def
definition
interference_amplitude
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IndisputableMonolith.Modal.ModalGeometry on GitHub at line 178.
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175 I(γ₁, γ₂) = √(W[γ₁] · W[γ₂]) · cos(Δφ)
176
177 where Δφ is the phase difference. -/
178noncomputable def interference_amplitude (path1 path2 : List Config) (phase_diff : ℝ) : ℝ :=
179 Real.sqrt (PathWeight path1 * PathWeight path2) * Real.cos phase_diff
180
181/-- **CONSTRUCTIVE INTERFERENCE**: When paths reinforce.
182
183 Occurs when phase_diff = 2πn (paths in phase). -/
184def constructive_interference (path1 path2 : List Config) : Prop :=
185 ∃ n : ℤ, interference_amplitude path1 path2 (2 * Real.pi * n) > 0
186
187/-- **DESTRUCTIVE INTERFERENCE**: When paths cancel.
188
189 Occurs when phase_diff = π(2n+1) (paths out of phase). -/
190def destructive_interference (path1 path2 : List Config) : Prop :=
191 ∃ n : ℤ, interference_amplitude path1 path2 (Real.pi * (2 * n + 1)) < 0
192
193/-- Constructive interference at phase 0. -/
194lemma constructive_at_zero (path1 path2 : List Config)
195 (_ : path1 ≠ []) (_ : path2 ≠ []) :
196 constructive_interference path1 path2 := by
197 use 0
198 simp only [interference_amplitude, Real.cos_zero, mul_one]
199 apply mul_pos
200 · apply Real.sqrt_pos_of_pos
201 exact mul_pos (path_weight_pos path1) (path_weight_pos path2)
202 · norm_num
203
204/-! ## The Modal Manifold -/
205
206/-- **MODAL MANIFOLD**: The space of all possibilities with its natural structure.
207
208 M_modal is the manifold whose points are configurations and whose