AP07_2-3.vp 1 Introduction and statement of results This 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 phys- ics, and also have inherent mathematical interest since they encode certain spectral properties of differential operators. Physically, such determinants arise, for example, in semi- classical approximations in quantum mechanics and quan- tum 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. Determinants of free Laplacians and free Dirac operators have been extensively studied [1–6], but much less is known about operators involving an arbitrary potential function. When the operator under consideration is an ordi- nary (i.e., one dimensional) differential operator, a beautiful general theory due to Gel’fand and Yaglom [7] has been de- veloped for defining and computing the determinant [8–11]. In this paper I discuss attempts to extend these results to par- tial differential operators. 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 differen- tial operators, leads to a divergence, as noted by Forman [9]. However, it turns out that this divergence has a clear phys- ical meaning and can be understood in the context of re- normalization in quantum field theory. This leads to finite, renormalized expressions [see Eqs. (6)–(8) below] for the de- terminant of such separable operators. The result for four dimensions was first found in [12] using radial WKB and an angular momentum cut-off regularization and renormal- ization [13], and then 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 mathe- matical generalization to arbitrary dimension should be clear. Consider the radially separable partial differential operators � � � �� V r( ) ;�free � �� (1) where � is the Laplace operator in|Rd , and V(r) is a radial po- tential vanishing at infinity as r�2�� for d � 2 and d � 3, and as r�4�� for d � 4. 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[ ] det[ ] ( ) ( ) � � � � � � � m m 2 2free free � � , (2) where [ ]� � �m2 0� , with initial value boundary conditions: �( )0 0� and � �� ( )0 1. The function �free is defined similarly in terms of the free operator: [ ]� free � m2 . The squared mass, m2, is important for physical applications, and plays the mathematical role of a spectral parameter. The result (4) is geometrically interesting, in addition to being com- putationally simple, as it means that the determinant is determined simply by the boundary values of the solutions of [ ]� � �m2 0� , and no detailed information is needed con- cerning the actual spectrum of eigenvalues. Now consider dimensions d 1. Since the potential is ra- dial, V=V(r), we can express the eigenfunctions of � as linear combinations of basis functions of the form �� �r r r Yd l l � � � � �) ( ) ( )( ) ( ) ( )� � 1 1 2 , where Y l( )( ) � � is a hyperspherical harmonic, labeled in part by a non-negative integer l, and the radial function �( )( )l r is an eigenfunction of the Schrödinger-like radial operator �( ) ( )l d dr l d l d r V r � � � �� � � � � � �� � � � � � 2 2 2 3 2 1 2 . (3) �( ) free l is defined similarly, with the potential omitted: V � 0. In dimension d �2, the radial eigenfunctions �(l) have degeneracy given by deg( ; ) ( )( ) ! !( ) ! l d l d l d l d � � � � � 2 2 3 2 . (4) Formally, for the separable operators in (1), the logarithm of the determinant ratio can be written as a sum over l (weighted with the degeneracy factor) of the logarithm of one-dimensional determinant ratios, ln det[ ] det[ ] deg( ; ) ln det[� � �� � � � � � � � � m m l d 2 2free ( l ll m m ) ( ) free � � � � � � � � � � � 2 2 0 ] det[ ]� (5) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 3 Acta Polytechnica Vol. 47 No. 2–3/2007 Functional Determinants for Radially Separable Partial Differential Operators G. V. Dunne 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. Keywords: quantum field theory, functional determinants, zeta functions, spectral theory, partial differential operators. Each term in the sum can be computed using the Sturm- -Liouville extension [10] of the Gel’fand-Yaglom result (2). However, the l sum in (5) is divergent, as noted by Forman [9] for the free Laplace operator in a two-dimensional disc. How- ever, it is possible to understand this divergence and define a finite and renormalized determinant ratio for the radially separable partial differential operators (1). Specifically, we have found [1] the following simple expressions, which gener- alize (2) to higher dimensions: ln det[ ] det[ ] ln ( )( ) ( ) � � � � � � � � � � � � � m m d 2 2 2 0 0 free � � free free d ( ) ln ( ) ( ) ( ) ( )� � � � � � � � � � � � � � � � �2 � � l l r r V ( ) ln r l r r V r l 0 1 2 � � � � � � � � �� � � � � � � � �� � � � � � � � � � � � � 2 d � �� � � � � 0 (6) ln det[ ] det[ ] ( ) ln (( ) � � � � � � � � � � � � � m m l d l 2 2 3 2 1 free � � � � � � � � � � � � � � � � � � � �� � � � �) ( ) ( ) ( )� l r r V r l free d 2 0 1 2� � � � � �� � � � � � � � l 0 (7) ln det[ ] det[ ] ( ) ln (( ) � � � � � � � � � � � � � m m l d l 2 2 4 21 free � ! " � � � � � � � �� � � � � � �) ( ) ( ) ( ( )� l r r V r l r r V V m free d 2 d 0 3 2 1 2 ! " ) ( ) ln 0 3 0 3 2 1 1 8 2 � � � � � � � � � � � � � � � � � � � � � � � � � � 8 d l r r V V m l �r 2 1 0 � � � � � � � � � � � � � � . (8) Here is Euler’s constant, and � is a renormalization scale (defined in the next section), which is essential for physical applications, and which arises naturally in even dimensions. A conventional renormalization choice is to take � � m in (6)–(8). In each of (6)–(8), the sum over l is convergent once the indicated subtractions are made. The function �(l)(r) is the solution to the radial equation [ ] ( ) ( ) ~ , . ( ) ( ) ( ) �( )l l l l d m r r r r � � # � � 2 1 2 0 0 � � as (9) The function �( ) ( )l rfree is defined similarly, with the same behavior as r # 0, in terms of the operator [ ]� ( ) free l m� 2 . Thus, in d dimensions, �( ) ( )l rfree is expressed as a Bessel function: �( ) ( ) (l l d l dr m l d r I mrfree � � � � � � �� � � � � � � � � 2 2 2 1 1 2 2 1 ). (10) Notice that the results (6)–(8) state once again that the de- terminant is determined by the boundary values of solutions of [ ]� � �m2 0� , with the only additional information be- ing a finite number of integrals involving the potential V(r). We also stress the computational simplicity of (6)–(8), as the initial value problem (9) is trivial to implement numerically. 2 Zeta function formalism The functional determinant can be defined in terms of a zeta function [1, 2, 5] for the operator �. For dimensional reasons, we define � � � � � � � [ ] [ ]( ) ( ) ( ) � �� � �� � ��m s m s ss s m2 2 2 2 2 2 , (11) where the sum is over the spectrum of �, and � is an arbi- trary parameter with dimension of a mass. Physically, � plays the role of a renormalization scale. Then the logarithm of the determinant is defined as [1, 2, 5] lndet[ ] ( ) ln( ) ( ) [ ] [ ] [ � � � � � � � � � � � � � � m m m 2 2 2 2 2 0 0 � � � � � m2 0]( ) . (12) To compute the determinant ratio, we define the zeta func- tion difference � � �( ) ( ) ( )[ ] [ ] s s sm m � � �� � 2 2free . (13) Thus we need to compute the zeta function and its de- rivative, each evaluated at s � 0. In general, the zeta function at s � 0 is related to the heat kernel coefficient, ad/2(�), as- sociated with the operator � [6]: �� �( ) ( )0 2� ad . For the operator � � � �� E, these heat kernel coefficients are known [6], and we find the standard results �( ) ( ), , ( ), . 0 2 0 3 2 4 1 2 0 1 16 3 2 0 � � � � � � � � � � d d r r V r d d r r V V m d � � � � � � � (14) This gives the first term on the RHS of (12). Now we turn to the second term, the derivative of the zeta function at s � 0, �� ( )0 . This can be evaluated using the relation to the familiar Jost functions of scattering theory [15]. Consider the radial eigenvalue equation �( ) ( ),( ),l l pl p p � 2 , (15) where �(l) is the Schrödinger-like radial operator defined in (3). A distinguished role is played by the so-called regular solu- tion, (l),p(r), which is defined to have the same behavior as r#0 as the solution without potential: ( ), ( ) ~ � ( )( )l p r l r j prd # � � 0 3 2 . (16) Here the spherical Bessel function is defined as � ( ) ( )( )j z z J z l ld d � � �� �3 2 22 1 � . 4 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 The asymptotic behavior of the regular solution, (l),p(r), as r #� defines theJost function, fl(p), [15] ( ), *( ) ~ ( ) � ( ) ( ) � (l p r l l l lr i f p h pr f p hd d #� � � � � � �� 2 3 2 3 2 pr)� �� � � . (17) Here � ( )h prl d� � �3 2 and � ( )h prl d� � �3 2 are the Riccati-Hankel functions � ( ) ( ) , � ( ) ( )h z i z H z h z i z H l l l d d d � � � � � � � � � � � 3 2 2 3 2 2 2 1 1� � l d z � �2 1 2( ) ( ) . (18) As is well known from scattering theory [15], the analytic properties of the Jost function fl(p) strongly depend on the properties of the potential V(r). Analyticity of the Jost function as a function of p for$ p 0 is guaranteed, if in addition to the aforementioned behavior as r #�, we impose V(r) ~ r�2�� for r # 0, and continuity of V(r) in 0