|
In functional analysis, a branch of mathematics, for a linear operator ''S'' mapping a function space ''V'' to itself, it is sometimes possible to define an infinite-dimensional generalization of the determinant. The corresponding quantity det(''S'') is called the functional determinant of ''S''. There are several formulas for the functional determinant. They are all based on the determinant of diagonalizable finite-dimensional matrices being equal to the product of its eigenvalues. A mathematically rigorous definition is via the zeta function of the operator, : where tr stands for the functional trace: the determinant is then defined by : where the zeta function in the point ''s'' = 0 is defined by analytic continuation. Another possible generalization, often used by physicists when using the Feynman path integral formalism in quantum field theory (QFT), uses a functional integration: : This path integral is only well defined up to some divergent multiplicative constant. To give it a rigorous meaning it must be divided by another functional determinant, thus effectively cancelling the problematic 'constants'. These are now, ostensibly, two different definitions for the functional determinant, one coming from quantum field theory and one coming from spectral theory. Each involves some kind of regularization: in the definition popular in physics, two determinants can only be compared with one another; in mathematics, the zeta function was used. have shown that the results obtained by comparing two functional determinants in the QFT formalism agree with the results obtained by the zeta functional determinant. ==Defining formulae== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「functional determinant」の詳細全文を読む スポンサード リンク
|