ice zeta functions, heat kernels, and the quantum …heat kernels 06, blaubeuren, 28.11-2.12.2006...

ICE

��

��

Zeta functions, heat kernels,and the quantum vacuum:some non-standard cases

EMILIO ELIZALDE

ICE/CSIC & IEEC, UAB, Barcelona, Spain

web: google → emilio elizalde

Heat Kernels 06, Blaubeuren, November 30, 2006

Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 1/32

Outline of this presentation

ΨDOs, Zeta Functions, Determinants, and Traces




Wodzicki Residue, Multiplicative (or Noncommutative)Anomaly, or Defect





The Chowla-Selberg Expansion Formula (CS)& Extended Expressions (ECS)






Singularities of ζA: Compact vs. Non-compact cases







Examples from Physics:− Non-commutative QFTs− Exponential Potentials








Proposal for extended zeta function regularization








Proposal for extended zeta function regularization

Relation with regularized products


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∣

∣

∣∂α

ξ ∂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∣

∣

∣∂α

ξ ∂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)f̂(ξ) dξ

f is a smooth function

f ∈ S ={

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

S ′ space of tempered distributions

f̂ is the Fourier transform of fHeat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 3/32

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 formula3. In QFT they appear in any analytical continuation process —as complex

powers of differential operators, like the Laplacian [Seeley, Gilkey, ...]4. Constitute nowadays the basic starting point of any rigorous formulationof QFT field theory through µlocalization (the most important step towardsthe understanding of linear PDEs since the invention of distributions)

[Fredenhagen, Rehren, Seiler, Wald, Brunetti, Verch, Radzikowski, ...][RW] The history and present status of QFT in curved spacetime, gr-qc/0608018

[F+R+S] QFT: where we are, hep-th/0603155 (Ans. Interact QFT at perturb level)Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 4/32

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)








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)








m ≡ s0




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








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 at

sk = (n− k)/m, k = 0, 1, 2, . . . , n− 1, n+ 1, . . .


Definition 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








i∈I lnλi

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



∏

i∈I λi ?! ln∏

i∈I λi =∑

i∈I lnλi








i∈I lnλi

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

Weierstrass definition: subtract leading behavior of λi in i, as

i→ ∞, until the series∑

i∈I lnλi converges

=⇒ non-local counterterms !!Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 6/32

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, Vishik]




Asymptotic expansion for the heat kernel:




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! [PPζA(−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}

PPφ = lims→p

[

φ(s) − Ress=p φ(s)s−p

]


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−1j=0 µj = O(logN), µ0 ≥ µ1 ≥ · · ·





σN (T ) ≡ ∑N−1j=0 µj = O(logN), µ0 ≥ µ1 ≥ · · ·

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

1log N σN (T )

provided that the Cesaro means M(σ)(N) of the sequence in N are

convergent as N → ∞ [remember: M(f)(λ) = 1ln λ

∫ λ

1f(u)du

u ]





σN (T ) ≡ ∑N−1j=0 µj = O(logN), µ0 ≥ µ1 ≥ · · ·

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

1log N σN (T )

provided that the Cesaro means M(σ)(N) of the sequence in N are

convergent as N → ∞ [remember: M(f)(λ) = 1ln 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)


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 an(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







res A =∫



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

n res A−1







res A =∫



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

n res A−1

The Wodzicki res makes sense for ΨDOs of arbitrary order. Even ifsymbols 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 the zetafunction in the complex plane can be 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∗Mtr a−sk

−n (x, ξ) dn−1ξ





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∗Mtr a−sk

−n (x, ξ) dn−1ξ

Proof. Homog component of degree −n of the corresp power of theprincipal symbol of A are obtained by the appropriate derivative of apower 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 (ordefect) 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 (ordefect) 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 AnomIn 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 ?



