Functional Determinants In Higher Dimensions Using . - MSRI
A Window Into Zeta and Modular PhysicsMSRI PublicationsVolume 57, 2010Functional determinants in higher dimensionsusing contour integralsKLAUS KIRSTENA BSTRACT. In this contribution we first summarize how contour integrationmethods can be used to derive closed formulae for functional determinantsof ordinary differential operators. We then generalize our considerations topartial differential operators. Examples are used to show that also in higherdimensions closed answers can be obtained as long as the eigenvalues of thedifferential operators are determined by transcendental equations. Examplesconsidered comprise of the finite temperature Casimir effect on a ball and thefunctional determinant of the Laplacian on a two-dimensional torus.1. IntroductionFunctional determinants of second-order differential operators are of greatimportance in many different fields. In physics, functional determinants provide the one-loop approximation to quantum field theories in the path integralformulation [21; 48]. In mathematics they describe the analytical torsion of amanifold [47].Although there are various ways to evaluate functional determinants, the zetafunction scheme seems to be the most elegant technique to use [9; 16; 17; 31].This is the method introduced by Ray and Singer to define analytical torsion[47]. In physics its origin goes back to ambiguities in dimensional regularizationwhen applied to quantum field theory in curved spacetime [11; 29].For many second-order ordinary differential operators surprisingly simple answers can be given. The determinants for these situations have been related toboundary values of solutions of the operators, see, e.g., [8; 10; 12; 22; 23;26; 36; 39; 40]. Recently, these results have been rederived with a simple andaccessible method which uses contour integration techniques [33; 34; 35]. Themain advantage of this approach is that it can be easily applied to general kinds307
308KLAUS KIRSTENof boundary conditions [35] and also to cases where the operator has zero modes[34; 35]; see also [37; 38; 42]. Equally important, for some higher dimensionalsituations the task of finding functional determinants remains feasible. Onceagain closed answers can be found but compared to one dimension technicalitiesare significantly more involved [13; 14]. It is the aim of this article to choose specific higher dimensional examples where technical problems remain somewhatconfined. The intention is to illustrate that also for higher dimensional situationsclosed answers can be obtained which are easily evaluated numerically.The outline of this paper is as follows. In Section 2 the essential ideas arepresented for ordinary differential operators. In Section 3 and 4 examples offunctional determinants for partial differential operators are considered. Thedeterminant in Section 3 describes the finite temperature Casimir effect of amassive scalar field in the presence of a spherical shell [24; 25]. The calculation in Section 4 describes determinants for strings on world-sheets that aretori [46; 50] and it gives an alternative derivation of known answers. Section 5summarizes the main results.2. Contour integral formulation of zeta functionsIn this section we review the basic ideas that lead to a suitable contour integralrepresentation of zeta functions associated with ordinary differential operators.This will form the basis of the considerations for partial differential operatorsto follow later.We consider the simple class of differential operatorsP WDd2C V .x/dx 2on the interval I D Œ0; 1 , where V .x/ is a smooth potential. For simplicity weconsider Dirichlet boundary conditions. From spectral theory [41] it is knownthat there is a spectral resolution f n ; n g1nD1 satisfyingP n .x/ D n n .x/;n .0/ D n .1/ D 0:The spectral zeta function associated with this problem is then defined by P .s/ D1X n s ;(2-1)nD1where by Weyl’s theorem about the asymptotic behavior of eigenvalues [49] thisseries is known to converge for Re s 21 .If the potential is not a very simple one, eigenfunctions and eigenvalues willnot be known explicitly. So how can the zeta function in equation (2-1), and in
FUNCTIONAL DETERMINANTS IN HIGHER DIMENSIONS VIA CONTOUR INTEGRALS 309particular the determinant of P defined via0 P.0/det P D e;be analyzed? From complex analysis it is known that series can often be evaluated with the help of the argument principle or Cauchy’s residue theorem byrewriting them as contour integrals. In the given context this can be achieved asfollows. Let 2 C be an arbitrary complex number. From the theory of ordinarydifferential equations it is known that the initial value problem.P /u .x/ D 0;u .0/ D 0;u0 .0/ D 1;(2-2)has a unique solution. The connection with the boundary value problem is madeby observing that the eigenvalues n follow as solutions to the equationu .1/ D 0I(2-3)note that u .1/ is an analytic function of .With the help of the argument principle, equation (2-3) can be used to writethe zeta function, equation (2-1), asZ1d P .s/ Dd sln u .1/:(2-4)2 id Here, is a counterclockwise contour and encloses all eigenvalues which weassume to be positive; see Figure 1. The pertinent remarks when finitely manyeigenvalues are nonpositive are given in [35].The asymptotic behavior of u .1/ as j j ! 1, namelypsin u .1/ p ; implies that this representation is valid for Re s 12 . To find the determinantof P we need to construct the analytical continuation of equation (2-4) to aneighborhood about s D 0. This is best done by deforming the contour to enclose6 -plane q q q q q q q q q q-Figure 1. Contourused in equation (2-4).
