We study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green's functions in an expansion around the light cone. For this, we rst expand all Feynman diagrams and then explicitly sum up the perturbation series. The light-cone expansion expresses the Green's functions as an innnit...

The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff functions. The result provides a validation of an existing singular perturbation expansion formula for the fast mean re...

We study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green’s functions in an expansion around the light cone. For this, we first expand all Feynman diagrams and then explicitly sum up the perturbation series. The light-cone expansion expresses the Green’s functions as an infi...

Online machine learning rules and many biological spike-timing-dependent plasticity (STDP) learning rules generate jump process Markov chains for the synaptic weights. We give a perturbation expansion for the dynamics that, unlike the usual approximation by a Fokker-Planck equation (FPE), is well justified. Our approach extends the related system size expansion by giving an expansion for the pr...

The causal perturbation expansion defines the Dirac sea in the presence of a time-dependent external field. It yields an operator whose image generalizes the vacuum solutions of negative energy and thus gives a canonical splitting of the solution space into two subspaces. After giving a self-contained introduction to the ideas and techniques, we show that this operator is in general not idempot...

The notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to quantum anharmonic oscill...