On cosmology, infinities, and zeta functions

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On cosmology, infinities, and zeta functions Emilio Elizalde ICE/CSIC & IEEC Pizza-Lunch Seminar ICE-IEEC, Bellaterra, 9 Oct 2015

Transcript of On cosmology, infinities, and zeta functions

Page 1: On cosmology, infinities, and zeta functions

On cosmology, infinities,

and zeta functionsEmilio Elizalde

ICE/CSIC & IEEC

Pizza-Lunch Seminar ICE-IEEC, Bellaterra, 9 Oct 2015

Page 2: On cosmology, infinities, and zeta functions

• 1917 Einstein found from his equations of GR a static model of the Universe, by introducing the cosmological constant Λ

• 1917, few months after, de Sitter found a solution which might explain Slipher’s nebular redshifts

• 1922 Friedmann found Einstein’s Eqs allowed a dynamic Universe, but did not connect it to astronomical observations

• 1922 Ernst Öpik, by an ingenious method, had already found a distance of 450 kpc to Andromeda, much closer to the real value (775 kpc) than Hubble’s later distance of 285 kpc

• 1927 Lemaître derived value of H using Hubble’s 1926 distances and Slipher’s redshifts; depending on choice of observations he arrived at 625 or 575 (compared to Hubble’s 500(km/s)/Mpc in 1929). He was fully aware of the significance of his discovery

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Ernst J. Öpik

Estonian astronomer and astrophys. (1893-1985) worked at the Armagh Observatory in Northern Ireland. In 1922 published a paper estimating the distance to Andromeda using an original method based on observed rotational velocities of the galaxy: 450 kpc. Was the first to calculate the density of a white dwarf.

His result was closer to recent estimates (775 kpc) than Hubble's result (285 kpc) of Nov 23, 1924; E Öpik, ApJ 55, 406, 1922.

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VestoSlipher

On September 17, 1912, obtained the first radial velocity of a "spiral nebula" - Andromeda. Using the 24-inch telescope at Lowell Observatory, AZ, he got more Doppler shifts, establishing that large velocities, usually in recession, were a general property of the spiral nebulae.

Slipher presented his results of the speed of 15 nebulae to the Am Astronomical Society in 1914, and received a standing ovation.

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• 1929 Hubble opted for K = 500(km/s)/Mpc as his favorite value, workingwith his own distances and Slipher’s redshifts, as tabulated inEddington’s The Mathematical Theory of Relativity (2nd Ed 1924)

• Hubble refrained from interpreting his observational discovery: “Theoutstanding feature, however, is the possibility that the velocity-distance relation may represent the de Sitter effect…”. In a later letter tode Sitter, Hubble wrote he would leave the interpretation to those“competent to discuss the matter with authority”. In none of the 7 pagesof Hubble’s paper is there a single word about an expanding universe; itis likely that Hubble never believed in such a thing

• It is not known when Einstein was converted to believe in the expandingUniverse; it probably happened when Eddington showed him that hisstatic solution was unstable

Stigler’s law of eponymy:‘No scientific discovery is named after its original discoverer’

Kragh, Smith ‘03; Livio ‘11; Luminet, ‘11; Nussbaumer, Bieri, CUP ‘09 ‘11 ‘13

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Annales de la Société Scientifique de Bruxelles, 1927: "Un Univers homogène Member of the Pontifical Academy of Sciences de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extragalactiques", G. Lemaître

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The accelerating Universe Evidence for the acceleration of the Universe expansion:

distant supernovae AG Riess e a ’98, S Perlmutter e a ’99

the cosmic

microwave

background

S. Dunkley

et

al.,

’99,

E.

Komatsu

et

al. ’99, B.D. Sherwin

et al. ’11, A. van Engelen et al. ’12

baryon acoustic oscillations BAO Martin White mwhite/bao/

the galaxy distribution AG Sanchez ea 12, Gaztanaga ea

correlations of galaxy distribs R Scranton ea ’03

Multiple sets of evidence: no systematics affect theconclusion that a > 0, a scale factor of the Universe

the age of the Universe Planck Collab ’14

Eli
Resaltado
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A masterful commentary on the history of science from the Greeks to modern times, by Nobel Prize-winning physicist Steven Weinberg—a thought-provoking and important book by one of the most distinguished scientists and intellectuals of our time.

