up-to date with developments in MathLib4.
|
||||
- Create github workflows of relevence to the high energy physics community.
|
||||
- Create gitHub workflows of relevance to the high energy physics community.
|
||||
|
||||
## Where to learn more
|
||||
|
||||
- Read the associated paper:
|
||||
|
||||
https://arxiv.org/abs/2405.08863
|
||||
|
||||
- The documentation for this project is at:
|
||||
|
||||
https://heplean.github.io/HepLean/
|
||||
|
|
@ -27,13 +35,14 @@ A project to digitalize high energy physics.
|
|||
|
||||
- A small example script relating to the three fermion anomaly cancellation condition can be found [here](https://live.lean-lang.org/#code=import%20Mathlib.Tactic.Polyrith%20%0A%0Atheorem%20threeFamily%20(a%20b%20c%20%3A%20ℚ)%20(h%20%3A%20a%20%2B%20b%20%2B%20c%20%3D%200)%20(h3%20%3A%20a%20%5E%203%20%2B%20b%20%5E%203%20%2B%20c%20%5E%203%20%3D%200)%20%3A%20%0A%20%20%20%20a%20%3D%200%20∨%20b%20%3D%200%20∨%20c%20%3D%200%20%20%3A%3D%20by%20%0A%20%20have%20h1%20%3A%20c%20%3D%20-%20(a%20%2B%20b)%20%3A%3D%20by%20%0A%20%20%20%20linear_combination%20h%20%0A%20%20have%20h4%20%3A%20%203%20*%20a%20*%20b%20*%20c%20%3D%200%20%3A%3D%20by%20%0A%20%20%20%20rw%20%5B←%20h3%2C%20h1%5D%0A%20%20%20%20ring%20%0A%20%20simp%20at%20h4%20%0A%20%20exact%20or_assoc.mp%20h4%0A%20%20%0A)
|
||||
|
||||
|
||||
|
||||
## Contributing
|
||||
|
||||
We follow here roughly the same contribution policies as MathLib4 (which can be found [here](https://leanprover-community.github.io/contribute/index.html)).
|
||||
|
||||
If you want permission to create a pull-request for this repository contact Joseph Tooby-Smith on the lean Zulip, or email.
|
||||
A guide to contributing can be found [here](https://github.com/HEPLean/HepLean/blob/master/CONTRIBUTING.md).
|
||||
|
||||
If you want permission to create a pull-request for this repository contact Joseph Tooby-Smith on the lean Zulip, or email.
|
||||
|
||||
## Installation
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue