In theoretical physics, background field method is a useful procedure to calculate the effective action of a quantum field theory by expanding a quantum field around a classical "background" value B: After this is done, the Green's functions are evaluated as a function of the background.
is the Lagrangian density of the system, d is the number of dimensions and
The first term does not depend on any fluctuating fields, so that it can be brought out of the path integral.
The result is The path integral which now remains is (neglecting the corrections in the dots) of Gaussian form and can be integrated exactly: where "det" signifies a functional determinant and C is a constant.
The power of minus one half will naturally be plus one for Grassmann fields.