∫

[dΦ] exp

{

−∫

dDx[

Φ†(x)( )

Φ(x) + · · ·]

}


A1 A2

A3 A4

−→

A

B


In a situation where a superselection rule exists, AB has no

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



∫

[dΦ] exp

{

−∫

dDx[

Φ†(x)( )

Φ(x) + · · ·]

}


A1 A2

A3 A4

−→

A

B


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)Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 12/32

The Chowla-Selberg Formula (CS)M. Lerch, Sur quelques formules relatives du nombre des classes, Bull. Sci.

Math. 21 (1897) 290-304



Math. 21 (1897) 290-304

S. Chowla and A. Selberg, On Epstein’s Zeta function (I), Proc. Nat. Acad.Sci. 35 (1949) 371-74



Math. 21 (1897) 290-304


A. Selberg and S. Chowla, On Epstein’s Zeta function, J. reine angew. Math.

(Crelle’s J.) 227 (1967) 86-110



Math. 21 (1897) 290-304



(Crelle’s J.) 227 (1967) 86-110

K. Ramachandra, Some applications of Kronecker’s limit formulas, Ann. Math.

80 (1964) 104-148



Math. 21 (1897) 290-304



(Crelle’s J.) 227 (1967) 86-110


80 (1964) 104-148

A. Weil, Elliptic functions according to Eisenstein and Kronecker (Springer,Berlin, 1976)



Math. 21 (1897) 290-304



(Crelle’s J.) 227 (1967) 86-110


80 (1964) 104-148


S. Iyanaga and Y. Kawada, Eds., Encyclopedic Dictionary of Mathematics, Vol.

II (The MIT Press, Cambridge, 1977), pp. 1378-79



Math. 21 (1897) 290-304



(Crelle’s J.) 227 (1967) 86-110


80 (1964) 104-148




B.H. Gross, On the periods of abelian integrals and a formula of Chowla andSelberg, Inv. Math. 45 (1978) 193-211



Math. 21 (1897) 290-304



(Crelle’s J.) 227 (1967) 86-110


80 (1964) 104-148




B.H. Gross, On the periods of abelian integrals and a formula of Chowla andSelberg, Inv. Math. 45 (1978) 193-211

P. Deligne, Valeurs de fonctions L et periodes d’integrales, PSPM 33 (1979)313-346 Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 13/32

HistoryLerch (1897):

|D|∑

λ=1

(

D

λ

)

log Γ

(

λ

D

)

= h log |D| − h

3log(2π) −

∑

(a,b,c)

log a

+2

3

∑

(a,b,c)

log[

θ′1(0|α)θ′1(0|β)]

D discriminant, θ′1 ∼ η3

h class number of binary quadratic forms (a, b, c)


HistoryLerch (1897):

|D|∑

λ=1

(

D

λ

)

log Γ

(

λ

D

)

= h log |D| − h

3log(2π) −

∑

(a,b,c)

log a

+2

3

∑

(a,b,c)

log[

θ′1(0|α)θ′1(0|β)]

D discriminant, θ′1 ∼ η3

h class number of binary quadratic forms (a, b, c)

Eta evaluations Dedekind eta function for Im (τ) > 0

η(τ) = q1/24∏∞

n=1(1 − qn), q := e2πiτ

It is a 24-th root of the discriminant func ∆(τ) of an elliptic

curve C/L from a lattice L = {aτ + b | a, b ∈ Z}

∆(τ) = (2π)12q

∞∏

n=1

(1 − qn)24


Properties & Recent Results=⇒ The C-S formula gives the value of a product of eta

functions



functions

=⇒ If there is only one form in the class, it yields the value ofa single eta function in terms of gamma functions



functions


=⇒ Long series of improvements [Kaneko (90),Nakajima and Taguchi (91), Williams et al. (95)]



functions



=⇒ In the last 5 years the C-S formula has been ‘broken’ toisolate the eta functions[Williams, van Poorten, Chapman, Hart]



functions




R. Chapman and W.B. Hart, Evaluation of the Dedekindeta function, Can. Math. Bull. (2005)



functions




R. Chapman and W.B. Hart, Evaluation of the Dedekindeta function, Can. Math. Bull. (2005)

W.B. Hart, PhD Thesis, 2004 (Macquarie U., Sidney)Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 15/32

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



θ3(z, τ) := 1 + 2∑∞

n=1 qn2


θ3(z, τ) = 1√−iτ

ez2/iπτ θ3

(

zτ|−1

τ

)

equivalently:

∞∑

n=−∞e−(n+z)2t =

√

π

t

∞∑

n=0

e−π2n2


Higher dimensions: Poisson summ formula (Riemann)∑

~n∈Zp

f(~n) =∑

~m∈Zp

f̃(~m)

f̃ Fourier transform [Gelbart + Miller, BAMS, Oct. ’03]



θ3(z, τ) := 1 + 2∑∞

n=1 qn2


θ3(z, τ) = 1√−iτ

ez2/iπτ θ3

(

zτ|−1

τ

)

equivalently:

∞∑

n=−∞e−(n+z)2t =

√

π

t

∞∑

n=0

e−π2n2


Higher dimensions: Poisson summ formula (Riemann)∑

~n∈Zp

f(~n) =∑

~m∈Zp

f̃(~m)

f̃ Fourier transform [Gelbart + Miller, BAMS, Oct. ’03]

Truncated sums−→ asymptotic series


Extended CS 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) + q̄



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) + q̄Case 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/4

Kp/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) + q̄Case 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/4

Kp/2−s

(

2π√

2q ~mTA−1~m)

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

Ress=p/2ζA,~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 (for now) 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 can 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 can 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/2aj/2−sp−j Γ

(

s− j

2

)

ζR(2s−j) +

4πsaj4− s

2p−j

∞∑

n=1

∑

~mj∈Zj

′nj/2−s(

~mtjA

−1j ~mj

)s/2−j/4Kj/2−s

(

2πn√

ap−j ~mtjA

−1j ~mj

)]

[ECS3d]Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 18/32

QFT in s-t with non-comm toroidal partD−dim non-commut manifold: M = R

1,d⊗

Tpθ, D = d+ p+ 1

Tpθ a p−dim non-commutative torus: [xj , xk] = iθσjk

σjk a real, nonsingular, antisymmetric matrix of ±1 entries

θ the non-commutative parameter [with Bytsenko, Zerbini]



1,d⊗

Tpθ, D = d+ p+ 1




Interest recently, in connection with M−theory & string theory

[Connes,Douglas,Seiberg,Cheung,Chu,Chomerus,Ardalan, . . . ]



1,d⊗

Tpθ, D = d+ p+ 1




Interest recently, in connection with M−theory & string theory

[Connes,Douglas,Seiberg,Cheung,Chu,Chomerus,Ardalan, . . . ]

Unified treatment: only one zeta function, nature of field

(bosonic, fermionic) as a parameter, together with # of

compact, noncompact, and noncommutative dimensions

ζα(s) =V Γ(s− (d+ 1)/2)

(4π)(d+1)/2 Γ(s)

∑

~n∈Zp

′Q(~n)(d+1)/2−s

[

1+Λθ2−2αQ(~n)−α](d+1)/2−s

α = 2 bos, α = 3 ferm, V = Vol (Rd+1) of non-compact part

Q(~n) =∑p

j=1 ajn2j a diag quadratic form, Rj = a

−1/2j compactific radii


After some calculations,

ζα(s) =V

(4π)(d+1)/2

∞∑

l=0

Γ(s+ l − d+12 )

l! Γ(s)(−Λθ2−2α)l ζQ,~0,0(s+αl−

d+ 1

2)

for all radii equal to R, with I(~n) =∑p

j=1 n2j ,

ζα(s) =V

(4π)(d+1)/2Rd+1−2s

∞∑

l=0

Γ(s+ l − d+12 )

l! Γ(s)(−Λθ2−2α)l ζE(s+αl−d+ 1

2)

where we use the notation ζE(s) ≡ ζI,~0,0(s)

e.g., the Epstein zeta function for the standard quadratic form



ζα(s) =V

(4π)(d+1)/2

∞∑

l=0

Γ(s+ l − d+12 )

l! Γ(s)(−Λθ2−2α)l ζQ,~0,0(s+αl−

d+ 1

2)


j=1 n2j ,

ζα(s) =V

(4π)(d+1)/2Rd+1−2s

∞∑

l=0

Γ(s+ l − d+12 )


2)



Rich pole structure: pole of Epstein zf at

s = p/2 − αk + (d+ 1)/2 = D/2 − αk, combined with poles of

Γ, yields a rich pattern of singul for ζα(s)



ζα(s) =V

(4π)(d+1)/2

∞∑

l=0

Γ(s+ l − d+12 )

l! Γ(s)(−Λθ2−2α)l ζQ,~0,0(s+αl−

d+ 1

2)


j=1 n2j ,

ζα(s) =V

(4π)(d+1)/2Rd+1−2s

∞∑

l=0

Γ(s+ l − d+12 )


2)



Rich pole structure: pole of Epstein zf at

s = p/2 − αk + (d+ 1)/2 = D/2 − αk, combined with poles of

Γ, yields a rich pattern of singul for ζα(s)

Classify the different possible cases according to the values of

d and D = d+ p+ 1. We obtain, at s = 0,Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 20/32

For d = 2k

if D 6= ˙2α =⇒ ζα(0) = 0

if D = ˙2α =⇒ ζα(0) = finite

For d = 2k − 1

if D 6= ˙2α

finite, for l ≤ k

0, for l > k

=⇒ ζα(0) = finite

if D = 2αl

pole, for l ≤ k

finite, for l > k

=⇒ ζα(0) = pole

Pole structure of the zeta function ζα(s), at s = 0, according to the

different possible values of d and D ( ˙2α means multiple of 2α)


For d = 2k

if D 6= ˙2α =⇒ ζα(0) = 0

if D = ˙2α =⇒ ζα(0) = finite

For d = 2k − 1

if D 6= ˙2α

finite, for l ≤ k

0, for l > k

=⇒ ζα(0) = finite

if D = 2αl

pole, for l ≤ k

finite, for l > k

=⇒ ζα(0) = pole

Pole structure of the zeta function ζα(s), at s = 0, according to the

different possible values of d and D ( ˙2α means multiple of 2α)

=⇒ Explicit analytic continuation of ζα(s), α = 2, 3,

& specific pole structureHeat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 21/32

ζα(s) =2s−d V

(2π)(d+1)/2Γ(s)

∞∑

l=0

Γ(s+ l − (d+ 1)/2)

l! Γ(s+ αl − (d+ 1)/2)(−2αΛθ2−2α)l

p−1∑

j=0

(detAj)− 1

2

×[

πj/2a−s−αl+(d+j+1)/2p−j Γ(s+ αl − (d+ j + 1)/2)ζR(2s+ 2αl − d− j − 1)

+4πs+αl−(d+1)/2a−(s+αl)/2−(d+j+1)/4p−j

∞∑

n=1

∑

~mj∈Zj

′n(d+j+1)/2−s−αl

×(

~mtjA

−1j ~mj

)(s+αl)/2−(d+j+1)/4K(d+j+1)/2−s−αl

(

2πn√

ap−j ~mtjA

−1j ~mj

)]


ζα(s) =2s−d V

(2π)(d+1)/2Γ(s)

∞∑

l=0

Γ(s+ l − (d+ 1)/2)

l! Γ(s+ αl − (d+ 1)/2)(−2αΛθ2−2α)l

p−1∑

j=0

(detAj)− 1

2

×[

πj/2a−s−αl+(d+j+1)/2p−j Γ(s+ αl − (d+ j + 1)/2)ζR(2s+ 2αl − d− j − 1)

+4πs+αl−(d+1)/2a−(s+αl)/2−(d+j+1)/4p−j

∞∑

n=1

∑

~mj∈Zj

′n(d+j+1)/2−s−αl

×(

~mtjA

−1j ~mj

)(s+αl)/2−(d+j+1)/4K(d+j+1)/2−s−αl

(

2πn√

ap−j ~mtjA

−1j ~mj

)]

p \ D even odd

odd (1a) pole / finite (l ≥ l1) (2a) pole / pole

even (1b) double pole / pole (l ≥ l1, l2) (2b) pole / double pole (l ≥ l2)

General pole structure of ζα(s), for the possible values of D and p being odd or

even. Magenta, type of behavior corresponding to lower values of l; behavior in

blue corresponds to larger values of lHeat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 22/32

Generalized zeta function regularizationLaplace type operators with discrete spectrum in

non compact domains [with Cognola, Zerbini]




A general theory is lacking: heat-kernel expansion

investigated by means of several examples






Class of exponential (in general, analytic) interactions:

non-compact of domain −→ logarithmic terms in heat-kernel








Analytic continuation of the zeta function not regular at origin:

displays a pole of higher order










For a physically meaningful evaluation of the functional

determinant, we propose a generalized zeta-f reg procedure










For a physically meaningful evaluation of the functional

determinant, we propose a generalized zeta-f reg procedure

One-loop approx in QFT: Euclidean 1ℓ effective action is sum of

classical action and a functional determinant of an elliptic diff

op: the fluctuation operator (needs to be regularized)Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 23/32

For self-adjoint, non-negative, 2nd-ord diff operator

L = −∆ + V

∆ the Laplace-Beltrami op, V a potential depending on theclassical background solution, with possibly a mass term



L = −∆ + V


One-loop eff act W ≡W [Φ] related to the functional det of L by

W = − lnZ = Sc +1

2ln det

L

µ2

Sc classical actionµ2 a renormalization parameter (for dimensional reasons)



L = −∆ + V


One-loop eff act W ≡W [Φ] related to the functional det of L by

W = − lnZ = Sc +1

2ln det

L

µ2

Sc classical actionµ2 a renormalization parameter (for dimensional reasons)

Zeta-function regularization:

W (ε) = S − 1

2

∫ ∞

0

dttε−1

Γ(1 + ε)Tr e−tL/µ2

= S − 1

2εζ(ε|L/µ2)

for the elliptic operator L the zf is def as a Mellin-like transform

ζ(s|L) =1

Γ(s)

∫ ∞

0

dt ts−1Tr e−tL, ζ(s|L/µ2) = µ2sζ(s|L)


Here the heat trace Tr e−tL plays import role. Recall that for a 2nd-ordelliptic non-neg op L in a compact d−dim manifold without boundary

Tr e−tL ≃∞∑

j=0

Aj(L) tj−d/2

with Aj(L) the Seeley-DeWitt coeffs (converge for Re s > d/2)



Tr e−tL ≃∞∑

j=0

Aj(L) tj−d/2


In the compact case, ζ(s|L) regular at origin and ζ(0|L) = Ad/2(L)

For odd dims without boundaries: ζ(0|L) = 0



Tr e−tL ≃∞∑

j=0

Aj(L) tj−d/2




Performing a Taylor expansion of the zeta function

W (ε) = S − 1

2εζ(0|L) +

ζ(0|L)

2lnµ2 +

ζ ′(0|L)

2+O(ε)

Thus, the 1ℓ divergences and finite contribs to the 1ℓ eff action areexpressed in terms of the zf and its deriv’s at the origin



Tr e−tL ≃∞∑

j=0

Aj(L) tj−d/2




Performing a Taylor expansion of the zeta function

W (ε) = S − 1

2εζ(0|L) +

ζ(0|L)

2lnµ2 +

ζ ′(0|L)

2+O(ε)

Thus, the 1ℓ divergences and finite contribs to the 1ℓ eff action areexpressed in terms of the zf and its deriv’s at the origin

