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The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematics of the Casimir Effect Thomas Prellberg School of Mathematical Sciences Queen Mary, University of London Annual Lectures February 19, 2007 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Casimir forces: still surprising after 60 years Physics Today, February 2007 The Casimir effect heats up AIP News Update, February 7, 2007 Scientists devise test for string theory EE Times, February 6, 2007 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Casimir forces: still surprising after 60 years Physics Today, February 2007 The Casimir effect heats up AIP News Update, February 7, 2007 Scientists devise test for string theory EE Times, February 6, 2007 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Casimir forces: still surprising after 60 years Physics Today, February 2007 The Casimir effect heats up AIP News Update, February 7, 2007 Scientists devise test for string theory EE Times, February 6, 2007 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Topic Outline 1 The Casimir Effect History Quantum Electrodynamics Zero-Point Energy Shift 2 Making Sense of Infinity - Infinity The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula 3 Conclusion Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Outline 1 The Casimir Effect History Quantum Electrodynamics Zero-Point Energy Shift 2 Making Sense of Infinity - Infinity 3 Conclusion Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift student of Ehrenfest, worked with Pauli and Bohr retarded Van-der-Waals-forces E= 23 α ~c 4π R 7 Casimir and Polder, 1948 Force between cavity walls F =− π 2 ~c A 240 d 4 Casimir, 1948 Thomas Prellberg Hendrik Brugt Gerhard Casimir, 1909 - 2000 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift student of Ehrenfest, worked with Pauli and Bohr retarded Van-der-Waals-forces E= 23 α ~c 4π R 7 Casimir and Polder, 1948 Force between cavity walls F =− π 2 ~c A 240 d 4 Casimir, 1948 Thomas Prellberg Hendrik Brugt Gerhard Casimir, 1909 - 2000 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift student of Ehrenfest, worked with Pauli and Bohr retarded Van-der-Waals-forces E= 23 α ~c 4π R 7 Casimir and Polder, 1948 Force between cavity walls F =− π 2 ~c A 240 d 4 Casimir, 1948 Thomas Prellberg Hendrik Brugt Gerhard Casimir, 1909 - 2000 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift The Electromagnetic Field The electromagnetic field, described by the Maxwell Equations, satisfies the wave equation 1 ∂2 ∆− 2 2 ~ A(~x , t) = 0 c ∂t Fourier-transformation (~x ↔ ~k) gives 2 ∂ 2 ~ ~ + ω A(k, t) = 0 with ω = c|~k| ∂t 2 which, for each ~k, describes a harmonic oscillator Quantising harmonic oscillators is easy... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift The Electromagnetic Field The electromagnetic field, described by the Maxwell Equations, satisfies the wave equation 1 ∂2 ∆− 2 2 ~ A(~x , t) = 0 c ∂t Fourier-transformation (~x ↔ ~k) gives 2 ∂ 2 ~ ~ + ω A(k, t) = 0 with ω = c|~k| ∂t 2 which, for each ~k, describes a harmonic oscillator Quantising harmonic oscillators is easy... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift The Electromagnetic Field The electromagnetic field, described by the Maxwell Equations, satisfies the wave equation 1 ∂2 ∆− 2 2 ~ A(~x , t) = 0 c ∂t Fourier-transformation (~x ↔ ~k) gives 2 ∂ 2 ~ ~ + ω A(k, t) = 0 with ω = c|~k| ∂t 2 which, for each ~k, describes a harmonic oscillator Quantising harmonic oscillators is easy... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 ~ Em (k) = m + ~ω with ω = c|~k| 2 Interpretation: m photons with energy ~ω and momentum ~~k In particular, the ground state energy 12 ~ω is non-zero! This leads to a zero-point energy density of the field Z E d 3k = 2 E0 (~k) V (2π)3 (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 ~ Em (k) = m + ~ω with ω = c|~k| 2 Interpretation: m photons with energy ~ω and momentum ~~k In particular, the ground state energy 12 ~ω is non-zero! This leads to a zero-point energy density of the field Z E d 3k = 2 E0 (~k) V (2π)3 (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 ~ Em (k) = m + ~ω with ω = c|~k| 2 Interpretation: m photons with energy ~ω and momentum ~~k In particular, the ground state energy 12 ~ω is non-zero! This leads to a zero-point energy density of the field Z E d 3k = 2 E0 (~k) V (2π)3 (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 ~ Em (k) = m + ~ω with ω = c|~k| 2 Interpretation: m photons with energy ~ω and momentum ~~k In particular, the ground state energy 12 ~ω is non-zero! This leads to a zero-point energy density of the field Z E d 3k = 2 E0 (~k) V (2π)3 (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 ~ Em (k) = m + ~ω with ω = c|~k| 2 Interpretation: m photons with energy ~ω and momentum ~~k In particular, the ground state energy 12 ~ω is non-zero! This leads to a zero-point energy density of the field Z E d 3k = 2 E0 (~k) V (2π)3 (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Making (Physical) Sense of Infinity The zero-point energy shifts due to a restricted geometry In the presence of the boundary X Ediscrete = E0,n n is a sum over discrete energies E0,n = 21 ~ωn In the absence of a boundary Z d 3k E = 2V E0 (~k) (2π)3 The difference of the infinite zero-point energies is finite! ∆E = Ediscrete − E = − π 2 ~c L2 720 d 3 for a box of size L × L × d with d L Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Making (Physical) Sense of Infinity The zero-point energy shifts due to a restricted geometry In the presence of the boundary X Ediscrete = E0,n n is a sum over discrete energies E0,n = 21 ~ωn In the absence of a boundary Z d 3k E = 2V E0 (~k) (2π)3 The difference of the infinite zero-point energies is finite! ∆E = Ediscrete − E = − π 2 ~c L2 720 d 3 for a box of size L × L × d with d L Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Making (Physical) Sense of Infinity The zero-point energy shifts due to a restricted geometry In the presence of the boundary X Ediscrete = E0,n n is a sum over discrete energies E0,n = 21 ~ωn In the absence of a boundary Z d 3k E = 2V E0 (~k) (2π)3 The difference of the infinite zero-point energies is finite! ∆E = Ediscrete − E = − π 2 ~c L2 720 d 3 for a box of size L × L × d with d L Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Making (Physical) Sense of Infinity The zero-point energy shifts due to a restricted geometry In the presence of the boundary X Ediscrete = E0,n n is a sum over discrete energies E0,n = 21 ~ωn In the absence of a boundary Z d 3k E = 2V E0 (~k) (2π)3 The difference of the infinite zero-point energies is finite! ∆E = Ediscrete − E = − π 2 ~c L2 720 d 3 for a box of size L × L × d with d L Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Wolfgang Pauli’s initial reaction: ‘absolute nonsense’ Experimental verification Sparnaay (1958): ‘not inconsistent with’ van Blokland and Overbeek (1978): experimental accuracy of 50% Lamoreaux (1997): experimental accuracy of 5% Theoretical extensions Geometry dependence Dynamical Casimir effect Real media: non-zero temperature, finite conductivity, roughness, . . . We are done with the physics. Let’s look at some mathematics! Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Wolfgang Pauli’s initial reaction: ‘absolute nonsense’ Experimental verification Sparnaay (1958): ‘not inconsistent with’ van Blokland and Overbeek (1978): experimental accuracy of 50% Lamoreaux (1997): experimental accuracy of 5% Theoretical extensions Geometry dependence Dynamical Casimir effect Real media: non-zero temperature, finite conductivity, roughness, . . . We are done with the physics. Let’s look at some mathematics! Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Wolfgang Pauli’s initial reaction: ‘absolute nonsense’ Experimental verification Sparnaay (1958): ‘not inconsistent with’ van Blokland and Overbeek (1978): experimental accuracy of 50% Lamoreaux (1997): experimental accuracy of 5% Theoretical extensions Geometry dependence Dynamical Casimir effect Real media: non-zero temperature, finite conductivity, roughness, . . . We are done with the physics. Let’s look at some mathematics! Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion History Quantum Electrodynamics Zero-Point Energy Shift Wolfgang Pauli’s initial reaction: ‘absolute nonsense’ Experimental verification Sparnaay (1958): ‘not inconsistent with’ van Blokland and Overbeek (1978): experimental accuracy of 50% Lamoreaux (1997): experimental accuracy of 5% Theoretical extensions Geometry dependence Dynamical Casimir effect Real media: non-zero temperature, finite conductivity, roughness, . . . We are done with the physics. Let’s look at some mathematics! Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Outline 1 The Casimir Effect 2 Making Sense of Infinity - Infinity The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula 3 Conclusion Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Spectral Theory Consider −∆ for a compact manifold Ω with a smooth boundary ∂Ω On a suitable function space, this operator is self-adjoint and positive with pure point spectrum One finds formally 1 Ediscrete = ~c Trace(−∆)1/2 2 This would be a different talk — let’s keep it simple for today Choose ∂2 ∂x 2 with Dirichlet boundary conditions f (0) = f (L) = 0. Ω = [0, L] and ∆ = Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Spectral Theory Consider −∆ for a compact manifold Ω with a smooth boundary ∂Ω On a suitable function space, this operator is self-adjoint and positive with pure point spectrum One finds formally 1 Ediscrete = ~c Trace(−∆)1/2 2 This would be a different talk — let’s keep it simple for today Choose ∂2 ∂x 2 with Dirichlet boundary conditions f (0) = f (L) = 0. Ω = [0, L] and ∆ = Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Spectral Theory Consider −∆ for a compact manifold Ω with a smooth boundary ∂Ω On a suitable function space, this operator is self-adjoint and positive with pure point spectrum One finds formally 1 Ediscrete = ~c Trace(−∆)1/2 2 This would be a different talk — let’s keep it simple for today Choose ∂2 ∂x 2 with Dirichlet boundary conditions f (0) = f (L) = 0. Ω = [0, L] and ∆ = Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Spectral Theory Consider −∆ for a compact manifold Ω with a smooth boundary ∂Ω On a suitable function space, this operator is self-adjoint and positive with pure point spectrum One finds formally 1 Ediscrete = ~c Trace(−∆)1/2 2 This would be a different talk — let’s keep it simple for today Choose ∂2 ∂x 2 with Dirichlet boundary conditions f (0) = f (L) = 0. Ω = [0, L] and ∆ = Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Spectral Theory Consider −∆ for a compact manifold Ω with a smooth boundary ∂Ω On a suitable function space, this operator is self-adjoint and positive with pure point spectrum One finds formally 1 Ediscrete = ~c Trace(−∆)1/2 2 This would be a different talk — let’s keep it simple for today Choose ∂2 ∂x 2 with Dirichlet boundary conditions f (0) = f (L) = 0. Ω = [0, L] and ∆ = Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Spectral Theory Consider −∆ for a compact manifold Ω with a smooth boundary ∂Ω On a suitable function space, this operator is self-adjoint and positive with pure point spectrum One finds formally 1 Ediscrete = ~c Trace(−∆)1/2 2 This would be a different talk — let’s keep it simple for today Choose ∂2 ∂x 2 with Dirichlet boundary conditions f (0) = f (L) = 0. Ω = [0, L] and ∆ = Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Casimir Effect in One Dimension The solutions are standing waves with wavelength λ satisfying λ =L 2 1 nπ = ~c 2 L n We therefore find E0,n The zero-point energies are given by Ediscrete = ∞ X π n ~c 2L n=0 Thomas Prellberg and E= π ~c 2L Z ∞ t dt 0 