A Gaussian path integral over a bosonic field with quadratic operator produces a factor proportional to . When depends on background fields, this functional determinant contributes to their quantum effective action.
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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.