refactor: Lorentz action on tensors
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jstoobysmith
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d385f72087
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757afbc60f
3 changed files with 298 additions and 119 deletions
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@ -38,7 +38,7 @@ def ofReal (d : ℕ) (r : ℝ) : RealLorentzTensor d Empty where
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def ofVecUp {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.up
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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coord := fun i => v <| i <| Sum.inr <| 0
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/-- A marked 1-tensor with a single down index constructed from a vector.
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@ -46,7 +46,7 @@ def ofVecUp {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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def ofVecDown {d : ℕ} (v : Fin 1 ⊕ Fin d → ℝ) :
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Marked d Empty 1 where
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color := fun _ => Colors.down
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coord := fun i => v <| i <| Sum.inr <| ⟨0, PUnit.unit⟩
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coord := fun i => v <| i <| Sum.inr <| 0
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/-- A tensor with two up indices constructed from a matrix.
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@ -54,7 +54,7 @@ Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatUpUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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coord := fun i => m (i (Sum.inr 0)) (i (Sum.inr 1))
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/-- A tensor with two down indices constructed from a matrix.
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@ -62,7 +62,7 @@ Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun _ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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coord := fun i => m (i (Sum.inr 0)) (i (Sum.inr 1))
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/-- A marked 2-tensor with the first index up and the second index down.
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@ -71,9 +71,9 @@ Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatUpDown {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.up
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.down
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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| Sum.inr 0 => Colors.up
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| Sum.inr 1 => Colors.down
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coord := fun i => m (i (Sum.inr 0)) (i (Sum.inr 1))
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/-- A marked 2-tensor with the first index down and the second index up.
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@ -81,9 +81,9 @@ Note: This is not the same as rising or lowering indices on other `ofMat...`. -/
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def ofMatDownUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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Marked d Empty 2 where
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color := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => Colors.down
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| Sum.inr ⟨1, PUnit.unit⟩ => Colors.up
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coord := fun i => m (i (Sum.inr ⟨0, PUnit.unit⟩)) (i (Sum.inr ⟨1, PUnit.unit⟩))
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| Sum.inr 0 => Colors.down
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| Sum.inr 1 => Colors.up
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coord := fun i => m (i (Sum.inr 0)) (i (Sum.inr 1))
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/-!
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@ -94,85 +94,85 @@ def ofMatDownUp {d : ℕ} (m : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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/-- Index values for `ofVecUp v` are equivalent to elements of `Fin 1 ⊕ Fin d`. -/
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def ofVecUpIndexValue (v : Fin 1 ⊕ Fin d → ℝ) :
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IndexValue d (ofVecUp v).color ≃ (Fin 1 ⊕ Fin d) where
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toFun i := i (Sum.inr ⟨0, PUnit.unit⟩)
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toFun i := i (Sum.inr 0)
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x
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| Sum.inr 0 => x
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr 0 => rfl
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right_inv x := rfl
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/-- Index values for `ofVecDown v` are equivalent to elements of `Fin 1 ⊕ Fin d`. -/
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def ofVecDownIndexValue (v : Fin 1 ⊕ Fin d → ℝ) :
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IndexValue d (ofVecDown v).color ≃ (Fin 1 ⊕ Fin d) where
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toFun i := i (Sum.inr ⟨0, PUnit.unit⟩)
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toFun i := i (Sum.inr 0)
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x
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| Sum.inr 0 => x
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr 0 => rfl
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right_inv x := rfl
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/-- Index values for `ofMatUpUp v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatUpUpIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatUpUp M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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toFun i := (i (Sum.inr 0), i (Sum.inr 1))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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| Sum.inr 0 => x.1
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| Sum.inr 1 => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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| Sum.inr 0 => rfl
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| Sum.inr 1 => rfl
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right_inv x := rfl
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/-- Index values for `ofMatDownDown v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatDownDownIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatDownDown M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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toFun i := (i (Sum.inr 0), i (Sum.inr 1))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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| Sum.inr 0 => x.1
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| Sum.inr 1 => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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| Sum.inr 0 => rfl
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| Sum.inr 1 => rfl
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right_inv x := rfl
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/-- Index values for `ofMatUpDown v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatUpDownIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatUpDown M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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toFun i := (i (Sum.inr 0), i (Sum.inr 1))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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| Sum.inr 0 => x.1
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| Sum.inr 1 => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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| Sum.inr 0 => rfl
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| Sum.inr 1 => rfl
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right_inv x := rfl
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/-- Index values for `ofMatDownUp v` are equivalent to elements of
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`(Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)`. -/
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def ofMatDownUpIndexValue (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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IndexValue d (ofMatDownUp M).color ≃ (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d) where
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toFun i := (i (Sum.inr ⟨0, PUnit.unit⟩), i (Sum.inr ⟨1, PUnit.unit⟩))
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toFun i := (i (Sum.inr 0), i (Sum.inr 1))
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invFun x := fun i => match i with
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| Sum.inr ⟨0, PUnit.unit⟩ => x.1
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| Sum.inr ⟨1, PUnit.unit⟩ => x.2
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| Sum.inr 0 => x.1
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| Sum.inr 1 => x.2
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left_inv i := by
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funext y
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match y with
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| Sum.inr ⟨0, PUnit.unit⟩ => rfl
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| Sum.inr ⟨1, PUnit.unit⟩ => rfl
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| Sum.inr 0 => rfl
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| Sum.inr 1 => rfl
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right_inv x := rfl
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/-!
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@ -235,7 +235,7 @@ lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d
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lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
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congrSet ((Equiv.sumEmpty (Empty ⊕ Fin 1) Empty))
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(mul (unmarkFirst $ ofMatUpDown M) (ofVecUp v) rfl) = ofVecUp (M *ᵥ v) := by
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refine ext' ?_ ?_
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· funext i
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@ -247,7 +247,7 @@ lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d)
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lemma mul_ofMatDownUp_ofVecDown_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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congrSet ((Equiv.sumEmpty (Empty ⊕ PUnit.{1}) Empty).trans equivPUnitToSigma)
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congrSet (Equiv.sumEmpty (Empty ⊕ Fin 1) Empty)
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(mul (unmarkFirst $ ofMatDownUp M) (ofVecDown v) rfl) = ofVecDown (M *ᵥ v) := by
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refine ext' ?_ ?_
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· funext i
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@ -326,7 +326,6 @@ lemma lorentzAction_ofMatUpUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
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rw [Fin.prod_univ_two]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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@ -346,7 +345,6 @@ lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
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rw [Fin.prod_univ_two]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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@ -366,7 +364,6 @@ lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
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rw [Fin.prod_univ_two]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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@ -387,7 +384,6 @@ lemma lorentzAction_ofMatDownUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [← Equiv.prod_comp (Equiv.sigmaPUnit (Fin 2)).symm]
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rw [Fin.prod_univ_two]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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