feat: Add Pauli-matrices as tensor.

HepLean/SpaceTime/PauliMatrices/Basic.lean

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