310KLAUS KIRSTEN6 q q q q q q q q q q--Figure 2. Contour -planeused in equation (2-4) after deformation.the branch cut along the negative real axis and then shrinking it to the negativereal axis; see Figure 2.The outcome isZsin s 1d P .s/ Dd sln u .1/:(2-5) d 0To see where this representation is well defined notice that for ! 1 thebehavior follows from [41]pp sin.i /e u .1/ D p 1 e 2 :pi 2 The integrand, to leading order in , therefore behaves like s 1 2 and convergence at infinity is established for Re s 21 . As ! 0 the behavior sfollows. Therefore, in summary, (2-5) is well defined for 21 Re s 1. To shiftthe range of convergence to the left we add and subtract the leading ! 1asymptotic behavior of u .1/. The whole point of this procedure will be toobtain one piece that at s D 0 is finite, and another piece for which the analyticalcontinuation can be easily constructed.Given we want to improve the ! 1 behavior without worsening the ! 0behavior, we split the integration range. In detail we write P .s/ D P;f .s/ C P;as .s/;(2-6)wheresin s P;f .s/ D sin s P;as .s/ D 1Z0Z1dln u .1/d Zsin s 1dCd sln u d 1d 1sp .1/2 ep ;(2-7)pd sde ln p :d 2 (2-8)
FUNCTIONAL DETERMINANTS IN HIGHER DIMENSIONS VIA CONTOUR INTEGRALS 311By construction, P;f .s/ is analytic about s D 0 and its derivative at s D 0 istrivially obtained,0 P;f.0/ D ln u1 .1/ln u0 .1/ln u1 .1/2e1 Dln2u0 .1/:e(2-9)Although the representation (2-8) is only defined for Re s 21 , the analyticcontinuation to a meromorphic function on the complex plane is found usingZ 11d Dfor Re 1: 11This shows that sin s1 P;as .s/ D2 s 1 2 1;sand furthermore0 P;as.0/ D 1:Adding up, the final answer reads P0 .0/ Dln.2u0 .1//:(2-10)For the numerical evaluation of the determinant, not even one eigenvalue isneeded. The only relevant information is the boundary value of the uniquesolution to the initial value problem d2C V .x/ u0 .x/ D 0;u0 .0/ D 0;u00 .0/ D 1:2dxGeneral boundary conditions can be dealt with as easily. The best formulation results by rewriting the second-order differential equation as a first-ordersystem in the usual way. Namely, we define v .x/ D du .x/ dx such that thedifferential equation (2-2) turns into d u .x/u .x/01D:(2-11)V .x/ 0v .x/dx v .x/Linear boundary conditions are given in the form u .0/u .1/0MCND;v .0/v .1/0(2-12)where M and N are 2 2 matrices whose entries characterize the nature of theboundary conditions. For example, the previously described Dirichlet boundaryconditions are obtained by choosing 1 00 0MD;ND:1 00 0
312KLAUS KIRSTENIn order to find an implicit equation for the eigenvalues like equation (2-3) we.1/.2/use the fundamental matrix of (2-11). Let u .x/ and u .x/ be linearly independent solutions of (2-11). Suitably normalized, these define the fundamentalmatrix!.1/.2/u .x/ u .x/H .x/ D;H .0/ D Id2 2 :.1/.2/v .x/ v .x/The solution of (2-11) with initial value .u .0/; v .0// is then obtained as u .x/u .0/D H .x/:v .x/v .0/The boundary conditions (2-12) can therefore be rewritten as u .0/0.M C NH .1//D:v .0/0(2-13)This shows that the condition for eigenvalues to exist isdet.M C NH .1// D 0;which replaces (2-3) in case of general boundary conditions. The zeta functionassociated with the boundary condition (2-12) therefore takes the formZ1dd sln det.M C NH .1// P .s/ D2 id and the analysis proceeds from here depending on M and N . If P representsa system of operators one can proceed along the same lines. Note that we havereplaced the task of evaluating