Taming Divergent with Zeta Functions: Lecture Notes

Eric Palmerduca Math 313: Functions of a Complex Variable Professor Chen

I. INTRODUCTION L[φ(x)] = λφ(x), (6) There are situations in math and physics in which appear, and yet finite results are and the spectrum of L is defined as the set of eigenvalues, required.[1]. There are a number of regularization tech- {λn}. One may naively try to carry over the definition niques that can be used to circumvent the issues pre- of the finite dimensional trace (equation 4) as well. How- sented by divergent series; one of which is zeta function ever, the operator L may have infinitely many eigenval- regularization. ues, and therefore the series may diverge. In fact, this sum only converges if the ordered λn is bounded and tends to zero sufficiently quickly. Many linear opera- II. THE TRACE tors, including the important class of Sturm-Liouville op- erators, do not satisfy this stringent condition.[3] Thus, The technique of zeta function regularization is best a broader definition of the trace is necessary to properly understood by examining the trace of linear operators. extend the notion of a trace to infinite dimensions. In finite dimensions, a linear operator can be expressed The example of the Dirichlet Laplacian can be used as a matrix in some basis. The trace is then defined as to motivate a new definition of the trace. The Dirichlet the sum of the diagonal components of that matrix. That Laplacian is defined as is, d2 X −∆D = − 2 (7) T r(A) = aii (1) dx i and the corresponding eigenvalue problem is where matrix A has components a . It is straightforward ij d2φ to show that the trace of a matrix is invariant under 2 = −λφ(x) (8) similarity transformations[2]: dx with boundary conditions ( tr(A0) = tr(A) (2) φ(0) = 0 (9) φ(L) = 0. if

A0 = BAB−1 (3) The eigenvalues are all positive[4], so the general solu- tions of equation 8 take the form for some invertible matrix B. It follows that the trace of p p a diagonalizable matrix may be equivalently defined as φn(x) = Ansin( λnx) + Bncos( λnx). (10) the sum of its eigenvalues, λn: The cosines do not satisfy the boundary condition at X x = 0, while the boundary condition at x = L constrains T r(A) = λn. (4) the eigenvalues. The resultant eigenfunctions and eigen- n values are Thus, there is no issue in calculating the trace of a finite ( φ (x) = sin nπ x dimensional linear operator. n L nπ 2 (11) The trouble arises in defining the trace for an infinite λn = L dimensional linear operator. An infinite dimensional lin- ear operator L maps functions to other functions: where n ∈ N. Attempting to calculate the trace of −∆D using equation (1) yields the divergent series,

L[θ(x)] = ψ(x). (5) ∞ ? X nπ 2 T r(−∆ ) =  (12) The definitions for eigenfunctions (or eigenvectors) and D L eigenvalues carry over to the infinite dimension case. φ(x) n=1 is an eigenfunction of L with eigenvalue λ if Clearly, this definition of the trace is ill-defined for −∆D. III. ZETA FUNCTION REGULARIZATION discrete spatial frequencies. Thus, the solutions are lin- ear combinations of the following modes: With the aim of defining a meaningful trace of an in- nπ E = e−iωnte−(ikxx+iky y)sin( z) (18) finite dimensional linear operator, we define the spectral kx,ky ,n L zeta function associated with L, where the temporal frequencies ωn are given by ∞ X −s r ζL(s) = A.C. λn . (13) nπ 2 ω = k2 + k2 +  (19) n=1 n x y L Assuming lim λ = ∞, the series in equation 13 con- n→∞ n where k , k ∈ and n ∈ . The zero-point energy of verges for Re(s) sufficiently large, that is, for Re(s) > c x y R N each mode stores an energy equal to[7] for some c > 0. Since the series is well-behaved for Re(s) > c, it can be differentiated term by term, and E = ω (20) therefore the series is analytic on that domain. It follows ωn ~ n that the series can then be (uniquely) analytically con- Thus, one may expect to obtain the total energy by sum- tinued into the region Re(s) < c[5]. ζL is defined as this ming over all modes (which amounts to integrating over . The trace is then redefined to be kx and ky since they are continuous):

T r(L) = ζL(−1). (14) Z ∞ E ? 1 X = ω dk dk (21) A 4π2 n x y This avoids the evaluation of a divergent series because K(x,y) n=1

∞ 2 X where the area of the plates A and the factor of 4π T r(L) 6= λn (15) are geometric factors from the integration. The ωn are n=1 unbounded, so the summation and integration clearly di- but rather the trace is the analytic continuation of the se- verge. However, a finite result can be obtained by using P∞ −s zeta function regularization. We define the zeta function ries n=1 λn at s = −1. Replacing the divergent series with this analytic continuation is known as zeta function associated with the 3D Dirichlet Laplacian as regularization. We now demonstrate how this technique ∞ is practically used in a well-known physical application. h 1 Z X i ζ (s) = A.C. ω−sdk dk ∆ 4π2 n x y K(x,y) n=1 " ∞ −s/2 # IV. THE CASIMIR EFFECT 1 X Z  nπ 2 = A.C. k2 +k2 + dk dk 4π2 x y L x y n=1 K(x,y) The Casimir effect refers to the phenomenon that two neutral infinite conducting plates existing in a vacuum and separated by some distance L will attract each other. (22) This effect can be understood by examining the zero- point energy of the electromagnetic field in the vacuum and identify subject to the relevant boundary conditions.[6] The elec- tric field Ein free space satisfies the homogeneous wave E = ζ (−1). (23) equation (in natural units): A ∆

∂2E ∂2E ∂2E ∂2E ∂2E The integral can be evaluated in closed form by switching − ∆E = − − − = 0 (16) to polar coordinates. If we let q = k2 + k2 then ∂t2 ∂t2 ∂x2 ∂y2 ∂z2 x y

Suppose the conducting plates are positioned at z = 0 " ∞ Z ∞  2−s/2 # 1 X 2 nπ  and z = L and extend infinitely in the x and y directions. ζ∆(s) = A.C. q + 2πqdq 4π2 L Electric fields cannot exist inside of perfect conductors, n=1 0 ∞ so the plates impose the Dirichlet boundary conditions π h X nπ 2−si = A.C. . (24) ( 4(1 − s/2) L E(t, x, y, 0) = 0 n=1 (17) E(t, x, y, L) = 0. The analytic continuation of the series in the last line is simply a constant multiple of the The PDE can be solved using separation of variables, of s − 2, thus the spectral zeta function is yielding oscillatory solutions in all variables. As in the case of the 1D Laplacian, the Dirichlet boundary condi- π1−s ζ (s) = − ζ(s − 2) (25) tions restrict the oscillations in z to sine functions with ∆ 2(2 − s)L2−s From equations 23 and 25 we see effect for parallel plates (including the L−4 dependence) was first predicted by Hendrik Casimir in 1948 using a E π2 π2 = − ~ ζ(−3) = − ~ . (26) more classical argument rather than the one based on A 6L3 720L3 quantum field theory presented here.[10] Casimir and his contemporaries believed the attractive force was due to E Note that limL→0 A = −∞. If we allow the plates to Van der Waals forces. These are attractive forces between move in the z in direction, the system will evolve towards neutral, yet polarization objects. However, the attractive states of lower energy; that is, the plates will move to- Van der Waals forces fail to explain the true complexity of gether. This is also apparent by calculating the pressure the Casimir force, namely, the fact that the Casimir force (force per area) on the plates: can be either attractive or repulsive. For example, if a spherical conductor is used instead of parallel plates, one F d E π2 = −  = − ~ (27) finds an outward pressure on the sphere. The sign of the A dL A 240L4 pressure depends in a highly non-trivial way on the ge- Thus, we see explicitly that there is an attractive force ometry, dimension, and curvature of the system at hand. between the plates that scales like L−4. Although the Currently, the only known way to resolve the sign of the zeta-function regularization may appear to be some dubi- pressure is to go through the procedure of zeta function ous slight-of-hand, these predictions have been observed regularization (which becomes increasingly difficult with experimentally.[8] less trivial configurations than the one presented here). This is a remarkable result, although it was not appre- Emilio Elizalde dubs this poorly-understood sign depen- ciated as such at the time of its discovery.[9] The Casimir dence as “the mystery of the Casimir force.”[11]

[1] Such situations frequently arise in quantum field the- z = c. ory calculations. See E. Zeidler’s Quantum Field Theory [6] The derivation of the Casimir energy and the Casimir II: Quantum Electrodynamics (2008) Ch. 7 for examples force in this section are based on the derivation given from the analysis of the free particle and harmonic oscil- in the Wikipedia article on the Casimir effect. See lator. https://en.wikipedia.org/wiki/Casimir effect. 1 [2] A short proof is contributed by A. Zimmer in the [7] Each mode actually contributes 2 ~ωn, but there are two open source book A First Course in Linear Alge- polarizations for each frequency so the total energy is bra (2004). See http://linear.ups.edu/jsmath/0200/fcla- ~ωn. jsmath-2.00li100.html [8] See S. K. Lamoreaux, ”Demonstration of the Casimir [3] Sturm-Liouville operators are those that take the form Force in the 0.6 to 6 m Range” (1997). For practical d d  L = dx p(x) dx + q(x) where p(x), q(x) > 0 and the reasons, this experiment used a geometrical variant of boundary conditions are homogeneous. These operators the parallel plate configuration discussed in these notes. always have unbounded spectra (see R. Haberman Ap- [9] This paragraph is based on information presented in E. plied Partial Differential Equations (2012) p. 155-57. Elizalde’s book, Ten Applications of Spectral Zeta Func- [4] This can be shown either by using the Reyleigh quotient, tions (2012), Ch. 5. or by simply solving the ODE for λ ≤ 0 and showing that [10] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. B 51, 793 φ(x) cannot satisfy the boundary conditions. (1948). [5] See the theorem in Complex Variables and Applications [11] E. Elizalde, Ten Applications of Spectral Zeta Functions §28 (Brown and Churchill). Technically, this theorem (2012)p. 101 does not preclude singularities in the analytic continu- ation. In fact, most zeta functions will have a pole at