Abstract
Functional determinants of differential operators play a prominent role in many fields of theoretical and mathematical physics, ranging from condensed matter physics, to atomic, molecular and particle physics. They are, however, difficult to compute reliably in non-trivial cases. In one dimensional problems (i.e. functional determinants of ordinary differential operators), a classic result of Gel’fand and Yaglom greatly simplifies the computation of functional determinants. Here I report some recent progress in extending this approach to higher dimensions (i.e., functional determinants of partial differential operators), with applications in quantum field theory.
Highlights
Introduction and statement of resultsThis paper considers the fundamental questions: what is the determinant of a partial differential operator, and how might one compute it?Determinants of differential operators occur naturally in many applications in mathematical and theoretical physics, and have inherent mathematical interest since they encode certain spectral properties of differential operators.Physically, such determinants arise, for example, in semiclassical approximations in quantum mechanics and quantum field theory, in grand canonical potentials in many-body theory and statistical mechanics, in gap equations in the mean-field approximation, in lattice gauge theory, and in gauge fixing (Faddeev-Popov determinant) for non-abelian gauge theory
The result for four dimensions was first found in [12] using radial WKB and an angular momentum cut-off regularization and renormalization [13], and in [14] using the zeta function approach to determinants
The primary motivation of this work is for applications in quantum field theory, so we concentrate on examples in two, three and four dimensions, but the mathematical generalization to arbitrary dimension should be clear
Summary
This paper considers the fundamental questions: what is the determinant of a partial differential operator, and how might one compute it?. Even for the simple radially separable case of the free Laplacian on a 2d disc, the naive extension via a sum over partial waves of ordinary differential operators, leads to a divergence, as noted by Forman [9]. It turns out that this divergence has a clear physical meaning and can be understood in the context of renormalization in quantum field theory This leads to finite, renormalized expressions [see Eqs. For d = 1, with Dirichlet boundary conditions on the interval [0, ¥), the results of Gel’fand and Yaglom [7] lead to the following simple expression for the determinant ratio: det[M + det[ M free m2 ] + m2 ]. It is possible to understand this divergence and define a finite and renormalized determinant ratio for the radially separable partial differential operators (1). We stress the computational simplicity of (6)–(8), as the initial value problem (9) is trivial to implement numerically
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