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Casimir Effect in One Dimension The solutions are standing waves with wavelength λ satisfying λ =L 2 1 nπ = ~c 2 L n We therefore find E0,n The zero-point energies are given by Ediscrete = ∞ X π n ~c 2L n=0 Thomas Prellberg and E= π ~c 2L Z ∞ t dt 0 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Casimir Effect in One Dimension The solutions are standing waves with wavelength λ satisfying λ =L 2 1 nπ = ~c 2 L n We therefore find E0,n The zero-point energies are given by Ediscrete = ∞ X π n ~c 2L n=0 Thomas Prellberg and E= π ~c 2L Z ∞ t dt 0 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Mathematical Problem We need to make sense of ∆E = Ediscrete ∞ X π −E = ~c 2L n− n=0 Z ∞ t dt 0 More generally, consider ∆(f ) = ∞ X f (n) − n=0 Thomas Prellberg Z ∞ f (t) dt 0 The Mathematics of the Casimir Effect ! The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Mathematical Problem We need to make sense of ∆E = Ediscrete ∞ X π −E = ~c 2L n− n=0 Z ∞ t dt 0 More generally, consider ∆(f ) = ∞ X f (n) − n=0 Thomas Prellberg Z ∞ f (t) dt 0 The Mathematics of the Casimir Effect ! The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Divergent Series On the Whole, Divergent Series are the Works of the Devil and it’s a Shame that one dares base any Demonstration upon them. You can get whatever result you want when you use them, and they have given rise to so many Disasters and so many Paradoxes. Can anything more horrible be conceived than to have the following oozing out at you: 0 = 1 − 2n + 3n − 4n + etc. where n is an integer number? Niels Henrik Abel Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Summing Divergent Series Some divergent series can be summed in a sensible way . . . S= ∞ X (−1)n = 1 − 1 + 1 − 1 + 1 − 1 + . . . n=0 Cesaro summation: let SN = PN S = lim N→∞ n=0 (−1) n and compute N 1 X 1 SN = N + 1 n=0 2 Abel summation: S = lim x →1− ∞ X n=0 (−1)n x n = 1 2 Borel summation, Euler summation, . . . : again S = . . . but some (such as ∞ X 1 2 n) cannot n=0 Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Summing Divergent Series Some divergent series can be summed in a sensible way . . . S= ∞ X (−1)n = 1 − 1 + 1 − 1 + 1 − 1 + . . . n=0 Cesaro summation: let SN = PN S = lim N→∞ n=0 (−1) n and compute N 1 X 1 SN = N + 1 n=0 2 Abel summation: S = lim x →1− ∞ X n=0 (−1)n x n = 1 2 Borel summation, Euler summation, . . . : again S = . . . but some (such as ∞ X 1 2 n) cannot n=0 Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Summing Divergent Series Some divergent series can be summed in a sensible way . . . S= ∞ X (−1)n = 1 − 1 + 1 − 1 + 1 − 1 + . . . n=0 Cesaro summation: let SN = PN S = lim N→∞ n=0 (−1) n and compute N 1 X 1 SN = N + 1 n=0 2 Abel summation: S = lim x →1− ∞ X n=0 (−1)n x n = 1 2 Borel summation, Euler summation, . . . : again S = . . . but some (such as ∞ X 1 2 n) cannot n=0 Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Summing Divergent Series Some divergent series can be summed in a sensible way . . . S= ∞ X (−1)n = 1 − 1 + 1 − 1 + 1 − 1 + . . . n=0 Cesaro summation: let SN = PN S = lim N→∞ n=0 (−1) n and compute N 1 X 1 SN = N + 1 n=0 2 Abel summation: S = lim x →1− ∞ X n=0 (−1)n x n = 1 2 Borel summation, Euler summation, . . . : again S = . . . but some (such as ∞ X 1 2 n) cannot n=0 Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Summing Divergent Series Some divergent series can be summed in a sensible way . . . S= ∞ X (−1)n = 1 − 1 + 1 − 1 + 1 − 1 + . . . n=0 Cesaro summation: let SN = PN S = lim N→∞ n=0 (−1) n and compute N 1 X 1 SN = N + 1 n=0 2 Abel summation: S = lim x →1− ∞ X n=0 (−1)n x n = 1 2 Borel summation, Euler summation, . . . : again S = . . . but some (such as ∞ X 1 2 n) cannot n=0 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series If a divergent series cannot be summed, physicists like to remove infinity Regularisation of ∞ X f (n) (in particular, f (n) = n) n=0 Heat kernel