Frome Bethe-Salpeter Equation To Schrodinger Equation

The major work of this research is reducing the Bethe-Salpeter equation of positronium（the bound state of e^-e⁺）to the Schr?dinger equation and get the high order corrections of the potential.The Bethe-Salpeter equation describes the bound states of two-body quantum field theoretical system in a relativistically covariant formalism.And the Schr?dinger equation is based on the quantum mechanics and is non-covariant.Some approximations are necessary by reducing the BS equation to the Schr?dinger equation.The only approximation of the research is the instantaneous approximation that reduces the4-dimensional BS kernel to the 3-dimensional.We study the 0^-+bound states.The formalism of the BS wave functions are 4×4 matrix.Because of the symmetry of 0^-+bound state,the wave functions can be expressed by four bases and the coefficients are scalar wave functions.The relationships of these wave functions are discussed.The leading order of the final potential is coulomb potential.The next leading order is corresponding to the Breit potential with some difference between them.The potential we get only use one approximation and have good accuracy.We can get discretionary orders by the result.Comparing with the calculation of the potential energy in the framework of quantum mechanics,the BS equation can obtain the potential energy with very few assumptions,simply and gracefully.And the result can be expanded to any order.Comparing with numerical method that the BS equations get the potential by matching way,the analytical potential we get is universal.