The Problem Zeta Function Semitransparent Pistons Questions
Semitransparent Pistons
Pedro Morales-Almazan
Department of MathematicsBaylor University
pedro [email protected]
April, 15th 2011
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Outline
1 The ProblemCasimir Effect
Mathematical Model
2 Zeta FunctionDefinition
3 Semitransparent PistonsEigenvalue Problem
Operator DeterminantCasimir Force
4 Questions
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Outline
1 The ProblemCasimir Effect
Mathematical Model
2 Zeta FunctionDefinition
3 Semitransparent PistonsEigenvalue Problem
Operator DeterminantCasimir Force
4 Questions
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Outline
1 The ProblemCasimir Effect
Mathematical Model
2 Zeta FunctionDefinition
3 Semitransparent PistonsEigenvalue Problem
Operator DeterminantCasimir Force
4 Questions
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Outline
1 The ProblemCasimir Effect
Mathematical Model
2 Zeta FunctionDefinition
3 Semitransparent PistonsEigenvalue Problem
Operator DeterminantCasimir Force
4 Questions
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
History
• Predicted theoretically in 1948 by Hendrik B. G. Casimir andDirk Polder when Casimir was trying to compute van derWaals forces between polarizable molecules.
• Confirmed experimentally in 1997 by S. K. Lamoreaux.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
History
• Predicted theoretically in 1948 by Hendrik B. G. Casimir andDirk Polder when Casimir was trying to compute van derWaals forces between polarizable molecules.
• Confirmed experimentally in 1997 by S. K. Lamoreaux.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Why is so important?
• Believed to explain the stability of an electron
• Very sensitive to the geometry of the space (Quantum andComsmological implications)
• Provides a better understanding of the zero-point energy
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Why is so important?
• Believed to explain the stability of an electron
• Very sensitive to the geometry of the space (Quantum andComsmological implications)
• Provides a better understanding of the zero-point energy
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Why is so important?
• Believed to explain the stability of an electron
• Very sensitive to the geometry of the space (Quantum andComsmological implications)
• Provides a better understanding of the zero-point energy
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Mathematical Model
In order to calculate the Casimir Energy of a system, consider aRiemannian manifold M possibly with boundary and theeigenvalue problem
(∆ + V )φ = λφ
where ∆ is the Laplacian on M, V is a potential and φ ∈ L2(M).If ∂M 6= ∅ boundary conditions must be imposed.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Mathematical Model
In order to calculate the Casimir Energy of a system, consider aRiemannian manifold M possibly with boundary and theeigenvalue problem
(∆ + V )φ = λφ
where ∆ is the Laplacian on M, V is a potential and φ ∈ L2(M).If ∂M 6= ∅ boundary conditions must be imposed.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Mathematical Model
In order to calculate the Casimir Energy of a system, consider aRiemannian manifold M possibly with boundary and theeigenvalue problem
(∆ + V )φ = λφ
where ∆ is the Laplacian on M, V is a potential and φ ∈ L2(M).If ∂M 6= ∅ boundary conditions must be imposed.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Definition of Casimir Energy
The self energy of the system is defined to be
E =1
2
∑λ
√λ
Since the self-adjointness of ∆, the eigenvalues λ are unboundedand hence, E is not well defined. Regularization methods to avoidinfinities are required.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Definition of Casimir Energy
The self energy of the system is defined to be
E =1
2
∑λ
√λ
Since the self-adjointness of ∆, the eigenvalues λ are unboundedand hence, E is not well defined.
Regularization methods to avoidinfinities are required.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Casimir Effect
Definition of Casimir Energy
The self energy of the system is defined to be
E =1
2
∑λ
√λ
Since the self-adjointness of ∆, the eigenvalues λ are unboundedand hence, E is not well defined. Regularization methods to avoidinfinities are required.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Definition
Zeta Function
Given a self-adjoint operator P with eigenvalues λn∞n=1, the zetafunction is defined by
ζP(s) =∞∑n=1
λ−sn
which is convergent for <s large enough.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Definition
Zeta Function
Given a self-adjoint operator P with eigenvalues λn∞n=1, the zetafunction is defined by
ζP(s) =∞∑n=1
λ−sn
which is convergent for <s large enough.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Definition
• Values at s = −1/2, 0 provide information of the Casimirenergy and the operator determinant
• An analytic continuation of the zeta function is required
• Lack of explicit eigenvalues requires an indirect method forcalculations
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Definition
• Values at s = −1/2, 0 provide information of the Casimirenergy and the operator determinant
• An analytic continuation of the zeta function is required
• Lack of explicit eigenvalues requires an indirect method forcalculations
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Definition
• Values at s = −1/2, 0 provide information of the Casimirenergy and the operator determinant
• An analytic continuation of the zeta function is required
• Lack of explicit eigenvalues requires an indirect method forcalculations
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Eigenvalue Problem
Consider the piston configuration modeled by
Pφ = λ2φ
where P is the Laplace-type differential operator defined on[0, L]×N
P = − ∂2
∂x2−∆N + σδ(x − a)
N is a compact Riemannian manifold and we have Dirichletboundary conditions φ(0) = φ(L) = 0.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Eigenvalue Problem
