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![The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Eigenvalue Problem
Consider the piston configuration modeled by
Pφ = λ2 φ
where P is the Laplace-type differential operator defined on
[0, L] × N
∂2
P=− − ∆N + σδ(x − a)
∂x 2
Pedro Morales-Almaz´n
a Math Department
Semitransparent Pistons](https://image.slidesharecdn.com/presentation001-110413150356-phpapp01/85/Semitransparent-Pistons-24-320.jpg)
![The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Eigenvalue Problem
Consider the piston configuration modeled by
Pφ = λ2 φ
where P is the Laplace-type differential operator defined on
[0, L] × N
∂2
P=− − ∆N + σδ(x − a)
∂x 2
N is a compact Riemannian manifold and we have Dirichlet
boundary conditions φ(0) = φ(L) = 0.
Pedro Morales-Almaz´n
a Math Department
Semitransparent Pistons](https://image.slidesharecdn.com/presentation001-110413150356-phpapp01/85/Semitransparent-Pistons-25-320.jpg)
![The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Separation of variables
Using separation of variables
λ2 = νk + η 2
k
2
where νk and η 2 are the eigenvalues for the Laplacian on [0, L] and
2
N respectively
Pedro Morales-Almaz´n
a Math Department
Semitransparent Pistons](https://image.slidesharecdn.com/presentation001-110413150356-phpapp01/85/Semitransparent-Pistons-27-320.jpg)
![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) d
dν (ν 2 − η 2 )−s [log F (ıν) − asymptotics]
π η dν
=1
Pedro Morales-Almaz´n
a Math Department
Semitransparent Pistons](https://image.slidesharecdn.com/presentation001-110413150356-phpapp01/85/Semitransparent-Pistons-37-320.jpg)
![The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Asymptotic Part
∞ ∞
sin(πs) d
ζ (as) (s) = dν (ν 2 − η 2 )−s [asymptotics]
π η dν
=1
Pedro Morales-Almaz´n
a Math Department
Semitransparent Pistons](https://image.slidesharecdn.com/presentation001-110413150356-phpapp01/85/Semitransparent-Pistons-38-320.jpg)
![The Problem Zeta Function Semitransparent Pistons Questions
Eigenvalue Problem
Piston Behavior
Piston Behavior
Given the second order differential operator
∂2
P = − 2 − ∆N + σδ(x − a) defined on [0, L] × N with Dirichlet
∂x
boundary conditions, the piston is then attracted to the closest
wall.
Pedro Morales-Almaz´n
a Math Department
Semitransparent Pistons](https://image.slidesharecdn.com/presentation001-110413150356-phpapp01/85/Semitransparent-Pistons-48-320.jpg)
Uploaded byPedro Morales
Semitransparent Pistons
The document discusses the Casimir effect, a quantum field phenomenon arising from vacuum fluctuations, and introduces a mathematical model for calculating Casimir energy using a Riemannian manifold. It explains the definitions of the zeta function, the determination of eigenvalues, and various mathematical techniques used to analyze the Casimir force and its implications. Finally, it explores the behavior of semitransparent pistons in relation to the Casimir effect and provides insights into the stability of electrons and the significance of geometry in quantum contexts.