Harper and Collins, 2015, ISBN-13: 978-0062346650, ISBN-10: 0062346652

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2mcE =

λΩΩΩΩΩ +++= kmrtot

TcGgRgR µνµνµνµνπλ 42

1 8−=− −

2rMmGF=

Isaac Newton (1642–1727)

Albert Einstein (1879-1955)

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Zero point energy

QFT vacuum to vacuum transition: 〈0|H|0〉Spectrum, normal ordering (harm oscill):

H =

(n+

1

2

)λn an a

†n

〈0|H|0〉 =~ c

2

n

λn =1

2tr H

gives ∞ physical meaning?

Regularization + Renormalization ( cut-off, dim, ζ )

Even then: Has the final value real sense ?

QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27

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Operator Zeta F’s in MΦ : OriginsThe Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, onestarts with the infinite series ∞∑

n=1

1

ns

which converges for all complex values of s with real Re s > 1, and then defines ζ(s) asthe analytic continuation, to the whole complex s−plane, of the function given, Re s > 1,by the sum of the preceding series.

Leonhard Euler already considered the above series in 1740, but for positive integer values of s, and later Чебышёв extended the definition to Re s > 1.

QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 4/27

Emilio
Resaltado
Emilio
Resaltado
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−3 −2 −1 0 1 2 3 4 5 −4

s

a.c.

(s) = −−1

nsζ Σ

(s)ζ

pole

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Operator Zeta F’s in M Φ: OriginsThe Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, onestarts with the infinite series ∞∑

n=1

1

ns

which converges for all complex values of s with real Re s > 1, and then defines ζ(s) asthe analytic continuation, to the whole complex s−plane, of the function given, Re s > 1,by the sum of the preceding series.

Leonhard Euler already considered the above series in 1740, but for positive integervalues of s, and later Chebyshev extended the definition to Re s > 1.

Godfrey H Hardy and John E Littlewood, “Contributions to the Theory of the RiemannZeta-Function and the Theory of the Distribution of Primes", Acta Math 41, 119 (1916)

Did much of the earlier work, by establishing the convergence and equivalence of seriesregularized with the heat kernel and zeta function regularization methods

G H Hardy, Divergent Series (Clarendon Press, Oxford, 1949)

Srinivasa I Ramanujan had found for himself the functional equation of the zeta function

Torsten Carleman, “Propriétés asymptotiques des fonctions fondamentales desmembranes vibrantes" (French), 8. Skand Mat-Kongr, 34-44 (1935)

Zeta function encoding the eigenvalues of the Laplacian of a compact Riemannianmanifold for the case of a compact region of the plane

QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 4/27

Emilio
Resaltado
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Robert T Seeley, “Complex powers of an elliptic operator. 1967Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966)pp. 288-307, Amer. Math. Soc., Providence, R.I.

Extended this to elliptic pseudo-differential operators A on compactRiemannian manifolds. So for such operators one can define thedeterminant using zeta function regularization

D B Ray, Isadore M Singer, “R-torsion and the Laplacian onRiemannian manifolds", Advances in Math 7, 145 (1971)

Used this to define the determinant of a positive self-adjoint operatorA (the Laplacian of a Riemannian manifold in their application) witheigenvalues a1, a2, ...., and in this case the zeta function is formallythe trace

ζA(s) = Tr (A)−s

the method defines the possibly divergent infinite product∞∏

n=1

an = exp[−ζA′(0)]

QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 5/27

Emilio
Resaltado
Emilio
Resaltado
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J. Stuart Dowker, Raymond Critchley

“Effective Lagrangian and energy-momentum tensor

in de Sitter space", Phys. Rev. D13, 3224 (1976)

Abstract

The effective Lagrangian and vacuum energy-momentum

tensor < Tµν > due to a scalar field in a de Sitter space

background are calculated using the dimensional-regularization

method. For generality the scalar field equation is chosen in the

form (2 + ξR+m2)ϕ = 0. If ξ = 1/6 and m = 0, the

renormalized < Tµν > equals gµν(960π2a4)−1, where a is the

radius of de Sitter space. More formally, a general zeta-function

method is developed. It yields the renormalized effective

Lagrangian as the derivative of the zeta function on the curved

space. This method is shown to be virtually identical to a

method of dimensional regularization applicable to any

Riemann space.QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 6/27

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Stephen W Hawking, “Zeta function regularization of path integralsin curved spacetime", Commun Math Phys 55, 133 (1977)

This paper describes a technique for regularizing quadratic pathintegrals on a curved background spacetime. One forms ageneralized zeta function from the eigenvalues of the differentialoperator that appears in the action integral. The zeta function is ameromorphic function and its gradient at the origin is defined to be thedeterminant of the operator. This technique agrees with dimensionalregularization where one generalises to n dimensions by adding extraflat dims. The generalized zeta function can be expressed as a Mellintransform of the kernel of the heat equation which describes diffusionover the four dimensional spacetime manifold in a fifth dimension ofparameter time. Using the asymptotic expansion for the heat kernel,one can deduce the behaviour of the path integral under scaletransformations of the background metric. This suggests that theremay be a natural cut off in the integral over all black hole backgroundmetrics. By functionally differentiating the path integral one obtains anenergy momentum tensor which is finite even on the horizon of ablack hole. This EM tensor has an anomalous trace.

QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 7/27

Emilio
Resaltado
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Basic strategiesJacobi’s identity for the θ−function

θ3(z, τ) := 1 + 2∑∞

n=1 qn2

cos(2nz), q := eiπτ , τ ∈ C

θ3(z, τ) = 1√−iτ

ez2/iπτ θ3

(zτ|−1

τ

)equivalently:

∞∑

n=−∞e−(n+z)2t =

√π

t

∞∑

n=0

e−π2n2

t cos(2πnz), z, t ∈ C, Ret > 0

Higher dimensions: Poisson summ formula (Riemann)∑

~n∈Zp

f(~n) =∑

~m∈Zp

f(~m)

f Fourier transform[Gelbart + Miller, BAMS ’03, Iwaniec, Morgan, ICM ’06]

Truncated sums −→ asymptotic series

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2nd Reading

December 22, 2014 15:18 WSPC/S0219-8878 IJGMMP-J043 1550019

International Journal of Geometric Methods in Modern PhysicsVol. 12 (2015) 1550019 (28 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021988781550019X

Zeta functions on tori using contour integration

Emilio Elizalde

Institute of Space Science, National Higher Research CouncilICE–CSIC, Facultat de Ciencies, Campus UAB

Torre C5-Par-2a, 08193 Bellaterra (Barcelona), [email protected]

[email protected]

Klaus Kirsten

GCAP–CASPER, Department of MathematicsBaylor University, One Bear Place, Waco TX 76798, USA

klaus [email protected]

Nicolas Robles∗

Institut fur Mathematik, Universitat ZurichWinterthurerstrasse 190, CH-8057 Zurich, Switzerland

[email protected]

Floyd Williams

Department of Mathematics and Statistics,Lederle Graduate Research Tower, 710 North Pleasant StreetUniversity of Massachusetts, Amherst MA 01003-9305, USA

[email protected]

Received 26 August 2013Accepted 29 October 2014

Published 24 December 2014

A new, seemingly useful presentation of zeta functions on complex tori is derived by usingcontour integration. It is shown to agree with the one obtained by using the Chowla–Selberg series formula, for which an alternative proof is thereby given. In addition,a new proof of the functional determinant on the torus results, which does not usethe Kronecker first limit formula nor the functional equation of the non-holomorphicEisenstein series. As a bonus, several identities involving the Dedekind eta function areobtained as well.

Keywords: Tori; zeta functions; spectral theory; functional determinants.

