Two-fold ground states of the Pauli-Fierz Hamiltonian including spin

The Pauli-Fierz Hamiltonian describes an interaction between a low energy electron and photons. Existence of ground states has been established. The purpose of this talk is to show that its ground states is exactly two-fold in a weak coupling region. 1 The Pauli-Fierz Hamiltonian This is a joint work with Herbert Spohn . The Hamiltonian in question is the PauliFierz Hamiltonian in nonrelativisitic QED with spin, which will be denoted byH acting on the Hilbert space H = L2(R3;C2)⊗ F . Here L(R;C) denotes the Hilbert space for the electron with spin σ, where σ = (σ1, σ2, σ3) denotes the Pauli spin 1/2 matrices, σ1 = ( 0 1 1 0 ) , σ2 = ( 0 −i i 0 ) , σ3 = ( 1 0 0 −1 ) . F is the symmetric Fock space for the photons given by F = ⊕n=0 (L(R × {1, 2}))nsym . Here (· · ·)sym denotes the n-fold symmetric tensor product of (· · ·) with (· · ·)sym = C. The photons live in R and have helicity ±1. The Fock vacuum is denoted by Ω. The photon field is represented in F by the two-component Bose field a(k, j), j = 1, 2, with commutation relations [a(k, j), a∗(k′, j′)] = δjj′δ(k − k′), 1 [12]. 2 Zentrum Mathematik, Technische Universität München, D80290, München, Germany.