feat: Add Pauli-matrices as tensor.
This commit is contained in:
jstoobysmith
parent
a60ade65f0
commit
691b7e112e
13 changed files with 989 additions and 154 deletions
|
|
@ -4,7 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.MinkowskiMetric
|
||||
import HepLean.SpaceTime.PauliMatrices.SelfAdjoint
|
||||
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
|
||||
import Mathlib.Tactic.Polyrith
|
||||
import LLMLean
|
||||
/-!
|
||||
# Lorentz vector as a self-adjoint matrix
|
||||
|
||||
|
|
@ -20,150 +23,79 @@ namespace SpaceTime
|
|||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
|
||||
/-- A 2×2-complex matrix formed from a Lorentz vector point. -/
|
||||
@[simp]
|
||||
def toMatrix (x : LorentzVector 3) : Matrix (Fin 2) (Fin 2) ℂ :=
|
||||
!![x.time + x.space 2, x.space 0 - x.space 1 * I; x.space 0 + x.space 1 * I, x.time - x.space 2]
|
||||
|
||||
/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
|
||||
lemma toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (toMatrix x) := by
|
||||
rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]
|
||||
intro i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
simp only [toMatrix, LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply,
|
||||
Fin.zero_eta, conjTranspose_apply, of_apply, cons_val', cons_val_zero, empty_val',
|
||||
cons_val_fin_one, star_add, RCLike.star_def, conj_ofReal, Fin.mk_one, cons_val_one,
|
||||
head_fin_const, star_mul', conj_I, mul_neg, head_cons, star_sub, sub_neg_eq_add]
|
||||
rfl
|
||||
|
||||
/-- A self-adjoint matrix formed from a Lorentz vector point. -/
|
||||
@[simps!]
|
||||
def toSelfAdjointMatrix' (x : LorentzVector 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
|
||||
⟨toMatrix x, toMatrix_isSelfAdjoint x⟩
|
||||
|
||||
/-- A self-adjoint matrix formed from a Lorentz vector point. -/
|
||||
@[simp]
|
||||
noncomputable def fromSelfAdjointMatrix' (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
|
||||
LorentzVector 3 := fun i =>
|
||||
match i with
|
||||
| Sum.inl 0 => 1/2 * (x.1 0 0 + x.1 1 1).re
|
||||
| Sum.inr 0 => (x.1 1 0).re
|
||||
| Sum.inr 1 => (x.1 1 0).im
|
||||
| Sum.inr 2 => 1/2 * (x.1 0 0 - x.1 1 1).re
|
||||
noncomputable section
|
||||
|
||||
/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint
|
||||
2×2-complex matrices. -/
|
||||
noncomputable def toSelfAdjointMatrix :
|
||||
LorentzVector 3 ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
|
||||
toFun := toSelfAdjointMatrix'
|
||||
invFun := fromSelfAdjointMatrix'
|
||||
left_inv x := by
|
||||
funext i
|
||||
match i with
|
||||
| Sum.inl 0 =>
|
||||
simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix', toMatrix,
|
||||
LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply,
|
||||
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
|
||||
head_fin_const, add_add_sub_cancel, add_re, ofReal_re]
|
||||
ring_nf
|
||||
| Sum.inr 0 =>
|
||||
simp [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, toMatrix_isSelfAdjoint]
|
||||
| Sum.inr 1 =>
|
||||
simp only [fromSelfAdjointMatrix', toSelfAdjointMatrix', toMatrix, LorentzVector.time,
|
||||
Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply, cons_val', cons_val_zero,
|
||||
empty_val', cons_val_fin_one, cons_val_one, head_fin_const, add_im, ofReal_im, mul_im,
