139 lines
3.4 KiB
Text
139 lines
3.4 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Mathematics.PiTensorProduct
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import Mathlib.RepresentationTheory.Rep
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import HepLean.Tensors.Basic
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import Mathlib.Logic.Equiv.TransferInstance
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import HepLean.SpaceTime.LorentzGroup.Basic
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import LLMLean
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/-!
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## Pauli matrices
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-/
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namespace PauliMatrix
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open Complex
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open Matrix
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/-- The zeroth Pauli-matrix as a `2 x 2` complex matrix. -/
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def σ0 : Matrix (Fin 2) (Fin 2) ℂ := !![1, 0; 0, 1]
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/-- The first Pauli-matrix as a `2 x 2` complex matrix. -/
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def σ1 : Matrix (Fin 2) (Fin 2) ℂ := !![0, 1; 1, 0]
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/-- The second Pauli-matrix as a `2 x 2` complex matrix. -/
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def σ2 : Matrix (Fin 2) (Fin 2) ℂ := !![0, -I; I, 0]
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/-- The third Pauli-matrix as a `2 x 2` complex matrix. -/
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def σ3 : Matrix (Fin 2) (Fin 2) ℂ := !![1, 0; 0, -1]
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@[simp]
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lemma σ0_selfAdjoint : σ0ᴴ = σ0 := by
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rw [eta_fin_two σ0ᴴ]
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simp [σ0]
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@[simp]
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lemma σ1_selfAdjoint : σ1ᴴ = σ1 := by
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rw [eta_fin_two σ1ᴴ]
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simp [σ1]
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@[simp]
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lemma σ2_selfAdjoint : σ2ᴴ = σ2 := by
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rw [eta_fin_two σ2ᴴ]
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simp [σ2]
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@[simp]
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lemma σ3_selfAdjoint : σ3ᴴ = σ3 := by
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rw [eta_fin_two σ3ᴴ]
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simp [σ3]
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/-!
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## Traces
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-/
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@[simp]
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lemma σ0_σ0_trace : Matrix.trace (σ0 * σ0) = 2 := by
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simp only [σ0, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd, vecMul_cons, head_cons, one_smul,
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tail_cons, zero_smul, empty_vecMul, add_zero, zero_add, empty_mul, Equiv.symm_apply_apply,
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trace_fin_two_of]
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norm_num
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@[simp]
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lemma σ0_σ1_trace : Matrix.trace (σ0 * σ1) = 0 := by
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simp [σ0, σ1]
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@[simp]
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lemma σ0_σ2_trace : Matrix.trace (σ0 * σ2) = 0 := by
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simp [σ0, σ2]
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@[simp]
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lemma σ0_σ3_trace : Matrix.trace (σ0 * σ3) = 0 := by
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simp [σ0, σ3]
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@[simp]
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lemma σ1_σ0_trace : Matrix.trace (σ1 * σ0) = 0 := by
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simp [σ1, σ0]
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@[simp]
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lemma σ1_σ1_trace : Matrix.trace (σ1 * σ1) = 2 := by
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simp only [σ1, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd, vecMul_cons, head_cons, one_smul,
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tail_cons, zero_smul, empty_vecMul, add_zero, zero_add, empty_mul, Equiv.symm_apply_apply,
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trace_fin_two_of]
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norm_num
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@[simp]
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lemma σ1_σ2_trace : Matrix.trace (σ1 * σ2) = 0 := by
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simp [σ1, σ2]
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@[simp]
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lemma σ1_σ3_trace : Matrix.trace (σ1 * σ3) = 0 := by
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simp [σ1, σ3]
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@[simp]
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lemma σ2_σ0_trace : Matrix.trace (σ2 * σ0) = 0 := by
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simp [σ2, σ0]
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@[simp]
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lemma σ2_σ1_trace : Matrix.trace (σ2 * σ1) = 0 := by
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simp [σ2, σ1]
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@[simp]
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lemma σ2_σ2_trace : Matrix.trace (σ2 * σ2) = 2 := by
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simp only [σ2, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd, vecMul_cons, head_cons, one_smul,
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tail_cons, zero_smul, empty_vecMul, add_zero, zero_add, empty_mul, Equiv.symm_apply_apply,
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trace_fin_two_of]
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norm_num
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@[simp]
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lemma σ2_σ3_trace : Matrix.trace (σ2 * σ3) = 0 := by
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simp [σ2, σ3]
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@[simp]
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lemma σ3_σ0_trace : Matrix.trace (σ3 * σ0) = 0 := by
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simp [σ3, σ0]
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@[simp]
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lemma σ3_σ1_trace : Matrix.trace (σ3 * σ1) = 0 := by
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simp [σ3, σ1]
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@[simp]
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lemma σ3_σ2_trace : Matrix.trace (σ3 * σ2) = 0 := by
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simp [σ3, σ2]
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@[simp]
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lemma σ3_σ3_trace : Matrix.trace (σ3 * σ3) = 2 := by
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simp only [σ3, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd, vecMul_cons, head_cons, one_smul,
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tail_cons, zero_smul, empty_vecMul, add_zero, zero_add, empty_mul, Equiv.symm_apply_apply,
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trace_fin_two_of]
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norm_num
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end PauliMatrix
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