Mathematics and Cosmology: on the Universe acceleration and the … · 2012. 4. 7. · Srinivasa I...

Mathematics and Cosmology: on the Universe acceleration

and the ζ function as a regularization tool

Emilio Elizalde National Research Council of Spain

Instituto de Ciencias del Espacio (ICE/CSIC)

Institut d’Estudis Espacials de Catalunya (IEEC), UAB, Spain

Dartmouth Mathematics Colloquium, April 5, 2012

2mcE =

λΩΩΩΩΩ +++= kmrtot

TcGgRgR µνµνµνµνπλ 42

1 8−=− −

2rMmGF=

• Karl Schwarzschild: Black Hole solution (22 Dec 1915) • Willem de Sitter: massless univer static sol (just cc,’17) • Alexander Friedmann: expanding universe sol (1922) • Georges Lemaître: expanding universe (MIT 1925, AF

solution); visited Vesto Slipher (Lowell Obs, Arizona, 1912 redshifts) and Edwin Hubble (Mount Wilson, Pasadena); Keeler-Slipher-Campbell, 1918

• Led to Big Bang theory (Fred Hoyle, BBC radio 3rd Programme, 18:30 GMT, 28 March 1949)

• Fred Hoyle, Thomas Gold, Hermann Bondi: SteadyState ’48, “C-field” with a negative pressure, to be consistent with energy conservation (anticipated inflation)

• Friedmann-Lemaître-Robertson-Walker (FLRW 1931-37) • A Riess B Schmidt ’98, S Perlmutter ’99: acceler expan!

Result: supernovae are dimmer than expected

The universe is not

decelerating It is accelerating

Cannot be explained by matter+radiation

(see before)

Riess, Schmidt et al. ‘98 Perlmutter et al. ‘99

Trying to solve this puzzle !The cc λ is indeed a peculiar quantity

has to do with cosmology Einstein’s eqs., FRW universe

has to do with the local structure of elementary particle physicsstress-energy density µ of the vacuum

Lcc =

∫d4x

√−g µ4 =1

8πG

∫d4x

√−gΛ

In other words: two contributions on the same footing[Pauli 20’s, Zel’dovich ’68]

Λ c2

8πG+

1

Vol~ c

2

∑

i

ωi

For elementary particle physicists: a great embarrassment

no way to get rid offColeman, Hawking, Weinberg, Polchinski, ... ’88-’89

THE COSMOLOGICAL CONSTANT PROBLEM

E Elizalde, Dartmouth Mathematics Colloquium, April 5, 2012 – p. 6/27

Zero point energyQFT 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 ?


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 integervalues of s, and later Chebyshev extended the definition to Re s > 1.



n=1

1

ns



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



n=1

1

ns







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



n=1

1

ns







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


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)




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)]


J. Stuart Dowker, Raymond Critchley

“Effective Lagrangian and energy-momentum tensor

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


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.E Elizalde, Dartmouth Mathematics Colloquium, April 5, 2012 – p. 10/27

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


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.


Pseudodifferential Operator (ΨDO)A ΨDO of order m Mn manifold

Symbol of A: a(x, ξ) ∈ Sm(Rn × Rn) ⊂ C∞ functions such that

for any pair of multi-indices α, β there exists a constant Cα,β so

that ∣∣∣∂αξ ∂

βxa(x, ξ)

∣∣∣ ≤ Cα,β(1 + |ξ|)m−|α|


Pseudodifferential Operator (ΨDO)A ΨDO of order m Mn manifold

Symbol of A: a(x, ξ) ∈ Sm(Rn × Rn) ⊂ C∞ functions such that

for any pair of multi-indices α, β there exists a constant Cα,β so

that ∣∣∣∂αξ ∂

βxa(x, ξ)

∣∣∣ ≤ Cα,β(1 + |ξ|)m−|α|

Definition of A (in the distribution sense)

Af(x) = (2π)−n

∫ei<x,ξ>a(x, ξ)f(ξ) dξ

f is a smooth function

f ∈ S =f ∈ C∞(Rn); supx|xβ∂αf(x)| <∞, ∀α, β ∈ Nn

S ′ space of tempered distributions

f is the Fourier transform of f


ΨDOs are useful toolsThe symbol of a ΨDO has the form:

a(x, ξ) = am(x, ξ) + am−1(x, ξ) + · · · + am−j(x, ξ) + · · ·being ak(x, ξ) = bk(x) ξk

a(x, ξ) is said to be elliptic if it is invertible for large |ξ| and if there exists aconstant C such that |a(x, ξ)−1| ≤ C(1 + |ξ|)−m, for |ξ| ≥ C

− An elliptic ΨDO is one with an elliptic symbol


ΨDOs are useful toolsThe symbol of a ΨDO has the form:

a(x, ξ) = am(x, ξ) + am−1(x, ξ) + · · · + am−j(x, ξ) + · · ·being ak(x, ξ) = bk(x) ξk

a(x, ξ) is said to be elliptic if it is invertible for large |ξ| and if there exists aconstant C such that |a(x, ξ)−1| ≤ C(1 + |ξ|)−m, for |ξ| ≥ C

− An elliptic ΨDO is one with an elliptic symbol

−− ΨDOs are basic tools both in Mathematics & in Physics −−

1. Proof of uniqueness of Cauchy problem [Calderón-Zygmund]

2. Proof of the Atiyah-Singer index formula

3. In QFT they appear in any analytical continuation process —as complexpowers of differential operators, like the Laplacian [Seeley, Gilkey, ...]