More general case with log terms in heat-trace asympts. Localheat-kernel exp of Laplace type op H = −∆ + V (x). If the potential isreal and non-negative, with an additional, rather mild hypothesis, theoperator H is essentially self-adjoint in C∞

0 (Rd)Heat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 25/32

Consider confining potentials, smooth functions giving rise to discrete

spectrum [L. Parker]. Local heat-kernel expansion can be partially summedover

Kt(x, x) =1

(4πt)d/2e−tV (x)

∞X

n=0

bn(x) tn

new coeffs bn(x) easily computed, depend on the derivatives of V (x)

b0(x) = 1, b1(x) = 0, b2(x) = −1

6∆V, b3(x) = −

∆2V

60+

∇kV ∇kV

12, . . .




Kt(x, x) =1

(4πt)d/2e−tV (x)

∞X

n=0

bn(x) tn


b0(x) = 1, b1(x) = 0, b2(x) = −1

6∆V, b3(x) = −

∆2V

60+

∇kV ∇kV

12, . . .

For smooth compact manifolds, the heat-kernel trace is obtained integratingterm by term over the coordinates (no logarithm appears)




Kt(x, x) =1

(4πt)d/2e−tV (x)

∞X

n=0

bn(x) tn


b0(x) = 1, b1(x) = 0, b2(x) = −1

6∆V, b3(x) = −

∆2V

60+

∇kV ∇kV

12, . . .


For non-smooth manifolds one may get logs, as Laplace op on higher-dim

cones [Bordag, Cognola], or in 4-dim spacetimes with a 3-dim, non-compact,hyp spatial section of finite vol [Bytsenko], and in general ΨDOs [Grubb].

→ More recently, in self-interacting scalar field theory on manifolds withnon-commut coord. Goes together with non-typical behaviour of corresp zf:

generically a simple pole at the origin & higher-order poles at other places




Kt(x, x) =1

(4πt)d/2e−tV (x)

∞X

n=0

bn(x) tn


b0(x) = 1, b1(x) = 0, b2(x) = −1

6∆V, b3(x) = −

∆2V

60+

∇kV ∇kV

12, . . .


For non-smooth manifolds one may get logs, as Laplace op on higher-dim

cones [Bordag, Cognola], or in 4-dim spacetimes with a 3-dim, non-compact,hyp spatial section of finite vol [Bytsenko], and in general ΨDOs [Grubb].

→ More recently, in self-interacting scalar field theory on manifolds withnon-commut coord. Goes together with non-typical behaviour of corresp zf:

generically a simple pole at the origin & higher-order poles at other places

Here Laplace-type self-adjoint ops on non-compact manifolds. For general

case of confining potent and discrete spectrum, no systematic theory [Nash].1-dim problems on real half-line [Voros] and Barnes zfs [Dowker]

→ Log terms appear in the abstract context of regularized productsHeat Kernels 06, Blaubeuren, 28.11-2.12.2006 – p. 26/32

Under certain conditions, the regularized product assoc with aninfinite sequence of non-zero complex numbers {λn} has arelated Dirichlet series

∑

n λ−sn (the zeta function)



∑


Interested in case when λn are eingenvalues of a non-negative diffop and the zf converges absolutely for Re s suffic large. When zfholomorphic at origin, the regularized product is def as exp [−ζ ′(0)]



∑



General theory [Illies, Jorgenson, Manin, Simon]. For non-compactdomains but scattering potentials (continuous spectrum exists), is wellunderstood and S−matrix or phase shift function enter game [Muller].In this context, delta-like potents considered [Solo, Nail]. If the pot issingular (eg proport to 1/x2), log terms can appear in the local heat-kernel expansion, their coeffs becoming distributions [Callias, Kirsten]



∑



General theory [Illies, Jorgenson, Manin, Simon]. For non-compactdomains but scattering potentials (continuous spectrum exists), is wellunderstood and S−matrix or phase shift function enter game [Muller].In this context, delta-like potents considered [Solo, Nail]. If the pot issingular (eg proport to 1/x2), log terms can appear in the local heat-kernel expansion, their coeffs becoming distributions [Callias, Kirsten]

Explicit model: massive scalar field on flat spacetime R × R3 in

external static field of confining pot which is asymptoticallyexponential in 2-dims. In Euclidean version, we compactify ‘time’coord and third spatial coord, with periods β and l, respect.


