feat: Field struct and creation and annihilation sections

HepLean/PerturbationTheory/FieldStruct/Basic.lean

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+/-
+Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.PerturbationTheory.Wick.OperatorMap
+import HepLean.Mathematics.Fin.Involutions
+import HepLean.Lorentz.RealVector.Basic
+import HepLean.SpaceTime.Basic
+/-!
+
+# Field structures
+
+-/
+
+/-- A field structure is a type of fields plus a specification of the
+  statistics (fermionic or bosonic) of each field. -/
+structure FieldStruct where
+  /-- The type of fields. -/
+  Fields : Type
+  /-- The specification if a field is bosonic or fermionic. -/
+  statistics : 𝓕 → FieldStatistic
+
+namespace FieldStruct
+variable (𝓕 : FieldStruct)
+
+/-- Negative asymptotic states are specified by a field and a momentum. -/
+def AsymptoticNegTime : Type := 𝓕.Fields × Lorentz.Contr 4
+
+/-- Positive asymptotic states are specified by a field and a momentum. -/
+def AsymptoticPosTime : Type := 𝓕.Fields × Lorentz.Contr 4
+
+/-- States specified by a field and a space-time position. -/
+def PositionStates : Type := 𝓕.Fields × SpaceTime
+
+/-- The combintation of asymptotic states and position states. -/
+inductive States (𝓕 : FieldStruct) where
+  | negAsymp : 𝓕.AsymptoticNegTime → 𝓕.States
+  | position : 𝓕.PositionStates → 𝓕.States
+  | posAsymp : 𝓕.AsymptoticPosTime → 𝓕.States
+
+/-- Taking a state to its underlying field. -/
+def statesToField : 𝓕.States → 𝓕.Fields
+  | States.negAsymp φ => φ.1
+  | States.position φ => φ.1
+  | States.posAsymp φ => φ.1
+
+/-- The statistics associated to a state. -/
+def statesStatistic : 𝓕.States → FieldStatistic := 𝓕.statistics ∘ 𝓕.statesToField
+
+/-- Returns true if `timeOrder a b` is true if `a` has time greater then or equal to `b`.
+  This will put the stats at the greatest time to left. -/
+def timeOrder : 𝓕.States → 𝓕.States → Prop
+  | States.posAsymp _, _ => True
+  | States.position φ0, States.position φ1 => φ1.2 0 ≤ φ0.2 0
+  | States.position _, States.negAsymp _ => True
+  | States.position _, States.posAsymp _ => False
+  | States.negAsymp _, States.posAsymp _ => False
+  | States.negAsymp _, States.position _ => False
+  | States.negAsymp _, States.negAsymp _ => True
+
+/-- Time ordering is total. -/
+instance : IsTotal 𝓕.States 𝓕.timeOrder where
+  total a b := by
+    cases a <;> cases b <;> simp [timeOrder]
+    exact LinearOrder.le_total _ _
+
+/-- Time ordering is transitive. -/
+instance : IsTrans 𝓕.States 𝓕.timeOrder where
+  trans a b c := by
+    cases a <;> cases b <;> cases c <;> simp [timeOrder]
+    exact fun h1 h2 => Preorder.le_trans _ _ _ h2 h1
+
+end FieldStruct