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refactor: Spelling

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jstoobysmith 2025-02-07 15:43:59 +00:00
parent b82791d671
commit f8f1e1757f
25 changed files with 80 additions and 80 deletions
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16
HepLean/Lorentz/PauliMatrices/SelfAdjoint.lean
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@ -12,7 +12,7 @@ import HepLean.Lorentz.PauliMatrices.Basic
namespace PauliMatrix
open Matrix
/-- The trace of `σ0` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ0` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ0_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ0 * A.1)).re = Matrix.trace (σ0 * A.1) := by
rw [eta_fin_two A.1]
@ -35,7 +35,7 @@ lemma selfAdjoint_trace_σ0_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
cons_val_fin_one, head_fin_const] at h11
exact Complex.conj_eq_iff_re.mp h11
/-- The trace of `σ1` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ1` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ1_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ1 * A.1)).re = Matrix.trace (σ1 * A.1) := by
rw [eta_fin_two A.1]
@ -55,7 +55,7 @@ lemma selfAdjoint_trace_σ1_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
simp only [Fin.isValue, Complex.ofReal_mul, Complex.ofReal_ofNat]
ring
/-- The trace of `σ2` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ2` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ2_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ2 * A.1)).re = Matrix.trace (σ2 * A.1) := by
rw [eta_fin_two A.1]
@ -79,7 +79,7 @@ lemma selfAdjoint_trace_σ2_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
simp
· ring
/-- The trace of `σ3` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ3` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ3_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ3 * A.1)).re = Matrix.trace (σ3 * A.1) := by
rw [eta_fin_two A.1]
@ -100,7 +100,7 @@ lemma selfAdjoint_trace_σ3_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
open Complex
/-- Two `2×2` self-adjiont matrices are equal if the (complex) traces of each matrix multiplied by
/-- Two `2×2` self-adjoint matrices are equal if the (complex) traces of each matrix multiplied by
each of the Pauli-matrices are equal. -/
lemma selfAdjoint_ext_complex {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
(h0 : Matrix.trace (PauliMatrix.σ0 * A.1) = Matrix.trace (PauliMatrix.σ0 * B.1))
@ -133,7 +133,7 @@ lemma selfAdjoint_ext_complex {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
| 1, 1 =>
linear_combination (norm := ring_nf) (h0 - h3) / 2
/-- Two `2×2` self-adjiont matrices are equal if the real traces of each matrix multiplied by
/-- Two `2×2` self-adjoint matrices are equal if the real traces of each matrix multiplied by
each of the Pauli-matrices are equal. -/
lemma selfAdjoint_ext {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
(h0 : ((Matrix.trace (PauliMatrix.σ0 * A.1))).re = ((Matrix.trace (PauliMatrix.σ0 * B.1))).re)
@ -154,7 +154,7 @@ lemma selfAdjoint_ext {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
noncomputable section
/-- An auxillary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
/-- An auxiliary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
Pauli-matrix as a self-adjoint matrix. -/
def σSA' (i : Fin 1 ⊕ Fin 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
match i with
@ -225,7 +225,7 @@ lemma σSA_span : ⊤ ≤ Submodule.span ℝ (Set.range σSA') := by
def σSA : Basis (Fin 1 ⊕ Fin 3) ℝ (selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :=
Basis.mk σSA_linearly_independent σSA_span
/-- An auxillary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
/-- An auxiliary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
Pauli-matrix as a self-adjoint matrix with a minus sign for `Sum.inr _`. -/
def σSAL' (i : Fin 1 ⊕ Fin 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
match i with