The Auxiliary Field Method as a Powerful Tool for Nonperturbative Study

The auxiliary field method, defined through introducing an auxiliary (also called as the Hubbard-Stratonovich or the Mean-) field and utilizing a loop-expansion, gives an excellent result for a wide range of a coupling constant. The analysis is made for Anharmonic-Oscillator and Double-Well examples in 0-(a simple integral) and 1-(quantum mechanics)dimension. It is shown that the result becomes more and more accurate by taking a higher loop into account in a weak coupling region, however, it is not the case in a strong coupling region. The 2-loop approximation is shown to be still insufficient for the Double-Well case in quantum mechanics.