Simple confining modelThe relevant operator [with Cognola, Zerbini]

L = −d2

dτ2−

d2

dz2+ H2 + M2, H2 = −∆2 + V (r), V (r) = g2eα2r2

g, α dimfull parameters. Poisson’s summ form and the heat-trace:

Tr e−tL = S e−tM2

4πtTr e−tH2 + · · · , S = β l, dots are exp small terms in t


Simple confining modelThe relevant operator [with Cognola, Zerbini]

L = −d2

dτ2−

d2

dz2+ H2 + M2, H2 = −∆2 + V (r), V (r) = g2eα2r2

g, α dimfull parameters. Poisson’s summ form and the heat-trace:

Tr e−tL = S e−tM2

4πtTr e−tH2 + · · · , S = β l, dots are exp small terms in t

Since the potential is defined everywhere in R2, one needs a factor e−tM2/t

ζ(s|L) ∼S

(4π)2Γ(s)

X

n

Z ∞

0

dt ts+n−3

Z

R2

dx b̃n(x) e−tV (r)

=X

n

Γ(s + n − 2)

(4π)2Γ(s)

Z

R2

dx b̃n(x) [V (r)]−(s+n−2)

b̃n =X

j+k=n

(−1)k bj M2k

k!, n ≥ 2, b̃0 = 1, b̃1 = −M2

b̃n have same structure as before, but now q can vanish

b̃n =X

pq

C̃npqr

paqeqbr2

, 0 ≤ p ≤ 2(n − 1), 0 ≤ q < n, n ≥ 0


With C̃n00 = (−1)nM2n

n!. First few non-trivial bn coefficients

b2 = −2gαeαr2

3(1 + αr2), b3 = −

4gα2eαr2

15(2 + 4αr2 + α2r4) +

g2α2e2αr2

3,

b4(x) = −∆3V

840+

(∆V )2

72+

∇i∇jV ∇i∇jV

90+

∇kV ∇k∆V

30, . . .

from which we get the Cnpq




b2 = −2gαeαr2

3(1 + αr2), b3 = −

4gα2eαr2

15(2 + 4αr2 + α2r4) +

g2α2e2αr2

3,

b4(x) = −∆3V

840+

(∆V )2

72+

∇i∇jV ∇i∇jV

90+

∇kV ∇k∆V

30, . . .


Integrating, the non-holomorphic contribution to the zf reads

ζ(s|L) =S

16πΓ(s)

X

n≥0;pq

C̃npq

Γ(s + n − 2)Γ(1 + p/2) a−(s+n−q−2)

b1+p/2 (s + n − q − 2)1+p/2

Since p even, the zf has only poles of order p/2. The pole structure at s = 0

ζ(s|L) =S

16πα

"

M4

2s+

6X

n=3

C̃n2,n−2Γ(n − 2)

αs+ 2

6X

n=3

C̃n4,n−2Γ(n − 2)

α2s2

#

+ · · ·




b2 = −2gαeαr2

3(1 + αr2), b3 = −

4gα2eαr2

15(2 + 4αr2 + α2r4) +

g2α2e2αr2

3,

b4(x) = −∆3V

840+

(∆V )2

72+

∇i∇jV ∇i∇jV

90+

∇kV ∇k∆V

30, . . .



