fix: Typo

HepLean/PerturbationTheory/Wick/Algebra.lean

@@ -0,0 +1,122 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.PerturbationTheory.Wick.Species
+/-!
+
+# Operator algebra
+
+Currently this file is only for an example of Wick strings, correpsonding to a
+theory with two complex scalar fields. The concepts will however generalize.
+
+This file is currently a stub.
+
+We will formally define the operator ring, in terms of the fields present in the theory.
+
+## Futher reading
+
+- https://physics.stackexchange.com/questions/258718/ and links therein
+- Ryan Thorngren (https://physics.stackexchange.com/users/10336/ryan-thorngren), Fermions,
+  different species and (anti-)commutation rules, URL (version: 2019-02-20) :
+  https://physics.stackexchange.com/q/461929
+-/
+
+namespace Wick
+open CategoryTheory
+open FeynmanDiagram
+open PreFeynmanRule
+
+informal_definition WickAlgebra where
+  math :≈ "
+    Modifications of this may be needed.
+    A structure with the following data:
+    - A ℤ₂-graded algebra A.
+    - A map from `ψ : 𝓔 × SpaceTime → A` where 𝓔 are field colors.
+    - A map `ψc : 𝓔 × SpaceTime → A`.
+    - A map `ψd : 𝓔 × SpaceTime → A`.
+    Subject to the conditions:
+    - The sum of `ψc` and `ψd` is `ψ`.
+    - Two fields super-commute if there colors are not dual to each other.
+    - The super-commutator of two fields is always in the
+      center of the algebra. "
+  physics :≈ "This is defined to be an
+    abstraction of the notion of an operator algebra."
+  ref :≈ "https://physics.stackexchange.com/questions/24157/"
+
+informal_definition WickMonomial where
+  math :≈ "The type of elements of the Wick algebra which is a product of fields."
+  deps :≈ [``WickAlgebra]
+
+namespace WickMonomial
+
+informal_definition toWickAlgebra where
+  math :≈ "A function from WickMonomial to WickAlgebra which takes a monomial and
+    returns the product of the fields in the monomial."
+  deps :≈ [``WickAlgebra, ``WickMonomial]
+
+informal_definition timeOrder where
+  math :≈ "A function from WickMonomial to WickAlgebra which takes a monomial and
+    returns the monomial with the fields time ordered, with the correct sign
+    determined by the Koszul sign factor.
+
+    If two fields have the same time, then their order is preserved e.g.
+    T(ψ₁(t)ψ₂(t)) = ψ₁(t)ψ₂(t)
+    and
+    T(ψ₂(t)ψ₁(t)) = ψ₂(t)ψ₁(t).
+    This allows us to make sense of the construction in e.g.
+    https://www.physics.purdue.edu/~clarkt/Courses/Physics662/ps/qftch32.pdf
+    which permits normal-ordering within time-ordering.
+    "
+  deps :≈ [``WickAlgebra, ``WickMonomial]
+
+informal_definition normalOrder where
+  math :≈ "A function from WickMonomial to WickAlgebra which takes a monomial and
+    returns the element in `WickAlgebra` defined as follows
+    - The ψd fields are move to the right.
+    - The ψc fields are moved to the left.
+    - Othewise the order of the fields is preserved."
+  ref :≈ "https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qft/handouts/qftwickstheorem.pdf"
+  deps :≈ [``WickAlgebra, ``WickMonomial]
+
+end WickMonomial
+
+informal_definition contraction where
+  math :≈ "Given two `i j : 𝓔 × SpaceTime`, the element of WickAlgebra
+    defined by subtracting the normal ordering of `ψ i ψ j` from the time-ordering of
+    `ψ i ψ j`."
+  deps :≈ [``WickAlgebra, ``WickMonomial]
+
+informal_lemma contraction_in_center where
+  math :≈ "The contraction of two fields is in the center of the algebra."
+  deps :≈ [``WickAlgebra, ``contraction]
+
+informal_lemma contraction_non_dual_is_zero where
+  math :≈ "The contraction of two fields is zero if the fields are not dual to each other."
+  deps :≈ [``WickAlgebra, ``contraction]
+
+informal_lemma timeOrder_single where
+  math :≈ "The time ordering of a single field is the normal ordering of that field."
+  proof :≈ "Follows from the definitions."
+  deps :≈ [``WickAlgebra, ``WickMonomial.timeOrder, ``WickMonomial.normalOrder]
+
+informal_lemma timeOrder_pair where
+  math :≈ "The time ordering of two fields is the normal ordering of the fields plus the
+    contraction of the fields."
+  proof :≈ "Follows from the definition of contraction."
+  deps :≈ [``WickAlgebra, ``WickMonomial.timeOrder, ``WickMonomial.normalOrder,
+    ``contraction]
+
+informal_definition WickMap where
+  math :≈ "A linear map `vev` from the Wick algebra `A` to the underlying field such that
+   `vev(...ψd(t)) = 0` and `vev(ψc(t)...) = 0`."
+  physics :≈ "An abstraction of the notion of a vacuum expectation value, containing
+    the necessary properties for lots of theorems to hold."
+  deps :≈ [``WickAlgebra, ``WickMonomial]
+
+informal_lemma normalOrder_wickMap where
+  math :≈ "Any normal ordering maps to zero under a Wick map."
+  deps :≈ [``WickMap, ``WickMonomial.normalOrder]
+
+end Wick