138 lines
6.7 KiB
Text
138 lines
6.7 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Lorentz.RealVector.Basic
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import HepLean.PerturbationTheory.FieldStatistics.ExchangeSign
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import HepLean.SpaceTime.Basic
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import HepLean.PerturbationTheory.FieldStatistics.OfFinset
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import HepLean.Meta.Remark.Basic
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/-!
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# Field specification
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In this module is the definition of a field specification.
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A field specification is a structure consisting of a type of fields and a
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the field statistics of each field.
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From each field we can create three different types of `FieldOp`.
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- Negative asymptotic states.
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- Position states.
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- Positive asymptotic states.
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These states carry the same field statistic as the field they are derived from.
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## Some references
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- https://particle.physics.ucdavis.edu/modernsusy/slides/slideimages/spinorfeynrules.pdf
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-/
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/--
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The structure `FieldSpecification` is defined to have the following content:
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- A type `Field` whose elements are the constituent fields of the theory.
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- For every field `f` in `Field`, a type `PositionLabel f` whose elements label the different
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position operators associated with the field `f`. For example,
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- For `f` a *real-scalar field*, `PositionLabel f` will have a unique element.
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- For `f` a *complex-scalar field*, `PositionLabel f` will have two elements, one for the field
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operator and one for its conjugate.
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- For `f` a *Dirac fermion*, `PositionLabel f` will have eight elements, one for each Lorentz
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index of the field and its conjugate.
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- For `f` a *Weyl fermion*, `PositionLabel f` will have four elements, one for each Lorentz
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index of the field and its conjugate.
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- For every field `f` in `Field`, a type `AsymptoticLabel f` whose elements label the different
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types of incoming asymptotic field operators associated with the
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field `f` (this also matches the types of outgoing asymptotic field operators).
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For example,
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- For `f` a *real-scalar field*, `AsymptoticLabel f` will have a unique element.
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- For `f` a *complex-scalar field*, `AsymptoticLabel f` will have two elements, one for the
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field operator and one for its conjugate.
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- For `f` a *Dirac fermion*, `AsymptoticLabel f` will have four elements, two for each spin.
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- For `f` a *Weyl fermion*, `AsymptoticLabel f` will have two elements, one for each spin.
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- For each field `f` in `Field`, a field statistic `statistic f` which classifies `f` as either
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`bosonic` or `fermionic`.
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-/
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structure FieldSpecification where
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/-- A type whose elements are the constituent fields of the theory. -/
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Field : Type
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/-- For every field `f` in `Field`, the type `PositionLabel f` has elements that label the
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different position operators associated with the field `f`. -/
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PositionLabel : Field → Type
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/-- For every field `f` in `Field`, the type `AsymptoticLabel f` has elements that label
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the different asymptotic based field operators associated with the field `f`. -/
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AsymptoticLabel : Field → Type
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/-- For every field `f` in `Field`, the field statistic `statistic f` classifies `f` as either
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`bosonic` or `fermionic`. -/
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statistic : Field → FieldStatistic
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namespace FieldSpecification
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variable (𝓕 : FieldSpecification)
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/-- For a field specification `𝓕`, the inductive type `𝓕.FieldOp` is defined
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to contain the following elements:
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- For every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
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element labelled `inAsymp f e p` corresponding to an incoming asymptotic field operator
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of the field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
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- For every `f` in `𝓕.Field`, element of `e` of `PositionLabel f` and space-time position `x`, an
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element labelled `position f e x` corresponding to a position field operator of the field `f`,
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of label `e` (e.g. specifying the Lorentz index), and position `x`.
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- For every `f` in `𝓕.Field`, element of `e` of `AsymptoticLabel f` and `3`-momentum `p`, an
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element labelled `outAsymp f e p` corresponding to an outgoing asymptotic field operator of the
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field `f`, of label `e` (e.g. specifying the spin), and momentum `p`.
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As an example, if `f` corresponds to a Weyl-fermion field, then
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- For `inAsymp f e p`, `e` would correspond to a spin `s`, and `inAsymp f e p` would, once
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represented in the operator algebra,
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be proportional to the creation operator `a(p, s)`.
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- `position f e x`, `e` would correspond to a Lorentz index `a`, and `position f e x` would,
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once represented in the operator algebra, be proportional to the operator
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`∑ s, ∫ d³p/(…) (xₐ(p,s) a(p, s) e ^ (-i p x) + yₐ(p,s) a†(p, s) e ^ (-i p x))`.
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- `outAsymp f e p`, `e` would correspond to a spin `s`, and `outAsymp f e p` would,
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once represented in the operator algebra, be proportional to the
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annihilation operator `a†(p, s)`.
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-/
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inductive FieldOp (𝓕 : FieldSpecification) where
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| inAsymp : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ) → 𝓕.FieldOp
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| position : (Σ f, 𝓕.PositionLabel f) × SpaceTime → 𝓕.FieldOp
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| outAsymp : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ) → 𝓕.FieldOp
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/-- The bool on `FieldOp` which is true only for position field operator. -/
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def statesIsPosition : 𝓕.FieldOp → Bool
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| FieldOp.position _ => true
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| _ => false
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/-- For a field specification `𝓕`, `𝓕.fieldOpToField` is defined to take field operators
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to their underlying field. -/
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def fieldOpToField : 𝓕.FieldOp → 𝓕.Field
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| FieldOp.inAsymp (f, _) => f.1
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| FieldOp.position (f, _) => f.1
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| FieldOp.outAsymp (f, _) => f.1
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/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`.
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The field statistic `fieldOpStatistic φ` is defined to be the statistic associated with
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the field underlying `φ`.
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The following notation is used in relation to `fieldOpStatistic`:
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- For `φ` an element of `𝓕.FieldOp`, `𝓕 |>ₛ φ` is `fieldOpStatistic φ`.
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- For `φs` a list of `𝓕.FieldOp`, `𝓕 |>ₛ φs` is the product of `fieldOpStatistic φ` over
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the list `φs`.
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- For a function `f : Fin n → 𝓕.FieldOp` and a finite set `a` of `Fin n`, `𝓕 |>ₛ ⟨f, a⟩` is the
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product of `fieldOpStatistic (f i)` for all `i ∈ a`. -/
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def fieldOpStatistic : 𝓕.FieldOp → FieldStatistic := 𝓕.statistic ∘ 𝓕.fieldOpToField
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@[inherit_doc fieldOpStatistic]
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scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => fieldOpStatistic 𝓕 φ
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@[inherit_doc fieldOpStatistic]
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scoped[FieldSpecification] notation 𝓕 "|>ₛ" φ => FieldStatistic.ofList
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(fieldOpStatistic 𝓕) φ
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@[inherit_doc fieldOpStatistic]
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scoped[FieldSpecification] notation 𝓕 "|>ₛ" "⟨" f ","a "⟩"=> FieldStatistic.ofFinset
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(fieldOpStatistic 𝓕) f a
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end FieldSpecification
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