refactor: Rename States to FieldOps
This commit is contained in:
jstoobysmith
parent
171e80fc04
commit
8f41de5785
36 changed files with 946 additions and 946 deletions
|
|
@ -42,13 +42,13 @@ scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca"
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca =
|
||||
ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs) := by
|
||||
rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
|
||||
simp only [superCommuteF, Basis.constr_basis]
|
||||
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
|
||||
ofCrAnOpF φ * ofCrAnOpF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnOpF φ' * ofCrAnOpF φ := by
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
|
||||
|
|
@ -57,7 +57,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnStates) :
|
|||
rw [ofCrAnListF_append]
|
||||
rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnFieldOp) (φs : List 𝓕.FieldOp) :
|
||||
[ofCrAnListF φcas, ofFieldOpListF φs]ₛca = ofCrAnListF φcas * ofFieldOpListF φs -
|
||||
𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofCrAnListF φcas := by
|
||||
conv_lhs => rw [ofFieldOpListF_sum]
|
||||
|
|
@ -70,7 +70,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (
|
|||
← Finset.sum_mul, ← ofFieldOpListF_sum]
|
||||
simp
|
||||
|
||||
lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
|
||||
[ofFieldOpListF φ, ofFieldOpListF φs]ₛca = ofFieldOpListF φ * ofFieldOpListF φs -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpListF φ := by
|
||||
conv_lhs => rw [ofFieldOpListF_sum]
|
||||
|
|
@ -83,165 +83,165 @@ lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs
|
|||
Algebra.smul_mul_assoc, Finset.sum_sub_distrib]
|
||||
rw [← Finset.sum_mul, ← Finset.smul_sum, ← Finset.mul_sum, ← ofFieldOpListF_sum]
|
||||
|
||||
lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
|
||||
[ofFieldOpF φ, ofFieldOpListF φs]ₛca = ofFieldOpF φ * ofFieldOpListF φs -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
|
||||
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
|
||||
simp
|
||||
|
||||
lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.States) (φ : 𝓕.States) :
|
||||
lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
|
||||
[ofFieldOpListF φs, ofFieldOpF φ]ₛca = ofFieldOpListF φs * ofFieldOpF φ -
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
|
||||
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
|
||||
simp
|
||||
|
||||
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
|
||||
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
|
||||
[anPartF φ, crPartF φ']ₛca = anPartF φ * crPartF φ' -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * anPartF φ := by
|
||||
match φ, φ' with
|
||||
| States.inAsymp φ, _ =>
|
||||
| FieldOp.inAsymp φ, _ =>
|
||||
simp
|
||||
| _, States.outAsymp φ =>
|
||||
| _, FieldOp.outAsymp φ =>
|
||||
simp only [crPartF_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
|
||||
sub_self]
|
||||
| States.position φ, States.position φ' =>
|
||||
| FieldOp.position φ, FieldOp.position φ' =>
|
||||
simp only [anPartF_position, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.outAsymp φ, States.position φ' =>
|
||||
| FieldOp.outAsymp φ, FieldOp.position φ' =>
|
||||
simp only [anPartF_posAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.position φ, States.inAsymp φ' =>
|
||||
| FieldOp.position φ, FieldOp.inAsymp φ' =>
|
||||
simp only [anPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp only [List.singleton_append, instCommGroup.eq_1, crAnStatistics,
|
||||
FieldStatistic.ofList_singleton, Function.comp_apply, crAnStatesToStates_prod, ←
|
||||
FieldStatistic.ofList_singleton, Function.comp_apply, crAnFieldOpToFieldOp_prod, ←
|
||||
ofCrAnListF_append]
|
||||
| States.outAsymp φ, States.inAsymp φ' =>
|
||||
| FieldOp.outAsymp φ, FieldOp.inAsymp φ' =>
|
||||
simp only [anPartF_posAsymp, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
|
||||
lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
|
||||
lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) :
|
||||
[crPartF φ, anPartF φ']ₛca = crPartF φ * anPartF φ' -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ := by
|
||||
match φ, φ' with
|
||||
| States.outAsymp φ, _ =>
|
||||
| FieldOp.outAsymp φ, _ =>
|
||||
simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
|
||||
mul_zero, sub_self]
|
||||
| _, States.inAsymp φ =>
|
||||
| _, FieldOp.inAsymp φ =>
|
||||
simp only [anPartF_negAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
|