4. Basic starting point of any rigorous formulation of QFT & gravitationalinteractions through µlocalization (the most important step towards theunderstanding of linear PDEs since the invention of distributions)

[K Fredenhagen, R Brunetti, . . . R Wald ’06, R Haag EPJH35 ’10]E Elizalde, Dartmouth Mathematics Colloquium, April 5, 2012 – p. 13/27

Existence of ζA 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)







∑j λ

−sj , Re s > n

m := s0


(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)







∑j λ

−sj , Re s > n

m := s0




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







∑j λ

−sj , Re s > n

m := s0




(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, . . .


Def nition of DeterminantH ΨDO operator ϕi, λi spectral decomposition



∏i∈I λi ?! ln

∏i∈I λi =

∑i∈I lnλi



∏i∈I λi ?! ln

∏i∈I λi =

∑i∈I lnλi

Riemann zeta func: ζ(s) =∑∞

n=1 n−s, Re s > 1 (& analytic cont)

Definition: 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 , Re s > s0

Derivative: ζ ′H(0) = −∑i∈I lnλi



∏i∈I λi ?! ln

∏i∈I λi =

∑i∈I lnλi








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



∏i∈I λi ?! ln

∏i∈I λi =

∑i∈I lnλi









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

until series∑

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



∏i∈I λi ?! ln

∏i∈I λi =

∑i∈I lnλi









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,...


PropertiesThe definition of the determinant detζ A only depends on thehomotopy 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 of complex 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) +∑

n6=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

]


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 on M (some authors put acoefficient in front of the integral: Adler-Manin residue)

If dim M = 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


Singularities of ζAA complete determination of the meromorphic structure of some zetafunctions in the complex plane can be also obtained by means of theDixmier trace and the Wodzicki residue

Missing for full descript of the singularities: residua of all poles

As for the regular part of the analytic continuation: specific methodshave to be used (see later)

Proposition. Under the conditions of existence of the zeta function ofA, given above, and being the symbol a(x, ξ) of the operator Aanalytic in ξ−1 at ξ−1 = 0:

Ress=skζA(s) = 1

m res A−sk = 1m

∫S∗M

tr a−sk

−n (x, ξ) dn−1ξ

Proof. The homog component of degree −n of the corresp power ofthe principal symbol of A is obtained by the appropriate derivative ofa power of the symbol with respect to ξ−1 at ξ−1 = 0 :

a−sk

−n (x, ξ) =(

∂∂ξ−1

)k [ξn−ka(k−n)/m(x, ξ)

]∣∣∣∣ξ−1=0

ξ−n


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 defined as

δ(A,B) = ln

[detζ(AB)

detζ A detζ B

]= −ζ ′AB(0) + ζ ′A(0) + ζ ′B(0)




The multiplicative (or noncommutative) anomaly (defect)is defined 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) = Aord BB−ord A


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 no

sense (much less its determinant): =⇒ detA · detB

But if diagonal form obtained after change of basis (diag.

process), the preserved quantity is: =⇒ det(AB)


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, Re t > 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 seriesImplicit spectrum −→ Cauchy’s formula (argum princ)


Extended CS Series Formulas (ECS)Consider the zeta function (Re s > p/2, A > 0,Re q > 0)

ζA,~c,q(s) =∑

~n∈Zp

′[1

2(~n+ ~c)

TA (~n+ ~c) + q

]−s

=∑

~n∈Zp

′[Q (~n+ ~c) + q]

−s

prime: point ~n = ~0 to be excluded from the sum(inescapable condition when c1 = · · · = cp = q = 0)

Q (~n+ ~c) + q = Q(~n) + L(~n) + q



ζA,~c,q(s) =∑

~n∈Zp

′[1

2(~n+ ~c)

TA (~n+ ~c) + q

]−s

=∑

~n∈Zp

′[Q (~n+ ~c) + q]

−s


Q (~n+ ~c) + q = Q(~n) + L(~n) + q

Case q 6= 0 (Re q > 0)

ζA,~c,q(s) =(2π)p/2qp/2−s

√detA

Γ(s− p/2)

Γ(s)+

2s/2+p/4+2πsq−s/2+p/4

√detA Γ(s)

×∑

~m∈Zp1/2

′ cos(2π~m · ~c)(~mTA−1~m

)s/2−p/4Kp/2−s

(2π

√2q ~mTA−1~m

)

[ECS1]



ζA,~c,q(s) =∑

~n∈Zp

′[1

2(~n+ ~c)

TA (~n+ ~c) + q

]−s

=∑

~n∈Zp

′[Q (~n+ ~c) + q]

−s


Q (~n+ ~c) + q = Q(~n) + L(~n) + q

Case q 6= 0 (Re q > 0)

