ICE
Cosmology, Vacuum Energy,
Zeta Functions
E MILIO E LIZALDE
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. 1/2
Outline
Motivation: trying to solve the puzzle of the
cosmological 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. 2/2
Trying to solve the puzzle of the CC
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 physics
stress-energy density µ of the vacuum
Z
Z
√
√
1
d4 x −g Λ
Lcc = d4 x −g µ4 =
8πG
In other words: two contributions on the same footing
[Pauli 20’s, Zel’dovich ’68]
Λ c2
1 ~c X
+
ωi
8πG
Vol 2 i
For elementary particle physicists: a great embarrassment
no way to get rid off
Coleman, Hawking, Weinberg, Polchinski, ... ’88-’89
THE COSMOLOGICAL CONSTANT PROBLEM
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 4/2
Zero point energy
QFT
vacuum to vacuum transition:
Spectrum, normal ordering (harm oscill):
1
λn an a†n
H = n+
2
h0|H|0i
~c X
1
h0|H|0i =
λn =
tr H
2 n
2
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. 5/2
The Casimir Effect
BC
BC e.g. periodic
=⇒ all kind of fields
=⇒ curvature or topology
Universal process:
Sonoluminiscence (Schwinger)
F
vacuum
Cond. matter (wetting 3 He alc.)
Optical cavities
Direct experim. confirmation
Casimir Effect
Dynamical CE ⇐
Van der Waals, Lifschitz theory
Lateral CE
Extract energy from vacuum
CE and the cosmological constant ⇐
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 6/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
Equivalent to a cosmological const Λ = 8πGE, ρc =
3H 2
8πG
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 7/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
Equivalent to a cosmological const Λ = 8πGE, ρc =
3H 2
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. 7/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
Equivalent to a cosmological const Λ = 8πGE, ρc =
3H 2
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. 7/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
Equivalent to a cosmological const Λ = 8πGE, ρc =
3H 2
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 a
physical 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. 7/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
Equivalent to a cosmological const Λ = 8πGE, ρc =
3H 2
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 a
physical 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. 7/2
Vacuum Fluct & the Equival Principle
The main issue:
S.A. Fulling et. al., hep-th/070209
energy ALWAYS gravitates therefore the energy density of the vacuum
appears on the rhs of Einstein’s equations:
1
Rµν − gµν R = −8πG(Teµν − Egµν )
2
Equivalent to a cosmological const Λ = 8πGE, ρc =
3H 2
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 a
physical 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. 7/2
Operator Zeta F’s in MΦ: Origins
The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one
∞
X
starts with the infinite series
1
n=1
ns
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.
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/2
Operator Zeta F’s in MΦ: Origins
The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one
∞
X
starts with the infinite series
1
n=1
ns
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/2
Operator Zeta F’s in MΦ: Origins
The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one
∞
X
starts with the infinite series
1
n=1
ns
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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/2
Operator Zeta F’s in MΦ: Origins
The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one
∞
X
starts with the infinite series
1
n=1
ns
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 8/2
Robert T Seeley, “Complex powers of an elliptic operator. 1967
Singular 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. 9/2
Robert T Seeley, “Complex powers of an elliptic operator. 1967
Singular 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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/2
Robert T Seeley, “Complex powers of an elliptic operator. 1967
Singular 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)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/2
Robert T Seeley, “Complex powers of an elliptic operator. 1967
Singular 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 a1 , a2 , ...., and in this case the zeta function is formally
the trace
ζA (s) = Tr (A)−s
the method defines the possibly divergent infinite product
∞
Y
an = exp[−ζA ′ (0)]
n=1