Mathematics Subject Classification 2010: Primary 11M41, 81T40; Secondary 81T25

∗Corresponding author.

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Emilio
Resaltado
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Existence ofζA for A a ΨDO1. A a positive-definite elliptic ΨDO of positive order m ∈ R+

2. A acts on the space of smooth sections of

3. E, n-dim vector bundle over

4. M closed n-dim manifold

(a) The zeta function is defined as:ζA(s) = tr A−s =

∑j λ

−sj , Re s > n

m := s0

λj ordered spect of A, s0 = dimM/ordA abscissa of converg of ζA(s)

(b) ζA(s) has a meromorphic continuation to the whole complex plane C

(regular at s = 0), provided the principal symbol of A, am(x, ξ), admits aspectral cut: Lθ = λ ∈ C;Argλ = θ, θ1 < θ < θ2, SpecA ∩ Lθ = ∅(the Agmon-Nirenberg condition)

(c) The definition of ζA(s) depends on the position of the cut Lθ

(d) The only possible singularities of ζA(s) are poles atsj = (n− j)/m, j = 0, 1, 2, . . . , n− 1, n+ 1, . . .

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Definition of DeterminantH ΨDO operator ϕi, λi spectral decomposition

∏i∈I λi ?! ln

∏i∈I λi =

∑i∈I lnλi

Riemann zeta func: ζ(s) =∑∞

n=1 n−s, Re s > 1 (& analytic cont)

Def nition: zeta function of H ζH(s) =∑

i∈I λ−si = tr H−s

As Mellin transform: ζH(s) = 1Γ(s)

∫ ∞0 dt ts−1 tr e−tH , Res > s0

Derivative: ζ ′H(0) = −∑i∈I lnλi

Determinant: Ray & Singer, ’67detζ H = exp [−ζ ′H(0)]

Weierstrass def: subtract leading behavior of λi in i, as i→ ∞,until series

∑i∈I lnλi converges =⇒ non-local counterterms !!

C. Soulé et al, Lectures on Arakelov Geometry, CUP 1992; A. Voros,...

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PropertiesThe definition of the determinant detζ A only depends on the homotopy class of the cut

A zeta function (and corresponding determinant) with the samemeromorphic structure in the complex s-plane and extending theordinary definition to operators ofcomplex order m ∈ C\Z (they do notadmit spectral cuts), has been obtained [Kontsevich and Vishik]

Asymptotic expansion for the heat kernel:

tr e−tA =∑′

λ∈Spec A e−tλ

∼ αn(A) +∑

n 6=j≥0 αj(A)t−sj +∑

k≥1 βk(A)tk ln t, t ↓ 0

αn(A) = ζA(0), αj(A) = Γ(sj)Ress=sjζA(s), sj /∈ −N

αj(A) = (−1)k

k! [PP ζA(−k) + ψ(k + 1)Ress=−k ζA(s)] ,

sj = −k, k ∈ N

βk(A) = (−1)k+1

k! Ress=−k ζA(s), k ∈ N\0

PP φ := lims→p

[φ(s) − Ress=p φ(s)

s−p

]

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“ Hi, Emilio. This is a question I have been trying to solve for years. With a bit of luck you could maybe provide me with a hint or two.

• Imagine I’ve got a functional integral and I perform a point transformation (doesn’t involvederivatives). Its Jacobian is a kind of functional determinant, but of a non-elliptic operator(it is simply infinite times multiplication by a function.) Did anybody study this seriously?