ζA,~c,q(s) =(2π)p/2qp/2−s

√detA

Γ(s− p/2)

Γ(s)+

2s/2+p/4+2πsq−s/2+p/4

√detA Γ(s)

×∑

~m∈Zp1/2

′ cos(2π~m · ~c)(~mTA−1~m

)s/2−p/4Kp/2−s

(2π

√2q ~mTA−1~m

)

[ECS1]Pole: s = p/2 Residue:

Ress=p/2 ζA,~c,q(s) =(2π)p/2

Γ(p/2)(detA)−1/2


Gives (analytic cont of) multidimensional zeta function in terms of anexponentially convergent multiseries, valid in the whole complex plane



Exhibits singularities (simple poles) of the meromorphic continuation—with the corresponding residua— explicitly




Only condition on matrix A: corresponds to (non negative) quadraticform, Q. Vector ~c arbitrary, while q is (to start) a non-neg constant





Kν modified Bessel function of the second kind and the subindex 1/2in Z

p1/2 means that only half of the vectors ~m ∈ Z

p participate in thesum. E.g., if we take an ~m ∈ Z

p we must then exclude −~m[simple criterion: one may select those vectors in Z

p\~0 whosefirst non-zero component is positive]





Kν modified Bessel function of the second kind and the subindex 1/2in Z

p1/2 means that only half of the vectors ~m ∈ Z

p participate in thesum. E.g., if we take an ~m ∈ Z

p we must then exclude −~m[simple criterion: one may select those vectors in Z

p\~0 whosefirst non-zero component is positive]

Case c1 = · · · = cp = q = 0 [true extens of CS, diag subcase]

ζAp(s) =21+s

Γ(s)

p−1∑

j=0

(detAj)−1/2

[πj/2a

j/2−sp−j Γ

(s− j

2

)ζR(2s−j) +

4πsaj4− s

2p−j

∞∑

n=1

∑

~mj∈Zj

′nj/2−s(~mt

jA−1j ~mj

)s/2−j/4Kj/2−s

(2πn

√ap−j ~mt

jA−1j ~mj

)]

EE, JPA34 (2001) 3025 [ECS3d]


The Sign of the Casimir ForceMany papers dealing on this issue: here just short account



Casimir calculation: attractive force




Boyer got repulsion [TH, Phys Rev, 174 (1968)] for a spherical shell. Itis a special case requiring stringent material properties of the sphereand a perfect geometry and BC





Systematic calculation, for different fields, BCs, and dimensionsJ Ambjørn, S Wolfram, Ann Phys NY 147, 1 (1983) attract, repuls






Possibly not relevant at lab scales, but very important for cosmologicalmodels






Possibly not relevant at lab scales, but very important for cosmologicalmodels

More general results: Kenneth, Klich, PRL 97, 160401 (2006)a mirror pair of dielectric bodies always attract each otherCP Bachas, J Phys A40, 9089 (2007) from a general property ofEuclidean QFT ‘reflection positivity’ (Osterwalder - Schrader 73, 75):∃ of positive Hilbert space and self-adjoint non-negative Hamiltonian


E.g. ∃ correlation inequality: 〈fΘ(f)〉 > 0

Θ reflection with respect to a 3-dim hyperplane in R4

the action of Θ on f is anti-unitary Θ(cf) = c∗Θ(f)





The existence of the reflection operator Θ is a consequence ofunitarity only, and makes no assumptions about the discreteC,P, T symmetries






Boyer’s result does not contradict the theorem, since cutting an elasticshell into two rigid hemispheres is a mathematically singular operation(which introduces divergent edge contributions)







Theorem does not apply for








mirror probes in a Fermi sea (chemical-potential term), eg whenelectron-gas fluctuations become important









periodic BCs for fermions









periodic BCs for fermions

Robin BCs in general ⇐


The Braneworld Case1. Braneworld may help to solve:

the hierarchy problem

the cosmological constant problem

2. Presumably, the bulk Casimir effect will play a role in the construction (radion

stabilization) of braneworlds [A Flachi]

Bulk Casimir effect (effective potential) for a conformal or massive scalar field

Bulk is a 5-dim AdS or dS space with 2/1 4-dim dS brane (our universe)

Consistent with observational data even for relatively large extra dimension

Previous work: −→ flat space brane−→ bulk conformal scalar field

−→ conclusion: no CE

We used zeta regularization at full power, with positive results!

EE, Nojiri, Odintsov, Ogushi, PRD67(2003)063515, hep-th/0209242 Casimir effect in

de Sitter and Anti-de Sitter braneworlds EE, Odintsov, Saharian PRD79(2009)065023,

0902.0717 Repulsive Casimir effect from extra dimensions and Robin BC: from branes to pistons


Some Books 2nd Ed June 201 2

http://books.google.com/books?id=PXkhYSqIeykC&hl=es&source=gbs_similarbooks

http://books.google.com/books?id=GKkEIyuITWQC&hl=es&source=gbs_similarbooks

http://books.google.com/books?id=tBmliSFW8eYC&dq=Cosmology%2C%20Quantum%20Vacuum%2C%20Zeta%20functions&hl=es&source=gbs_similarbooks

Thank You

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