|
||||
ofReal_re, I_im, mul_one, I_re, mul_zero, add_zero, zero_add]
|
||||
| Sum.inr 2 =>
|
||||
simp only [fromSelfAdjointMatrix', one_div, toSelfAdjointMatrix', toMatrix,
|
||||
LorentzVector.time, Fin.isValue, LorentzVector.space, Function.comp_apply, of_apply,
|
||||
cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, head_cons,
|
||||
head_fin_const, add_sub_sub_cancel, add_re, ofReal_re]
|
||||
ring
|
||||
right_inv x := by
|
||||
simp only [toSelfAdjointMatrix', toMatrix, LorentzVector.time, fromSelfAdjointMatrix', one_div,
|
||||
Fin.isValue, add_re, ofReal_mul, ofReal_inv, ofReal_ofNat, ofReal_add, LorentzVector.space,
|
||||
Function.comp_apply, sub_re, ofReal_sub, re_add_im]
|
||||
ext i j
|
||||
fin_cases i <;> fin_cases j <;>
|
||||
field_simp [fromSelfAdjointMatrix', toMatrix, conj_ofReal]
|
||||
· exact conj_eq_iff_re.mp (congrArg (fun M => M 0 0) $ selfAdjoint.mem_iff.mp x.2)
|
||||
· have h01 := congrArg (fun M => M 0 1) $ selfAdjoint.mem_iff.mp x.2
|
||||
simp only [Fin.isValue, star_apply, RCLike.star_def] at h01
|
||||
rw [← h01, RCLike.conj_eq_re_sub_im]
|
||||
rfl
|
||||
· exact conj_eq_iff_re.mp (congrArg (fun M => M 1 1) $ selfAdjoint.mem_iff.mp x.2)
|
||||
map_add' x y := by
|
||||
ext i j : 2
|
||||
simp only [toSelfAdjointMatrix'_coe, add_apply, ofReal_add, of_apply, cons_val', empty_val',
|
||||
cons_val_fin_one, AddSubmonoid.coe_add, AddSubgroup.coe_toAddSubmonoid, Matrix.add_apply]
|
||||
fin_cases i <;> fin_cases j
|
||||
· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
|
||||
rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
|
||||
simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero]
|
||||
simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
|
||||
LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
|
||||
head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_zero, empty_val',
|
||||
cons_val_fin_one]
|
||||
ring
|
||||
· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
|
||||
rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
|
||||
simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
|
||||
cons_val_zero]
|
||||
simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
|
||||
LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
|
||||
head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_one, empty_val',
|
||||
cons_val_fin_one, cons_val_zero]
|
||||
ring
|
||||
· rw [show (x + y) (Sum.inr 0) = x (Sum.inr 0) + y (Sum.inr 0) from rfl]
|
||||
rw [show (x + y) (Sum.inr 1) = x (Sum.inr 1) + y (Sum.inr 1) from rfl]
|
||||
simp only [Fin.isValue, ofReal_add, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
|
||||
head_fin_const]
|
||||
simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
|
||||
LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
|
||||
head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_zero, empty_val',
|
||||
cons_val_fin_one, cons_val_one, head_fin_const]
|
||||
ring
|
||||
· rw [show (x + y) (Sum.inl 0) = x (Sum.inl 0) + y (Sum.inl 0) from rfl]
|
||||
rw [show (x + y) (Sum.inr 2) = x (Sum.inr 2) + y (Sum.inr 2) from rfl]
|
||||
simp only [Fin.isValue, ofReal_add, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
|
||||