||||
sub_self]
|
||||
| States.position φ, States.position φ' =>
|
||||
| FieldOp.position φ, FieldOp.position φ' =>
|
||||
simp only [crPartF_position, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.position φ, States.outAsymp φ' =>
|
||||
| FieldOp.position φ, FieldOp.outAsymp φ' =>
|
||||
simp only [crPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.inAsymp φ, States.position φ' =>
|
||||
| FieldOp.inAsymp φ, FieldOp.position φ' =>
|
||||
simp only [crPartF_negAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.inAsymp φ, States.outAsymp φ' =>
|
||||
| FieldOp.inAsymp φ, FieldOp.outAsymp φ' =>
|
||||
simp only [crPartF_negAsymp, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
|
||||
lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
|
||||
lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.FieldOp) :
|
||||
[crPartF φ, crPartF φ']ₛca = crPartF φ * crPartF φ' -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ := by
|
||||
match φ, φ' with
|
||||
| States.outAsymp φ, _ =>
|
||||
| FieldOp.outAsymp φ, _ =>
|
||||
simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
|
||||
mul_zero, sub_self]
|
||||
| _, States.outAsymp φ =>
|
||||
| _, FieldOp.outAsymp φ =>
|
||||
simp only [crPartF_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
|
||||
sub_self]
|
||||
| States.position φ, States.position φ' =>
|
||||
| FieldOp.position φ, FieldOp.position φ' =>
|
||||
simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.position φ, States.inAsymp φ' =>
|
||||
| FieldOp.position φ, FieldOp.inAsymp φ' =>
|
||||
simp only [crPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.inAsymp φ, States.position φ' =>
|
||||
| FieldOp.inAsymp φ, FieldOp.position φ' =>
|
||||
simp only [crPartF_negAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.inAsymp φ, States.inAsymp φ' =>
|
||||
| FieldOp.inAsymp φ, FieldOp.inAsymp φ' =>
|
||||
simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
|
||||
lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.States) :
|
||||
lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.FieldOp) :
|
||||
[anPartF φ, anPartF φ']ₛca =
|
||||
anPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ := by
|
||||
match φ, φ' with
|
||||
| States.inAsymp φ, _ =>
|
||||
| FieldOp.inAsymp φ, _ =>
|
||||
simp
|
||||
| _, States.inAsymp φ =>
|
||||
| _, FieldOp.inAsymp φ =>
|
||||
simp
|
||||
| States.position φ, States.position φ' =>
|
||||
| FieldOp.position φ, FieldOp.position φ' =>
|
||||
simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.position φ, States.outAsymp φ' =>
|
||||
| FieldOp.position φ, FieldOp.outAsymp φ' =>
|
||||
simp only [anPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.outAsymp φ, States.position φ' =>
|
||||
| FieldOp.outAsymp φ, FieldOp.position φ' =>
|
||||
simp only [anPartF_posAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
| States.outAsymp φ, States.outAsymp φ' =>
|
||||
| FieldOp.outAsymp φ, FieldOp.outAsymp φ' =>
|
||||
simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [crAnStatistics, ← ofCrAnListF_append]
|
||||
|
||||
lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
|
||||
[crPartF φ, ofFieldOpListF φs]ₛca =
|
||||
crPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs *
|
||||
crPartF φ := by
|
||||
match φ with
|
||||
| States.inAsymp φ =>
|
||||
| FieldOp.inAsymp φ =>
|
||||
simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
simp [crAnStatistics]
|
||||
| States.position φ =>
|
||||
| FieldOp.position φ =>
|
||||
simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
simp [crAnStatistics]
|
||||
| States.outAsymp φ =>
|
||||
| FieldOp.outAsymp φ =>
|
||||
simp
|
||||
|
||||
lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
|
||||
[anPartF φ, ofFieldOpListF φs]ₛca =
|
||||
anPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
|
||||
ofFieldOpListF φs * anPartF φ := by
|
||||
match φ with
|
||||
| States.inAsymp φ =>
|
||||
| FieldOp.inAsymp φ =>
|
||||
simp
|
||||
| States.position φ =>
|
||||
| FieldOp.position φ =>
|
||||
simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
simp [crAnStatistics]
|
||||
| States.outAsymp φ =>
|
||||
| FieldOp.outAsymp φ =>
|
||||
simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
simp [crAnStatistics]
|
||||
|
||||
lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.States) :
|
||||
lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
|
||||
[crPartF φ, ofFieldOpF φ']ₛca =
|
||||
crPartF φ * ofFieldOpF φ' -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * crPartF φ := by
