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ICE
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Cosmology, Vacuum Energy,
Zeta Functions
EMILIO ELIZALDE
ICE/CSIC & IEEC, UAB, Barcelona
MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 1/26
Outline
Motivation: trying to solve the puzzle of thecosmological constant
Vacuum Fluctuations and the Equivalence Principle
The Zero Point Energy
The Casimir Effect
Operator Zeta Functions in Math Physics: Origins
ζA for A a ΨDO, Determinant
Wodzicki Residue, Multiplicative Anomaly or Defect
Extended CS Series Formulas (ECS)
The Sign of the Vacuum Forces
Repulsion from Higher Dimensions and BCs
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 2/26
Trying to solve the puzzle of the CCThe 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, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 4/26
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 ?
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 5/26
The Casimir Effect
vacuum
BC
F
Casimir Effect
BC e.g. periodic=⇒ all kind of fields=⇒ curvature or topology
Universal process:
Sonoluminiscence (Schwinger)
Cond. matter (wetting 3He alc.)
Optical cavities
Direct experim. confirmation
Van der Waals, Lifschitz theoryDynamical CE ⇐Lateral CE
Extract energy from vacuum
CE and the cosmological constant ⇐E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 6/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
Equivalent to a cosmological const Λ = 8πGE , ρc = 3H2
8πG
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
Equivalent to a cosmological const Λ = 8πGE , ρc = 3H2
8πG
Observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
Λ = (2.14 ± 0.13 × 10−3 eV)4 ∼ 4.32 × 10−9 erg/cm3
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
Equivalent to a cosmological const Λ = 8πGE , ρc = 3H2
8πG
Observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
Λ = (2.14 ± 0.13 × 10−3 eV)4 ∼ 4.32 × 10−9 erg/cm3
Question: how finite Casimir energy of pair of plates couples to gravity?
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
Equivalent to a cosmological const Λ = 8πGE , ρc = 3H2
8πG
Observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
Λ = (2.14 ± 0.13 × 10−3 eV)4 ∼ 4.32 × 10−9 erg/cm3
Question: how finite Casimir energy of pair of plates couples to gravity?
Two ways to proceed. Gauge-invariant procedure:
energy-momentum tensor of the phys sys must be conserved, so include aphysical mechanism holding the plates apart against the Casimir force
−→ Leads to complicated model-dependent calculations
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
Equivalent to a cosmological const Λ = 8πGE , ρc = 3H2
8πG
Observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
Λ = (2.14 ± 0.13 × 10−3 eV)4 ∼ 4.32 × 10−9 erg/cm3
Question: how finite Casimir energy of pair of plates couples to gravity?
Two ways to proceed. Gauge-invariant procedure:
energy-momentum tensor of the phys sys must be conserved, so include aphysical mechanism holding the plates apart against the Casimir force
−→ Leads to complicated model-dependent calculations
Alternative: find a physically natural coordinate system, more realistic than
another: Fermi coord system [Marzlin ’94]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
Vacuum Fluct & the Equival PrincipleThe main issue: S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuumappears on the rhs of Einstein’s equations:
Rµν −1
2gµνR = −8πG(T̃µν − Egµν)
Equivalent to a cosmological const Λ = 8πGE , ρc = 3H2
8πG
Observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
Λ = (2.14 ± 0.13 × 10−3 eV)4 ∼ 4.32 × 10−9 erg/cm3
Question: how finite Casimir energy of pair of plates couples to gravity?
Two ways to proceed. Gauge-invariant procedure:
energy-momentum tensor of the phys sys must be conserved, so include aphysical mechanism holding the plates apart against the Casimir force
−→ Leads to complicated model-dependent calculations
Alternative: find a physically natural coordinate system, more realistic than
another: Fermi coord system [Marzlin ’94]
Calculations done also in Rindler coord (uniform accel obs)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/26
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.
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/26
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.
