 Functional determinant

A Gaussian path integral over a bosonic field with quadratic operator $A$ produces a factor proportional to $(\det A{)}^{-1/2}$. When $A$ depends on background fields, this functional determinant contributes to their quantum effective action.

The term "functional determinant" typically refers to the determinant of an operator in the context of functional analysis, particularly in the study of linear operators on infinite-dimensional spaces. This concept extends the classical notion of determinant from finite-dimensional linear algebra to the realm of infinite-dimensional spaces, where one often deals with unbounded operators, such as differential operators.