Pauli Basis' repr inr 0
PauliMatrix.pauliBasis'_repr_inr_0
Plain-language statement
The component of a self-adjoint matrix in the direction -σ1 under the basis formed by the covariant Pauli matrices.
Exact Lean statement
@[simp]
lemma pauliBasis'_repr_inr_0 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
pauliBasis'.repr M (Sum.inr 0) = - 1 / 2 * Matrix.trace (σ1 * M.1)Formal artifact
Lean source
@[simp]lemma pauliBasis'_repr_inr_0 (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) : pauliBasis'.repr M (Sum.inr 0) = - 1 / 2 * Matrix.trace (σ1 * M.1) := by have hM : M = ∑ i, pauliBasis'.repr M i • pauliBasis' i := (Basis.sum_repr pauliBasis' M).symm simp only [Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, Fin.sum_univ_three] at hM have h0 := congrArg (fun A => - Matrix.trace (σ1 * A.1)/ 2) hM simp only [Fin.isValue, pauliBasis', Basis.mk_repr, Basis.coe_mk, pauliSelfAdjoint', AddSubgroup.coe_add, selfAdjoint.val_smul, smul_neg, mul_add, Algebra.mul_smul_comm, mul_neg, trace_add, trace_smul, σ1_σ0_trace, smul_zero, trace_neg, σ1_σ1_trace, real_smul, σ1_σ2_trace, neg_zero, add_zero, σ1_σ3_trace, zero_add, neg_neg, isUnit_iff_ne_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, IsUnit.mul_div_cancel_right] at h0 linear_combination (norm := ring_nf) -h0 simp [pauliBasis']- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/PauliMatrices/SelfAdjoint.lean:268-282
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