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Physlib.Relativity.Tensors.LeviCivita.Contractions

Euclidean contraction identities for the Levi-Civita tensor

i. Overview

This file proves the "epsilon-epsilon" contraction identities for the rank-four Levi-Civita tensor `leviCivita` (notation `ε4`) in `d = 3`, stated in terms of the standard-basis components of `ε4` itself (`realLorentzTensor.leviCivita_basis_repr_apply`).

The underlying facts about the `generalizedKroneckerDelta` alone, with no tensor content — lives in `Physlib.Mathematics.KroneckerDelta.Contraction`, next to the definition of `generalizedKroneckerDelta`. Here we specialise those facts to the components of `ε4`, where `(ε4)_b = (Tensor.basis _).repr ε4 b` is the standard-basis component of `ε4`, an integer Levi-Civita symbol carried to the reals, and the sums run over the remaining (uncontracted) component slots.

ii. Key results

- `leviCivita_symbol_contract_zero` : `∑_b (ε4)_b · (ε4)_b = 24` (full Euclidean contraction). - `leviCivita_symbol_contract_one` : `∑_h (ε4)_{a,h} · (ε4)_{b,h} = 6 · δ[a,b]`. - `leviCivita_symbol_contract_two` : `∑_h (ε4)_{r,s,h} · (ε4)_{t,w,h} = 2 · (δ[r,t]·δ[s,w] - δ[r,w]·δ[s,t])`.

iii. Table of contents

  • A. The combinatorial bridge lemma
  • B. Euclidean epsilon-epsilon contraction identities

iv. References

A. Euclidean epsilon-epsilon contraction identities

3 declarations

theorem

Full contraction of the 4D Euclidean Levi-Civita symbol equals 24

The sum of the squares of all components of the 4-dimensional Euclidean Levi-Civita symbol ϵi0i1i2i3\epsilon_{i_0 i_1 i_2 i_3} over all possible indices i0,i1,i2,i3∈{0,1,2,3}i_0, i_1, i_2, i_3 \in \{0, 1, 2, 3\} is equal to 24: ∑i0,i1,i2,i3=03ϵi0i1i2i32=24\sum_{i_0, i_1, i_2, i_3 = 0}^3 \epsilon_{i_0 i_1 i_2 i_3}^2 = 24 The Levi-Civita symbol ϵi0i1i2i3\epsilon_{i_0 i_1 i_2 i_3} is defined to be the sign of the permutation (i0,i1,i2,i3)(i_0, i_1, i_2, i_3) if the indices are a permutation of (0,1,2,3)(0, 1, 2, 3), and zero otherwise. This result represents the full Euclidean contraction of the tensor components, effectively counting the 4!=244! = 24 possible permutations.

theorem

Triple Contraction of the 4D Levi-Civita Symbol Equals 6δab6 \delta_{ab}

Let ϵijkl\epsilon_{ijkl} denote the 4-dimensional Euclidean Levi-Civita symbol for indices in {0,1,2,3}\{0, 1, 2, 3\}, and let δab\delta_{ab} denote the Kronecker delta. For any indices a,b∈{0,1,2,3}a, b \in \{0, 1, 2, 3\}, the triple contraction over the remaining three indices is given by: ∑h1,h2,h3=03ϵah1h2h3ϵbh1h2h3=6δab\sum_{h_1, h_2, h_3 = 0}^3 \epsilon_{a h_1 h_2 h_3} \epsilon_{b h_1 h_2 h_3} = 6 \delta_{ab}

theorem

Double Contraction of the 4D Euclidean Levi-Civita Symbol

For any indices r,s,t,w∈{0,1,2,3}r, s, t, w \in \{0, 1, 2, 3\}, the double contraction of the 4-dimensional Euclidean Levi-Civita symbol ϵijkl\epsilon_{ijkl} over its last two indices satisfies: ∑h1,h2=03ϵrsh1h2ϵtwh1h2=2(δrtδsw−δrwδst)\sum_{h_1, h_2 = 0}^{3} \epsilon_{r s h_1 h_2} \epsilon_{t w h_1 h_2} = 2 (\delta_{rt}\delta_{sw} - \delta_{rw}\delta_{st}) where δij\delta_{ij} denotes the Kronecker delta.