|
||||
rw [← ofFieldOpListF_singleton, superCommuteF_crPartF_ofFieldOpListF]
|
||||
simp
|
||||
|
||||
lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.States) :
|
||||
lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
|
||||
[anPartF φ, ofFieldOpF φ']ₛca =
|
||||
anPartF φ * ofFieldOpF φ' -
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * anPartF φ := by
|
||||
|
|
@ -256,44 +256,44 @@ Lemmas which rewrite a multiplication of two elements of the algebra as their co
|
|||
multiplication with a sign plus the super commutor.
|
||||
|
||||
-/
|
||||
lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
ofCrAnListF φs * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnListF φs
|
||||
+ [ofCrAnListF φs, ofCrAnListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [ofCrAnListF_append]
|
||||
|
||||
lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
|
||||
lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.CrAnFieldOp) :
|
||||
ofCrAnOpF φ * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnOpF φ
|
||||
+ [ofCrAnOpF φ, ofCrAnListF φs']ₛca := by
|
||||
rw [← ofCrAnListF_singleton, ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF]
|
||||
simp
|
||||
|
||||
lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.States) :
|
||||
lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.FieldOp) :
|
||||
ofFieldOpListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
|
||||
+ [ofFieldOpListF φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
|
||||
simp
|
||||
|
||||
lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
|
||||
lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
|
||||
ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ
|
||||
+ [ofFieldOpF φ, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofFieldOpF_ofFieldOpFsList]
|
||||
simp
|
||||
|
||||
lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
|
||||
lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
|
||||
ofFieldOpListF φs * ofFieldOpF φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs
|
||||
+ [ofFieldOpListF φs, ofFieldOpF φ]ₛca := by
|
||||
rw [superCommuteF_ofFieldOpListF_ofFieldOpF]
|
||||
simp
|
||||
|
||||
lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
|
||||
lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||||
crPartF φ * anPartF φ' =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ +
|
||||
[crPartF φ, anPartF φ']ₛca := by
|
||||
rw [superCommuteF_crPartF_anPartF]
|
||||
simp
|
||||
|
||||
lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
|
||||
lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||||
anPartF φ * crPartF φ' =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||||
crPartF φ' * anPartF φ +
|
||||
|
|
@ -301,20 +301,20 @@ lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
|
|||
rw [superCommuteF_anPartF_crPartF]
|
||||
simp
|
||||
|
||||
lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
|
||||
lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||||
crPartF φ * crPartF φ' =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ +
|
||||
[crPartF φ, crPartF φ']ₛca := by
|
||||
rw [superCommuteF_crPartF_crPartF]
|
||||
simp
|
||||
|
||||
lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
|
||||
lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||||
anPartF φ * anPartF φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ +
|
||||
[anPartF φ, anPartF φ']ₛca := by
|
||||
rw [superCommuteF_anPartF_anPartF]
|
||||
simp
|
||||
|
||||
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
|
||||
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp) (φs' : List 𝓕.FieldOp) :
|
||||
ofCrAnListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
|
||||
+ [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
|
|
@ -326,7 +326,7 @@ lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnState
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca =
|
||||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnListF φs', ofCrAnListF φs]ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF, smul_sub]
|
||||
|
|
@ -338,7 +338,7 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnState
|
|||
simp only [one_smul]
|
||||
abel
|
||||
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
|
||||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOpF φ', ofCrAnOpF φ]ₛca := by
|
||||
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, superCommuteF_ofCrAnOpF_ofCrAnOpF]
|
||||
|
|
