122 lines
4.8 KiB
Text
122 lines
4.8 KiB
Text
/-
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Copyright (c) 2024 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.PerturbationTheory.Wick.Species
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/-!
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# Operator algebra
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Currently this file is only for an example of Wick strings, correpsonding to a
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theory with two complex scalar fields. The concepts will however generalize.
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This file is currently a stub.
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We will formally define the operator ring, in terms of the fields present in the theory.
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## Futher reading
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- https://physics.stackexchange.com/questions/258718/ and links therein
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- Ryan Thorngren (https://physics.stackexchange.com/users/10336/ryan-thorngren), Fermions,
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different species and (anti-)commutation rules, URL (version: 2019-02-20) :
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https://physics.stackexchange.com/q/461929
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-/
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namespace Wick
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open CategoryTheory
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open FeynmanDiagram
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open PreFeynmanRule
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informal_definition WickAlgebra where
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math :≈ "
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Modifications of this may be needed.
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A structure with the following data:
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- A ℤ₂-graded algebra A.
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- A map from `ψ : 𝓔 × SpaceTime → A` where 𝓔 are field colors.
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- A map `ψc : 𝓔 × SpaceTime → A`.
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- A map `ψd : 𝓔 × SpaceTime → A`.
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Subject to the conditions:
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- The sum of `ψc` and `ψd` is `ψ`.
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- Two fields super-commute if there colors are not dual to each other.
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- The super-commutator of two fields is always in the
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center of the algebra. "
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physics :≈ "This is defined to be an
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abstraction of the notion of an operator algebra."
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ref :≈ "https://physics.stackexchange.com/questions/24157/"
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informal_definition WickMonomial where
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math :≈ "The type of elements of the Wick algebra which is a product of fields."
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deps :≈ [``WickAlgebra]
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namespace WickMonomial
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informal_definition toWickAlgebra where
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math :≈ "A function from WickMonomial to WickAlgebra which takes a monomial and
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returns the product of the fields in the monomial."
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deps :≈ [``WickAlgebra, ``WickMonomial]
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informal_definition timeOrder where
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math :≈ "A function from WickMonomial to WickAlgebra which takes a monomial and
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returns the monomial with the fields time ordered, with the correct sign
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determined by the Koszul sign factor.
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If two fields have the same time, then their order is preserved e.g.
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T(ψ₁(t)ψ₂(t)) = ψ₁(t)ψ₂(t)
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and
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T(ψ₂(t)ψ₁(t)) = ψ₂(t)ψ₁(t).
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This allows us to make sense of the construction in e.g.
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https://www.physics.purdue.edu/~clarkt/Courses/Physics662/ps/qftch32.pdf
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which permits normal-ordering within time-ordering.
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"
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deps :≈ [``WickAlgebra, ``WickMonomial]
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informal_definition normalOrder where
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math :≈ "A function from WickMonomial to WickAlgebra which takes a monomial and
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returns the element in `WickAlgebra` defined as follows
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- The ψd fields are move to the right.
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- The ψc fields are moved to the left.
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- Othewise the order of the fields is preserved."
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ref :≈ "https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qft/handouts/qftwickstheorem.pdf"
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deps :≈ [``WickAlgebra, ``WickMonomial]
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end WickMonomial
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informal_definition contraction where
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math :≈ "Given two `i j : 𝓔 × SpaceTime`, the element of WickAlgebra
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defined by subtracting the normal ordering of `ψ i ψ j` from the time-ordering of
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`ψ i ψ j`."
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deps :≈ [``WickAlgebra, ``WickMonomial]
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informal_lemma contraction_in_center where
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math :≈ "The contraction of two fields is in the center of the algebra."
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deps :≈ [``WickAlgebra, ``contraction]
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informal_lemma contraction_non_dual_is_zero where
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math :≈ "The contraction of two fields is zero if the fields are not dual to each other."
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deps :≈ [``WickAlgebra, ``contraction]
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informal_lemma timeOrder_single where
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math :≈ "The time ordering of a single field is the normal ordering of that field."
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proof :≈ "Follows from the definitions."
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deps :≈ [``WickAlgebra, ``WickMonomial.timeOrder, ``WickMonomial.normalOrder]
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informal_lemma timeOrder_pair where
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math :≈ "The time ordering of two fields is the normal ordering of the fields plus the
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contraction of the fields."
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proof :≈ "Follows from the definition of contraction."
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deps :≈ [``WickAlgebra, ``WickMonomial.timeOrder, ``WickMonomial.normalOrder,
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``contraction]
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informal_definition WickMap where
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math :≈ "A linear map `vev` from the Wick algebra `A` to the underlying field such that
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`vev(...ψd(t)) = 0` and `vev(ψc(t)...) = 0`."
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physics :≈ "An abstraction of the notion of a vacuum expectation value, containing
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the necessary properties for lots of theorems to hold."
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deps :≈ [``WickAlgebra, ``WickMonomial]
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informal_lemma normalOrder_wickMap where
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math :≈ "Any normal ordering maps to zero under a Wick map."
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deps :≈ [``WickMap, ``WickMonomial.normalOrder]
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end Wick
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