• I do know, from at least one paper I did with Luis AG, that in some cases (T duality) one isbound to define something like

• det f(x) ~ det [f(x) O] / det O where O es an elliptic operator (e.g. the Laplacian)

• This is what Schwarz and Tseytlin did in order to obtain the dilaton transformation• And LAG and I did also proceed in a basically similar way• As I know, Konsevitch, too, uses a related method involving the multiplicative anomaly

Tell me what you know about, please. Thanks so much.- Hugs, Enrique “

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The Dixmier TraceIn order to write down an action in operator language one needs afunctional that replaces integration

For the Yang-Mills theory this is the Dixmier trace

It is the unique extension of the usual trace to the ideal L(1,∞) of thecompact operators T such that the partial sums of its spectrumdiverge logarithmically as the number of terms in the sum:

σN (T ) :=∑N−1

j=0 µj = O(logN), µ0 ≥ µ1 ≥ · · ·

Definition of the Dixmier trace of T:Dtr T = limN→∞

1log N σN (T )

provided that the Cesaro meansM(σ)(N) of the sequence in N areconvergent as N → ∞ [remember: M(f)(λ) = 1

ln λ

∫ λ

1f(u)du

u ]

The Hardy-Littlewood theorem can be stated in a way that connectsthe Dixmier trace with the residue of the zeta function of the operatorT−1 at s = 1 [Connes] Dtr T = lims→1+(s− 1)ζT−1(s)

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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of theDixmier trace to ΨDOs which are not in L(1,∞)

Only trace one can define in the algebra of ΨDOs (up to multipl const)

Definition: res A = 2 Ress=0 tr(A∆−s), ∆ Laplacian

Satisfies the trace condition: res (AB) = res (BA)

Important!: it can be expressed as an integral (local form)

res A =∫

S∗Mtr a−n(x, ξ) dξ

with S∗M ⊂ T ∗M the co-sphere bundle onM (some authors put acoefficient in front of the integral: Adler-Manin residue)

If dimM = n = − ord A (M compact Riemann, A elliptic, n ∈ N)it coincides with the Dixmier trace, and Ress=1ζA(s) = 1

n res A−1

The Wodzicki residue makes sense for ΨDOs of arbitrary order.Even if the symbols aj(x,ξ), j < m, are not coordinate invariant, the integral is, and defines a trace

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Multipl or N-Comm Anomaly, or DefectGiven A, B, and AB ψDOs, even if ζA, ζB, and ζAB exist,it turns out that, in general,

detζ(AB) 6= detζA detζB

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Multipl or N-Comm Anomaly, or DefectGiven A, B, and AB ψDOs, even if ζA, ζB, and ζAB exist,it turns out that, in general,

detζ(AB) 6= detζA detζB

The multiplicative (or noncommutative) anomaly (defect)is def ned as

δ(A,B) = ln

[detζ(AB)

detζ A detζ B

]= −ζ ′AB(0) + ζ ′A(0) + ζ ′B(0)

Wodzicki formula

δ(A,B) =res

[ln σ(A,B)]2

2 ordA ordB (ordA+ ordB)

where σ(A,B) = AordBB−ordA

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Consequences of the Multipl AnomalyIn the path integral formulation

∫[dΦ] exp

∫dDx

[Φ†(x)

( )Φ(x) + · · ·

]

Gaussian integration: −→ det( )±

A1 A2

A3 A4

−→

A

B

det(AB) or detA · detB ?

In a situation where a superselection rule exists, AB has nosense (much less its determinant): =⇒ detA · detB

But if diagonal form obtained after change of basis (diag.process), the preserved quantity is: =⇒ det(AB)

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Cosmological attractor inflation from the RG-improved Higgs sector of finite gauge theory ArXiv ePrint: 1509.08817

Emilio Elizalde, Sergei D. Odintsov, Ekaterina O. Pozdeeva, Sergey Yu. Vernov

Abstract

The possibility to construct an inflationary scenario for renormalization-group improved potentials corresponding to the Higgs sector of finite gauge models is investigated. Taking into account quantum corrections to the renormalization-group potential which sums all leading logs of perturbation theory is essential for a successful realization of the inflationary scenario, with very reasonable parameters values. The inflationary models thus obtained are seen to be in good agreement with the most recent and accurate observational data. More specifically, the values of the relevant inflationary parameters, ns and r, are close to the corresponding ones in the R2 and Higgs-driven inflation scenarios. It is shown that the model here constructed and Higgs-driven inflation belong to the same class of cosmological attractors.

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Thank You

Gràcies