feat: Effective potential for Weyl fermions (Stacked on #1404)#1415
feat: Effective potential for Weyl fermions (Stacked on #1404)#1415jstoobysmith wants to merge 39 commits into
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Prompt: I want you to: 1. Fill out the sorries in this file. 2. Define a instance of a `Star` on `potentialAlgebra` defined by conjugating Left and Right weyl fermions, and using CliffordAlgebra.reverse 3. Define a representation of the Lorentz group on potentialAlgebra. 4. Define the majorana mass for a LeftHandedWeyl. 5. Show it is Lorentz invariant. Co-Authored-By: Claude <claude@users.noreply.github.com>
Your definition of majoranaMass is wrong. In particular it does not even take in a fermion as an argument. I wouldn't not define leftBilinear as it does not make any sense - again also no fermions there. > Really you should be using contractions. and the dual maps Co-Authored-By: Claude <claude@users.noreply.github.com>
Co-Authored-By: Claude <claude@users.noreply.github.com>
Co-Authored-By: Claude <claude@users.noreply.github.com>
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I believe this PR is still being worked on, just some comments I saw while trying to understand the content
One general comment would be that the documentation should get a final pass before merging, some points that could be included:
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The overview still refers to PotentialAlgebra and says that no reality condition has been defined, although the type is now EffectivePotential and Section E defines IsReal and proves isInvariant_iff_of_isReal
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Could this say “the representation of SL(2, ℂ) on EffectivePotential” rather than “the representation of the Lorentz group on PotentialAlgebra”? This matches the actual group and type names used in the definition, and it makes even_of_isInvariant easier to follow
I'm not fluent in the subject matter, so if I've misunderstood any key points please feel free to ignore the comments.
| all_goals | ||
| simp [rep_apply_ψ_eq_sum, rep_apply_barψ_eq_sum, d, adjugate_fin_two, Λ, | ||
| neg_smul, Complex.conj_ofNat] | ||
| try module |
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Some of the intermediate suffices in the proof above look like useful general results. Would it be worth extracting those parts into separate lemmas and leaving this result as the specialization?
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Mainly the general eigenvalue argument in Λ_basis and Λ_basis_two
…T Lagrangian
Fill all eleven remaining sorries in
Physlib/Particles/PureFermionic/EFTLagrangianExclDeriv.lean:
- coeff_fermionic_selection_rule: via termOfList_zero_of_not_nodup. The
IsInvariant hypothesis is dropped: the rule holds for any potential.
- coeff_odd_selection_rule: fermion parity, via the new rep_neg_one_apply
(the special case z = -1 of the diagonal family below).
- mem_allTermsWithIrrepContent_iff: via multisetsOfCard.
- eq_sum_irrepCoeff, eq_sum_massDimCoeff: via a new generic regrouping
lemma eq_sum_fiber_coeff (decomposition of the support sum along any
classifying map with specified fibers).
- irrepCoeff_rep, irrepCoeff_rep_apply_fieldSpecification: via
rep_termOfList_eq_sum_of_toIrrep, through a shared monomial helper.
- irrepCoeff_ψ_barψ_eq_zero_of_isInvariant: no Lorentz-invariant operator
mixes ψ with barψ. Proved with the new diagonal family
diagSL (z : ℂˣ) : SL(2,ℂ) at z = 2i: the scaling factors of the four
mixed pairs are 1/4, -1, -1 and 4, all ≠ 1, so
coeff_U1_selection_rule applies. (Real diagonal elements, i.e. pure
boosts, are insufficient: they scale the off-diagonal pairs by 1.)
A TODO records that diagSL, diagSL_inv, diagSL_neg_one and twoI should
move to Physlib.Relativity.SL2C.Basic when this development is split.
Repair the mass-dimension section, which did not compile:
- allTermsWithMassDimension is now the computable multisetsOfMassDim
from the new file Physlib/Mathematics/MultisetsOfMassDim.lean
(Finset.sym enumeration bounded by ⌊m / d_min⌋, with an ℕ-scaled
decide-friendly variant and a denominator-clearing bridge lemma, as
discussed on Zulip). upperBoundNumberOfFields is superseded by
massDimCardBound.
- massDimCoeff is defined as a sum of coeff over this fixed finset, so
linearity holds by construction (removing the map_add'/map_smul'
sorries).
- decide examples: {ψ 0, ψ 1} has mass dimension 3; the quartic does not.
Also:
- fix the stale import Physlib.Particles.PureFermionic.EffectivePotential
in Physlib.lean (renamed to EFTLagrangianExclDeriv), which broke the
full-library build;
- add docstrings to all previously undocumented definitions;
- add simp evaluation lemmas so definitions stay opaque downstream
(toIrrep_ψ, toIrrep_barψ, massDimension_eq, massDimensionNat_eq,
diagSL_inv, diagSL_neg_one, diagScale_neg_one, diagScale_twoI_ψ,
diagScale_twoI_barψ) and the structural fact sum_map_massDimension.
The changes passed a ten-angle rubric review (correctness, reuse, scope,
attribution, api-design, generality, placement, naming, documentation,
proof-quality); the review's requested changes (private bound lemmas,
authors header, defeq and simp-annotation hygiene, structure comments)
are incorporated, with the SL2C placement recorded as a TODO.
Co-authored-by: Claude Fable 5 <noreply@anthropic.com>
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claude Fable was used to help proof some results. All other content was, in the end, human written. I did experiment with getting claude to write things initially, but these were reverted.
This adds the effective potential for a left-handed Weyl fermion written as an element of the suitable Exterior algebra. It also proves the general form of an effective potential which is invariant under the Lorentz group, this is related to the Majorana mass.