Zeta Functions: Symmetry and Applications

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 Stephen W Hawking, “Zeta function regularization of path integrals in curved spacetime", Commun Math Phys 55, 133 (1977) This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dims. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fifth dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This EM tensor has an anomalous trace. QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 7/27 Name or Title or Xtra Institute of Space Sciences Symmetry 2017, Barcelona, Oct 2017 Zeta Functions and Symmetry


Zero point energy
QFT vacuum to vacuum transition: 0|H|0 Spectrum, normal ordering (harm oscill): which converges for all complex values of s with real Re s > 1, and then defines ζ(s) as the 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 Chebyshev extended the definition to Re s > 1.
Godfrey H Hardy and John E Littlewood, "Contributions to the Theory of the Riemann Zeta-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 series regularized 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 des membranes vibrantes" (French), 8. Skand Mat-Kongr, 34-44 (1935) Zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold 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 compact Riemannian manifolds.So for such operators one can define the determinant using zeta function regularization D B Ray, Isadore M Singer, "R-torsion and the Laplacian on Riemannian manifolds", Advances in Math 7, 145 (1971) Used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a 1 , a 2 , ...., and in this case the zeta function is formally the trace the method defines the possibly divergent infinite product Stephen W Hawking, "Zeta function regularization of path integrals in curved spacetime", Commun Math Phys 55, 133 (1977) This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime.One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral.The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator.This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dims.The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fifth dimension of parameter time.Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric.This suggests that there may be a natural cut off in the integral over all black hole background metrics.By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole.This EM tensor has an anomalous trace.

Institute of Space Sciences
Symmetry 2017, Barcelona, Oct 2017

Zeta Functions and Symmetry
Name or Title or Xtra

Institute of Space Sciences
Symmetry 2017, Barcelona, Oct 2017

Basic strategies
Jacobi's identity for the θ−function Higher dimensions: Poisson summ formula (Riemann) A acts on the space of smooth sections of 3. E, n-dim vector bundle over The zeta function is defined as: has a meromorphic continuation to the whole complex plane C (regular at s = 0), provided the principal symbol of A, a m (x, ξ), admits a spectral cut:

Definition of Determinant
H ΨDO operator {ϕ i , λ i } spectral decomposition Asymptotic expansion for the heat kernel: " 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 involve derivatives).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 is bound to define something like 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

The Dixmier Trace
In order to write down an action in operator language one needs a functional 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 the compact operators T such that the partial sums of its spectrum diverge logarithmically as the number of terms in the sum: Definition of the Dixmier trace of T: The Hardy-Littlewood theorem can be stated in a way that connects the Dixmier trace with the residue of the zeta function of the operator The Wodzicki Residue The Wodzicki (or noncommutative) residue is the only extension of the Dixmier trace to ΨDOs which are not in L (1,∞)   Only The Wodzicki residue makes sense for ΨDOs of arbitrary order.Even if the symbols a j (x,ξ), j < m, are not coordinate invariant, the integral is, and defines a trace

Consequences of the Multipl Anomaly
In the path integral formulation In a situation where a superselection rule exists, AB has no sense (much less its determinant): But if diagonal form obtained after change of basis (diag.process), the preserved quantity is: =⇒ det(AB)

History
Lerch (1897): It is a 24-th root of the discriminant func ∆(τ ) of an elliptic curve C/L from a lattice Extended CS Series Formulas (ECS) Consider the zeta function (Res > p/2, A > 0, Req > 0) −s prime: point n = 0 to be excluded from the sum (inescapable condition when Pole: s = p/2 Residue: Gives (analytic cont of) multidimensional zeta function in terms of an exponentially 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) quadratic form, Q. Vector c arbitrary, while q is (to start) a non-neg constant  The OR scheme is governed by the identity: Renormalization ( cut-off, dim, ζ ) Even then: Has the final value real sense ?QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 -p.3The Riemann zeta function ζ(s) is a function of a complex variable, s.To define it, one starts with the infinite series The definition of ζ A (s) depends on the position of the cut L θ (d) The only possible singularities of ζ A (s) are poles at s j = (n − j)/m, j = 0, 1, 2, . . ., n − 1, n + 1, . . .
Weierstrass def: subtract leading behavior of λ i in i, as i → ∞, until series i∈I ln λ i converges =⇒ non-local counterterms !! C. Soulé et al, Lectures on Arakelov Geometry, CUP 1992; A. Voros,... Properties The definition of the determinant det ζ A only depends on the homotopy class of the cut A zeta function (and corresponding determinant) with the same meromorphic structure in the complex s-plane and extending the ordinary definition to operators ofcomplex order m ∈ C\Z (they do not admit spectral cuts), has been obtained [Kontsevich and Vishik]

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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 " Multipl or N-Comm Anomaly, or Defect Given A, B, and AB ψDOs, even if ζ A , ζ B , and ζ AB exist, it turns out that, in general, det ζ (AB) = det ζ A det ζ B Multipl or N-Comm Anomaly, or Defect Given A, B, and AB ψDOs, even if ζ A , ζ B , and ζ AB exist, it turns out that, in general, det ζ (AB) = det ζ A det ζ B The multiplicative (or noncommutative) anomaly (defect) is def ned as δ(A, B) = ln det ζ (AB) det ζ A det ζ B = −ζ ′ AB (0) + ζ ′ A (0) + ζ ′ B (0) Wodzicki formula δ(A, B) = res [ln σ(A, B)] 2 2 ord A ord B (ord A + ord B) where σ(A, B) = A ord B B −ord A trace one can define in the algebra of ΨDOs (up to multipl const) Definition: res A = 2 Res s=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 * M tr a −n (x, ξ) dξ with S * M ⊂ T * M the co-sphere bundle on M (some authors put a coefficient 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 Res s=1 ζ A (s) = 1 n res A −1

K
ν modified Bessel function of the second kind and the subindex 1/2 in Z p 1/2 means that only half of the vectors m ∈ Z p participate in the sum.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} whose first non-zero component is positive] Case c 1 = • • • = c p = q = 0 [true extens of CS, diag subcase]

Future
McKeon and T N Sherry, Phys Rev Lett 59, 532 (1987); Phys Rev D35, 3854 (1987) Distinct advantage: it can be used with formally non-renormalizable theories: R B Mann, L Tarasov, D G C McKeon and T Steele, Nucl Phys B311, 630 (1989); A Y Shiekh, Can J Phys 74, 172 (1996) Divergences are not reabsorbed, each is removed and replaced by an arbitrary factor OR does not cure the non-predictability problem of non-renormalizability, but advantage that the initial Lagrangian need not be extended with addition of extra terms s are arbitrary, and it is enough that the degree of regularization is equal to the loop order, n QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 -p.24/2 Two separate aspects of the procedure: 1st the regularization, 2nd analytical continuation (divergences are replaced by arbitrary factors) Effect of OR: replace the divergent poles by arbitrary constants 1 ǫ n −→ α n to yield the finite expressionH −m = α n c −n + • • • + α 1 c −1 + c 0GeneralizationOR can be generalized to multiple operators, as in multi-loop casesH −m 1 • • • H −m r = lim ǫ→0 d n dǫ n 1 + 1 + α 1 ǫ + α 2 ǫ 2 + • • • + α n ǫ n × ǫ n n! H −ǫ−m 1 • • • H −ǫ−m rFurther Extension OR was first introduced in the context of the Schwinger approach, which is known to be equivalent to the Feynman one QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 -p.25/2 OR of the logarithm in the Schwinger approach ln H = − lim The Schwinger form can be transformed into the Feynman one H −m = (−1) m−1 (m − 1)! d m dH m ln H Equivalence with dimensional regularization in many cases Not always, problems (main one, unitarity), may appear A Rebhan, Phys Rev D39, 3101 (1989) its naive application to obtain finite amplitudes breaks unitarity QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 -p.26/2 No symmetry-breaking regulating parameter is ever inserted into the initial Lagrangian L Culumovic, M Leblanc, R B Mann, D G C McKeon and T N Sherry, Phys Rev D41, 514 (1990) actually use Bogoliubov's recursion formula to show how to construct a consistent OR operator Unitarity is upheld by employing a generalized evaluator consistently including lower-order quantum corrections to the quantities of interest