simp only [Fin.isValue, toSelfAdjointMatrix', toMatrix, LorentzVector.time,
|
||||
LorentzVector.space, Function.comp_apply, AddMemClass.mk_add_mk, of_add_of, add_cons,
|
||||
head_cons, tail_cons, empty_add_empty, of_apply, cons_val', cons_val_one, empty_val',
|
||||
cons_val_fin_one, head_fin_const]
|
||||
ring
|
||||
map_smul' r x := by
|
||||
ext i j : 2
|
||||
simp only [toSelfAdjointMatrix'_coe, Fin.isValue, of_apply, cons_val', empty_val',
|
||||
cons_val_fin_one, RingHom.id_apply, selfAdjoint.val_smul, smul_apply, real_smul]
|
||||
fin_cases i <;> fin_cases j
|
||||
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
|
||||
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
|
||||
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero]
|
||||
ring
|
||||
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
|
||||
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
|
||||
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
|
||||
cons_val_zero]
|
||||
ring
|
||||
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
|
||||
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
|
||||
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
|
||||
head_fin_const]
|
||||
ring
|
||||
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
|
||||
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
|
||||
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
|
||||
ring
|
||||
`2×2`-complex matrices. -/
|
||||
def toSelfAdjointMatrix : LorentzVector 3 ≃ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
|
||||
(Finsupp.linearEquivFunOnFinite ℝ ℝ (Fin 1 ⊕ Fin 3)).symm ≪≫ₗ PauliMatrix.σSAL.repr.symm
|
||||
|
||||
lemma toSelfAdjointMatrix_apply (x : LorentzVector 3) : toSelfAdjointMatrix x =
|
||||
x (Sum.inl 0) • ⟨PauliMatrix.σ0, PauliMatrix.σ0_selfAdjoint⟩
|
||||
- x (Sum.inr 0) • ⟨PauliMatrix.σ1, PauliMatrix.σ1_selfAdjoint⟩
|
||||
- x (Sum.inr 1) • ⟨PauliMatrix.σ2, PauliMatrix.σ2_selfAdjoint⟩
|
||||
- x (Sum.inr 2) • ⟨PauliMatrix.σ3, PauliMatrix.σ3_selfAdjoint⟩ := by
|
||||
simp only [toSelfAdjointMatrix, PauliMatrix.σSAL, LinearEquiv.trans_apply, Basis.repr_symm_apply,
|
||||
Basis.coe_mk, Fin.isValue]
|
||||
rw [Finsupp.linearCombination_apply_of_mem_supported ℝ (s := Finset.univ)]
|
||||
· change (∑ i : Fin 1 ⊕ Fin 3, x i • PauliMatrix.σSAL' i) = _
|
||||
simp [Fin.sum_univ_three, PauliMatrix.σSAL']
|
||||
apply Subtype.ext
|
||||
simp only [Fin.isValue, AddSubgroup.coe_add, selfAdjoint.val_smul, smul_neg,
|
||||
AddSubgroupClass.coe_sub]
|
||||
simp only [neg_add, add_assoc, sub_eq_add_neg]
|
||||
· simp_all only [Finset.coe_univ, Finsupp.supported_univ, Submodule.mem_top]
|
||||
|
||||
lemma toSelfAdjointMatrix_apply_coe (x : LorentzVector 3) : (toSelfAdjointMatrix x).1 =
|
||||
x (Sum.inl 0) • PauliMatrix.σ0
|
||||
- x (Sum.inr 0) • PauliMatrix.σ1
|
||||
- x (Sum.inr 1) • PauliMatrix.σ2
|
||||
- x (Sum.inr 2) • PauliMatrix.σ3 := by
|
||||
rw [toSelfAdjointMatrix_apply]
|
||||
simp only [Fin.isValue, AddSubgroupClass.coe_sub, selfAdjoint.val_smul]
|
||||
|
||||
lemma toSelfAdjointMatrix_stdBasis (i : Fin 1 ⊕ Fin 3) :
|
||||
toSelfAdjointMatrix (LorentzVector.stdBasis i) = PauliMatrix.σSAL i := by
|
||||
rw [toSelfAdjointMatrix_apply]
|
||||
match i with
|
||||
| Sum.inl 0 =>
|
||||
simp [LorentzVector.stdBasis]
|
||||
