7.6 N≥2 identical particles - symmetric group For understanding systems of many identical particles, Symmetric group Sₙ of permutations on N elements is an essential tool. Permutation group Sₙ Given N ordered objects a, b, c, ... a permutation is a general rearrangement of the objects' ordering ex. P: a → b b → c c → d d → a e → e [Diagram: A diagram showing the action of a permutation P on five objects labeled a, b, c, d, and e. Arrows connect each object to its image under the permutation.] action of P depends on positions of objects, not labels P: a → b b → c c → d d → a e → e Can describe any permutation by cycle structure write (1 → 3 → 4 → 2) (5) [often drop cycles of length 1 ⇒ (1342)] N! permutations on N objects form group Sₙ Sₙ is a nonabelian group, P₁P₂ ≠ P₂P₁ in general. ex. P₁ = (123) P₂ = (12) [Diagram: Two diagrams showing the actions of permutations P₁ and P₂ on a set of three objects (a, b, c). The diagrams demonstrate that the order of applying the permutations affects the final result.] Transpositions Pᵢⱼ switch (i,j). (ij) All permutations can be written as a product of Pᵢⱼ's. Parity of a permutation δp = (-1)^k where k = # of transpositions needed to make P. Representation Theory of Sₙ Consider N! - dimensional vector space spanned by all permutations of {1, 2, ..., N} ex. for N=3, {123}, {132}, {213}, {231}, {312}, {321} Any permutation acts on this basis as perm. matrix (one 1 in each row, other entries = 0) ex. P₁ = (123) (0 1 0) (0 0 1) (1 0 0) This is regular representation. Contains all irreps. Young diagrams: A) reps of Sₙ irreps ↔ Y.D.'s. dim = Dλ = # standard tableaux in N!-dim. reg. rep. mult = Dλ N! = ∑λ Dλ² B) Sₙ reps in (ℂᵏ)⊗ⁿ dim = Dλ^k mult = Dλ^k = # standard k-tab. ∑λ Dλ^k Dλ C) SU(k) reps in (ℂᵏ)⊗ⁿ dim = Dλ^k mult = Dλ

N>2 Identical Particles & the Symmetric Group

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