@ -357,7 +357,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnStates) :
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnListF φs, ofCrAnListF (φ :: φs')]ₛca =
|
||||
[ofCrAnListF φs, ofCrAnOpF φ]ₛca * ofCrAnListF φs' +
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
|
||||
|
|
@ -377,8 +377,8 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnStates) (φs φs
|
|||
rw [← ofCrAnListF_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||||
simp only [instCommGroup, map_mul, mul_comm]
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛca =
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.CrAnFieldOp)
|
||||
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛca =
|
||||
[ofCrAnListF φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
|
|
@ -402,8 +402,8 @@ Within the creation and annihilation algebra, we have that
|
|||
`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
|
||||
where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
|
||||
-/
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnStates) :
|
||||
(φs' : List 𝓕.CrAnStates) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
|
||||
(φs' : List 𝓕.CrAnFieldOp) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofCrAnListF (φs'.take n) * [ofCrAnListF φs, ofCrAnOpF (φs'.get n)]ₛca *
|
||||
ofCrAnListF (φs'.drop (n + 1))
|
||||
|
|
@ -417,7 +417,7 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnStates)
|
|||
· simp [Finset.mul_sum, smul_smul, ofCrAnListF_cons, mul_assoc,
|
||||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnFieldOp) : (φs' : List 𝓕.FieldOp) →
|
||||
[ofCrAnListF φs, ofFieldOpListF φs']ₛca =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛca *
|
||||
|
|
@ -436,7 +436,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnState
|
|||
· simp [Finset.mul_sum, smul_smul, ofFieldOpListF_cons, mul_assoc,
|
||||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||||
|
||||
lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
|
||||
lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnListF φs1, [ofCrAnListF φs2, ofCrAnListF φs3]ₛca]ₛca =
|
||||
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
|
||||
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnListF φs3, [ofCrAnListF φs1, ofCrAnListF φs2]ₛca]ₛca -
|
||||
|
|
@ -706,7 +706,7 @@ lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.FieldOpFreeAlge
|
|||
superCommuteF_fermionic_fermionic (by simp) (by simp)]
|
||||
abel
|
||||
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic := by
|
||||
by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
|
||||
|
|
@ -743,13 +743,13 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : Lis
|
|||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
|
||||
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule fermionic := by
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
|
||||
exact superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [φ']
|
||||
|
||||
lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
|
||||
lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
|
||||
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
|
||||
rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ2 φ3 with hs | hs
|
||||
|
|
@ -779,7 +779,7 @@ lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 :
|
|||
rw [h]
|
||||
apply superCommuteF_grade h1 hs
|
||||
|
||||
lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
|
||||
lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
|
||||
(ha : a ∈ statisticSubmodule bosonic) :
|
||||
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length),
|
||||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
|
||||
|
|
@ -808,7 +808,7 @@ lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φ
|
|||
simp_all [p, Finset.smul_sum]
|
||||
· exact ha
|
||||
|
||||
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
|
||||
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
|
||||
(ha : a ∈ statisticSubmodule fermionic) :
|
||||
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
|
||||
|
|
@ -842,7 +842,7 @@ lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (
|
|||
simp [smul_smul, mul_comm]
|
||||
· exact ha
|
||||
|
||||
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnStates}
|
||||
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnFieldOp}
|
||||
(h : [ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic) :
|
||||
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0 := by
|
||||
by_cases h0 : [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue