49 lines
1.4 KiB
Text
49 lines
1.4 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.SpaceTime.LorentzTensor.Basic
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/-!
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# Lorentz tensors indexed by Fin n
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In physics, in many (if not all) cases, we index our Lorentz tensors by `Fin n`.
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In this file we set up the machinery to deal with Lorentz tensors indexed by `Fin n`
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in a way that is convenient for physicists, and general caculation.
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## Note
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This file is currently a stub.
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-/
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open TensorProduct
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noncomputable section
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namespace TensorStructure
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variable {n m : ℕ}
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variable {R : Type} [CommSemiring R] (𝓣 : TensorStructure R)
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variable {d : ℕ} {X Y Y' Z W : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
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[Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z] [Fintype W] [DecidableEq W]
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{cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color}
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{cW : W → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
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{cn : Fin n → 𝓣.Color} {cm : Fin m → 𝓣.Color}
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@[simp]
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def finCast (h : n = m) (hc : cn = cm ∘ Fin.castOrderIso h) : 𝓣.Tensor cn ≃ₗ[R] 𝓣.Tensor cm :=
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𝓣.mapIso (Fin.castOrderIso h) hc
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@[simp]
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def finSumEquiv : 𝓣.Tensor cn ⊗[R] 𝓣.Tensor cm ≃ₗ[R]
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𝓣.Tensor (Sum.elim cn cm ∘ finSumFinEquiv.symm) :=
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(𝓣.tensoratorEquiv cn cm).trans (𝓣.mapIso finSumFinEquiv (by funext a; simp))
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end TensorStructure
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end
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