erw [Pi.basisFun_apply]
|
||||
simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
|
||||
| Sum.inr 0 =>
|
||||
simp [LorentzVector.stdBasis]
|
||||
erw [Pi.basisFun_apply]
|
||||
simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
|
||||
refine Eq.symm (PauliMatrix.selfAdjoint_ext rfl rfl rfl rfl)
|
||||
| Sum.inr 1 =>
|
||||
simp [LorentzVector.stdBasis]
|
||||
erw [Pi.basisFun_apply]
|
||||
simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
|
||||
refine Eq.symm (PauliMatrix.selfAdjoint_ext rfl rfl rfl rfl)
|
||||
| Sum.inr 2 =>
|
||||
simp [LorentzVector.stdBasis]
|
||||
erw [Pi.basisFun_apply]
|
||||
simp [PauliMatrix.σSAL, PauliMatrix.σSAL' ]
|
||||
refine Eq.symm (PauliMatrix.selfAdjoint_ext rfl rfl rfl rfl)
|
||||
|
||||
@[simp]
|
||||
lemma toSelfAdjointMatrix_symm_basis (i : Fin 1 ⊕ Fin 3) :
|
||||
toSelfAdjointMatrix.symm (PauliMatrix.σSAL i) = (LorentzVector.stdBasis i) := by
|
||||
refine (LinearEquiv.symm_apply_eq toSelfAdjointMatrix).mpr ?_
|
||||
rw [toSelfAdjointMatrix_stdBasis]
|
||||
|
||||
open minkowskiMetric in
|
||||
lemma det_eq_ηLin (x : LorentzVector 3) : det (toSelfAdjointMatrix x).1 = ⟪x, x⟫ₘ := by
|
||||
simp only [toSelfAdjointMatrix, LinearEquiv.coe_mk, toSelfAdjointMatrix'_coe, Fin.isValue,
|
||||
det_fin_two_of, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,
|
||||
rw [toSelfAdjointMatrix_apply_coe]
|
||||
simp only [Fin.isValue, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,
|
||||
PiLp.inner_apply, Function.comp_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_three,
|
||||
ofReal_sub, ofReal_mul, ofReal_add]
|
||||
simp only [Fin.isValue, PauliMatrix.σ0, smul_of, smul_cons, real_smul, mul_one, smul_zero,
|
||||
smul_empty, PauliMatrix.σ1, of_sub_of, sub_cons, head_cons, sub_zero, tail_cons, zero_sub,
|
||||
sub_self, zero_empty, PauliMatrix.σ2, smul_neg, sub_neg_eq_add, PauliMatrix.σ3, det_fin_two_of]
|
||||
ring_nf
|
||||
simp only [Fin.isValue, I_sq, mul_neg, mul_one]
|
||||
ring
|
||||
|
||||
end
|
||||
end SpaceTime
|
||||
|
|
|
|||
|
|
@ -10,6 +10,7 @@ import HepLean.SpaceTime.LorentzVector.Modules
|
|||
import HepLean.Meta.Informal
|
||||
import Mathlib.RepresentationTheory.Rep
|
||||
import HepLean.Tensors.Basic
|
||||
import HepLean.SpaceTime.PauliMatrices.SelfAdjoint
|
||||
/-!
|
||||
|
||||
# Complex Lorentz vectors
|
||||
|
|
@ -49,6 +50,11 @@ lemma complexContrBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
|
|||
change (((LorentzGroup.toComplex (SL2C.toLorentzGroup M))) *ᵥ (Pi.single j 1)) i = _
|
||||
simp only [mulVec_single, transpose_apply, mul_one]
|
||||
|
||||
lemma complexContrBasis_ρ_val (M : SL(2,ℂ)) (v : complexContr) :
|
||||
((complexContr.ρ M) v).val =
|
||||
LorentzGroup.toComplex (SL2C.toLorentzGroup M) *ᵥ v.val := by
|
||||
rfl
|
||||
|
||||
/-- The standard basis of complex covariant Lorentz vectors. -/
|
||||
def complexCoBasis : Basis (Fin 1 ⊕ Fin 3) ℂ complexCo := Basis.ofEquivFun
|
||||
(Equiv.linearEquiv ℂ CoℂModule.toFin13ℂFun)
|
||||
|
|
@ -62,5 +68,63 @@ lemma complexCoBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
|
|||
change ((LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ *ᵥ (Pi.single j 1)) i = _
|
||||
simp only [mulVec_single, transpose_apply, mul_one]
|
||||
|
||||
/-!