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/26
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)]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/26
J. Stuart Dowker, Raymond Critchley
“Effective Lagrangian and energy-momentum tensor
in de Sitter space", Phys. Rev. D13, 3224 (1976)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 10/26
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, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 10/26
Stephen W Hawking, “Zeta function regularization of path integralsin curved spacetime", Commun Math Phys 55, 133 (1977)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 11/26
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.
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 11/26
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|
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 12/26
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)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 f
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 12/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 13/26
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, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 13/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/26
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, . . .
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/26
Definition of DeterminantH ΨDO operator {ϕi, λi} spectral decomposition
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/26
Definition of DeterminantH ΨDO operator {ϕi, λi} spectral decomposition
∏i∈I λi ?! ln
∏i∈I λi =
∑i∈I lnλi
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/26
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)
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 , Res > s0
Derivative: ζ ′H(0) = −∑i∈I lnλi
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/26
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)
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 , Res > s0
Derivative: ζ ′H(0) = −∑i∈I lnλi
Determinant: Ray & Singer, ’67detζ H = exp [−ζ ′H(0)]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/26
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)
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 , 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 !!
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/26
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)
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 , 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,...
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/26
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
]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 16/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 17/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 18/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 19/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 19/26
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)
Wodzicki formula
δ(A,B) =res
{[ln σ(A,B)]2
}
2 ordA ordB (ordA+ ordB)
where σ(A,B) = Aord BB−ord A
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 19/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 20/26
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
f̃(~m)
f̃ Fourier transform[Gelbart + Miller, BAMS ’03, Iwaniec, Morgan, ICM ’06]
Truncated sums −→ asymptotic series
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 21/26
Extended CS Series Formulas (ECS)Consider the zeta function (Res > p/2, A > 0,Req > 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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 22/26
Extended CS Series Formulas (ECS)Consider the zeta function (Res > p/2, A > 0,Req > 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
Case q 6= 0 (Req > 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]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 22/26
Extended CS Series Formulas (ECS)Consider the zeta function (Res > p/2, A > 0,Req > 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
Case q 6= 0 (Req > 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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 22/26
Gives (analytic cont of) multidimensional zeta function in terms of anexponentially convergent multiseries, valid in the whole complex plane
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/26
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]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/26
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]
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
)]
[ECS3d]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/26
The Sign of the Casimir ForceMany papers dealing on this issue: here just short account
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/26
The Sign of the Casimir ForceMany papers dealing on this issue: here just short account
Casimir calculation: attractive force
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/26
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
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 Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
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
Robin BCs in general ⇐
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/26
Casimir eff in brworl’s w large extra dim
Casimir energy for massive scalar field with an arbitrary curvature coupling,
obeying Robin BCs on two codim-1 parallel plates embedded in backgroundspacetime R(D1−1,1) × Σ, Σ compact internal space
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 26/26
Casimir eff in brworl’s w large extra dim
Casimir energy for massive scalar field with an arbitrary curvature coupling,
obeying Robin BCs on two codim-1 parallel plates embedded in backgroundspacetime R(D1−1,1) × Σ, Σ compact internal space
Most general case: constants in the BCs different for the two plates
It is shown that Robin BCs with different coefficients are necessaryto obtain repulsive Casimir forces
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 26/26
Casimir eff in brworl’s w large extra dim
Casimir energy for massive scalar field with an arbitrary curvature coupling,
obeying Robin BCs on two codim-1 parallel plates embedded in backgroundspacetime R(D1−1,1) × Σ, Σ compact internal space
Most general case: constants in the BCs different for the two plates
It is shown that Robin BCs with different coefficients are necessaryto obtain repulsive Casimir forces
Robin type BCs are an extension of Dirichlet and Neumann’s
=⇒ most suitable to describe physically realistic situations
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 26/26
Casimir eff in brworl’s w large extra dim
Casimir energy for massive scalar field with an arbitrary curvature coupling,
obeying Robin BCs on two codim-1 parallel plates embedded in backgroundspacetime R(D1−1,1) × Σ, Σ compact internal space
Most general case: constants in the BCs different for the two plates
It is shown that Robin BCs with different coefficients are necessaryto obtain repulsive Casimir forces
Robin type BCs are an extension of Dirichlet and Neumann’s
=⇒ most suitable to describe physically realistic situations
Genuinely appear in: → vacuum effects for a confined charged scalar field
in external fields [Ambjørn ea 83],→ spinor and gauge field theories,
→ quantum gravity and supergravity [Luckock ea 91]Can be made conformally invariant, purely-Neumann conditions cannot
=⇒ needed for conformally invariant theories with BC, to preserve cf invar
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 26/26
'
&
$
%
A. A VERY SIMPLE MODEL: large & small dimensions
• Space-time:
Rd+1 ×Tp ×Tq
Rd+1 ×Tp × Sq
...
• Scalar field, φ, pervading the universe
• ρφ contribution to ρV from this field
ρφ =1
2
∑
i
λi
=1
2
∑
k
1
µ
(k2 + M2
)1/2
• ∑i and
∑k are generalized sums
• µ mass-dim parameter, h̄ = c = 1
• M mass of the field arbitrarily small
(a tiny mass for the field can never be excluded, see
L. Parker & A. Raval, PRL86 749 (2001); PRD62
083503 (2000))
'
&
$
%
For d–open, (p, q)–toroidal universe:
ρφ =π−d/2
2dΓ(d/2)∏p
j=1 aj∏q
h=1 bh
∫ ∞
0dk kd−1
∞∑
np=−∞
∞∑
mq=−∞
p∑
j=1
(2πnj
aj
)2
+q∑
h=1
(2πmh
bh
)2
+ k2d + M2
1/2
' 1
apbq
∞∑
np,mq=−∞
1
a2
p∑
j=1
n2j +
1
b2
q∑
h=1
m2h + M2
d+12
+1
For d–open, (p–toroidal, q–spherical) universe:
ρφ =π−d/2
2dΓ(d/2)∏p
j=1 aj bq
∫ ∞
0dk kd−1
∞∑
np=−∞
∞∑
l=1
Pq−1(l)
p∑
j=1
(2πnj
aj
)2
+Q2(l)
b2+ k2
d + M2
1/2
'
1
apbq
∞∑
np=−∞
∞∑
l=1
Pq−1(l)
4π2
a2
p∑
j=1
n2j +
l(l + q)
b2+
M2
4π2
d+12
+1
• Pq−1(l) is a polynomial in l of degree q − 1
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Regularize: zeta function
ζ(s) =1
2
∑
i
λ−si
=1
2
∑
k
(k2 + M2
µ2
)−s/2
ρφ = ζ(−1)
• No further subtraction or renormalization needed here
[E.E., J. Math. Phys. 35, 3308 , 6100 (1994)]
MATHEMATICAL TOOLS
Spectrum of the Hamiltonian op. known explicitly
• Generalized Chowla-Selberg expression
[E.E., Commun. Math. Phys. 198, 83 (1998)
E.E., J. Phys. A30, 2735 (1997)]
Consider the zeta function (<s > p/2):
ζA,~c,q(s) =∑
~n∈Zp
′ [1
2(~n + ~c)T A (~n + ~c) + q
]−s
≡ ∑
~n∈Zp
′[Q (~n + ~c) + q]−s
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• Prime means that point ~n = ~0 is excluded from sum
(not important).
• Gives (analytic continuation of) multidimensional
zeta function in terms of an exponentially convergent
multiseries, valid in the whole complex plane
• Exhibits singularities (simple poles) of meromorphic
continuation —with corresponding residua—
explicitly
Only condition on matrix A: corresponds to (non
negative) quadratic form, Q. Vector ~c arbitrary, while q
will (for the moment) be a positive constant.
ζA,~c,q(s) =(2π)p/2qp/2−s
√det A
Γ(s− p/2)
Γ(s)+
2s/2+p/4+2πsq−s/2+p/4
√det A Γ(s)
∑
~m∈Zp
1/2
′cos(2π~m · ~c)
(~mT A−1 ~m
)s/2−p/4Kp/2−s
(2π
√2q ~mT A−1 ~m
)
Kν modified Bessel function of the second kind and the
subindex 1/2 in Zp1/2 means that only half of the vectors
~m ∈ Zp intervene in the sum. That is, if we take an
~m ∈ Zp we must then exclude −~m (as simple criterion
one can, for instance, select those vectors in Zp\{~0}whose first non-zero component is positive).
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PRECISE CALCULATION OF
THE VACUUM ENERGY DENSITY
Analytic continuation of the zeta function:
ζ(s) =2πs/2+1
ap−(s+1)/2bq−(s−1)/2Γ(s/2)
∞∑
mq=−∞
p∑
h=0
(ph
)2h
×∞∑
nh=1
( ∑hj=1 n2
j∑qk=1 m2
k + M2
)(s−1)/4
×K(s−1)/2
2πa
b
h∑
j=1
n2j
1/2 ( q∑
k=1
m2k + M2
)1/2
Yields the vacuum energy density:
ρφ = − 1
apbq+1
p∑
h=0
(ph
)2h
∞∑
nh=1
∞∑
mq=−∞
√√√√∑q
k=1 m2k + M2
∑hj=1 n2
j
×K1
2πa
b
h∑
j=1
n2j
1/2 ( q∑
k=1
m2k + M2
)1/2
Now, from
Kν(z) ∼ 1
2Γ(ν)(z/2)−ν , z → 0
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when M is very small
ρφ = − 1
apbq+1
{M K1
(2πa
bM
)+
p∑
h=0
(ph
)2h
×∞∑
nh=1
M√∑hj=1 n2
j
K1
2πa
bM
√√√√√h∑
j=1
n2j
+ O[q√
1 + M2K1
(2πa
b
√1 + M2
)]}
Inserting the h̄ and c factors
ρφ = − h̄c
2πap+1bq
[1 +
p∑
h=0
(ph
)2hα
]+O
[qK1
(2πa
b
)]
α some finite constant (explicit geometrical sum in the
limit M → 0).
• Sign may change with BC (e.g., Dirichlet).
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MATCHING THE OBSERVATIONAL RESULTS
FOR THE COSMOLOGICAL CONSTANT
b ∼ lP (lanck)
a ∼ RU(niverse)
a/b ∼ 1060
ρφ p = 0 p = 1 p = 2 p = 3
b = lP 10−13 10−6 1 105
b = 10lP 10−14 [10−8] 10−3 10
b = 102lP 10−15 (10−10) 10−6 10−3
b = 103lP 10−16 10−12 [10−9] (10−7)
b = 104lP 10−17 10−14 10−12 10−11
b = 105lP 10−18 10−16 10−15 10−15
Table 1: The vacuum energy density contribution, in units oferg/cm3, for p large compactified dimensions a, and q = p + 1 smallcompactified dimensions b, p = 0, . . . , 3, for different values of b, pro-portional to lP . In brackets, the values that more exactly match theone for the cosmological constant coming from observations, and inparenthesis the otherwise closest approximations.
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RESULTS
* Precise coincidence with the observational
value for the cosmological constant with
ρφ −→ [ ]
* Approx coincidence −→ ( )
* b ∼ 10 to 1000 times the Planck length lP
* q = p + 1: (1,2) and (2,3) compactified
dimensions, respectively
* Everything dictated by two basic lengths:
Planck value and radius of our Universe
* Both situations correspond to a marginally
closed universe
To examine → couplings in GR
→ alternative theories