Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Coe Add Monoid Hom apply eq bosonic plus fermionic

FieldSpecification.WickAlgebra.coeAddMonoidHom_apply_eq_bosonic_plus_fermionic

physics quantum field theory relativity grading

Project documentation

The projection of 𝓕.WickAlgebra to statSubmodule (𝓕 := 𝓕) fermionic. -/ def fermionicProj : 𝓕.WickAlgebra →ₗ[ℂ] statSubmodule (𝓕 := 𝓕) fermionic where toFun := Quotient.lift fermionicProjFree fermionicProjFree_eq_of_equiv map_add' x y := by obtain ⟨x, hx⟩ := ι_surjective x obtain ⟨y, hy⟩ := ι_surjective y subst hx hy rw [← map_add, ι_apply, ι_ap...

Exact Lean statement

lemma coeAddMonoidHom_apply_eq_bosonic_plus_fermionic
    (a : DirectSum FieldStatistic (fun i => (statSubmodule (𝓕 := 𝓕) i))) :
    DirectSum.coeAddMonoidHom statSubmodule a = a.1 bosonic + a.1 fermionic

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma coeAddMonoidHom_apply_eq_bosonic_plus_fermionic    (a : DirectSum FieldStatistic (fun i => (statSubmodule (𝓕 := 𝓕) i))) :    DirectSum.coeAddMonoidHom statSubmodule a = a.1 bosonic + a.1 fermionic := by  let C : DirectSum FieldStatistic (fun i => (statSubmodule (𝓕 := 𝓕) i)) → Prop :=    fun a => DirectSum.coeAddMonoidHom statSubmodule a = a.1 bosonic + a.1 fermionic  change C a  apply DirectSum.induction_on  · simp [C]  · intro i x    simp only [DFinsupp.toFun_eq_coe, DirectSum.coeAddMonoidHom_of, C]    rw [DirectSum.of_apply, DirectSum.of_apply]    match i with    | bosonic => simp    | fermionic => simp  · intro x y hx hy    simp_all only [C, DFinsupp.toFun_eq_coe, map_add, DirectSum.add_apply, Submodule.coe_add]    abel

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/QFT/PerturbationTheory/WickAlgebra/Grading.lean:346-362

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Deriv Within mean Energy Beta eq neg variance

CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance

Plain-language statement

(∂U/∂β) = -Var(E) for finite systems.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record