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4π|Xk − Xl | VC (x1 , . . , xN ) = VCnuc (X1 , . . 63)) will be studied in the next section. 67) where Pa denotes the projection onto the subspace of totally anti-symmetric spinor wave functions. 67). , Vωi is ‘associated ’ to Vσj ). Spin and Quantum Statistics 29 ⊗N N The spin space C2 C2 carries the N -fold tensor product representa1 tion, ρ, of the spin s = 2 representation of SU(2). This representation is a direct sum of irreducible representations with spin s = s0 , s0 + 1, . . , N2 , where s0 = 0 if N is even and s0 = 12 if N is odd.

Sig(π), π ∈ SN . , Δ must be the trivial representation of SN . , s = N2 (for N electrons). The 30 J¨ urg Fr¨ ohlich expectation described here is at the core of explanations of Hund’s ﬁrst rule and of ferromagnetism. While, in many situations, this expectation is quite plausible it is still poorly understood, mathematically. What is missing? Well, maybe, a few comments on Dirac’s relativistic electron equation. But I will cut this short, since everybody is familiar with it! A nice approach to the Dirac equation can be extracted from the theory of projective, ↑ unitary, irreducible representations of the Poincar´e group P+ , which is the semidirect product of the group of proper, orthochronous Lorentz transformations of Minkowski space M4 and the group of space-time translations.

31) which is a linear operator on Γ(S). Locally, in a coordinate chart of M, with coordinates x1 , . . 32) i=1 with c(dxi ), c(dxj ) = g ij (x) . 33) an operator acting on Γ(S). The bundles S and S are equipped with a natural hermitian structure. Let dvolg denote the Riemannian volume form on M. By He we denote the Hilbertspace completion of Γ(S) in the scalar product on Γ(S) determined by the hermitian structure of S and dvolg ; Hp is deﬁned similarly, with S replaced by S. We note, in passing, that the closures of DA , D −A are selfadjoint, elliptic operators densely deﬁned on He , Hp , respectively.