Weyl Law at 101. Masoud Khalkhali Colloquium, UWO, 2012

Transcription

1 Weyl Law at 101 Masoud Khalkhali Colloquium, UWO, 2012

2 Figure: Sun s Spectrum; notice the black lines AX = λx

3 Dramatis Personae: Physics: Planck, Einstein, Lorentz, Sommerfeld, among others; quantum mechanics classical mechanics Mathematics: Hilbert, Weyl. spectrum geometry In 1910 H. A. Lorentz gave a series of lectures in Göttingen under the title old and new problems of physics. Weyl and Hilbert were in attendance. In particular he mentioned attempts to drive Planck s radiation formula in a mathematically satisfactory way and remarked:

4 It is here that there arises the mathematical problem to prove that the number of sufficiently high overtones which lie between ν and ν + dν is independent of the shape of the enclosure and is simply proportional to its volume....there is no doubt that it holds in general even for multiply connected spaces. Hilbert was not very optimistic to see a solution in his lifetime. His bright student Hermann Weyl solved this conjecture of Lorentz and Sommerfeld within a year and announced a proof in 1911! All he needed was Hilbert s theory of integral equations and compact operators developed by Hilbert and his students in

5 Figure: Hermann Weyl in Göttingen

6 Figure: H. A. Lorentz

7 Black body radiation Figure: Black body spectrum

8 Planck s Radiation Law From 1859 (Kirchhoff) till 1900 (Planck) a great effort went into finding the right formula for spectral energy density function of a radiating black body ρ(ν, T ) Kirchhoff predicted: ρ will be independent of the shape of the cavity and should only depend on its volume. Planck s formula: ρ(ν, T ) = 8πhν3 c 3 1 e hν/kt 1

9 Limits of Planck s Law Quantum Limit (high-frequency or low temperature regime; hν/kt 1) ρ(ν, T ) Aν 3 e Bν/T (T 0) Semiclassical Limit (low frequency or high temperature; hν/kt 1) ρ(ν, T ) = 8πν2 (kt )(1 + O(h)) (T ) c3 RHS is the Rayleigh-Jeans-Einstein radiation formula. It can be established, assuming Weyl s Law: For high frequencies there are approximately V (8πν 3 dν/c 3 ) modes of oscillations in the frequency interval v, ν + dν.

10 Moral: To relate classical and quantum worlds, Weyl s law is needed: One can hear the volume of a cavity. But the ultimate question is What else can one hear about the shape of a cavity?

11 Dirichlet eigenvalues and Weyl Law Let Ω R 2 be a compact connected domain with a piecewise smooth boundary. { u = λu u Ω = 0 0 < λ 1 λ 2 λ 3 u i, u j = δ ij o.n. basis for L 2 (Ω)

12 Weyl Law for planar domains Ω R 2 N(λ) Area(Ω) λ 4π λ where N(λ) is the eigenvalue counting function. In general, for Ω R n N(λ) ω nvol(ω) (2π) n λ n 2 λ

13 Weyl Law and acoustics Figure: One can hear the area of a drum

14 One can hear the Volume and dimension of a compact Riemannian manifold. In his book, The Theory of Sound (1877), Lord Rayleigh writes: Some of the natural notes of the air contained within a room may generally be detected on singing the scale. Probably it is somewhat in this way that blind people are able to estimate the size of rooms... A remarkable instance is in Darwin s Zoonomia: The late blind Justice Fielding walked for the first time into my room, when he once visited me, and after speaking a few words said, This room is about 22 feet long, 18 wide, and 12 high; all which he guessed by the ear with great accuracy

15 Weyl Law with remainder; oscillations Weyl s conjecture for planar domains Ω R 2 N(λ) Area(Ω) 4π λ L( Ω) λ + o( λ) 4π λ Weyl s conjecture for spatial domains Ω R 3 N(λ) Vol(Ω) 6π 2 λ 3 2 A( Ω) 16π λ + o(λ) λ As innocent as they look, they were only proved in early 1980 s after a long time effort by many mathematicians.

16 Wild oscillations! To understand the nature of the spectrum, beyond Weyl s term, one needs to understand the highly oscillatory remainder term R(λ) = N(λ) ω nvol(ω) (2π) n λ n 2 Hormander-Avakumovic R(λ) = O(λ n 1 2 ) (An improvement over Weyl s result o(λ n 2 ) x log x Compare with Prime number theorem π(x) and the Riemann hypothesis which gives the best estimate for π(x) Li(x). In fact the two problems are quite related, thanks to Connes trace formula.

17 Figure: RH is probably true!

18 Gauss circle problem For M = T 2 = (R/2πZ) 2 with flat metric, the spectrum is λ m,n = m 2 + n 2, m, n Z N(λ) = #{(m, n) : m 2 + n 2 λ} is the number of integral lattice points in a circle of radius λ. Gauss: N(λ) = πλ + O λ Circle problem: let R(λ) = N(λ) πλ Find α 0 = inf (α) such that R(λ) = O(λ α )

19 Hardy s conjecture: α 0 = 1 4. He showed that α van der Corput (1923): α = Iwaniec and Mozzochi (1988): α = Huxley (1992): R(x) = O(x (log x) ) We are still far off from Hardy s conjectured value of 1 4 for α 0!

20 F. Fathizadeh and M. Khalkhali, The Gauss-Bonnet Theorem for noncommutative two tori with a general conformal structure, May F. Fathizadeh and M. Khalkhali, Scalar Curvature for the Noncommutative Two Torus, Oct F. Fathizadeh and M. Khalkhali, Weyl s law and Connes trace theorem for the Noncommutative Two Torus, Nov A. Connes and P. Tretkoff, The Gauss-Bonnet Theorem for the noncommutative two torus, 1992, and Sept A. Connes and H. Moscovici, Modular curvature for noncommutative two-tori, Oct

21 A simple example in spectral geometry: flat tori Let Γ R m be a cocompact lattice. M = R m /Γ flat torus = m i=1 2 xi 2 : Dom( ) L 2 (M) L 2 (M) flat Laplacian is an unbounded, s.a., positive operator with pure point spectrum: spec( ) = {4π 2 γ 2 ; γ Γ }, Heat equation: ( t + )ϕ = 0

22 Let k(t, x, y) = fundamental solution of the heat equation= kernel of e t. Then: Tre t = 1 K(t, x, y) = (4πt) m/2 M γ Γ k(t, x, x) = e 4π2 x y+γ 2 /4t e 4π 2 γ 2 t γ Γ = Vol(M) (4πt) m/2 γ Γ e 4π2 γ 2 /4t And from this we obtain the asymptotic expansion of the heat trace near t = 0 Tre t Vol(M) (4πt) m/2 (t 0)

23 Can one hear the shape of a drum? Cor: One can hear the dimension and volume of M. In fact more is true: a Tauberian theorem + asymptotic expansion Weyl s law: N(λ) Vol (M) (4π) m/2 Γ(1 + m/2) λm/2 λ Cor: One can hear the total scalar curvature of M (= 0).

24 From heat trace and zeta functions to spectral invariants (M, g) = closed Riemannian manifold. Laplacian on functions = d d : C (M) C (M) is an unbounded positive operator with pure point spectrum 0 λ 1 λ 2 The spectrum contains a wealth of geometric and topological informations about M. In particular the dimension, volume, total scalar curvature, Betti numbers, and hence the Euler characteristic of M are fully determined by the spectrum of (on all p-forms).

25 The heat engine Let k(t, x, y) = kernel of e t. Restrict to the diagonal: as t 0, we have (Minakshisundaram-Plejel; MacKean-Singer) k(t, x, x) 1 (4πt) m/2 (a 0(x, ) + a 1 (x, )t + a 2 (x, )t 2 + ) a i (x, ): the Seeley-De Witt-Schwinger coefficients.

26 Functions a i (x, ): expressed by universal polynomials in curvature tensor R and its covariant derivatives: a 0 (x, ) = 1 a 1 (x, ) = 1 S(x) 6 scalar curvature a 2 (x, ) = (2 R(x) 2 2 Ric(x) S(x) 2 ) a 3 (x, ) = For flat torus a i (x, ) = 0 for all i 1.

27 Short time asymptotics of the heat trace So Trace(e t ) = e tλ i = a j = (4πt) m 2 are manifestly spectral invariants. a 0 = dvol x = Vol(M), a 1 = M 6 S(x)dvol x, M M M k(t, x, x)dvol x a j t j (t 0) j=0 a j (x, )dvol x, = Weyl s law total scalar curvature

28 Abelian-Tauberian Theorem Assume 1 e λnt is convergent for all t > 0. TFAE: lim t 0 + tr e λnt = a, 1 N(λ) a lim λ λ r = Γ(r + 1)

29 Spectral zeta functions ζ (s) := λ j 0 λ s j, Re(s) > m 2 Mellin transform + asymptotic expansion: λ s = 1 Γ(s) 0 e t t s 1 dt Re(s) > 0 ζ (s) = = 1 Γ(s) 1 Γ(s) { 0 c 0 (Trace(e t ) Dim Ker )t s 1 dt + c } The second term defines an entire function, while the first term has a meromorphic extension to C with simple poles within the set

30 m j, j = 0, 1, 2 Also: 0 is always a regular point. Simplest example: For M = S 1 with round metric, we have ζ (s) = 2ζ(2s) Riemann zeta function

31 The spectral invariants a i in the heat asymptotic expansion Trace(e t ) (4πt) m 2 a j t j (t 0) j=0 are related to residues of spectral zeta function by Res s=α ζ (s) = (4π) m 2 a m 2 α Γ(α), α = m 2 j > 0 Focusing on subleading pole s = m 2 1 and using a 1 = 1 6 we obtain a formula for scalar curvature density. M S(x)dvol x,

32 Let ζ f (s) := Tr (f s ), f C (M). Res ζ f (s) s= m 2 1 = (4π) m/2 fs(x)dvol x, m 3 Γ(m/2 1) M ζ f (s) s=0 = 1 fs(x)dvol x Tr(fP) m = 2 4π M log det( ) = ζ (0), Ray-Singer regularized determinant

33 Spectral Triples: (A, H, D) A= involutive unital algebra, H = Hilbert space, π : A L(H), D : H H D has compact resolvent and all commutators [D, π(a)] are bounded. An asymptotic expansion holds Trace (e td2 ) a α t α (t 0) Let = D 2. Spectral zeta function ζ D (s) = Tr ( D s ) = Tr ( s/2 ), Re(s) 0. The metric dimension and dimension spectrum of (A, H, D).

34 Noncommutative Torus Fix θ R. A θ = C -algebra generated by unitaries U and V satisfying VU = e 2πiθ UV. Dense subalgebra of smooth functions : A θ A θ, a A θ iff a = a mn U m V n where (a mn ) S(Z 2 ) is rapidly decreasing: for all k N. sup (1 + m 2 + n 2 ) k a mn < m,n

35 Differential operators on A θ δ 1, δ 2 : A θ A θ, Infinitesimal generators of the action Analogues of 1 i α s (U m V n ) = e is.(m,n) U m V n s R 2. x, 1 i y on 2-torus. Canonical trace t : A θ C on smooth elements: t( a m,nu m V n ) = a 0,0. m,n Z

36 Complex structures on A θ Let H 0 = L 2 (A θ )= GNS completion of A θ w.r.t. t. Fix τ = τ 1 + iτ 2, τ 2 = I(τ) > 0, and define := δ 1 + τδ 2, := δ 1 + τδ 2. Hilbert space of (1, 0)-forms: H (1,0) := completion of finite sums a b, a, b A θ, w.r.t. a b, a b := t((a b ) a b). Flat Dolbeault Laplacian: = δ τ 1δ 1 δ 2 + τ 2 δ 2 2.

37 Conformal perturbation of metric Fix h = h A θ. Replace the volume form t by ϕ : A θ C, ϕ(a) := t(ae h ), a A θ. It is a KMS state with modular group σ t (x) = e ith xe ith, and modular automorphism (Tomita-Takesaki theory) σ i (x) = (x) = e h xe h. ϕ(ab) = ϕ(b (a)), a, b A θ. Warning: and are very different operators!

38 Connes-Tretkoff spectral triple Hilbert space H ϕ := GNS completion of A θ w.r.t., ϕ, a, b ϕ := ϕ(b a), a, b A θ View ϕ = = δ 1 + τδ 2 : H ϕ H (1,0). and let ϕ : H (1,0) H ϕ H = H ϕ H (1,0), ( ) 0 D = ϕ : H H. ϕ 0

39 Full perturbed Laplacian: ( := D 2 = ϕ ϕ 0 0 ϕ ϕ ) : H H. Lemma: ϕ ϕ : H ϕ H ϕ, and ϕ ϕ : H (1,0) H (1,0) are anti-unitarily equivalent to k k : H 0 H 0, where k = e h/2. k 2 : H (1,0) H (1,0), The Tomita anti-unitary map J is used.

40 Scalar curvature for A θ The scalar curvature of the curved nc torus (T 2 θ, τ, k) is the unique element R A θ satisfying Trace (a s ) s=0 + Trace (ap) = t (ar), a A θ, where P is the projection onto the kernel of. In practice this is done by finding an asymptotic expansin for the kernel of the operator e t, using Connes pseudodifferential calculus.

41 Connes pseudodifferential calculus Symbols: ρ : R 2 A θ. ΨDO s: P ρ : A θ A θ, P ρ (a) = (2π) 2 R 2 R 2 e is.ξ ρ(ξ)α s (a)dsdξ. For example: ρ(ξ 1, ξ 2 ) = a ij ξ i 1ξ j 2, a ij A θ P ρ = a ij δ i 1δ j 2. Multiplication of symbol. σ(pq) l 1,l l 1!l 2! l1 1 l2 2 (ρ(ξ))δl1 1 δl2 2 (ρ (ξ)).

42 Local expression for the scalar curvature Cauchy integral formula: e t = 1 e tλ ( λ) 1 dλ. 2πi C B λ ( λ) 1 : σ(b λ ) b 0 (ξ, λ) + b 1 (ξ, λ) + b 2 (ξ, λ) +, each b j (ξ, λ) is a symbol of order 2 j, and σ(b λ ( λ)) 1. (Note: λ is considered of order 2.)

43 Proposition: The scalar curvature of the spectral triple attached to (A θ, τ, k) is equal to 1 e λ b 2 (ξ, λ) dλ dξ, 2πi R 2 C where b 2 is defined as above.

44 Final formula for the scalar curvature (Connes-Moscovici, Fathizadeh-K, Oct. 2011) Theorem: The scalar curvature of (A θ, τ, k), up to an overall factor of, is equal to π τ 2 R 1 (log ) ( 0 (log k) ) + ( R 2 (log (1), log (2) ) δ 1 (log k) 2 + τ 2 δ 2 (log k) 2 { +τ 1 δ1 (log k), δ 2 (log k) }) + [ iw (log (1), log (2) ) (τ 2 δ1 (log k), δ 2 (log k) ])

45 where R 2 (s, t) = (1 + cosh((s + t)/2)) 1 2 R 1 (x) = sinh(x/2) x sinh 2 (x/4), t(s + t) cosh s + s(s + t) cosh t (s t)(s + t + sinh s + sinh t sinh(s + t)), st(s + t) sinh(s/2) sinh(t/2) sinh 2 ((s + t)/2) W (s, t) = ( s t + t cosh s + s cosh t + sinh s + sinh t sinh(s + t)). st sinh(s/2) sinh(t/2) sinh((s + t)/2)

46 The limiting case In the commutative case, the above modular curvature reduces to a constant multiple of the formula of Gauss: 1 τ 2 δ 2 1(log k) + τ 2 τ 2 δ 2 2(log k) + 2 τ 1 τ 2 δ 1 δ 2 (log k).

47 First application: conformal anomaly (Connes-Moscovici) Polyakov s conformal anomaly formula: log det( ) = ζ (0) log det( ) log det( 0 ) = log ϕ 0 (e h ) + ϕ 0 ( R).

48 Second application: the Gauss-Bonnet theorem for A θ How to relate geometry (short term asymptotics) to topology (long term asymptotics)? MacKean-Singer formula: m ( 1) p Tr (e t p ) = p=0 m ( 1) p β p = χ(m) t > 0 p=0 Spectral formulation of the Gauss-Bonnet theorem: ζ(0) + 1 = 1 R vol(g) = 1 12π Σ 6 χ(σ)

49 Theorem (Connes-Tretkoff; Fathizadeh-K.): Let θ R, τ C \ R, k A θ be a positive invertible element. Then Trace( s ) s=0 + 2 = t (R) = 0, where is the Laplacian and R is the scalar curvature of the spectral triple attached to (A θ, τ, k).

50 Third application: Weyl Law for A θ Equivalently: This suggests: N(λ) π ϕ(1)λ as λ. I(τ) λ j I(τ) j as j. πϕ(1) Vol(T 2 θ) := 4π2 4π2 ϕ(1) = I(τ) I(τ) ϕ 0(k 2 ).

51 The geometry in noncommutative geometry Geometry starts with metric and curvature. While there are a good number of soft topological tools in NCG, like cyclic cohomology, K and KK-theory, and index theory, a truly noncommutative theory of curvature is still illusive. The situation is better with scalar curvature, but computations are quite tough at the moment. Metric aspects of NCG are informed by Spectral Geometry. Spectral invariants are the only means by which we can formulate metric ideas of NCG.