ζ(s|L) =S

16πΓ(s)

X

n≥0;pq

C̃npq

Γ(s + n − 2)Γ(1 + p/2) a−(s+n−q−2)

b1+p/2 (s + n − q − 2)1+p/2


ζ(s|L) =S

16πα

"

M4

2s+

6X

n=3

C̃n2,n−2Γ(n − 2)

αs+ 2

6X

n=3

C̃n4,n−2Γ(n − 2)

α2s2

#

+ · · ·

ζ(s|L) is not regular at the origin: pole of second order appears

Within a physical context (ΨDOs in compact domains), this is a veryunusual behavior for the zf




b2 = −2gαeαr2

3(1 + αr2), b3 = −

4gα2eαr2

15(2 + 4αr2 + α2r4) +

g2α2e2αr2

3,

b4(x) = −∆3V

840+

(∆V )2

72+

∇i∇jV ∇i∇jV

90+

∇kV ∇k∆V

30, . . .



ζ(s|L) =S

16πΓ(s)

X

n≥0;pq

C̃npq

Γ(s + n − 2)Γ(1 + p/2) a−(s+n−q−2)

b1+p/2 (s + n − q − 2)1+p/2


ζ(s|L) =S

16πα

"

M4

2s+

6X

n=3

C̃n2,n−2Γ(n − 2)

αs+ 2

6X

n=3

C̃n4,n−2Γ(n − 2)

α2s2

#

+ · · ·

ζ(s|L) is not regular at the origin: pole of second order appears

Within a physical context (ΨDOs in compact domains), this is a veryunusual behavior for the zf

Zeta function regularization procedure: −→ needs to be modified


Proposal for extended zf regularizationIntroduce an additional spectral function depending on order of

pole at the origin of the initial zf. For a pole of order N

ω(s) = sNζ(s|L)

and the definition of the regularized determinant is

ln detL

µ2= − 1

(N + 1)!lims→0

dN+1

dsN+1

[

µ2sω(s)]

with the normalization chosen so that when ζ(s|L) is regular at

the origin, we recover the ordinary def (essential to preserve

the properties of zf reg)




ω(s) = sNζ(s|L)


ln detL

µ2= − 1

(N + 1)!lims→0

dN+1

dsN+1

[

µ2sω(s)]




Back to example: a 2nd-ord pole generically appears. New

spectral function (regular at origin)ω(s) = s2ζ(s|L)




ω(s) = sNζ(s|L)


ln detL

µ2= − 1

(N + 1)!lims→0

dN+1

dsN+1

[

µ2sω(s)]




Back to example: a 2nd-ord pole generically appears. New

spectral function (regular at origin)ω(s) = s2ζ(s|L)

We correspondingly define

ln detL

µ2= − 1

3!lims→0

d3

ds3[

µ2sω(s)]


Relation with regularized infinite prod’sWithin the context of a general theory of regularized products

[Illies 01], in cases when the zf is not holomorphic at the origin

but has a first-order pole a new def of regularized product was

proposed recently [Hirano 03]

∞∏

k=1

λk ≡ exp

[

−Ress=0ζ(s)

s2

]

, ζ(s) =

∞∑

k=1

λ−sk






∞∏

k=1

λk ≡ exp

[

−Ress=0ζ(s)

s2

]

, ζ(s) =

∞∑

k=1

λ−sk

This prescription is equivalent to ours (a further consistency

check, inscribes our result in a very general context)






∞∏

k=1

λk ≡ exp

[

−Ress=0ζ(s)

s2

]

, ζ(s) =

∞∑

k=1

λ−sk

This prescription is equivalent to ours (a further consistency

check, inscribes our result in a very general context)

Shows power and flexibility of the zeta function method to deal

with non-standard situations, while always fulfilling the most

fundamental condition:

results obtained should reproduce measuredexperimental values


Collaborators

Andrei Bytsenko

Guido Cognola

Klaus Kirsten

Sergio Leseduarte

August Romeo

Miguel Tierz

Sergio Zerbini


ice zeta functions, heat kernels, and the quantum …heat kernels 06, blaubeuren, 28.11-2.12.2006...

Documents