the determinant of a differential operator by oneof computing the determinant of a finite matrix.The procedure just outlined is by no means confined to be applied to ordinary differential operators only. In fact, the zeta function associated with manyboundary value problems allowing for a separation of variables can be analyzedusing this contour integral technique. In more detail, starting off with somecoordinate system (see [43], for example), eigenvalues are often determined byFj . j ;n / D 0;where j is a suitable quantum number depending on the coordinate system considered and Fj is a given special function depending on the coordinate system;e.g. for ellipsoidal coordinate systems the relevant special function is the Mathieu function. Continuing along the lines described above, denoting by dj anappropriate degeneracy that might be present, we write somewhat symbolicallyZX1d P .s/ Ddjd sln Fj . /;(2-14)2 id j
FUNCTIONAL DETERMINANTS IN HIGHER DIMENSIONS VIA CONTOUR INTEGRALS 313the task being to construct the analytical continuation of this object to s D 0. Thedetails of the procedure will depend very much on the properties of the specialfunction Fj that enters. For example, on balls Bessel functions are relevant [4; 6;7], the spherical suspension [3], or sphere-disc configurations [27; 32], involveLegendre functions, ellipsoidal boundaries involve Mathieu functions etc. Formany examples relevant properties of Fj . / are not available in the literatureand need to be derived using techniques of asymptotic analysis [41; 44; 45]. Forquite common coordinate systems like the polar coordinates this is not necessary.When the asymptotics is known, the relevant integrals resulting in (2-14) needto be evaluated and closed expressions representing the determinant of partialdifferential operators are found. Although the remaining sums in general cannotbe explicitly performed, the results obtained are very suitable for numericalevaluation.3. Finite temperature Casimir energy on the ballLet us now apply the above remarks about higher dimensions using the general formalism described in [14]. As a concrete example we consider the finitetemperature theory of a massive scalar field on the three dimensional ball. Usingthe zeta function scheme we have to consider the eigenvalue problem d22P . ; x/E WD C m . ; x/E D 2 . ; x/;E(3-1)d 2where is the imaginary time and xE 2 B 3 WD fxE 2 R 3 j jxjE 1g. We have written2 for the eigenvalues to avoid the occurrence of square roots in arguments ofBessel functions later on.For finite temperature theory we impose periodic boundary conditions in theimaginary time, . ; x/E D . C ˇ; x/;Ewhere ˇ is the inverse temperature, and for simplicity we choose Dirichletboundary conditions on the boundary of the ball,ˇD 0: . ; x/E ˇjxjD1EThe zeta function associated with this boundary value problem is thenX P .s/ D 2s ;(3-2) and the energy of the system is defined byE WD1 @ 0 2 .0/;2 @ˇ P (3-3)
314KLAUS KIRSTENwhere is an arbitrary parameter with dimension of a mass introduced in orderto get the correct dimension for the energy. For a full discussion of its relevancein the renormalization process in this model at zero temperature see [5]. Thatdiscussion remains completely unchanged at finite temperature and we will put D 1 henceforth.Given the radial symmetry of the problem we separate variables in polar coordinates according to1 . ; r; ; '/ D p e i.2 n ˇ/ J C1 2 .! j r /Y m . ; '/;rwith the spherical surface