Semitransparent Pistons
- 1. The Problem Zeta Function Semitransparent Pistons Questions Semitransparent Pistons Pedro Morales-Almaz´n a Department of Mathematics Baylor University pedro morales@baylor.edu April, 15th 2011 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 2. The Problem Zeta Function Semitransparent Pistons Questions Outline 1 The Problem Casimir Effect Mathematical Model Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 3. The Problem Zeta Function Semitransparent Pistons Questions Outline 1 The Problem Casimir Effect Mathematical Model 2 Zeta Function Definition Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 4. The Problem Zeta Function Semitransparent Pistons Questions Outline 1 The Problem Casimir Effect Mathematical Model 2 Zeta Function Definition 3 Semitransparent Pistons Eigenvalue Problem Operator Determinant Casimir Force Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 5. The Problem Zeta Function Semitransparent Pistons Questions Outline 1 The Problem Casimir Effect Mathematical Model 2 Zeta Function Definition 3 Semitransparent Pistons Eigenvalue Problem Operator Determinant Casimir Force 4 Questions Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 6. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Casimir Effect Is a quantum field effect that arises when considering vacuum fluctuations Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 7. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect History • Predicted theoretically in 1948 by Hendrik B. G. Casimir and Dirk Polder when Casimir was trying to compute van der Waals forces between polarizable molecules. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 8. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect History • Predicted theoretically in 1948 by Hendrik B. G. Casimir and Dirk Polder when Casimir was trying to compute van der Waals forces between polarizable molecules. • Confirmed experimentally in 1997 by S. K. Lamoreaux. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 9. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Why is so important? • Believed to explain the stability of an electron Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 10. 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 and Comsmological implications) Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 11. 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 and Comsmological implications) • Provides a better understanding of the zero-point energy Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 12. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Mathematical Model In order to calculate the Casimir Energy of a system, consider a Riemannian manifold M possibly with boundary and the eigenvalue problem Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 13. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Mathematical Model In order to calculate the Casimir Energy of a system, consider a Riemannian manifold M possibly with boundary and the eigenvalue problem (∆ + V )φ = λφ Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 14. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Mathematical Model In order to calculate the Casimir Energy of a system, consider a Riemannian manifold M possibly with boundary and the eigenvalue problem (∆ + V )φ = λφ where ∆ is the Laplacian on M, V is a potential and φ ∈ L2 (M). If ∂M = ∅ boundary conditions must be imposed. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 15. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Definition of Casimir Energy The self energy of the system is defined to be 1 √ E= λ 2 λ Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 16. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Definition of Casimir Energy The self energy of the system is defined to be 1 √ E= λ 2 λ Since the self-adjointness of ∆, the eigenvalues λ are unbounded and hence, E is not well defined. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 17. The Problem Zeta Function Semitransparent Pistons Questions Casimir Effect Definition of Casimir Energy The self energy of the system is defined to be 1 √ E= λ 2 λ Since the self-adjointness of ∆, the eigenvalues λ are unbounded and hence, E is not well defined. Regularization methods to avoid infinities are required. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 18. The Problem Zeta Function Semitransparent Pistons Questions Definition Zeta Function Given a self-adjoint operator P with eigenvalues {λn }∞ , the zeta n=1 function is defined by ∞ ζP (s) = λ−s n n=1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 19. The Problem Zeta Function Semitransparent Pistons Questions Definition Zeta Function Given a self-adjoint operator P with eigenvalues {λn }∞ , the zeta n=1 function is defined by ∞ ζP (s) = λ−s n n=1 which is convergent for s large enough. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 20. The Problem Zeta Function Semitransparent Pistons Questions Definition • Values at s = −1/2, 0 provide information of the Casimir energy and the operator determinant Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 21. The Problem Zeta Function Semitransparent Pistons Questions Definition • Values at s = −1/2, 0 provide information of the Casimir energy and the operator determinant • An analytic continuation of the zeta function is required Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 22. The Problem Zeta Function Semitransparent Pistons Questions Definition • Values at s = −1/2, 0 provide information of the Casimir energy and the operator determinant • An analytic continuation of the zeta function is required • Lack of explicit eigenvalues requires an indirect method for calculations Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 23. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Eigenvalue Problem Consider the piston configuration modeled by Pφ = λ2 φ Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 24. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Eigenvalue Problem Consider the piston configuration modeled by Pφ = λ2 φ where P is the Laplace-type differential operator defined on [0, L] × N ∂2 P=− − ∆N + σδ(x − a) ∂x 2 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 25. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Eigenvalue Problem Consider the piston configuration modeled by Pφ = λ2 φ where P is the Laplace-type differential operator defined on [0, L] × N ∂2 P=− − ∆N + σδ(x − a) ∂x 2 N is a compact Riemannian manifold and we have Dirichlet boundary conditions φ(0) = φ(L) = 0. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 26. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Configuration Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 27. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Separation of variables Using separation of variables λ2 = νk + η 2 k 2 where νk and η 2 are the eigenvalues for the Laplacian on [0, L] and 2 N respectively Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 28. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Zeta Function ∞ ∞ ∞ ∞ ζ(s) = λ−2s = k (νk + η 2 )−s 2 k=1 =1 k=1 =1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 29. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Zeta Function ∞ ∞ ∞ ∞ ζ(s) = λ−2s = k (νk + η 2 )−s 2 k=1 =1 k=1 =1 2 Remark The eigenvalues νk cannot be calculated explicitly, an indirect way of finding the zeta function is required Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 30. 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 connected region Ω of the complex plane and let {ak }n be its poles on Ω. k=1 Let γ be a closed curve in Ω, then n 1 f (z)dz = I (ak , γ) Res(f (z))|z=ak 2πı γ k=1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 31. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Integral Representation ∞ 1 d ζ(s) = dν (ν 2 + η 2 )−s log F (ν) 2πı γ dν =1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 32. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Integral Representation ∞ 1 d ζ(s) = dν (ν 2 + η 2 )−s log F (ν) 2πı γ dν =1 where σ sin(ν(L − a) sin(νa)) sin(νL) F (ν) = + ν2 ν where γ is a contour enclosing {νk }∞ k=1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 33. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Contour Deformation After deforming the contours γ to the imaginary axis, the zeta function becomes Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 34. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Contour Deformation After deforming the contours γ to the imaginary axis, the zeta function becomes ∞ ∞ sin(πs) d ζ(s) = dν (ν 2 − η 2 )−s log F (ıν) π η dν =1 which converges for s big enough Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 35. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Analytic Continuation In order to extend analytically ζ(s) to the left in the complex plane, we subtract the asymptotic behavior of log F (ıν), ∞ (−1)n+1 σ n log F (ıν) ∼ Lν − 2ν + n 2ν n=1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 36. 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) Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 37. 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) d dν (ν 2 − η 2 )−s [log F (ıν) − asymptotics] π η dν =1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 38. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Asymptotic Part ∞ ∞ sin(πs) d ζ (as) (s) = dν (ν 2 − η 2 )−s [asymptotics] π η dν =1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 39. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem The operator determinant for the differential operator P is defined as Operator Determinant exp(ζ (0)) Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 40. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem The operator determinant for the differential operator P is defined as Operator Determinant exp(ζ (0)) which after some algebra, is computed to be... Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 41. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem ζ (0) = ∞ N n (−1)n+1 σ log F (ıη ) − Lη + log(2η ) + n 2η =1 n=1 1 1 1 −L FPζN (− ) − ResζN (− )(−2 + log 4) − ζN (0) 2 2 2 N (−1)n σ n n n n +2 FPζN + ResζN (γ + ψ ) n 2 2 2 2 n=1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 42. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Casimir Force The casimir force is defined to be Casimir Force 1 ∂ 1 − ζ − 2 ∂a 2 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 43. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Casimir Force The casimir force is defined to be Casimir Force 1 ∂ 1 − ζ − 2 ∂a 2 which after some small algebra, is computed to be... Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 44. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Casimir Force ∞ ∞ 1 ∂ ∂ dν (ν 2 − η 2 )1/2 log F (ıν) 2π η ∂a ∂ν =1 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 45. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Casimir Force ∞ ∞ 1 ∂ ∂ dν (ν 2 − η 2 )1/2 log F (ıν) 2π η ∂a ∂ν =1 which after a lot of algebra, is computed to be... Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 46. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Casimir Force ∞ ∞ 1 ∂ ∂ dν (ν 2 − η 2 )1/2 log F (ıν) 2π η ∂a ∂ν =1 which after a lot of algebra, is computed to be... negative for 0 < a < L/2 Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 47. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Casimir Force ∞ ∞ 1 ∂ ∂ dν (ν 2 − η 2 )1/2 log F (ıν) 2π η ∂a ∂ν =1 which after a lot of algebra, is computed to be... negative for 0 < a < L/2 and positive for L/2 < a < L. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 48. The Problem Zeta Function Semitransparent Pistons Questions Eigenvalue Problem Piston Behavior Piston Behavior Given the second order differential operator ∂2 P = − 2 − ∆N + σδ(x − a) defined on [0, L] × N with Dirichlet ∂x boundary conditions, the piston is then attracted to the closest wall. Pedro Morales-Almaz´n a Math Department Semitransparent Pistons
- 49. The Problem Zeta Function Semitransparent Pistons Questions Questions Thanks Pedro Morales-Almaz´n a Math Department Semitransparent Pistons