Chapter BosonicCCR

import Mathlib


/-!
# Chapter *"On the physical parity transformation and antiparticles"*, §*"Majorana spinors in
canonical quantization and antiparticles"*: the **bosonic** canonical commutation variant

This file formalizes the self-contained algebraic content of the **bosonic**
paragraph of the book section *"Majorana spinors in canonical quantization and
antiparticles"* (`book.tex` line ~7700), the companion of the fermionic
Clifford/CAR construction already formalized in `ChapterMajoranaClifford.lean`.

There the book writes, for the canonical quantization of a **real symplectic
space** `V` (with a complex structure `J`, `J² = -1`, so that
`ω(v,w) = ⟪v, J w⟫` is the symplectic product):

> *"For bosons, we have a similar situation, except that a commutation relation
> holds `[a(v), a(w)] = ⟪v, J w⟫ i` instead of `{a(v), a(w)} = ⟪v, w⟫ 1`; a
> symplectic product `⟪v, J w⟫ i` replaces the inner product `⟪v, w⟫ 1` … The
> operators `a(v) = a(v + iJ v) + a(v − iJ v)` are again self-adjoint and
> represent a particle which is its own antiparticle."*

The Weyl / CCR algebra realizing this has **no finite-dimensional
representation** (the trace of a commutator vanishes while the trace of a nonzero
scalar does not), and Mathlib has no ready-made Weyl algebra, so — exactly as the
`GhostCAR` hypothesis structure of `ChapterBRSTNilpotent.lean` does for the
fermionic ghosts — we package the two book relations as a **hypothesis
structure** `BosonicCCR` on an abstract complex `*`-algebra `R`:

* `selfAdjoint` : each field operator `a(v)` is **self-adjoint**
  (`star (a v) = a v`) — *"a particle which is its own antiparticle"*;
* `ccr` : the **canonical commutation relation**
  `a(v)·a(w) − a(w)·a(v) = (i · ⟪v, J w⟫)·1`.

Here `a : V →ₗ[ℝ] R` is the **real-linear** field map (a real representation), and
`R` is a complex `*`-algebra. From these two relations we derive:

* `symplectic_antisymm` / `symplectic_self` — the symplectic form `ω(v,w) = ⟪v,J w⟫`
  built from a skew `J` is **antisymmetric** and **alternating** (`ω(v,v) = 0`);
* `field_commute_self_scalar` — the CCR scalar for `v = w` **vanishes**, i.e. a
  field operator commutes with itself (consistency with `ω(v,v) = 0`);
* `commutator_antiSelfAdjoint` — the commutator `[a(v),a(w)]` is
  **anti-self-adjoint**, which is exactly the self-adjointness constraint on
  `i·ω(v,w)·1` (a real multiple of `i` is anti-self-adjoint);
* `field_comp_selfAdjoint` — **real representations preserve self-adjointness**:
  for *any* real-linear `T : V → V`, `a(T v)` is again self-adjoint
  (the book's `a(v) → a(T v)` with `T` a real operator);
* `ccr_symplectic_invariant` — a **symplectic symmetry** `T` (one preserving
  `ω`) transports the CCR unchanged: `a ∘ T` satisfies the same commutation
  relation;
* `ann` / `cre`, `star_ann` / `star_cre` — the **creation/annihilation split**
  `a(v ± iJv)`: the involution `*` **swaps** annihilation and creation
  (`star (ann v) = cre v`), and `ann_add_cre` recovers the self-adjoint field
  `ann v + cre v = a v + a v`;
* `commutator_field_Jfield` — `[a(v), a(J v)] = -(i · ‖v‖²)·1` (using `J² = -1`);
* `commutator_cre_ann` — the resulting **number-operator CCR**
  `[c(v), a(v)] = (2‖v‖²)·1`, the bosonic analogue of `[a, a†] = 1`.

The C\*-completion (the natural norm) is prose in the book and is not formalized.
-/

open RealInnerProductSpace

namespace BookProof.Bosonic

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
variable {R : Type*} [Ring R] [StarRing R] [Algebra ℂ R] [Algebra ℝ R]
  [IsScalarTower ℝ ℂ R] [StarModule ℂ R]

/-- **Bosonic canonical commutation relations.**  For a real symplectic space
`V` with complex structure `J` (`ω(v,w) = ⟪v, J w⟫`), a real-linear field map
`a : V →ₗ[ℝ] R` into a complex `*`-algebra `R` such that every `a(v)` is
self-adjoint and `[a(v), a(w)] = (i · ⟪v, J w⟫)·1`.  This is the bosonic (Weyl /
CCR) companion of `ChapterMajoranaClifford`'s fermionic Clifford relation. -/
structure BosonicCCR (J : V →ₗ[ℝ] V) (a : V →ₗ[ℝ] R) : Prop where
  /-- Each field operator is **self-adjoint** (the Majorana / own-antiparticle
  property). -/
  selfAdjoint : ∀ v, star (a v) = a v
  /-- The **canonical commutation relation** `[a(v),a(w)] = (i·⟪v,J w⟫)·1`. -/
  ccr : ∀ v w, a v * a w - a w * a v = algebraMap ℂ R (Complex.I * (⟪v, J w⟫ : ℂ))

variable {J : V →ₗ[ℝ] V} {a : V →ₗ[ℝ] R}













/-- The **annihilation** operator `a(v + iJ v) = a(v) + i·a(J v)`. -/
noncomputable def ann (a : V →ₗ[ℝ] R) (J : V →ₗ[ℝ] V) (v : V) : R :=
  a v + Complex.I • a (J v)

/-- The **creation** operator `a(v − iJ v) = a(v) − i·a(J v)`. -/
noncomputable def cre (a : V →ₗ[ℝ] R) (J : V →ₗ[ℝ] V) (v : V) : R :=
  a v - Complex.I • a (J v)













end BookProof.Bosonic