harmonics Y m . ; '/ [20] solving@2sin2 @' 211 @@sin Y m . ; '/ D . C 1/Y m . ; '/;sin @ @ and with the Bessel function J .z/, which is the regular solution of the differential equation [28] 2d 2 J .z/ 1 dJ .z/C 1CJ .z/ D 0:z dzdz 2z2Imposing the boundary condition on the unit sphere,J C1 2 .! j / D 0;determines the eigenvalues. Namely, 2 n 222C ! jC m2 ; n j Dˇ(3-4)n 2 Z ; 2 N0 ; j 2 N:(3-5)This leads to the analysis of the zeta function P .s/ D11 X1XX2.2 C 1/ pn2 C ! jC m2 s;(3-6)nD 1 D0 j D1where we have used the standard abbreviation pn D 2 n ˇ. The factor 2 C 1represents the multiplicity of eigenvalues for angular momentum .The zeroes ! j of the Bessel functions J C1 2 .! j / are not known in closedform and thus we represent the j -summation using contour integrals. Startingwith equation (3-4) and following the argumentation of the previous section,this gives the identityZ11XX s dd 2pn C 2 Cm2ln J C1 2 . /; (3-7) P .s/ D.2 C1/2 id nD 1 D0
FUNCTIONAL DETERMINANTS IN HIGHER DIMENSIONS VIA CONTOUR INTEGRALS 315valid for Re s 2. The contour runs counterclockwise and must enclose all thesolutions of (3-4) on the positive real axis. The next step is to shift the contourand place it along the imaginary axis. As ! 0 we observe that to leadingorder J . / .2 . C 1// such that the integrand diverges in this limit.Therefore, we include an additional factor 1 2 in the logarithm in order toavoid contributions coming from the origin. Because there is no additional poleenclosed, this does not change the result. Furthermore we should note that theintegrand has branch cuts starting at D i .pn2 C m2 /. Leaving out the n; summations for the moment and considering the -integration alone, we thenobtain, with D C 12 , d 2d.pn C 2 Cm2 / sln J . /2 id Z 1dsin sln kd k .k 2 pn2 m2 / sDp 2 dkpn Cm2 P;n .s/ WDZ I .k/ ; (3-8)where J .i k/ D e i J . i k/ and I .k/ D e i 2 J .i k/ has been used [28].The next step is to add and subtract the asymptotic behavior of the integrandin (3-8). The relevant uniform asymptotics, after substituting k D z in theintegral, is the Debye expansion of the Bessel functions [1]. We have 1Xe uk .t/I . z/ p1C; k2 .1 C z 2 /1 41(3-9)kD1ppp with t D 1 1 C z 2 and D 1 C z 2 C ln z .1 C 1 C z 2 / . The first fewcoefficients are listed in [1], higher coefficients are immediately obtained byusing the recursion [1]1ukC1 .t/ D t 2 .12t 2 /u0k .t/ C18Z0td .15 2 /uk . /;(3-10)starting with u0 .t/ D 1. As is clear, all the uk .t/ are polynomials in t. The sameholds for the coefficients Dn .t/ defined by X11Xuk .t/Dn .t/ln 1 C :k n kD1(3-11)nD1The polynomials uk .t/ as well as Dn .t/ are easily found with the help of asimple computer program. As we will see below, we need the first three terms
316KLAUS KIRSTENin the expansion (3-11). Explicitly,D1 .t/ D 81 t5 324 t ;D2 .t/ D1 216 tD3 .t/ D25 3384 t3 45 68 t C 16 t ;(3-12)531 5221 7640 t C 128 t1105 91152 t :Adding and subtracting these terms in (3-8) allows us to rewrite the zeta functionas P .s/ D P;f .s/ C P;as .s/;whereZ 111 ssin s X X P;f .s/ Ddz z 2 2 pn2 m2.2 C1/ p nD 1pn2 Cm2 D0 z e D1 .t/ D2 .t/ D3 .t/dln z I . z/ ln p; (3-13) dz 2 32 .1Cz 2 /1 4Z 111 ssin s X X P;as .s/ D.2 C1/ pdz z 2 2 pn2 m22 nD 1pn Cm2 D0 D1 .t/ D2 .t/ D3 .t/z e dCCln pC: (3-14) dz 2 32 .1Cz 2 /1 4The number of terms subtracted in (3-13) is chosen so that P;f .s/ is analyticabout s D 0. The contributions from the asymptotics collected in (3-14) aresimple enough for an analytical continuation to be