feat: Properties of 2HDM

HepLean/StandardModel/HiggsBoson/PointwiseInnerProd.lean

@@ -66,6 +66,14 @@ lemma innerProd_right_zero (φ : HiggsField) : ⟪φ, 0⟫_H = 0 := by
   funext x
   simp [innerProd]
 
+/-- Expands the inner product on Higgs fields in terms of complex components of the
+  Higgs fields. -/
+lemma innerProd_expand' (φ1 φ2 : HiggsField) (x : SpaceTime) :
+    ⟪φ1, φ2⟫_H x = conj (φ1 x 0) * φ2 x 0 + conj (φ1 x 1) * φ2 x 1 := by
+  simp [innerProd]
+
+/-- Expands the inner product on Higgs fields in terms of real components of the
+  Higgs fields. -/
 lemma innerProd_expand (φ1 φ2 : HiggsField) :
     ⟪φ1, φ2⟫_H = fun x => equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
     + (φ1 x 1).re * (φ2 x 1).re+ (φ1 x 0).im * (φ2 x 0).im + (φ1 x 1).im * (φ2 x 1).im),