Consider the piston configuration modeled by
Pφ = λ2φ
where P is the Laplace-type differential operator defined on[0, L]×N
P = − ∂2
∂x2−∆N + σδ(x − a)
N is a compact Riemannian manifold and we have Dirichletboundary conditions φ(0) = φ(L) = 0.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Eigenvalue Problem
Consider the piston configuration modeled by
Pφ = λ2φ
where P is the Laplace-type differential operator defined on[0, L]×N
P = − ∂2
∂x2−∆N + σδ(x − a)
N is a compact Riemannian manifold and we have Dirichletboundary conditions φ(0) = φ(L) = 0.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Configuration
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Separation of variables
Using separation of variables
λ2k` = ν2
k + η2`
where ν2k and η2
` are the eigenvalues for the Laplacian on [0, L] andN respectively
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Zeta Function
ζ(s) =∞∑k=1
∞∑`=1
λ−2sk` =
∞∑k=1
∞∑`=1
(ν2k + η2
` )−s
Remark The eigenvalues ν2k cannot be calculated explicitly, an
indirect way of finding the zeta function is required
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Zeta Function
ζ(s) =∞∑k=1
∞∑`=1
λ−2sk` =
∞∑k=1
∞∑`=1
(ν2k + η2
` )−s
Remark The eigenvalues ν2k cannot be calculated explicitly, an
indirect way of finding the zeta function is required
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Contour Integration
Cauchy’s residue Theorem
Let f be a meromorphic function defined on a simply connectedregion Ω of the complex plane and let aknk=1 be its poles on Ω.Let γ be a closed curve in Ω, then
1
2πı
∫γf (z)dz =
n∑k=1
I (ak , γ) Res(f (z))|z=ak
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Integral Representation
ζ(s) =1
2πı
∞∑`=1
∫γ`
dν (ν2 + η2` )−s
d
dνlog F (ν)
where
F (ν) =σ sin(ν(L− a) sin(νa))
ν2+
sin(νL)
ν
where γ` is a contour enclosing νk∞k=1
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Integral Representation
ζ(s) =1
2πı
∞∑`=1
∫γ`
dν (ν2 + η2` )−s
d
dνlog F (ν)
where
F (ν) =σ sin(ν(L− a) sin(νa))
ν2+
sin(νL)
ν
where γ` is a contour enclosing νk∞k=1
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Contour Deformation
After deforming the contours γ` to the imaginary axis, the zetafunction becomes
ζ(s) =sin(πs)
π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )−s
d
dνlog F (ıν)
which converges for <s big enough
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Contour Deformation
After deforming the contours γ` to the imaginary axis, the zetafunction becomes
ζ(s) =sin(πs)
π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )−s
d
dνlog F (ıν)
which converges for <s big enough
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Analytic Continuation
In order to extend analytically ζ(s) to the left in the complexplane, we subtract the asymptotic behavior of log F (ıν),
log F (ıν) ∼ Lν − 2ν +∞∑n=1
(−1)n+1
n
( σ2ν
)n
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Finite Part
Subtracting the asymptotic terms enlarges the convergence region
ζ(s) = ζ(f )(s) + ζ(as)(s)
where
ζ(f )(s) =
sin(πs)
π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )−s
d
dν[log F (ıν)− asymptotics]
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Finite Part
Subtracting the asymptotic terms enlarges the convergence region
ζ(s) = ζ(f )(s) + ζ(as)(s)
where
ζ(f )(s) =
sin(πs)
π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )−s
d
dν[log F (ıν)− asymptotics]
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Asymptotic Part
ζ(as)(s) =sin(πs)
π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )−s
d
dν[asymptotics]
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
The operator determinant for the differential operator P is definedas
Operator Determinant
exp(ζ ′(0))
which after some algebra, is computed to be...
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
The operator determinant for the differential operator P is definedas
Operator Determinant
exp(ζ ′(0))
which after some algebra, is computed to be...
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
ζ ′(0) =∞∑`=1
(log F (ıη`)− Lη` + log(2η`) +
N∑n=1
(−1)n+1
n
(σ
2η`
)n)
−L(
FPζN (−1
2)− ResζN (−1
2)(−2 + log 4)
)− 1
2ζ ′N (0)
+2N∑
n=1
(−1)n
n
(σ2
)n [FPζN
(n2
)+ ResζN
(n2
(γ + ψ(n
2
)))]
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Casimir Force
The casimir force is defined to be
Casimir Force
−1
2
∂
∂aζ
(−1
2
)
which after some small algebra, is computed to be...
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Casimir Force
The casimir force is defined to be
Casimir Force
−1
2
∂
∂aζ
(−1
2
)which after some small algebra, is computed to be...
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Casimir Force
1
2π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )1/2 ∂
∂a
∂
∂νlog F (ıν)
which after a lot of algebra, is computed to be... negative for0 < a < L/2 and positive for L/2 < a < L.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Casimir Force
1
2π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )1/2 ∂
∂a
∂
∂νlog F (ıν)
which after a lot of algebra, is computed to be...
negative for0 < a < L/2 and positive for L/2 < a < L.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Casimir Force
1
2π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )1/2 ∂
∂a
∂
∂νlog F (ıν)
which after a lot of algebra, is computed to be... negative for0 < a < L/2
and positive for L/2 < a < L.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Casimir Force
1
2π
∞∑`=1
∫ ∞η`
dν (ν2 − η2` )1/2 ∂
∂a
∂
∂νlog F (ıν)
which after a lot of algebra, is computed to be... negative for0 < a < L/2 and positive for L/2 < a < L.
Pedro Morales-Almazan Math Department
Semitransparent Pistons
The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Piston Behavior
Piston Behavior
Given the second order differential operator
P = − ∂2
∂x2−∆N + σδ(x − a) defined on [0, L]×N with Dirichlet
boundary conditions, the piston is then attracted to the closestwall.
Pedro Morales-Almazan Math Department
Semitransparent Pistons