found. Although it would bepossible to proceed just with the contribution from inside the ball, in order tomake the calculation as transparent and unambiguous as possible (as far as theinterpretation of results goes) let us add the contribution from outside the ball.The exterior of the ball, once the free Minkowski space contribution is subtracted, yields the starting point (3-8) with the replacement k I ! k K [5].In this case the relevant uniform asymptotics is [1]r 1X e k uk .t/K . z/ 1C. 1/;(3-15)2 .1 C z 2 /1 4 kkD1where the notation is as in (3-9). This produces the analogous splitting of thezeta function for the exterior space. Due to the characteristic sign changes inthe asymptotics of I and K , adding up the interior and exterior contributionsseveral cancellations take place. As a result, the zeta function for the total spacehas the form tot .s/ D tot;f .s/ C tot;as .s/
FUNCTIONAL DETERMINANTS IN HIGHER DIMENSIONS VIA CONTOUR INTEGRALS 317withZ 111 ssin s X X.2 C1/ pdz z 2 2 pn2 m2 tot;f .s/ D nD 1pn2 Cm2 D0 d ln I . z/K . z/ C ln.2 / C 12 ln.1 C z 2 / 2 2 D2 .t/ ; (3-16) dzZ 111 ssin s X X tot;as .s/ D.2 C1/ pdz z 2 2 pn2 m2 nD 1pn2 Cm2 D0 d ln.2 / 21 ln.1 C z 2 / C 2 2 D2 .t/ : (3-17) dzBy construction, tot;f .s/ is analytic about s D 0 and one immediately finds0 tot;f.0/ D1 X1X .2 C 1/ ln I . z/K . z/ C ln.2 /nD 1 D0C 21 ln.1Cz 2 / ˇˇ22 D2 .t/ ˇˇ(3-18)zDp;pn2 Cm2 pwith t D 1 1 C z 2 as defined earlier. Although one could use (3-18) fornumerical evaluation, further simplifications are possible. Following [14] werewrite this expression according to pn2 C m2pn2m221Cz D 1CD 1C 21C 2:(3-19) 2 C pn2The advantage of the right-hand side is that it can be expanded further for 2 ! 1 or pn2 ! 1 or both. This will allow us to subtract exactly the behaviorthat makes the double series convergent; the oversubtraction immanent in (3-18)can then be avoided. It is expected that expanding the rightmost factor further for0 2 C pn2 1 leads to considerable cancellations when combined with tot;as.0/[14].We split the asymptotic terms in (3-18) into those strictly needed to make thesums convergent and those that ultimately will not contribute. For example, weexpand according toˇˇln.1 C z 2 /ˇp 2pn Cm2 pn2 C m2pn2m2D ln 1 CD ln 1 C 2 C ln 1 C 2 2 C pn2 pn2m2m2m2D ln 1 C 2 C 2C ln 1 C 2: C pn2 C pn2 2 C pn2
318KLAUS KIRSTENThe first two terms have to be subtracted in (3-18) in order to make the summations convergent. The terms in brackets are of the order O.1 . 2 C pn2 /2 /and even after performing the summations in (3-18) a finite result follows. Thusthe first two terms represent a minimal set of terms to be subtracted in (3-18)in order to make thepsums finite. This minimal set of necessary terms will beasym;.1/.i pn2 Cm2 /. The last two terms can be summedpseparatelycalled ln f asym;.2/.i pn2 Cm2 /.yielding a finite answer; they are summarized under ln f One can proceed along the same lines for all other terms. With the definitionasym pln f .i pn2 Cm2 /ˇˇD ln.2 / 12 ln.1 C z 2 / C 2 2 D2 .t/ˇ p 2 2zD pn Cm pasym;.2/asym;.1/ p 2.i pn2 Cm2 / (3-20)D ln f .i pn Cm2 / C ln f the splitting isasym;.1/ln f asym;.2/ln f p1 pn2 1 m2.i pn2 Cm2 / D ln.2 /ln 1C 222 2 Cpn2 2 15 pn2 3pn2 1 3pn2 2C 21C 21C 21C 2C;816 16 11 m2m2ln 1 C 2 2 C22 2 Cpn2 Cpn 2 1p2 1m2C 21C 2n1C 2 2 Cpn 16 3 pn2 2 m21C 21C 2 28 Cpn 5p2 3m2C1C 2n1C 2 216 Cpn(3-21)p.i pn2 Cm2 / D 11 213 1 :(3-22)We have used the given n
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