|
||||
|
||||
## Relation to real
|
||||
|
||||
-/
|
||||
|
||||
def inclCongrRealLorentz : LorentzVector 3 →ₛₗ[Complex.ofReal] complexContr where
|
||||
toFun v := {val := ofReal ∘ v}
|
||||
map_add' x y := by
|
||||
apply Lorentz.ContrℂModule.ext
|
||||
rw [Lorentz.ContrℂModule.val_add]
|
||||
funext i
|
||||
simp only [Function.comp_apply, ofReal_eq_coe, Pi.add_apply]
|
||||
change ofReal (x i + y i) = _
|
||||
simp only [ofReal_eq_coe, ofReal_add]
|
||||
map_smul' c x := by
|
||||
apply Lorentz.ContrℂModule.ext
|
||||
rw [Lorentz.ContrℂModule.val_smul]
|
||||
funext i
|
||||
simp only [Function.comp_apply, ofReal_eq_coe, Pi.smul_apply]
|
||||
change ofReal (c • x i) = _
|
||||
simp only [smul_eq_mul, ofReal_eq_coe, ofReal_mul]
|
||||
|
||||
lemma inclCongrRealLorentz_val (v : LorentzVector 3) :
|
||||
(inclCongrRealLorentz v).val = ofReal ∘ v := rfl
|
||||
|
||||
lemma complexContrBasis_of_real (i : Fin 1 ⊕ Fin 3) :
|
||||
(complexContrBasis i) = inclCongrRealLorentz (LorentzVector.stdBasis i) := by
|
||||
apply Lorentz.ContrℂModule.ext
|
||||
simp [complexContrBasis, inclCongrRealLorentz, LorentzVector.stdBasis, ]
|
||||
ext j
|
||||
simp
|
||||
erw [Pi.basisFun_apply]
|
||||
change (Pi.single i 1) j = _
|
||||
exact Eq.symm (Pi.apply_single (fun _ => ofReal') (congrFun rfl) i 1 j)
|
||||
|
||||
lemma inclCongrRealLorentz_ρ (M : SL(2, ℂ)) (v : LorentzVector 3) :
|
||||
(complexContr.ρ M) (inclCongrRealLorentz v) =
|
||||
inclCongrRealLorentz (SL2C.repLorentzVector M v) := by
|
||||
apply Lorentz.ContrℂModule.ext
|
||||
rw [complexContrBasis_ρ_val, inclCongrRealLorentz_val, inclCongrRealLorentz_val]
|
||||
rw [LorentzGroup.toComplex_mulVec_ofReal]
|
||||
apply congrArg
|
||||
simp
|
||||
rw [SL2C.repLorentzVector_apply_eq_mulVec]
|
||||
rfl
|
||||
|
||||
|
||||
lemma SL2CRep_ρ_basis (M : SL(2, ℂ)) (i : Fin 1 ⊕ Fin 3) :
|
||||
(complexContr.ρ M) (complexContrBasis i) =
|
||||
∑ j, (SL2C.toLorentzGroup M).1 j i •
|
||||
complexContrBasis j := by
|
||||
rw [complexContrBasis_of_real, inclCongrRealLorentz_ρ, SL2C.repLorentzVector_stdBasis, map_sum]
|
||||
apply congrArg
|
||||
funext j
|
||||
simp only [LinearMap.map_smulₛₗ, ofReal_eq_coe, coe_smul]
|
||||
rw [complexContrBasis_of_real]
|
||||
|
||||
end Lorentz
|
||||
end
|
||||
|
|
|
|||
|
|
@ -32,6 +32,7 @@ structure ContrℂModule where
|
|||
|
||||
namespace ContrℂModule
|
||||
|
||||
|
||||
/-- The equivalence between `ContrℂModule` and `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||||
def toFin13ℂFun : ContrℂModule ≃ (Fin 1 ⊕ Fin 3 → ℂ) where
|
||||
toFun v := v.val
|
||||
|
|
@ -51,6 +52,19 @@ instance : AddCommGroup ContrℂModule := Equiv.addCommGroup toFin13ℂFun
|
|||
with `Fin 1 ⊕ Fin 3 → ℂ`. -/
|
||||
instance : Module ℂ ContrℂModule := Equiv.module ℂ toFin13ℂFun
|
||||
|
||||
@[ext]
|
||||
lemma ext (ψ ψ' : ContrℂModule) (h : ψ.val = ψ'.val) : ψ = ψ' := by
|
||||
cases ψ
|
||||
cases ψ'
|
||||
subst h
|
||||
simp_all only
|
||||
|
||||
@[simp]
|
||||
lemma val_add (ψ ψ' : ContrℂModule) : (ψ + ψ').val = ψ.val + ψ'.val := rfl
|
||||
|
||||
@[simp]
|
||||
lemma val_smul (r : ℂ) (ψ : ContrℂModule) : (r • ψ).val = r • ψ.val := rfl
|
||||
|
||||
/-- The linear equivalence between `ContrℂModule` and `(Fin 1 ⊕ Fin 3 → ℂ)`. -/
|
||||
@[simps!]
|
||||
def toFin13ℂEquiv : ContrℂModule ≃ₗ[ℂ] (Fin 1 ⊕ Fin 3 → ℂ) where
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue