Quantum Vacuum Energy as Spectral Theory. I shall introduce vacuum energy as a purely mathematical problem, suppressing or postponing physics issues.

Transcription

1 Quantum Vacuum Energy as Spectral Theory I shall introduce vacuum energy as a purely mathematical problem, suppressing or postponing physics issues. The setting Let H be a second-order, elliptic, self-adjoint PDO, on scalar functions, in a d-dimensional region Ω. Prototype: A billiard. H = 2, Ω R d, boundary conditions (say Dirichlet, u =on Ω). Generalizations: electromagnetic field (vector functions) other boundary conditions Riemannian manifold (Laplace Beltrami operator) potential: 2 + V (x) Technical assumptions: smoothness as needed self-adjointness (spectral decomposition of L 2 (Ω)) positivity for simplicity (H ; if is an eigenvalue, the eigenfunction is constant) 1

2 Spectral decomposition and functional calculus Hϕ n = λ n ϕ n, ϕ n 2 = ϕ n (x) 2 dx =1. Define ω n = λ n. f(h)u f(λ n ) ϕ n,u ϕ n. n=1 At least formally, f(h)u(x) = Ω Ω G(x, x)u( x)d x, G(x, y) = f(λ n )ϕ n (x)ϕ n (y). n=1 Trace: Tr G Ω G(x, x) dx = f(λ n ). n=1 Prototype: Heat kernel. f t (λ) =e tλ, G K, u t = Hu, Tr K = n=1 e tλ n u(,x)=u (x). b s t ( d+s)/2. s= 2

3 Total energy and the cylinder kernel A finite total energy is expected when Spectrum is discrete. Ω is compact (or V is confining). Cylinder (Poisson) kernel Let f t (λ) =e t λ. f t (H)u is the solution of 2 u t 2 = Hu, u(,x)=u (x), that is well-behaved as t +. (T is a boundary value of a derivative of the fundamental solution of the elliptic operator H 2 t in 2 Ω R. It is the ( imaginary time ) analytic continuation of the time derivative of the Wightman function, a certain fundamental solution of the hyperbolic operator H + 2 t.) 2 T (t, x, y) = Tr T = Ω e tω n ϕ n (x)ϕ n (y). n=1 T (t, x, x) dx = e tω n. n=1 3

4 Asymptotics: (t ) Tr T e s t d+s + s= s=d+1 s d odd f s t d+s ln t. Gilkey & Grubb, Commun. PDEs 23 (1998), 777. Fulling & Gustafson, Electr. J. DEs 1999, #6. Bär & Moroianu, Internat. J. Math. 14 (23), 397. Total vacuum energy Define the vacuum energy as E = 1 2 e 1+d (modulo local terms to be determined by physical considerations). Formally, E is the finite part of 1 2 ω n = 1 2 n=1 d dt n e ω nt. t= 23 QFExt proceedings: Quantum Field Theory Under the Influence of External Conditions, ed.byk.a. Milton, Rinton, Princeton, 24. 4

5 The prototype example: Ω=(,L), H = d2 dx 2 ω n = nπ ( nπx ) L, ϕ n(x)=sin. L For L = π, T (t, x, y) = 2 sin(kx)sin(ky)e kt π = t π N= = 1 2π [ [ k=1 1 (x y 2Nπ) 2 +t 2 1 (x+y 2Nπ) 2 +t 2 (image sum = sum over classical paths) ] sinh t cosh t cos(x y) sinh t. cosh t cos(x + y) So (reverting to general L) Tr T = 1 sinh(πt/l) 2 cosh(πt/l) L πt πt 12L + O(t3 ). Thus E = π (O(t) term times 1 24L 2 ). (There are no logarithms in this problem.) 1 nπ Note: 2 L = π 2L ζ( 1) = π 24L. ] 5

6 Another example: Vector bundle over Ω=(,L) u(l)=e iθ u() ϕ n (x) =e i(2πn+θ)x/l) (n Z); ω ± j E(θ) = π L B 2 = 2πj ± θ L ( ) θ 2π = π 12L (j N or Z + ). [ 2 6 θ π +3 ( θ π ) 2 ] E() = π 6L,E(π)=+ π 12L ; here the two sequences coincide, so the gaps are largest. E(.42π) = ; the spectrum is equally spaced when θ = π/2. For an individual eigenvalue sequence, the sign and magnitude of its energy contribution are determined by the phase of the spectral oscillation (here, θ; in billiards etc., the Maslov index). (Is there a vacuum-energy signature of chaos?) Fulling, Gorbar, & Romero, Electr. J. DEs Conf. 4 (2), 87. Fulling, in 23 QFExt proceedings. Schaden, Phys.Rev.A73 (26) Note: E = 1 2 e 1+d is sensitive to the size of the manifold (L) and the structure of the bundle (θ); the heat kernel coefficients are not!. 6

7 Energy density (remains meaningful when Ω is noncompact and H has some continuous spectrum) Leave out the integration in the trace: T (t, x, x) = e t λ dp (λ, x, x) e s (x)t d+s + s= s=d+1 s d odd f s (x)t d+s ln t. Define E(x) = 1 2 e 1+d(x) (a.k.a. T (x)). In quantum field theory (with ξ = 1 4 ) [ ( u E(x) = finite part of 1 ) 2 + uhu]. 2 t Example: The (Dirichlet) half-line Ω=(, ), H = d2, u() =. dx2 P (λ, x, y) = λ (Fourier sine transform). 2 π sin(kx)sin(ky) dk 7

8 T (t, x, y) = t π [ ] 1 (x y) 2 +t 2 1 (x+y) 2 +t 2, T(t, x, x) 1 πt t π(2x) 2 ( ) 2k t ( 1) k as t, 2x k= so E(x) = 1 8πx 2. Contrast heat kernel: K(t, x, x) (4πt) d/2 +O(t ) (for fixed x/ Ω) regardless of boundary conditions! Back to interval: E(x) = π 24L 2 + π ( πx ) 8L 2 csc2 L π ( πx 8L 2 csc2 L ). 1 as x, similar as x L. 8πx2 E(x) = bulk (true Casimir) energy + boundary energy. L E(x) dx = E +! The physicist says: Two kinds of renormalization. The mathematician says: Nonuniform convergence. Fulling, J. Phys. A 36 (23) 6857; 23 QFExt proceedings: 8

9 Boundary energy density for Ω = (, 1) Regularized energy density E(t, x) for Ω = (, ) E(t, x) = T(t, x, x) = t 2π t 2 4x 2 (t 2 +4x 2 ) 2. This effect is overlooked in zeta calculations. This regularization method may have no special physical significance. But similar results are found by physical modeling of softer boundaries. Ford & Svaiter, Phys. Rev. D 58 (1998) 657. Graham & Olum, Phys. Rev. D 67 (23)

10 Spectral density, counting function, etc. Tr T = Tr K = e tω dn, e te dn, T (t, x, x) = K(t, x, x) = e tω dp (x, x), e te dp (x, x). N(λ) = N(ω 2 ) = number of eigenvalues λ, P (λ, x, y) = projection kernel onto spectrum λ. Tr T e s t d+s + s= s=d+1 s d odd f s t d+s ln t, Tr K b s t ( d+s)/2, s= and similarly for the local quantities. Recall: Semiclassical approximation reveals oscillatory structures in N and P correlated with periodic and closed classical orbits. Schaden & Spruch, Phys. Rev. A 58 (1998) 935. Mazzitelli et al., Phys. Rev. A 67 (23) Jaffe & Scardicchio, Nucl. Phys. B 74 (25)

11 Theorem. The b s are proportional to coefficients in the high-frequency asymptotics of Riesz means of N (or P ) with respect to λ. Thee s and f s are proportional to coefficients in the asymptotics of Riesz means with respect to ω. Ifd sis even or positive, e s = π 1/2 2 d s Γ((d s +1)/2)b s. If d s is odd and negative, f s = ( 1)(s d+1)/2 2 d s+1 π Γ((s d +1)/2) b s, but e s is undetermined by the b r. Fulling & Gustafson, Electr. J. DEs 1999, #6; Estrada & Fulling, Electr. J. DEs 1999, #7. These new e s (of which the first is the vacuum energy) are a new set of moments of the spectral distribution. What are they good for, mathematically? Unlike the old ones, they are nonlocal in their dependence on the geometry of Ω (and the coefficients of H). Thus they embody (at least partially) the global dynamical structure of the system; they are a half-way house between the heat-kernel coefficients and a full semiclassical closed-orbit analysis. 11

12 But what about the zeta function? Let f s (H) =H s, ζ(s, H) Tr f s (H). Then ζ(s, H) =ζ(2s, H). Zeta functions are related to integral kernels by t s 1 T (t, H) dt =Γ(s)ζ(s, H), etc. Thus b n and e n are residues at poles of Γ(s)ζ(s, H) (at s = 1 2 (d n)) and Γ(s)ζ(s, H)(ats=d n), respectively. So (when there s no logarithm) ( d n Γ 2 ) 1 b n = 1 2 Γ(d n) 1 e n. Γ(d n) may have a pole where Γ ( 1 2 (d n)) does not; the information in the corresponding e n is thereby expunged from the heat-kernel expansion. That quantity is not a residue of the zeta function but a value of zeta at a regular point a more subtle object to calculate. (Logarithmic terms give rise to coinciding poles of ζ and Γ.) Gilkey, Duke Math. J. 47 (198),

13 A corollary Recall for the half-line E(t, x) = 1 1 T(t, x, x) = 2 t 2π t 2 4x 2 (t 2 +4x 2 ) 2. (1) E(x) lim E(t, x) = 1 t 8πx, 2 which implies a divergent total energy, but (2) E(t) E(t, x) dx =fort>. This disappearing divergence is a general property of energy densities that behave like x 2 near a boundary (confirmed for the corners of a rectangle). By dimensional analysis any correlate of such a term in E(t) mustbe t 1 and hence must come from a term f ln t in Tr T e s t d+s + f s t d+s ln t, s= s=d+1 s d odd but no such term can exist. (It would match a ln t term in the heat kernel.) Fulling, J. Phys. A 36 (23) Bernasconi, Graf, & Hasler, Ann. H. Poincaré 4 (23)

14 A model gravitational equation (23 QFExt proceedings and further work with R. Estrada, L. Kaplan, K. Kirsten, & K. Milton) Still on the half-line, consider d 2 y dx = 2πE bdry(t, x) = t2 4x 2 2 (t 2 +4x 2 ) θ(x). 2 y(t, x) = 1 ( ) 8 ln 1+ 4x2 t 2 θ(x) + homog. soln. As t in solution, y 1 4 ln ( 2x t ) 14 ( ) ln. xx As t in equation, d 2 ( ) y θ(x) dx 2 = F.P. 4x (1 + ln x )δ (x). (The finite part operation is not scale-invariant.) These are consistent! One may interpret x as a finite but arbitrary parameter. More generally (e.g., at edges of a rectangle) we expect δ as well as δ terms. 14

15 Summary and additional remarks 1. Vacuum energy [density] is the first new term in the asymptotic expansion of the [un]traced cylinder kernel at small t. These moments contain global information not present in the heat-kernel expansion. They are associated with spectral oscillations and periodic/closed orbits (cf. Balian & Bloch, Colin de Verdière, Gutzwiller, Duistermaat & Guillemin,... ). They are in correspondence with Riesz means of fractional order of the eigenvalue distribution [or spectral projection kernel]. They arise in functional calculus of a function that is not analytic at λ =. 2. The signs and magnitudes of vacuum energies reflect Maslovian phases in the associated eigenvaluedistribution oscillations. 3. Renormalization (or removal of cutoff) does not commute with integration over space, because of nonuniform convergence. In particular, energy densities that behave like x 2 near a boundary after the cutoff is removed integrate to when the cutoff is present. 15

16 4. Physics issues Traditional Casimir calculations (with renormalization) are noncontroversial for attraction between rigid bodies. Boundary divergences of vacuum energy density are disturbing. * gravitational effect? * For deformable bodies, the internal structure of the materials must be considered. In ultraviolet-cutoff regularizations (e.g., cylinderkernel calculations) these divergences are overt. In analytic regularizations (e.g., zeta-function calculations) they are hidden. In cylinder-kernel regularization a plausible universal response is to leave the cutoff parameter finite (of order the interatomic separation). This permits a renormalization in the Einstein equation. Cosmological applications (dark energy?) arguably are independent of these boundary issues. 16

17 5. Models studied recently and currently rectangle and parallelepiped (with L. Kaplan, K. Kirsten, & K. Milton) Robin boundary condition * Bondurant & Fulling, J. Phys. A 38 (25) 155. * Liu & Fulling, New J. Phys., in press. quantum graphs, yielding an example of a repulsive Casimir piston (25 Snowbird proceedings and further work with J. Wilson). effect of a Klein-Gordon mass in terms of the cylinder kernel for the massless case (Phys. Lett. B 624 (25) 281). 6. Distribution-theory issues Conditional convergence of sums over classical orbits affects only the spectral distribution at λ = (Bondurant paper, appendix). Ambiguity of factor 1 2 when a Dirac δ distribution appears at the end of an interval of integration can be treated rigorously (Estrada and Fulling, Int. J. Appl. Math. Stat., in press). 17