emilio elizalde- cosmological casimir effect and beyond
TRANSCRIPT
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ICE
Cosmological Casimir effect
and beyond
EMILIO ELIZALDE
ICE/CSIC & IEEC, UAB, Barcelona
web: google emilio elizalde
DSU2006, UAM, Madrid, June 20, 2006
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Outline of the talk
Zeta Function A(s) and Determinant detA
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
Multiplicative (or Noncommutative) Anomaly
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
Multiplicative (or Noncommutative) Anomaly
Vacuum Fluctuations and the Cosmol Const
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
Multiplicative (or Noncommutative) Anomaly
Vacuum Fluctuations and the Cosmol Const
New CC Theories & Cosmo-topol Casimir Effect
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
Multiplicative (or Noncommutative) Anomaly
Vacuum Fluctuations and the Cosmol Const
New CC Theories & Cosmo-topol Casimir Effect
A Simple Model with large & small dimensions
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
Multiplicative (or Noncommutative) Anomaly
Vacuum Fluctuations and the Cosmol Const
New CC Theories & Cosmo-topol Casimir Effect
A Simple Model with large & small dimensions
Realistic Models: dS & AdS Braneworlds Supergraviton Theories
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Outline of the talk
Zeta Function A(s) and Determinant detA
The Wodzicki Residue
Multiplicative (or Noncommutative) Anomaly
Vacuum Fluctuations and the Cosmol Const
New CC Theories & Cosmo-topol Casimir Effect
A Simple Model with large & small dimensions
Realistic Models: dS & AdS Braneworlds Supergraviton Theories
Conclusions
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Existence ofA for A a DO
A a positive-definite elliptic DO of positive order m
R
A acts on the space of smooth sections of E, n-dim vector bundle over
M closed n-dim manifold
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Existence ofA for A a DO
A a positive-definite elliptic DO of positive order m
R
A acts on the space of smooth sections of E, n-dim vector bundle over
M closed n-dim manifold(a) The zeta function is defined as A(s) = tr A
s =
j sj , Re s >
nm s0
{j
}ordered spect of A, s0 = dim M/ord A abscissa of converg of A(s)
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Existence ofA for A a DO
A a positive-definite elliptic DO of positive order m
R
A acts on the space of smooth sections of E, n-dim vector bundle over
M closed n-dim manifold(a) The zeta function is defined as A(s) = tr A
s =
j sj , Re s >
nm s0
{j
}ordered spect of A, s0 = dim M/ord A abscissa of converg of A(s)
(b) A(s) has a meromorphic continuation to the whole complex plane C
(regular at s = 0), provided the principal symbol of A, am(x, ), admits a
spectral cut: L = { C | Arg = , 1 < < 2}, Spec A L = (Agmon-Nirenberg condition)
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Existence ofA for A a DO
A a positive-definite elliptic DO of positive order m
R
A acts on the space of smooth sections of E, n-dim vector bundle over
M closed n-dim manifold(a) The zeta function is defined as A(s) = tr A
s =
j sj , Re s >
nm s0
{j
}ordered spect of A, s0 = dim M/ord A abscissa of converg of A(s)
(b) A(s) has a meromorphic continuation to the whole complex plane C
(regular at s = 0), provided the principal symbol of A, am(x, ), admits a
spectral cut: L = { C | Arg = , 1 < < 2}, Spec A L = (Agmon-Nirenberg condition)
(c) The definition of A(s) depends on the position of the cut L
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A
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Existence ofA for A a DO
A a positive-definite elliptic DO of positive order m
R
A acts on the space of smooth sections of E, n-dim vector bundle over
M closed n-dim manifold(a) The zeta function is defined as A(s) = tr A
s =
j sj , Re s >
nm s0
{j
}ordered spect of A, s0 = dim M/ord A abscissa of converg of A(s)
(b) A(s) has a meromorphic continuation to the whole complex plane C
(regular at s = 0), provided the principal symbol of A, am(x, ), admits a
spectral cut: L = { C | Arg = , 1 < < 2}, Spec A L = (Agmon-Nirenberg condition)
(c) The definition of A(s) depends on the position of the cut L
(d) The only possible singularities of A(s) are simple poles at
sk = (n k)/m, k = 0, 1, 2, . . . , n 1, n + 1, . . .5thICMMP, CBPF, Rio, April 2006 p. 3/
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Definition of DeterminantH DO operator
{i, i
}spectral decomposition
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Definition of DeterminantH DO operator
{i, i
}spectral decomposition
iI i ?! ln
iI i =
iI ln i
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Definition of DeterminantH DO operator
{i, i
}spectral decomposition
iI i ?! ln
iI i =
iI ln i
Riemann zeta func: (s) = n=1 ns, Re s > 1 (& analytic cont)= Definition: zeta function of H H(s) =
iI
si = tr H
s
As Mellin transform: H(s) =1
(s)0
dt ts1 tr etH, Re s > D/2
Derivative: H(0) =
iI ln i
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D fi i i f D i
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Definition of DeterminantH DO operator
{i, i
}spectral decomposition
iI i ?! ln
iI i =
iI ln i
Riemann zeta func: (s) = n=1 ns, Re s > 1 (& analytic cont)= Definition: zeta function of H H(s) =
iI
si = tr H
s
As Mellin transform: H(s) =1
(s)0
dt ts1 tr etH, Re s > D/2
Derivative: H(0) =
iI ln i
=
Determinant: Ray & Singer, 67
det H = exp [H(0)]
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D fi iti f D t i t
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Definition of DeterminantH DO operator
{i, i
}spectral decomposition
iI i ?! ln
iI i =
iI ln i
Riemann zeta func: (s) = n=1 ns, Re s > 1 (& analytic cont)= Definition: zeta function of H H(s) =
iI
si = tr H
s
As Mellin transform: H(s) =1
(s)0
dt ts1 tr etH, Re s > D/2
Derivative: H(0) =
iI ln i
=
Determinant: Ray & Singer, 67
det H = exp [H(0)]Weierstrass definition: subtract leading behavior of i in i, as
i
, until the series i
Iln i converges
= non-local counterterms !!5thICMMP, CBPF, Rio, April 2006 p. 4/
P ti
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PropertiesThe definition of the determinant det A
only depends on the homotopy class of the cut
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Properties
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PropertiesThe definition of the determinant det 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 donot admit spectral cuts), has been obtained [Kontsevich and Vishik]
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Properties
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PropertiesThe definition of the determinant det 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 donot admit spectral cuts), has been obtained [Kontsevich and Vishik]
Asymptotic expansion for the heat kernel
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Properties
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PropertiesThe definition of the determinant det 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 donot admit spectral cuts), has been obtained [Kontsevich and Vishik]
Asymptotic expansion for the heat kernel
tr etA =Spec A e
t
n(A) +
n=j0 j(A)tsj +
k1 k(A)t
k ln t, t 0n
(A) = A
(0), j
(A) = (sj
) Ress=sj
A
(s), sj
/
N
j(A) =(1)k
k! [PP A(k) + (k + 1) Ress=k A(s)] ,sj = k, k N
k(A) =(1)k+1
k! Ress=k A(s), k
N
\{0
}PP = limsp
(s) Ress=p (s)sp
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)Only trace one can define in the algebra of DOs (up to multipl const)
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)Only trace one can define in the algebra of DOs (up to multipl const)
Definition: res A = 2 Ress=0 tr (As), Laplacian
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)Only trace one can define in the algebra of DOs (up to multipl const)
Definition: res A = 2 Ress=0 tr (As), Laplacian
Satisfies the trace condition: res (AB) = res (BA)
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)Only trace one can define in the algebra of DOs (up to multipl const)
Definition: res A = 2 Ress=0 tr (As), Laplacian
Satisfies the trace condition: res (AB) = res (BA)
Important ! it can be expressed as an integral (local form)
res A =SM tr an(x, ) d
with SM TM the co-sphere bundle on M (some authors put acoefficient in front of the integral: Adler-Manin residue)
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)Only trace one can define in the algebra of DOs (up to multipl const)
Definition: res A = 2 Ress=0 tr (As), Laplacian
Satisfies the trace condition: res (AB) = res (BA)
Important ! it can be expressed as an integral (local form)
res A =SM tr an(x, ) d
with SM TM 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) itcoincides with the Dixmier trace, and Ress=1A(s) =
1n res A
1
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The Wodzicki Residue
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The Wodzicki ResidueThe Wodzicki (or noncommutative) residue is the only extension of
the Dixmier trace to DOs which are not in L(1,)Only trace one can define in the algebra of DOs (up to multipl const)
Definition: res A = 2 Ress=0 tr (As), Laplacian
Satisfies the trace condition: res (AB) = res (BA)
Important ! it can be expressed as an integral (local form)
res A =SM tr an(x, ) d
with SM TM 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) itcoincides with the Dixmier trace, and Ress=1A(s) =
1n res A
1
The Wodzicki res makes sense for DOs of arbitrary order. Even if
symbols aj(x, ), j < m, are not coordinate invariant, the integral is,
and defines a trace5thICMMP, CBPF, Rio, April 2006 p. 6/
Multiplicative Anomaly (or Defect)
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Multiplicative Anomaly (or Defect)
Given A, B, and AB DOs, even if A
, B
, and AB
exist,
it turns out that, in general,
det(AB)
= detA detB
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Multiplicative Anomaly (or Defect)
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Multiplicative Anomaly (or Defect)
Given A, B, and AB DOs, even if A
, B
, and AB
exist,
it turns out that, in general,
det(AB)
= detA detB
The multiplicative (or noncommutative) anomaly (or
defect) is defined as
(A, B) = ln
det(AB)det A det B
= AB(0) + A(0) + B(0)
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Multiplicative Anomaly (or Defect)
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Multiplicative Anomaly (or Defect)
Given A, B, and AB DOs, even if A
, B
, and AB
exist,
it turns out that, in general,
det(AB)
= detA detB
The multiplicative (or noncommutative) anomaly (or
defect) is defined as
(A, B) = ln
det(AB)det A det B
= AB(0) + A(0) + B(0)
Wodzicki formula
(A, B) =res
[ln (A, B)]2
2 ord A ord B (ord A + ord B)
where (A, B) = Aord BBord A
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Consequences of the mult. anom.
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C q aL(x) = 1
2(x)(x) 1
2m21(x)
2 + V1((x))
12
(x)(x) 12
m22(x)2 + V2((x)) + V3((x), (x))
Partition function
eZ =
D D exp
dd+1x
1
2(2 + m21) + 12 (
2 + m22)
V1() V2() V3(, )]}
where the x-dependence is no more explicitly depicted (for simplification)
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qL(x) = 1
2(x)(x) 1
2m21(x)
2 + V1((x))
12
(x)(x) 12
m22(x)2 + V2((x)) + V3((x), (x))
Partition function
eZ =
D D exp
dd+1x
1
2(2 + m21) + 12 (
2 + m22)
V1() V2() V3(, )]}
where the x-dependence is no more explicitly depicted (for simplification)
Background quantization: = 0 + , = 0 +
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qL(x) = 1
2(x)(x) 1
2m21(x)
2 + V1((x))
12
(x)(x) 12
m22(x)2 + V2((x)) + V3((x), (x))
Partition function
eZ =
D D exp
dd+1x
12
(2 + m21) + 12 (2 + m22)
V1() V2() V3(, )]}
where the x-dependence is no more explicitly depicted (for simplification)
Background quantization: = 0 + , = 0 +
Gaussian approximation (one loop)
eZ1 =
DD exp
dd+1x
( ) A C
C B
=
det A CC B
1/2
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with A
2 + m21
V1 (0)
21V3(0, 0),
B 2 + m22 V2 (0) 22V3(0, 0),C 12V3(0, 0).
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with A
2 + m21
V1 (0)
21V3(0, 0),
B 2 + m22 V2 (0) 22V3(0, 0),C 12V3(0, 0).
After some calculations it turns out that the determinant can beexpressed as:
det AB C2 = det (2 + )(2 + ) ,
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with A
2 + m21
V1 (0)
21V3(0, 0),
B 2 + m22 V2 (0) 22V3(0, 0),C 12V3(0, 0).
After some calculations it turns out that the determinant can beexpressed as:
det AB C2 = det (2 + )(2 + ) ,with
a
b2 + c2, a +
b2 + c2,
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with A
2 + m21
V1 (0)
21V3(0, 0),
B 2 + m22 V2 (0) 22V3(0, 0),C 12V3(0, 0).
After some calculations it turns out that the determinant can beexpressed as:
det AB C2
= det (2 + )(2 + ) ,
with
a
b2 + c2, a +
b2 + c2,
and
a c1 + c22
, b c1 c22
,
c1 m21 V1 (0) 21V3(0, 0), c2 m22 V2 (0) 22V3(0, 0),c 12V3(0, 0).
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This situation has been considered by us before, in much detail.
It gives rise to the multiplicative (or non commutative) anomaly
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It gives rise to the multiplicative (or non-commutative) anomaly
(or defect), namely the fact that, generically,
det
(2 + )(2 + ) = det(2 + ) det(2 + )
5thICMMP, CBPF, Rio, April 2006 p. 10/
This situation has been considered by us before, in much detail.
It gives rise to the multiplicative (or non commutative) anomaly
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It gives rise to the multiplicative (or non-commutative) anomaly
(or defect), namely the fact that, generically,
det
(2 + )(2 + ) = det(2 + ) det(2 + )The anomaly
(logarithm of the quotient) is expressed as a
perturbative series of the difference = 2b2 + c2
= n
an(
)n =
n
(
2)n an (b
2 + c2)n/2
5thICMMP, CBPF, Rio, April 2006 p. 10/
This situation has been considered by us before, in much detail.
It gives rise to the multiplicative (or non commutative) anomaly
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It gives rise to the multiplicative (or non-commutative) anomaly
(or defect), namely the fact that, generically,
det
(2 + )(2 + ) = det(2 + ) det(2 + )The anomaly
(logarithm of the quotient) is expressed as a
perturbative series of the difference = 2b2 + c2
= n
an(
)n =
n
(
2)n an (b
2 + c2)n/2
= Note that b depends on the mass difference, the difference inthe second derivatives (at the background) of the individual
potentials and on the difference between the two partial, second-
order derivatives of the double potential, while c is given by the
cross derivative of this potential.
5thICMMP, CBPF, Rio, April 2006 p. 10/
This situation has been considered by us before, in much detail.
It gives rise to the multiplicative (or non-commutative) anomaly
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It gives rise to the multiplicative (or non-commutative) anomaly
(or defect), namely the fact that, generically,
det
(2 + )(2 + ) = det(2 + ) det(2 + )The anomaly
(logarithm of the quotient) is expressed as a
perturbative series of the difference = 2b2 + c2
= n
an(
)n =
n
(
2)n an (b
2 + c2)n/2
= Note that b depends on the mass difference, the difference inthe second derivatives (at the background) of the individual
potentials and on the difference between the two partial, second-
order derivatives of the double potential, while c is given by the
cross derivative of this potential.
= In some cases the anomaly proves to be genuine: it persistsafter renormalization is performed.
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Zero point energy
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QFT vacuum to vacuum transition: 0|H|0
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Zero point energy
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QFT vacuum to vacuum transition: 0|H|0Spectrum, normal ordering:
H =
n + 12
n an an
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Zero point energy
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QFT vacuum to vacuum transition: 0|H|0Spectrum, normal ordering:
H =
n + 12
n an an
0|H|0 = (c)2
n
n =12
tr H
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Zero point energy
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QFT vacuum to vacuum transition: 0|H|0Spectrum, normal ordering:
H =
n + 12
n an an
0|H|0 = (c)2
n
n =12
tr H
gives physical meaning?
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Zero point energy
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QFT vacuum to vacuum transition: 0|H|0Spectrum, normal ordering:
H =
n + 12
n an an
0|H|0 = (c)2
n
n =12
tr H
gives physical meaning?Regularization + Renormalization ( cut-off, dim, )
5thICMMP, CBPF, Rio, April 2006 p. 11/
Zero point energy
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QFT vacuum to vacuum transition: 0|H|0Spectrum, normal ordering:
H =
n + 12
n an an
0|H|0 = (c)2
n
n =12
tr H
gives physical meaning?Regularization + Renormalization ( cut-off, dim, )
Even then: Has the final value any meaning??5thICMMP, CBPF, Rio, April 2006 p. 11/
The Casimir Effect
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5thICMMP, CBPF, Rio, April 2006 p. 12/
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The Casimir EffectBC i di
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vacuum
BC
F
CasimirEffect
BC e.g. periodic
= all kind of fields
5thICMMP, CBPF, Rio, April 2006 p. 12/
The Casimir EffectBC e g periodic
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vacuum
BC
F
CasimirEffect
BC e.g. periodic
= all kind of fields= curvature or topology
5thICMMP, CBPF, Rio, April 2006 p. 12/
The Casimir EffectBC e g periodic
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vacuum
BC
F
CasimirEffect
BC e.g. periodic
= all kind of fields= curvature or topology
Universal process:
5thICMMP, CBPF, Rio, April 2006 p. 12/
The Casimir EffectBC e g periodic
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vacuum
BC
F
CasimirEffect
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
5thICMMP, CBPF, Rio, April 2006 p. 12/
The Casimir EffectBC e g periodic
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vacuum
BC
F
CasimirEffect
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, Lipschitz theory
5thICMMP, CBPF, Rio, April 2006 p. 12/
The Casimir EffectBC e g periodic
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vacuum
BC
F
CasimirEffect
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, Lipschitz theory
Dynamical CELateral CE
Extract energy from vacuum
CE and the cosmological constant5thICMMP, CBPF, Rio, April 2006 p. 12/
Vacuum energy density and the CCThe main issue:
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The main issue:
energy ALWAYS gravitates, therefore the energy density of thevacuum, more precisely, the vacuum expectation value of the
stress-energy tensor T Eg
5thICMMP, CBPF, Rio, April 2006 p. 13/
Vacuum energy density and the CCThe main issue:
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The main issue:
energy ALWAYS gravitates, therefore the energy density of thevacuum, more precisely, the vacuum expectation value of the
stress-energy tensor T Eg
Appears on the rhs of Einsteins equations:
R 12
gR = 8G(T Eg)
It affects cosmology: T excitations above the vacuum
5thICMMP, CBPF, Rio, April 2006 p. 13/
Vacuum energy density and the CCThe main issue:
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energy ALWAYS gravitates, therefore the energy density of thevacuum, more precisely, the vacuum expectation value of the
stress-energy tensor T Eg
Appears on the rhs of Einsteins equations:
R 12
gR = 8G(T Eg)
It affects cosmology: T excitations above the vacuum
Equivalent to a cosmological constant = 8GE
5thICMMP, CBPF, Rio, April 2006 p. 13/
Vacuum energy density and the CCThe main issue:
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energy ALWAYS gravitates, therefore the energy density of thevacuum, more precisely, the vacuum expectation value of the
stress-energy tensor T Eg
Appears on the rhs of Einsteins equations:
R 12
gR = 8G(T Eg)
It affects cosmology: T excitations above the vacuum
Equivalent to a cosmological constant = 8GERecent observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
= (2.14 0.13 103 eV)4 4.32 109 erg/cm3
5thICMMP, CBPF, Rio, April 2006 p. 13/
Vacuum energy density and the CCThe main issue:
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energy ALWAYS gravitates, therefore the energy density of thevacuum, more precisely, the vacuum expectation value of the
stress-energy tensor T Eg
Appears on the rhs of Einsteins equations:
R 12
gR = 8G(T Eg)
It affects cosmology: T excitations above the vacuum
Equivalent to a cosmological constant = 8GERecent observations: M. Tegmark et al. [SDSS Collab.] PRD 2004
= (2.14 0.13 103 eV)4 4.32 109 erg/cm3
Idea: zero point fluctuations do contribute to the
cosmological constant
5thICMMP, CBPF, Rio, April 2006 p. 13/
Cosmo-Topological Casimir Effect (I)Zeta techniques used in calcul the contribution of the vacuum energy
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of quantum fields pervading the universe to the cosmolog const [cc]
5thICMMP, CBPF, Rio, April 2006 p. 14/
Cosmo-Topological Casimir Effect (I)Zeta techniques used in calcul the contribution of the vacuum energy
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of quantum fields pervading the universe to the cosmolog const [cc]
Direct calculations of the absolute contributions of the known fields
(all couple to gravity) lead to a value which is off by roughly
120 orders of magnitude [obs
/(Pl)4
10120
]= kind of a modern (and very thick!) aether
5thICMMP, CBPF, Rio, April 2006 p. 14/
Cosmo-Topological Casimir Effect (I)Zeta techniques used in calcul the contribution of the vacuum energy
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of quantum fields pervading the universe to the cosmolog const [cc]
Direct calculations of the absolute contributions of the known fields
(all couple to gravity) lead to a value which is off by roughly
120 orders of magnitude [obs
/(Pl)4
10120
]= kind of a modern (and very thick!) aether
Observational tests see nothing (or very little) of it:
= (new) cosmological constant problem.
5thICMMP, CBPF, Rio, April 2006 p. 14/
Cosmo-Topological Casimir Effect (I)Zeta techniques used in calcul the contribution of the vacuum energy
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of quantum fields pervading the universe to the cosmolog const [cc]
Direct calculations of the absolute contributions of the known fields
(all couple to gravity) lead to a value which is off by roughly
120 orders of magnitude [
obs
/(Pl)
4
10120
]= kind of a modern (and very thick!) aether
Observational tests see nothing (or very little) of it:
= (new) cosmological constant problem.Very difficult to solve and we do not address this question directly
[Baum, Hawking, Coleman, Polchinsky, Weinberg,...]
5thICMMP, CBPF, Rio, April 2006 p. 14/
Cosmo-Topological Casimir Effect (I)Zeta techniques used in calcul the contribution of the vacuum energy
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of quantum fields pervading the universe to the cosmolog const [cc]
Direct calculations of the absolute contributions of the known fields
(all couple to gravity) lead to a value which is off by roughly
120 orders of magnitude [
obs
/(Pl)
4
10120
]= kind of a modern (and very thick!) aether
Observational tests see nothing (or very little) of it:
= (new) cosmological constant problem.Very difficult to solve and we do not address this question directly
[Baum, Hawking, Coleman, Polchinsky, Weinberg,...]
What we do consider with relative success in some differentapproaches is the additional contribution to the cc coming from the
non-trivial topology of space or from specific boundary conditions
imposed on braneworld models:
= kind of cosmological Casimir effect5thICMMP, CBPF, Rio, April 2006 p. 14/
Cosmo-Topological Casimir Effect (II)Assuming one will be able to prove (in the future) that the ground
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value of the cc is zero (as many had suspected until recently), we willbe left with this incremental value coming from the topology or BCs
* L. Parker & A. Raval, VCDM, vacuum energy density
* C.P. Burgess, hep-th/0606020 & 0510123: Susy Large Extra Dims
(SLED), two 102m dims, bulk vs brane Susy breaking scales
* T. Padmanabhan, gr-qc/0606061: Holographic Perspective, CC is an
intg const, no response of gravity to changes in bulk vac energy dens
5thICMMP, CBPF, Rio, April 2006 p. 15/
Cosmo-Topological Casimir Effect (II)Assuming one will be able to prove (in the future) that the ground
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value of the cc is zero (as many had suspected until recently), we willbe left with this incremental value coming from the topology or BCs
* L. Parker & A. Raval, VCDM, vacuum energy density
* C.P. Burgess, hep-th/0606020 & 0510123: Susy Large Extra Dims
(SLED), two 102m dims, bulk vs brane Susy breaking scales
* T. Padmanabhan, gr-qc/0606061: Holographic Perspective, CC is an
intg const, no response of gravity to changes in bulk vac energy dens
We show (with different examples) that this value acquires
the correct order of magnitude corresponding to the one
coming from the observed acceleration in the expansion of
our universe in some reasonable models involving:
5thICMMP, CBPF, Rio, April 2006 p. 15/
Cosmo-Topological Casimir Effect (II)Assuming one will be able to prove (in the future) that the ground
-
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value of the cc is zero (as many had suspected until recently), we willbe left with this incremental value coming from the topology or BCs
* L. Parker & A. Raval, VCDM, vacuum energy density
* C.P. Burgess, hep-th/0606020 & 0510123: Susy Large Extra Dims
(SLED), two 102m dims, bulk vs brane Susy breaking scales
* T. Padmanabhan, gr-qc/0606061: Holographic Perspective, CC is an
intg const, no response of gravity to changes in bulk vac energy dens
We show (with different examples) that this value acquires
the correct order of magnitude corresponding to the one
coming from the observed acceleration in the expansion of
our universe in some reasonable models involving:(a) small and large compactified scales
5thICMMP, CBPF, Rio, April 2006 p. 15/
Cosmo-Topological Casimir Effect (II)Assuming one will be able to prove (in the future) that the ground
-
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value of the cc is zero (as many had suspected until recently), we willbe left with this incremental value coming from the topology or BCs
* L. Parker & A. Raval, VCDM, vacuum energy density
* C.P. Burgess, hep-th/0606020 & 0510123: Susy Large Extra Dims
(SLED), two 102m dims, bulk vs brane Susy breaking scales
* T. Padmanabhan, gr-qc/0606061: Holographic Perspective, CC is an
intg const, no response of gravity to changes in bulk vac energy dens
We show (with different examples) that this value acquires
the correct order of magnitude corresponding to the one
coming from the observed acceleration in the expansion of
our universe in some reasonable models involving:(a) small and large compactified scales
(b) dS & AdS worldbranes
5thICMMP, CBPF, Rio, April 2006 p. 15/
Cosmo-Topological Casimir Effect (II)Assuming one will be able to prove (in the future) that the ground
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value of the cc is zero (as many had suspected until recently), we willbe left with this incremental value coming from the topology or BCs
* L. Parker & A. Raval, VCDM, vacuum energy density
* C.P. Burgess, hep-th/0606020 & 0510123: Susy Large Extra Dims
(SLED), two 102m dims, bulk vs brane Susy breaking scales
* T. Padmanabhan, gr-qc/0606061: Holographic Perspective, CC is an
intg const, no response of gravity to changes in bulk vac energy dens
We show (with different examples) that this value acquires
the correct order of magnitude corresponding to the one
coming from the observed acceleration in the expansion of
our universe in some reasonable models involving:(a) small and large compactified scales
(b) dS & AdS worldbranes
(c) supergraviton theories (discret dims, deconstr)5thICMMP, CBPF, Rio, April 2006 p. 15/
A. Simple model: large & small dimsSpace-time:
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Rd+1 Tp Tq, Rd+1 Tp Sq,
5thICMMP, CBPF, Rio, April 2006 p. 16/
A. Simple model: large & small dimsSpace-time:
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Rd+1 Tp Tq, Rd+1 Tp Sq,
Scalar field, , pervading the universe
5thICMMP, CBPF, Rio, April 2006 p. 16/
A. Simple model: large & small dimsSpace-time:
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Rd+1 Tp Tq, Rd+1 Tp Sq,
Scalar field, , pervading the universe
contribution to V from this field
=1
2
i
i
=1
2
k
1
k2 + M21/2
5thICMMP, CBPF, Rio, April 2006 p. 16/
A. Simple model: large & small dimsSpace-time:
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Rd+1 Tp Tq, Rd+1 Tp Sq,
Scalar field, , pervading the universe
contribution to V from this field
=1
2
i
i
=1
2
k
1
k2 + M21/2
i and
k are generalized sums, mass-dim parameter,
= c = 1
5thICMMP, CBPF, Rio, April 2006 p. 16/
A. Simple model: large & small dimsSpace-time:
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Rd+1 Tp Tq, Rd+1 Tp Sq,
Scalar field, , pervading the universe
contribution to V from this field
=1
2
i
i
=1
2
k
1
k2 + M21/2
i and
k are generalized sums, mass-dim parameter,
= 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)5thICMMP, CBPF, Rio, April 2006 p. 16/
For dopen, (p,q)toroidal universe: =
d/2
2d(d/2) pj=1 ajqh=1 bh
0dk kd1
1/2
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np=
mq=
pj=1
2nj
aj
2
+q
h=1
2mh
bh
2
+ k2d + M2
1apbq
np,mq=
1a2
pj=1
n2j +1b2
qh=1
m2h + M2
d+12
+1
5thICMMP, CBPF, Rio, April 2006 p. 17/
For dopen, (p,q)toroidal universe: =
d/2
2d(d/2) pj=1 ajqh=1 bh
0dk kd1
1/2
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np=
mq=
pj=1
2nj
aj
2
+qh=1
2mh
bh
2
+ k2d + M2
1apbq
np,mq=
1a2
pj=1
n2j +1b2
qh=1
m2h + M2
d+12
+1
For dopen, (ptoroidal, qspherical) universe:
=d/2
2d(d/2)
pj=1 aj b
q
0dk kd1
np=
l=1
Pq1(l)
pj=1
2nj
aj
2
+Q2(l)
b2+ k2d + M
21/2
[Pq1(l) poly in l deg q 1]
1
apbq
np=
l=1
Pq1(l)42
a2
pj=1
n2j +l(l + q)
b2 +M2
42
d+12
+1
5thICMMP, CBPF, Rio, April 2006 p. 17/
Regularize: zeta function
(s) =1
2si =
1
2
k2 + M2
2
s/2
= = (1)
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i
k
5thICMMP, CBPF, Rio, April 2006 p. 18/
Regularize: zeta function
(s) =1
2si =
1
2
k2 + M2
2
s/2
= = (1)
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i
k
No further subtraction or renormalization needed here
[E.E., J. Math. Phys. 35, 3308, 6100 (1994)]
5thICMMP, CBPF, Rio, April 2006 p. 18/
Regularize: zeta function
(s) =1
2si =
1
2
k2 + M2
2
s/2
= = (1)
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i
k
No further subtraction or renormalization needed here
[E.E., J. Math. Phys. 35, 3308, 6100 (1994)]
Here spectrum of the Hamiltonian op. known explicitly.
Corresponds to generalized Chowla-Selberg expression
[E.E., Commun. Math. Phys. 198, 83 (1998)
E.E., J. Phys. A30, 2735 (1997)]
5thICMMP, CBPF, Rio, April 2006 p. 18/
Regularize: zeta function
(s) =1
2si =
1
2
k2 + M2
2
s/2
= = (1)
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i
k
No further subtraction or renormalization needed here
[E.E., J. Math. Phys. 35, 3308, 6100 (1994)]
Here spectrum of the Hamiltonian op. known explicitly.
Corresponds to generalized Chowla-Selberg expression
[E.E., Commun. Math. Phys. 198, 83 (1998)
E.E., J. Phys. A30, 2735 (1997)]
For the zeta function (Re s > p/2):
A,c,q(s) =
nZp
1
2(n + c)T A (n + c) + q
s
nZp
[Q (n + c) + q]
s
5thICMMP, CBPF, Rio, April 2006 p. 18/
Prime means that point n = 0 is excluded from sum (not important)
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5thICMMP, CBPF, Rio, April 2006 p. 19/
Prime means that point n = 0 is excluded from sum (not important)
Gives (analyt cont of) multidim zeta function in terms of an
exponentially convergent multiseries, valid in the whole complex plane
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5thICMMP, CBPF, Rio, April 2006 p. 19/
Prime means that point n = 0 is excluded from sum (not important)
Gives (analyt cont of) multidim zeta function in terms of an
exponentially convergent multiseries, valid in the whole complex plane
-
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Exhibits singularities (simple poles) of meromorphic continuation
with corresponding residua explicitly
5thICMMP, CBPF, Rio, April 2006 p. 19/
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Prime means that point n = 0 is excluded from sum (not important)
Gives (analyt cont of) multidim zeta function in terms of an
exponentially convergent multiseries, valid in the whole complex plane
E hibit i l iti ( i l l ) f hi ti ti
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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 (for the moment) positive constant
A,c,q(s) =(2)p/2qp/2s
det A
(s p/2)
(s)
+2s/2+p/4+2sqs/2+p/4
det A (s)mZ
p1/2
cos(2 m c) mTA1ms/2p/4 Kp/2s 22q mTA1 m
K modified Bessel function of the second kind and the subindex 1/2in Zp1/2 means that only half of the vectors m Zp are summed over.That is, if we take an m Zp we must then exclude m (as simplecriterion one can, for instance, select those vectors in Zp
\{0}
whose
first non-zero component is positive).5thICMMP, CBPF, Rio, April 2006 p. 19/
Calcul. of vacuum energy densityAnalytic continuation of the zeta function:
2 s/2+1 p h 2 s14
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(s) =2s/2+1
ap(s+1)/2bq(s1)/2(s/2)
mq=
ph=0
ph
2h
nh=1
hj=1 n2jq
k=1 m2k + M
2
4
K(s1)/22a
b
hj=1
n2j
1/2 q
k=1
m2k + M21/2
5thICMMP, CBPF, Rio, April 2006 p. 20/
Calcul. of vacuum energy densityAnalytic continuation of the zeta function:
2 s/2+1 p h 1 n2j s14
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(s) =2s/2+1
ap(s+1)/2bq(s1)/2(s/2)
mq=
ph=0
ph
2h
nh=1
hj=1 n2jq
k=1 m2k + M
2
4
K(s1)/22a
b
hj=1
n2j
1/2 qk=1
m2k + M21/2
Yields the vacuum energy density:
= 1apbq+1
p
h=0
ph 2h
nh=1
mq=
qk=1 m
2k + M
2
hj=1 n
2
j
K1
2a
b
h
j=1
n2j
1/2
q
k=1
m2k + M2
1/2
5thICMMP, CBPF, Rio, April 2006 p. 20/
Now, from K(z) 12()(z/2), z 0
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5thICMMP, CBPF, Rio, April 2006 p. 21/
Now, from K(z) 12()(z/2), z 0when M is very small
=
1
apbq+1
M K12a
b M
+
pph
2h
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= apbq+1 M K1 b M+h=0
h
2
nh=1
Mh
j=1 n2j
K12ab
Mh
j=1
n2
j
+ O q1 + M2K1
2a
b 1 + M2
5thICMMP, CBPF, Rio, April 2006 p. 21/
Now, from K(z) 12()(z/2), z 0when M is very small
=
1
apbq+1
M K12a
b M
+
pph
2h
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= apbq+1 M K1 b M+h=0
h
2
nh=1
Mh
j=1 n2j
K12ab
Mh
j=1
n2j+ O q1 + M
2K12a
b 1 + M2
Inserting the and c factors
= c
2ap+1bq
1 +
ph=0
ph
2
h
+ O qK12a
b
5thICMMP, CBPF, Rio, April 2006 p. 21/
Now, from K(z) 12()(z/2), z 0when M is very small
= 1
apbq+1
M K12a
b M
+
pph
2h
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= apbq+1 M K1 b M+h=0
h
2
nh=1
Mh
j=1 n2j
K12ab
Mh
j=1
n2j+ O q1 + M
2K12a
b 1 + M2
Inserting the and c factors
= c
2ap+1bq
1 +
ph=0
ph
2
h
+ O qK12a
b
some finite constant (explicit geometrical sum in the limit M 0)
5thICMMP, CBPF, Rio, April 2006 p. 21/
Now, from K(z) 12()(z/2), z 0when M is very small
= 1
apbq+1
M K12a
b M
+
pph
2h
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= apbq+
M K
b M
+h=0
h
2
nh=1
Mh
j=1 n2j
K12ab
Mh
j=1
n2j+ O q1 + M
2K12a
b 1 + M2
Inserting the and c factors
= c
2ap+1bq
1 +
ph=0
ph
2
h
+ O qK12a
b
some finite constant (explicit geometrical sum in the limit M 0)
= Sign may change with BC (e.g., Dirichlet): a problem5thICMMP, CBPF, Rio, April 2006 p. 21/
Matching the obs. results for the CCb lP(lanck) a RU(niverse) a/b 1060
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5thICMMP, CBPF, Rio, April 2006 p. 22/
Matching the obs. results for the CCb lP(lanck) a RU(niverse) a/b 1060
p 0 p 1 p 2 p 3
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p = 0 p = 1 p = 2 p = 3
b = lP 1013 106 1 105
b = 10lP 1014 [108] 103 10
b = 102lP 1015 (1010) 106 103
b = 103lP 1016 1012 [109] (107)
b = 104lP 1017 1014 1012 1011
b = 105lP 1018 1016 1015 1015
Table 2: Vacuum energy density in units of erg/cm3, for p large com-
pactified dimensions a, and q = p + 1 small compactified dimensions b,
p = 0, . . . , 3, for different values of b, proportional to the Planck length lP
5thICMMP, CBPF, Rio, April 2006 p. 22/
Results of this simple model* Precise (in absolute value) coincidence with the observational
value for the cosmological constant with
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a ue o t e cos o og ca co sta t t [ ]
5thICMMP, CBPF, Rio, April 2006 p. 23/
Results of this simple model* Precise (in absolute value) coincidence with the observational
value for the cosmological constant with
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g [ ]
* Approximate coincidence
( )
5thICMMP, CBPF, Rio, April 2006 p. 23/
Results of this simple model* Precise (in absolute value) coincidence with the observational
value for the cosmological constant with
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g [ ]
* Approximate coincidence
( )
* b 10 to 1000 times the Planck length lP
5thICMMP, CBPF, Rio, April 2006 p. 23/
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Results of this simple model* Precise (in absolute value) coincidence with the observational
value for the cosmological constant with[ ]
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[ ]* Approximate 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 observable Universe
5thICMMP, CBPF, Rio, April 2006 p. 23/
Results of this simple model* Precise (in absolute value) coincidence with the observational
value for the cosmological constant with[ ]
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[ ]* Approximate 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 observable Universe
* Both situations correspond to a marginally closed universe
5thICMMP, CBPF, Rio, April 2006 p. 23/
Results of this simple model* Precise (in absolute value) coincidence with the observational
value for the cosmological constant with[ ]
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[ ]* Approximate 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 observable Universe
* Both situations correspond to a marginally closed universe
=
To examine
couplings in GR
alternative theories5thICMMP, CBPF, Rio, April 2006 p. 23/
B. Realistic models: dS & AdS BWElizalde, Nojiri, Odintsov, PRD70 (2004) 043539
Elizalde, Nojiri, Odintsov, Ogushi, PRD67 (2003) 063515
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5thICMMP, CBPF, Rio, April 2006 p. 24/
B. Realistic models: dS & AdS BWElizalde, Nojiri, Odintsov, PRD70 (2004) 043539
Elizalde, Nojiri, Odintsov, Ogushi, PRD67 (2003) 063515
Bulk Casimir effect may play a role in radion stabilization of BW
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y p y
Bulk Casimir effect (eff. pot.) for a conformal or massive scalar field
5thICMMP, CBPF, Rio, April 2006 p. 24/
B. Realistic models: dS & AdS BWElizalde, Nojiri, Odintsov, PRD70 (2004) 043539
Elizalde, Nojiri, Odintsov, Ogushi, PRD67 (2003) 063515
Bulk Casimir effect may play a role in radion stabilization of BW
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y p y
Bulk Casimir effect (eff. pot.) for a conformal or massive scalar field
Bulk: 5-dim AdS or dS space with 2/1 4-dim dS brane (our universe)
5thICMMP, CBPF, Rio, April 2006 p. 24/
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B. Realistic models: dS & AdS BWElizalde, Nojiri, Odintsov, PRD70 (2004) 043539
Elizalde, Nojiri, Odintsov, Ogushi, PRD67 (2003) 063515
Bulk Casimir effect may play a role in radion stabilization of BW
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Bulk Casimir effect (eff. pot.) for a conformal or massive scalar field
Bulk: 5-dim AdS or dS space with 2/1 4-dim dS brane (our universe)
Consistent with observational data even for large extra dimension
Two dS4 branes in dS5 background; becomes a one-brane as L
5thICMMP, CBPF, Rio, April 2006 p. 24/
B. Realistic models: dS & AdS BWElizalde, Nojiri, Odintsov, PRD70 (2004) 043539
Elizalde, Nojiri, Odintsov, Ogushi, PRD67 (2003) 063515
Bulk Casimir effect may play a role in radion stabilization of BW
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Bulk Casimir effect (eff. pot.) for a conformal or massive scalar field
Bulk: 5-dim AdS or dS space with 2/1 4-dim dS brane (our universe)
Consistent with observational data even for large extra dimension
Two dS4 branes in dS5 background; becomes a one-brane as L
= Casimir energy density and effective potential for a de Sitter (dS)brane in a five-dimensional anti-de Sitter (AdS) background
5thICMMP, CBPF, Rio, April 2006 p. 24/
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= Euclidean metric of the 5-dim AdS bulkds2 = gdx
dx =2
sinh2 z
dz2 + d24
d24 = d
2 + sin2 d23
is the AdS radius, related to cc of AdS bulk
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is the AdS radius, related to cc of AdS bulk
d3 metric on the 3-sphere
5thICMMP, CBPF, Rio, April 2006 p. 25/
= Euclidean metric of the 5-dim AdS bulkds2 = gdx
dx =2
sinh2 z
dz2 + d24
d24 = d
2 + sin2 d23
is the AdS radius, related to cc of AdS bulk
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,
d3 metric on the 3-sphere
CASIMIR ENERGY
5thICMMP, CBPF, Rio, April 2006 p. 25/
= Euclidean metric of the 5-dim AdS bulkds2 = gdx
dx =2
sinh2 z
dz2 + d24
d24 = d
2 + sin2 d23
is the AdS radius, related to cc of AdS bulk
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,
d3 metric on the 3-sphere
CASIMIR ENERGY
(a) One-brane Casimir energy = 0
5thICMMP, CBPF, Rio, April 2006 p. 25/
= Euclidean metric of the 5-dim AdS bulkds2 = gdx
dx =2
sinh2 z
dz2 + d24
d24 = d
2 + sin2 d23
is the AdS radius, related to cc of AdS bulk
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d3 metric on the 3-sphere
CASIMIR ENERGY
(a) One-brane Casimir energy = 0
(b) Bulk Casimir energy (L brane separation, R brane radius)(s|L5) =
2s
6
n,l=1
(l + 1)(l + 2)(2l + 3)
nL
2+ R2
l2 + 3l +
9
4
s
5thICMMP, CBPF, Rio, April 2006 p. 25/
= Euclidean metric of the 5-dim AdS bulkds2 = gdx
dx =2
sinh2 z
dz2 + d24
d24 = d
2 + sin2 d23
is the AdS radius, related to cc of AdS bulk
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d3 metric on the 3-sphere
CASIMIR ENERGY
(a) One-brane Casimir energy = 0
(b) Bulk Casimir energy (L brane separation, R brane radius)(s|L5) =
2s
6
n,l=1
(l + 1)(l + 2)(2l + 3)
nL
2+ R2
l2 + 3l +
9
4
s
The Casimir energy density (pressure) follows (non-trivially!)
5thICMMP, CBPF, Rio, April 2006 p. 25/
= Euclidean metric of the 5-dim AdS bulkds2 = gdx
dx =2
sinh2 z
dz2 + d24
d24 = d
2 + sin2 d23
is the AdS radius, related to cc of AdS bulk
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d3 metric on the 3-sphere
CASIMIR ENERGY
(a) One-brane Casimir energy = 0
(b) Bulk Casimir energy (L brane separation, R brane radius)(s|L5) =
2s
6
n,l=1
(l + 1)(l + 2)(2l + 3)
nL
2+ R2
l2 + 3l +
9
4
s
The Casimir energy density (pressure) follows (non-trivially!)
ECas =c
2LR
4
1
2|L5 =
c3
36L6 2
315 1
240 L
R2
+ OL
R4
5thICMMP, CBPF, Rio, April 2006 p. 25/
C. Supergraviton Theories= Cognola, Elizalde, Zerbini, PLB624 (2005) 70
=Cognola, Elizalde, Nojiri, Odintsov, Zerbini, MPLA19 (2004) 1435
= Boulanger, Damour, Gualtieri, Henneaux, NPB597 (2001) 127S t G C l 9 (2003) 91
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= Sugamoto, Grav. Cosmol. 9 (2003) 91=
Arkani-Hamed, Cohen, Georgi, PRL86 (2001) 4757
= Arkani-Hamed, Georgi, Schwartz, Ann. Phys. (NY) 305 (2003) 96= Hill, Pokorski, Wang; Damour, Kogan, Papazoglou; Deffayet, Mourad
Effective potential for a multi-graviton model with
supersymmetry (discretized dims, deconstruction)
5thICMMP, CBPF, Rio, April 2006 p. 26/
C. Supergraviton Theories= Cognola, Elizalde, Zerbini, PLB624 (2005) 70=
Cognola, Elizalde, Nojiri, Odintsov, Zerbini, MPLA19 (2004) 1435
= Boulanger, Damour, Gualtieri, Henneaux, NPB597 (2001) 127 Sugamoto Grav Cosmol 9 (2003) 91
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= Sugamoto, Grav. Cosmol. 9 (2003) 91= Arkani-Hamed, Cohen, Georgi, PRL86 (2001) 4757= Arkani-Hamed, Georgi, Schwartz, Ann. Phys. (NY) 305 (2003) 96= Hill, Pokorski, Wang; Damour, Kogan, Papazoglou; Deffayet, Mourad
Effective potential for a multi-graviton model with
supersymmetry (discretized dims, deconstruction)
The bulk is a flat manifold with torus topology R T3= shown the induced cosmological constant could bepositive due to topological contributions
5thICMMP, CBPF, Rio, April 2006 p. 26/
C. Supergraviton Theories= Cognola, Elizalde, Zerbini, PLB624 (2005) 70=
Cognola, Elizalde, Nojiri, Odintsov, Zerbini, MPLA19 (2004) 1435
= Boulanger, Damour, Gualtieri, Henneaux, NPB597 (2001) 127 Sugamoto Grav Cosmol 9 (2003) 91
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= Sugamoto, Grav. Cosmol. 9 (2003) 91= Arkani-Hamed, Cohen, Georgi, PRL86 (2001) 4757= Arkani-Hamed, Georgi, Schwartz, Ann. Phys. (NY) 305 (2003) 96= Hill, Pokorski, Wang; Damour, Kogan, Papazoglou; Deffayet, Mourad
Effective potential for a multi-graviton model with
supersymmetry (discretized dims, deconstruction)
The bulk is a flat manifold with torus topology R T3= shown the induced cosmological constant could bepositive due to topological contributions
Previously considered in R4
5thICMMP, CBPF, Rio, April 2006 p. 26/
= Allow for non-nearest-neighbor couplings
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5thICMMP, CBPF, Rio, April 2006 p. 27/
= Allow for non-nearest-neighbor couplings= Multi-graviton model defined by taking Ncopies of fields, with
graviton hn and Stckelberg fields An, n
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5thICMMP, CBPF, Rio, April 2006 p. 27/
= Allow for non-nearest-neighbor couplings= Multi-graviton model defined by taking Ncopies of fields, with
graviton hn and Stckelberg fields An, n
= Lagrangian of theory generalization of Kan-ShiraishiN1
1 1
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L =n=01
2hn
hn + hnh
n hn hn +
1
2hn
hn
12
m2hnh
n (hn)2
2 mAn + n (hn hn)
12
(An An) (An An), difference operators
5thICMMP, CBPF, Rio, April 2006 p. 27/
= Allow for non-nearest-neighbor couplings= Multi-graviton model defined by taking Ncopies of fields, with
graviton hn and Stckelberg fields An, n
= Lagrangian of theory generalization of Kan-ShiraishiN1
1 1
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L =n=01
2hn
hn + hnh
n hn hn +
1
2hn
hn
12
m2hnh
n (hn)2
2 mAn + n (hn hn)
12
(An An) (An An), difference operators
operate on the indices n as
n
N1k=0 akn+k,
n
N1k=0 aknk, N1k=0 ak = 0,
ak are N constants and n can be identified with periodic fields on a lattice
with N sites if periodic boundary conditions, n+N = n, are imposed
5thICMMP, CBPF, Rio, April 2006 p. 27/
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In the multi-graviton model the induced cosmological constant can bepositive only if # of massive gravitons is sufficiently large
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5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can bepositive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
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5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can bepositive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
Topological effects discussed may also be relevant in the study of
l k b ki i d l i h i il f
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electroweak symmetry breaking in models with a similar type of
non-nearest-neighbour couplings, for the deconstruction issue
5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can bepositive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
Topological effects discussed may also be relevant in the study of
l t k t b ki i d l ith i il t f
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electroweak symmetry breaking in models with a similar type of
non-nearest-neighbour couplings, for the deconstruction issue
Case of the torus topology: top. contributions to the eff. potential have
always a fixed sign, depending on the BC one imposes
5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can bepositive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
Topological effects discussed may also be relevant in the study of
l t k t b ki i d l ith i il t f
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132/139
electroweak symmetry breaking in models with a similar type of
non-nearest-neighbour couplings, for the deconstruction issue
Case of the torus topology: top. contributions to the eff. potential have
always a fixed sign, depending on the BC one imposes
They are negative for periodic fields
5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can be
positive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
Topological effects discussed may also be relevant in the study of
electroweak symmetry breaking in models with a similar type of
-
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133/139
electroweak symmetry breaking in models with a similar type of
non-nearest-neighbour couplings, for the deconstruction issue
Case of the torus topology: top. contributions to the eff. potential have
always a fixed sign, depending on the BC one imposes
They are negative for periodic fieldsThey are positive for anti-periodic fields.
5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can be
positive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
Topological effects discussed may also be relevant in the study of
electroweak symmetry breaking in models with a similar type of
-
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134/139
electroweak symmetry breaking in models with a similar type of
non-nearest-neighbour couplings, for the deconstruction issue
Case of the torus topology: top. contributions to the eff. potential have
always a fixed sign, depending on the BC one imposes
They are negative for periodic fieldsThey are positive for anti-periodic fields.
Topology provides a natural mechanism which permits to have a
positive cc in the multi-supergravity model with anti-periodic fermions
5thICMMP, CBPF, Rio, April 2006 p. 28/
In the multi-graviton model the induced cosmological constant can be
positive only if # of massive gravitons is sufficiently large
In the supersymmetric case the cosmological constant can be positive
if one imposes anti-periodic BC in the fermionic sector
Topological effects discussed may also be relevant in the study of
electroweak symmetry breaking in models with a similar type of
-
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135/139
electroweak symmetry breaking in models with a similar type of
non-nearest-neighbour couplings, for the deconstruction issue
Case of the torus topology: top. contributions to the eff. potential have
always a fixed sign, depending on the BC one imposes
They are negative for periodic fieldsThey are positive for anti-periodic fields.
Topology provides a natural mechanism which permits to have a
positive cc in the multi-supergravity model with anti-periodic fermions
The value of the cc is regulated by the corresponding size of the torus
(one can most naturally use the minimum number, N = 3, of copies of
bosons and fermions), and are not far from observational values5thICMMP, CBPF, Rio, April 2006 p. 28/
Conclusions
At terrestrial scales, vacuum energy
fluctuations are becoming important
(MEMs NEMs nanotubes)
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(MEMs, NEMs, nanotubes)
5thICMMP, CBPF, Rio, April 2006 p. 29/
Conclusions
At terrestrial scales, vacuum energy
fluctuations are becoming important
(MEMs NEMs nanotubes)
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137/139
(MEMs, NEMs, nanotubes)
Topology + Vacuum energy may prove
to be a fundamental driving force at
cosmological scale
5thICMMP, CBPF, Rio, April 2006 p. 29/
Conclusions
At terrestrial scales, vacuum energy
fluctuations are becoming important
(MEMs NEMs nanotubes)
-
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138/139
(MEMs, NEMs, nanotubes)
Topology + Vacuum energy may prove
to be a fundamental driving force at
cosmological scale
Zeta functions provide a fine, precise, and
powerful tool to perform the calculations
5thICMMP, CBPF, Rio, April 2006 p. 29/
Conclusions
At terrestrial scales, vacuum energy
fluctuations are becoming important
(MEMs NEMs nanotubes)
-
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139/139
(MEMs, NEMs, nanotubes)
Topology + Vacuum energy may prove
to be a fundamental driving force at
cosmological scale
Zeta functions provide a fine, precise, and
powerful tool to perform the calculations
Thank You !5thICMMP, CBPF, Rio, April 2006 p. 29/