Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches

Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler-Lagrange equations are derived. Fractional equations of motion are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives.Comment: 21 pages, LaTe

Path Integral for Quantum Operations

In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation). We consider the path integral for quantum operation with a simple infinitesimal generator.Comment: 24 pages, LaTe

Analytical on-shell QED results: 3-loop vacuum polarization, 4-loop beta-function and the muon anomaly

We present the results of analytical calculations of the 3-loop contributions to the asymptotic photon vacuum polarization function, in the on shell scheme, and of the 4-loop contributions to the on shell QED beta-function. These are used to evaluate various 4-loop and 5-loop contributions to the muon anomaly. Our analytical contributions to (g-2)_\mu differ significantly from previous numerical results. A very recent numerical re-evaluation of 4-loop muon-anomaly contributions has yielded results much closer to ours.Comment: LaTex, 11 pages, figures available from O.V.T. CERN--TH. 6602/92, OUT--4102--39, BI--TP--92/3

Quark loop contribution to \pi^0 \to 4\gamma

We find the contribution of constituent quark loop mechanism to the branching ratio B_{4\gamma} = \Gamma_{4\gamma}/\Gamma_{4\gamma} \sim 5.45 \cdot 10^{-16} for the reasonable choice of constituent quark mass m \sim 280 MeV. This result is in agreement with vector-dominance approach result obtained years ago. Thus the main contribution arises from QED mechanism \pi^0 \to \gamma (\gamma^*) \to \gamma (3\gamma) including light-light scattering block with electron loop. This contribution was investigated in paper of one of us and gave B_{4\gamma} \sim 2.6 \cdot 10^{-11}.Comment: 7 pages, 2 figure