Functional integral representations of the Pauli-Fierz model with spin 1/2 (Q2426504)

The authors derive a Feynman-Kac-type formula for a Lévy process and an infinite-dimensional Gaussian random process associated with a quantized radiation field. In particular, they study by this mean the Pauli-Fierz Hamiltonian with spin \(1/2\) \[ H_{\text{PF}}=\tfrac12(\vec{\sigma}\cdot(-i\nabla-e{\mathcal A}))^2 + V + H_{\text{rad}}, \] where \(\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)\) are the Pauli matrices, \({\mathcal A}\) is the vector potential, \(H_{\text{rad}}\) the photon field and \(V\) is the external potential acting on the electron. A functional integral representation of \(e^{-tH_{\text{PF}}}\) is constructed by means of a \((3+1)\)-dimensional random process \((B_t,\sigma_t)_{t\geq 0}\) where \((B_t)_{t\geq 0}\) is a Wiener process and \(\sigma_t=\sigma (-1)^{N_t}\) jumps between the possible values of the spin variable \(\sigma\) driven by a Poisson process \((N_t)_{t\geq 0}\). When the external potential vanishes, \(H_{\text{PF}}\) is translation-invariant and is it decomposed as a direct integral \(H_{\text{PF}}= \int_{{\mathbb R}^3}^\oplus H_{\text{PF}}(P)\,dP\) with \[ H_{\text{PF}}(P)= \tfrac12 (\vec{\sigma}\cdot(P-P_f-e{\mathcal A}(0)))^2+ H_{\text{rad}}, \] where \(P_f\) denotes the momentum field operator. The functional integral representation of \(e^{-tH_{\text{PF}}(P)}\) with a scalar kernel is also given and some energy comparison inequalities for \(H_{\text{PF}}\) and \(H_{\text{PF}}(P)\) are derived.