Emilio Elizalde- Cosmological Casimir effect and beyond

8/3/2019 Emilio Elizalde- Cosmological Casimir effect and beyond

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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

5thICMMP, CBPF, Rio, April 2006 p. 1/


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Outline of the talk

Zeta Function A(s) and Determinant detA



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Outline of the talk


The Wodzicki Residue



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Outline of the talk



Multiplicative (or Noncommutative) Anomaly



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Outline of the talk




Vacuum Fluctuations and the Cosmol Const



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Outline of the talk





New CC Theories & Cosmo-topol Casimir Effect



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Outline of the talk






A Simple Model with large & small dimensions



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Outline of the talk







Realistic Models: dS & AdS Braneworlds Supergraviton Theories



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Outline of the talk







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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R


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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R



s =

j sj , Re s >

nm s0

{j


(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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R



s =

j sj , Re s >

nm s0

{j





(c) The definition of A(s) depends on the position of the cut L


A


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R



s =

j sj , Re s >

nm s0

{j





(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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{i, i


iI i ?! ln

iI i =

iI ln i



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{i, i


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


D fi i i f D i


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{i, i


iI i ?! ln

iI i =

iI ln i


iI

si = tr H

s


(s)0


Derivative: H(0) =

iI ln i

=

Determinant: Ray & Singer, 67

det H = exp [H(0)]


D fi iti f D t i t


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{i, i


iI i ?! ln

iI i =

iI ln i


iI

si = tr H

s


(s)0


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


Properties


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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]


Properties


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Asymptotic expansion for the heat kernel


Properties


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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 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 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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Definition: res A = 2 Ress=0 tr (As), Laplacian




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Satisfies the trace condition: res (AB) = res (BA)




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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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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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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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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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, B

, and AB

exist,


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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, B

, and AB

exist,


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(, )]}


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(, )]}


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).


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).


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).


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 + )





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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





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det




= 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.





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det




= 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.


Zero point energy


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QFT vacuum to vacuum transition: 0|H|0


Zero point energy


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QFT vacuum to vacuum transition: 0|H|0Spectrum, normal ordering:

H =

n + 12

n an an


Zero point energy


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H =

n + 12

n an an

0|H|0 = (c)2

n

n =12

tr H


Zero point energy


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H =

n + 12

n an an

0|H|0 = (c)2

n

n =12

tr H

gives physical meaning?


Zero point energy


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H =

n + 12

n an an

0|H|0 = (c)2

n

n =12

tr H

gives physical meaning?Regularization + Renormalization ( cut-off, dim, )


Zero point energy


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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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The Casimir EffectBC i di


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vacuum

BC

F

CasimirEffect

BC e.g. periodic

= all kind of fields


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




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vacuum

BC

F

CasimirEffect

BC e.g. periodic


Universal process:




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vacuum

BC

F

CasimirEffect

BC e.g. periodic


Universal process:

Sonoluminiscence (Schwinger)

Cond. matter (wetting 3He alc.)

Optical cavities

Direct experim. confirmation




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vacuum

BC

F

CasimirEffect

BC e.g. periodic


Universal process:



Optical cavities


Van der Waals, Lipschitz theory




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vacuum

BC

F

CasimirEffect

BC e.g. periodic


Universal process:



Optical cavities


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




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The main issue:



Appears on the rhs of Einsteins equations:

R 12

gR = 8G(T Eg)

It affects cosmology: T excitations above the vacuum




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R 12

gR = 8G(T Eg)


Equivalent to a cosmological constant = 8GE




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R 12

gR = 8G(T Eg)


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




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R 12

gR = 8G(T Eg)


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


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]




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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




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120 orders of magnitude [obs

/(Pl)4

10120


Observational tests see nothing (or very little) of it:

= (new) cosmological constant problem.




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120 orders of magnitude [

obs

/(Pl)

4

10120



= (new) cosmological constant problem.Very difficult to solve and we do not address this question directly

[Baum, Hawking, Coleman, Polchinsky, Weinberg,...]




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120 orders of magnitude [

obs

/(Pl)

4

10120



= (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




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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:




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our universe in some reasonable models involving:(a) small and large compactified scales




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(b) dS & AdS worldbranes




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(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,




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Scalar field, , pervading the universe




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contribution to V from this field

=1

2

i

i

=1

2

k

1

k2 + M21/2




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=1

2

i

i

=1

2

k

1

k2 + M21/2

i and

k are generalized sums, mass-dim parameter,

= c = 1




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=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


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


Regularize: zeta function

(s) =1

2si =

1

2

k2 + M2

2

s/2

= = (1)


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i

k



(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)]



(s) =1

2si =

1

2

k2 + M2

2

s/2

= = (1)


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i

k


[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)]



(s) =1

2si =

1

2

k2 + M2

2

s/2

= = (1)


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i

k


[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


Prime means that point n = 0 is excluded from sum (not important)


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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



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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


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


Now, from K(z) 12()(z/2), z 0


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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



=

1

apbq+1

M K12a

b M

+

pph

2h


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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



= 1

apbq+1

M K12a

b M

+

pph

2h


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h

2

nh=1

Mh

j=1 n2j

K12ab

Mh

j=1

n2j+ O q1 + M

2K12a

b 1 + M2


= c

2ap+1bq

1 +

ph=0

ph

2

h

+ O qK12a

b

some finite constant (explicit geometrical sum in the limit M 0)



= 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


= 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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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


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 [ ]





101/139

g [ ]

* Approximate coincidence

( )





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g [ ]

* Approximate coincidence

( )

* b 10 to 1000 times the Planck length lP



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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





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( )


respectively



* Both situations correspond to a marginally closed universe





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( )


respectively



* 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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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






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y p y


Bulk: 5-dim AdS or dS space with 2/1 4-dim dS brane (our universe)



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Consistent with observational data even for large extra dimension

Two dS4 branes in dS5 background; becomes a one-brane as L






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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



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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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d3 metric on the 3-sphere



dx =2

sinh2 z

dz2 + d24

d24 = d

2 + sin2 d23



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,


CASIMIR ENERGY



dx =2

sinh2 z

dz2 + d24

d24 = d

2 + sin2 d23



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,


CASIMIR ENERGY

(a) One-brane Casimir energy = 0



dx =2

sinh2 z

dz2 + d24

d24 = d

2 + sin2 d23



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CASIMIR ENERGY


(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



dx =2

sinh2 z

dz2 + d24

d24 = d

2 + sin2 d23



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CASIMIR ENERGY



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!)



dx =2

sinh2 z

dz2 + d24

d24 = d

2 + sin2 d23



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CASIMIR ENERGY



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


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)


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


121/139

= 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



The bulk is a flat manifold with torus topology R T3= shown the induced cosmological constant could bepositive due to topological contributions


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


122/139

= 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



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


= Allow for non-nearest-neighbor couplings


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= 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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= 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




= 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



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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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In the supersymmetric case the cosmological constant can be positive

if one imposes anti-periodic BC in the fermionic sector


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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






l t k t b ki i d l ith i il t f


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Case of the torus topology: top. contributions to the eff. potential have

always a fixed sign, depending on the BC one imposes






l t k t b ki i d l ith i il t f


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They are negative for periodic fields


In the multi-graviton model the induced cosmological constant can be

positive only if # of massive gravitons is sufficiently large






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They are negative for periodic fieldsThey are positive for anti-periodic fields.









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Topology provides a natural mechanism which permits to have a

positive cc in the multi-supergravity model with anti-periodic fermions









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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)


Conclusions





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Topology + Vacuum energy may prove

to be a fundamental driving force at

cosmological scale


Conclusions





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cosmological scale

Zeta functions provide a fine, precise, and

powerful tool to perform the calculations


Conclusions





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cosmological scale

Zeta functions provide a fine, precise, and

powerful tool to perform the calculations

Thank You !5thICMMP, CBPF, Rio, April 2006 p. 29/

Documents

Emilio Elizalde- Cosmological Casimir effect and beyond