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 9/2
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. 10/2
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 < T µν > 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 < T µν > equals g µν (960π 2 a4 )−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. 10/2
Stephen W Hawking, “Zeta function regularization of path integrals
in curved spacetime", Commun Math Phys 55, 133 (1977)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 11/2
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.
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 11/2
Pseudodifferential Operator (ΨDO)
A ΨDO of order m
Mn manifold
Symbol of A: a(x, ξ) ∈ S m (Rn × Rn ) ⊂ C ∞ functions such that
for any pair of multi-indices α, β there exists a constant Cα,β so
that
α β
∂
∂
a(x,
ξ)
ξ x
≤ Cα,β (1 + |ξ|)m−|α|
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 12/2
Pseudodifferential Operator (ΨDO)
A ΨDO of order m
Mn manifold
Symbol of A: a(x, ξ) ∈ S m (Rn × Rn ) ⊂ C ∞ functions such that
for any pair of multi-indices α, β there exists a constant Cα,β so
that
α β
∂
∂
a(x,
ξ)
ξ x
≤ Cα,β (1 + |ξ|)m−|α|
Definition of A (in the distribution sense)
Z
Af (x) = (2π)−n ei<x,ξ> a(x, ξ)fˆ(ξ) dξ
f is a smooth function
f ∈ S = f ∈ C ∞ (Rn ); supx |xβ ∂ α f (x)| < ∞, ∀α, β ∈ Nn }
S ′ space of tempered distributions
fˆ is the Fourier transform of f
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 12/2
ΨDOs are useful tools
The 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) ξ k
a(x, ξ) is said to be elliptic if it is invertible for large |ξ| and if there exists a
constant 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. 13/2
ΨDOs are useful tools
The 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) ξ k
a(x, ξ) is said to be elliptic if it is invertible for large |ξ| and if there exists a
constant 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 complex
powers of differential operators, like the Laplacian [Seeley, Gilkey, ...]
4. Basic starting point of any rigorous formulation of QFT & gravitational
interactions through µlocalization (the most important step towards the
understanding 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. 13/2
Existence of ζA for A a ΨDO
1. 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. 14/2
Existence of ζA for A a ΨDO
1. 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:
P
ζA (s) = tr A−s = j λ−s
j ,
Re s >
n
m
:= s0
{λj } ordered spect of A, s0 = dim M/ord A abscissa of converg of ζA (s)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/2
Existence of ζA for A a ΨDO
1. 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:
P
ζA (s) = tr A−s = j λ−s
j ,
Re s >
n
m
:= s0
{λj } ordered spect of A, s0 = dim M/ord A 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 a
spectral cut: Lθ = {λ ∈ C; Arg λ = θ, θ1 < θ < θ2 }, Spec A ∩ Lθ = ∅
(the Agmon-Nirenberg condition)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/2
Existence of ζA for A a ΨDO
1. 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:
P
ζA (s) = tr A−s = j λ−s
j ,
Re s >
n
m
:= s0
{λj } ordered spect of A, s0 = dim M/ord A 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 a
spectral cut: Lθ = {λ ∈ C; Arg λ = θ, θ1 < θ < θ2 }, Spec A ∩ 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. 14/2
Existence of ζA for A a ΨDO
1. 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:
P
ζA (s) = tr A−s = j λ−s
j ,
Re s >
n
m
:= s0
{λj } ordered spect of A, s0 = dim M/ord A 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 a
spectral cut: Lθ = {λ ∈ C; Arg λ = θ, θ1 < θ < θ2 }, Spec A ∩ 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 at
sj = (n − j)/m,
j = 0, 1, 2, . . . , n − 1, n + 1, . . .
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 14/2
Definition of Determinant
H
ΨDO operator
{ϕi , λi } spectral decomposition
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/2
Definition of Determinant
H
ΨDO operator
Q
i∈I λi ?!
{ϕi , λi } spectral decomposition
Q
P
ln i∈I λi =
i∈I ln λi
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/2
Definition of Determinant
H
ΨDO operator
Q
i∈I λi ?!
Riemann zeta func: ζ(s) =
{ϕi , λi } spectral decomposition
Q
P
ln i∈I λi =
i∈I ln λi
P∞
−s , Re s > 1 (& analytic cont)
n
n=1
Definition: zeta function of H
1
Γ(s)
ζH (s) =
R∞
P
−s
−s
λ
=
tr
H
i∈I i
s−1 tr e−tH , Res > s
dt
t
As Mellin transform: ζH (s) =
0
0
P
′
Derivative: ζH (0) = − i∈I ln λi
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/2
Definition of Determinant
H
ΨDO operator
Q
i∈I λi ?!
Riemann zeta func: ζ(s) =
{ϕi , λi } spectral decomposition
Q
P
ln i∈I λi =
i∈I ln λi
P∞
−s , Re s > 1 (& analytic cont)
n
n=1
Definition: zeta function of H
1
Γ(s)
ζH (s) =
R∞
P
−s
−s
λ
=
tr
H
i∈I i
s−1 tr e−tH , Res > s
dt
t
As Mellin transform: ζH (s) =
0
0
P
′
Derivative: ζH (0) = − i∈I ln λi
Determinant:
Ray & Singer, ’67
′ (0)]
detζ H = exp [−ζH
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/2
Definition of Determinant
H
ΨDO operator
Q
i∈I λi ?!
Riemann zeta func: ζ(s) =
{ϕi , λi } spectral decomposition
Q
P
ln i∈I λi =
i∈I ln λi
P∞
−s , Re s > 1 (& analytic cont)
n
n=1
Definition: zeta function of H
1
Γ(s)
ζH (s) =
R∞
P
−s
−s
λ
=
tr
H
i∈I i
s−1 tr e−tH , Res > s
dt
t
As Mellin transform: ζH (s) =
0
0
P
′
Derivative: ζH (0) = − i∈I ln λi
Determinant:
Ray & Singer, ’67
′ (0)]
detζ H = exp [−ζH
Weierstrass def: subtract leading behavior of λi in i, as i → ∞,
P
until series i∈I ln λi converges
=⇒ non-local counterterms !!
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 15/2
Definition of Determinant
H
ΨDO operator
Q
i∈I λi ?!
Riemann zeta func: ζ(s) =
{ϕi , λi } spectral decomposition
Q
P
ln i∈I λi =
i∈I ln λi
P∞
−s , Re s > 1 (& analytic cont)
n
n=1
Definition: zeta function of H
1
Γ(s)
ζH (s) =
R∞
P
−s
−s
λ
=
tr
H
i∈I i
s−1 tr e−tH , Res > s
dt
t
As Mellin transform: ζH (s) =
0
0
P
′
Derivative: ζH (0) = − i∈I ln λi
Determinant:
Ray & Singer, ’67
′ (0)]
detζ H = exp [−ζH
Weierstrass def: subtract leading behavior of λi in i, as i → ∞,
P
until series i∈I ln λi 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. 15/2
Properties
The definition of the determinant detζ 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 of complex order m ∈ C\Z (they do not
admit spectral cuts), has been obtained [Kontsevich and Vishik]
Asymptotic expansion for the heat kernel:
P′
−tA
tr e
= λ∈Spec A e−tλ
P
P
−sj
∼ αn (A) + n6=j≥0 αj (A)t
+ k≥1 βk (A)tk ln t, t ↓ 0
αn (A) = ζA (0),
αj (A) =
αj (A) = Γ(sj ) Ress=sj ζA (s), sj ∈
/ −N
(−1)k
k!
[PP ζA (−k) + ψ(k + 1) Ress=−k ζA (s)] ,
sj = −k, k ∈ N
(−1)k+1
βk (A) =
Ress=−k ζA (s), k ∈ N\{0}
k!
i
h
Ress=p φ(s)
PP φ := lims→p φ(s) −
s−p
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 16/2
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 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)
R
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 Ress=1 ζA (s) = n1 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. 17/2
Singularities of ζA
A complete determination of the meromorphic structure of some zeta
functions in the complex plane can be also obtained by means of the
Dixmier 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 methods
have to be used (see later)
Proposition. Under the conditions of existence of the zeta function of
A, given above, and being the symbol a(x, ξ) of the operator A
analytic in ξ −1 at ξ −1 = 0:
R
1
1
−sk
n−1
k
Ress=sk ζA (s) = m res A
= m S ∗ M tr a−s
ξ
−n (x, ξ) d
Proof. The homog component of degree −n of the corresp power of
the principal symbol of A is obtained by the appropriate derivative of
a power of the symbol with respect to ξ −1 at ξ −1 = 0 :
k −sk
∂
n−k (k−n)/m
ξ
a
(x, ξ) a−n (x, ξ) = ∂ξ−1
ξ −n
−1
ξ
=0
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 18/2
Multipl or N-Comm Anomaly, or Defect
Given A, B, and AB ψDOs, even if ζA , ζB , and ζAB exist,
it turns out that, in general,
detζ (AB) 6= detζ A detζ B
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 19/2
Multipl or N-Comm Anomaly, or Defect
Given A, B, and AB ψDOs, even if ζA , ζB , and ζAB exist,
it turns out that, in general,
detζ (AB) 6= detζ A detζ B
The multiplicative (or noncommutative) anomaly (defect)
is defined as
detζ (AB)
′
= −ζAB
(0) + ζA′ (0) + ζB′ (0)
δ(A, B) = ln
detζ A detζ B
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 19/2
Multipl or N-Comm Anomaly, or Defect
Given A, B, and AB ψDOs, even if ζA , ζB , and ζAB exist,
it turns out that, in general,
detζ (AB) 6= detζ A detζ B
The multiplicative (or noncommutative) anomaly (defect)
is defined as
detζ (AB)
′
= −ζAB
(0) + ζA′ (0) + ζB′ (0)
δ(A, B) = ln
detζ A detζ B
Wodzicki formula
2
res [ln σ(A, B)]
δ(A, B) =
2 ord A ord B (ord A + ord B)
where
σ(A, B) = Aord B B −ord A
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 19/2
Consequences of the Multipl Anomaly
In the path integral formulation
Z
Z
h
[dΦ] exp − dD x Φ† (x)
Φ(x) + · · ·
±
i
Gaussian integration: −→ det




A A2
A
 1



−→
A3 A4
B
det(AB)
or
det A · det B
?
In a situation where a superselection rule exists, AB has no
sense (much less its determinant): =⇒ det A · det B
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. 20/2
Basic strategies
Jacobi’s identity for the θ−function
P ∞ n2
θ3 (z, τ ) := 1 + 2 n=1 q cos(2nz),
q := eiπτ , τ ∈ C
2 /iπτ
1
z
−1
z
θ3 (z, τ ) = √−iτ e
θ3 τ | τ
equivalently:
r ∞
∞
X π 2 n2
X
π
2
e− t cos(2πnz), z, t ∈ C, Ret > 0
e−(n+z) t =
t n=0
n=−∞
Higher dimensions: Poisson summ formula (Riemann)
X
X
f (~n) =
fe(m)
~
~
n∈Zp
p
m∈Z
~
fe 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. 21/2
Extended CS Series Formulas (ECS)
Consider the zeta function (Res > p/2, A > 0, Req > 0)
−s
X ′ 1
X′
T
−s
ζA,~c,q (s) =
(~n + ~c) A (~n + ~c) + q
=
[Q (~n + ~c) + q]
2
p
p
~
n∈Z
~
n∈Z
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. 22/2
Extended CS Series Formulas (ECS)
Consider the zeta function (Res > p/2, A > 0, Req > 0)
−s
X ′ 1
X′
T
−s
ζA,~c,q (s) =
(~n + ~c) A (~n + ~c) + q
=
[Q (~n + ~c) + q]
2
p
p
~
n∈Z
~
n∈Z
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)
(2π)p/2 q p/2−s Γ(s − p/2) 2s/2+p/4+2 π s q −s/2+p/4
√
√
ζA,~c,q (s) =
+
Γ(s)
det A
det A Γ(s)
p
X
s/2−p/4
′
×
cos(2π m
~ · ~c) m
~ T A−1 m
~
Kp/2−s 2π 2q m
~ T A−1 m
~
p
m∈Z
~
1/2
[ECS1]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 22/2
Extended CS Series Formulas (ECS)
Consider the zeta function (Res > p/2, A > 0, Req > 0)
−s
X ′ 1
X′
T
−s
ζA,~c,q (s) =
(~n + ~c) A (~n + ~c) + q
=
[Q (~n + ~c) + q]
2
p
p
~
n∈Z
~
n∈Z
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)
(2π)p/2 q p/2−s Γ(s − p/2) 2s/2+p/4+2 π s q −s/2+p/4
√
√
ζA,~c,q (s) =
+
Γ(s)
det A
det A Γ(s)
p
X
s/2−p/4
′
×
cos(2π m
~ · ~c) m
~ T A−1 m
~
Kp/2−s 2π 2q m
~ T A−1 m
~
p
m∈Z
~
1/2
Pole:
[ECS1]
s = p/2
Residue:
(2π)p/2
Ress=p/2 ζA,~c,q (s) =
(det A)−1/2
Γ(p/2)
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 22/2
Gives (analytic cont of) multidimensional zeta function in terms of an
exponentially convergent multiseries, valid in the whole complex plane
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/2
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/2
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
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/2
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
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 participate in the
sum. E.g., if we take an m
~ ∈ Zp we must then exclude −m
~
[simple criterion: one may select those vectors in Zp \{~0} whose
first non-zero component is positive]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/2
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
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 participate in the
sum. E.g., if we take an m
~ ∈ Zp we must then exclude −m
~
[simple criterion: one may select those vectors in Zp \{~0} whose
first non-zero component is positive]
Case c1 = · · · = cp = q = 0 [true extens of CS, diag subcase]
p−1
1+s X
h
2
j
−1/2
j/2 j/2−s
ζAp (s) =
(det Aj )
π ap−j Γ s −
ζR (2s−j) +
Γ(s) j=0
2
q
∞
j
s X X
s/2−j/4
s 4−2
′ j/2−s
t −1
~ tj A−1
~j
4π ap−j
n
m
~ j Aj m
~j
Kj/2−s 2πn ap−j m
j m
n=1m
~ j ∈Zj
[ECS3d]
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 23/2
The Sign of the Casimir Force
Many papers dealing on this issue: here just short account
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/2
The Sign of the Casimir Force
Many 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. 24/2
The Sign of the Casimir Force
Many 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. It
is a special case requiring stringent material properties of the sphere
and a perfect geometry and BC
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/2
The Sign of the Casimir Force
Many 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. It
is a special case requiring stringent material properties of the sphere
and a perfect geometry and BC
Systematic calculation, for different fields, BCs, and dimensions
J Ambjørn, S Wolfram, Ann Phys NY 147, 1 (1983)
attract, repuls
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/2
The Sign of the Casimir Force
Many 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. It
is a special case requiring stringent material properties of the sphere
and a perfect geometry and BC
Systematic calculation, for different fields, BCs, and dimensions
J Ambjørn, S Wolfram, Ann Phys NY 147, 1 (1983)
attract, repuls
Possibly not relevant at lab scales, but very important for cosmological
models
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 24/2
The Sign of the Casimir Force
Many 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. It
is a special case requiring stringent material properties of the sphere
and a perfect geometry and BC
Systematic calculation, for different fields, BCs, and dimensions
J Ambjørn, S Wolfram, Ann Phys NY 147, 1 (1983)
attract, repuls
Possibly not relevant at lab scales, but very important for cosmological
models
More general results: Kenneth, Klich, PRL 97, 160401 (2006)
a mirror pair of dielectric bodies always attract each other
CP Bachas, J Phys A40, 9089 (2007) from a general property of
Euclidean 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. 24/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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. 25/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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 of
unitarity only, and makes no assumptions about the discrete
C, P, T symmetries
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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 of
unitarity only, and makes no assumptions about the discrete
C, P, T symmetries
Boyer’s result does not contradict the theorem, since cutting an elastic
shell 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. 25/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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 of
unitarity only, and makes no assumptions about the discrete
C, P, T symmetries
Boyer’s result does not contradict the theorem, since cutting an elastic
shell 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. 25/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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 of
unitarity only, and makes no assumptions about the discrete
C, P, T symmetries
Boyer’s result does not contradict the theorem, since cutting an elastic
shell 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 when
electron-gas fluctuations become important
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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 of
unitarity only, and makes no assumptions about the discrete
C, P, T symmetries
Boyer’s result does not contradict the theorem, since cutting an elastic
shell 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 when
electron-gas fluctuations become important
periodic BCs for fermions
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 25/2
E.g. ∃ correlation inequality: hf Θ(f )i > 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 of
unitarity only, and makes no assumptions about the discrete
C, P, T symmetries
Boyer’s result does not contradict the theorem, since cutting an elastic
shell 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 when
electron-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. 25/2
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 background
spacetime R(D1 −1,1) × Σ, Σ compact internal space
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 26/2
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 background
spacetime 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 necessary
to obtain repulsive Casimir forces
E Elizalde, MiniCursos d’Astrofisica i Cosmologia, Nov 30 - Dec 1, 2011 – p. 26/2
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 background
spacetime 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 necessary
to 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. 26/2
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 background
spacetime 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 necessary
to 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. 26/2
'
$
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
•
P
P
k are generalized sums
• µ mass-dim parameter, h̄ = c = 1
i
and
1X
=
λi
2 i
´
1X1³ 2
2 1/2
=
k +M
2
µ
k
• 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
Z ∞
∞
∞
X
X
π −d/2
d−1
= d
dk k
Q
Q
2 Γ(d/2) pj=1 aj qh=1 bh 0
np =−∞ mq =−∞


p Ã
X
j=1
2πnj
aj
!2
µ
q
X
+
h=1

∞
X
1
1

' p q
a b np ,mq =−∞ a2
2πmh
bh
p
X
¶2
1
n2j + 2
b
j=1
1/2
+ k2d + M 2 
q
X
 d+1 +1
2
m2h + M 2 
h=1
For d–open, (p–toroidal, q–spherical) universe:
ρφ
Z ∞
∞
∞
X
X
π −d/2
d−1
dk
k
Pq−1 (l)
= d
Qp
q
2 Γ(d/2) j=1 aj b 0
np =−∞ l=1


Ã
p
X
j=1
2πnj
aj
!2
1/2
Q2 (l)
+ 2 + k2d + M 2 
b
'

 d+1 +1
p
∞
∞
2
2 X
X
X
1
l(l
+
q)
M
4π
2
nj +
+ 2
Pq−1 (l)  2
p
q
2
a b np =−∞ l=1
a j=1
b
4π
2
• Pq−1 (l) is a polynomial in l of degree q − 1
&
%
Regularize: zeta function
'
$
1 X −s
ζ(s) =
λi
2 i
1X
=
2
k
Ã
2
k +M
µ2
2
!−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) =
X 0 ·1
~
n∈Z
≡
X 0
~
n∈Z
&
p
2
(~n + ~c)T A (~n + ~c) + q
¸−s
[Q (~n + ~c) + q]−s
p
%
'
$
• 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.
(2π)p/2 q p/2−s Γ(s − p/2) 2s/2+p/4+2 π s q −s/2+p/4
√
√
ζA,~c,q (s) =
+
Γ(s)
det A
det A Γ(s)
X 0
³
´s/2−p/4
cos(2π m
~ · ~c) m
~ T A−1 m
~
µ
q
¶
~ T A−1 m
~
Kp/2−s 2π 2q m
p
m∈
~ Z1/2
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).
&
%
'
$
PRECISE CALCULATION OF
THE VACUUM ENERGY DENSITY
Analytic continuation of the zeta function:
p ³ ´
∞
X
X
2π s/2+1
p h
ζ(s) = p−(s+1)/2 q−(s−1)/2
2
h
a
b
Γ(s/2) mq =−∞ h=0
×
Ã
∞
X
nh =1
Ph
2
j=1 nj
Pq
2
2
k=1 mk + M


!(s−1)/4
1/2 Ã
h
 2πa X 2 
×K(s−1)/2 
nj
b
j=1
q
X
!1/2
m2k + M 2



k=1
Yields the vacuum energy density:
ρφ
1
= − p q+1
ab

p
X
³
h=0

p´ h
h 2
h
∞
X
nh =1 mq =−∞
1/2 Ã
 2πa X 2 
×K1 
nj
b
j=1
∞
X
q
X
vP
u q
u k=1 m2k + M 2
t
Ph
2
j=1
!1/2
m2k + M 2

nj


k=1
Now, from
1
Kν (z) ∼ Γ(ν)(z/2)−ν ,
2
&
z→0
%
'
$
when M is very small
(
ρφ
µ
¶
p ³ ´
X
1
2πa
p h
= − p q+1 M K1
M +
2
h
ab
b
h=0
v

u
u h
 2πa uX 2 
K1 
M t nj 

×
∞
X
nh =1
M
qP
· √
h
j=1
n2j
+ O q 1 + M 2 K1
b
µ
j=1
2πa √
1 + M2
b
¶¸¾
Inserting the h̄ and c factors
"
p ³ ´
X
p
#
·
µ
2πa
h̄c
h
2
α
+
O
qK
1
+
ρφ = −
1
h
2πap+1 bq
b
h=0
¶¸
α some finite constant (explicit geometrical sum in the
limit M → 0).
• Sign may change with BC (e.g., Dirichlet).
&
%
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 = 102 lP
10−15 (10−10 )
10−6
10−3
b = 103 lP
10−16
10−12
[10−9 ]
(10−7 )
b = 104 lP
10−17
10−14
10−12
10−11
b = 105 lP
10−18
10−16
10−15
10−15
Table 1: The vacuum energy density contribution, in units of
erg/cm3 , for p large compactified dimensions a, and q = p + 1 small
compactified dimensions b, p = 0, . . . , 3, for different values of b, proportional to lP . In brackets, the values that more exactly match the
one for the cosmological constant coming from observations, and in
parenthesis the otherwise closest approximations.
&
%
'
$
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
&
%