regularisation ˜ f (s) = ∞ X f (n) e −sn n=0 ∞ X es 1 1 ne = s in particular, = 2 − + O(s 2 ) 2 (e − 1) s 12 n=0 Z ∞ 1 Compare with t e −st dt = 2 : divergent terms cancel s 0 ∞ X f (n) n−s Zeta function regularisation f˜(s) = −sn n=0 in particular, ∞ X nn −s = ζ(s − 1) , n=0 Thomas Prellberg ζ(−1) = − 1 12 The Mathematics of the Casimir Effect Digression The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series Regularisation result should be independent of the method used In particular, for a reasonable class of cutoff functions g (t; s) with lim g (t; s) = 0 and lim g (t; s) = 1 s→0+ t→∞ replacing f (t) by f (t)g (t; s) should give the same result for s → 0 + We need to study ! Z ∞ ∞ X lim+ ∆(fg ) = lim+ f (n)g (n; s) − f (t)g (t; s) dt s→0 s→0 0 n=0 Two mathematically sound approaches are Euler-Maclaurin Formula Abel-Plana Formula Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series Regularisation result should be independent of the method used In particular, for a reasonable class of cutoff functions g (t; s) with lim g (t; s) = 0 and lim g (t; s) = 1 s→0+ t→∞ replacing f (t) by f (t)g (t; s) should give the same result for s → 0 + We need to study ! Z ∞ ∞ X lim+ ∆(fg ) = lim+ f (n)g (n; s) − f (t)g (t; s) dt s→0 s→0 0 n=0 Two mathematically sound approaches are Euler-Maclaurin Formula Abel-Plana Formula Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series Regularisation result should be independent of the method used In particular, for a reasonable class of cutoff functions g (t; s) with lim g (t; s) = 0 and lim g (t; s) = 1 s→0+ t→∞ replacing f (t) by f (t)g (t; s) should give the same result for s → 0 + We need to study ! Z ∞ ∞ X lim+ ∆(fg ) = lim+ f (n)g (n; s) − f (t)g (t; s) dt s→0 s→0 0 n=0 Two mathematically sound approaches are Euler-Maclaurin Formula Abel-Plana Formula Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Regularising Divergent Series Regularisation result should be independent of the method used In particular, for a reasonable class of cutoff functions g (t; s) with lim g (t; s) = 0 and lim g (t; s) = 1 s→0+ t→∞ replacing f (t) by f (t)g (t; s) should give the same result for s → 0 + We need to study ! Z ∞ ∞ X lim+ ∆(fg ) = lim+ f (n)g (n; s) − f (t)g (t; s) dt s→0 s→0 0 n=0 Two mathematically sound approaches are Euler-Maclaurin Formula Abel-Plana Formula Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Euler-Maclaurin Formula Leonhard Euler, 1707 - 1783 Thomas Prellberg Colin Maclaurin, 1698 - 1746 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Euler-Maclaurin Formula A formal derivation (Hardy, Divergent Series, 1949) Denoting Df (x) = f 0 (x), the Taylor series can be written as f (x + n) = e nD f (x) It follows that N−1 X f (x + n) e ND − 1 1 f (x) = D (f (x + N) − f (x)) D e −1 e −1 ! ∞ X Bk k−1 −1 D (f (x + N) − f (x)) D + k! = n=0 = k=1 Skip precise statement ∞ X n=0 f (x + n) − Z ∞ f (x + t) dt = − 0 Thomas Prellberg ∞ X Bk k=1 k! D k−1 f (x) The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Euler-Maclaurin Formula A formal derivation (Hardy, Divergent Series, 1949) Denoting Df (x) = f 0 (x), the Taylor series can be written as f (x + n) = e nD f (x) It follows that N−1 X f (x + n) e ND − 1 1 f (x) = D (f (x + N) − f (x)) D e −1 e −1 ! ∞ X Bk k−1 −1 D (f (x + N) − f (x)) D + k! = n=0 = k=1 Skip precise statement ∞ X n=0 f (x + n) − Z ∞ f (x + t) dt = − 0 Thomas Prellberg ∞ X Bk k=1 k! D k−1 f (x) The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Euler-Maclaurin Formula A formal derivation (Hardy, Divergent Series, 1949) Denoting Df (x) = f 0 (x), the Taylor series can be written as f (x + n) = e nD f (x) It follows that N−1 X f (x + n) e ND − 1 1 f (x) = D (f (x + N) − f (x)) D e −1 e −1 ! ∞ X Bk k−1 −1 D (f (x + N) − f (x)) D + k! = n=0 = k=1 Skip precise statement ∞ X n=0 f (x + n) − Z ∞ f (x + t) dt = − 0 Thomas Prellberg ∞ X Bk k=1 k! D k−1 f (x) The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Euler-Maclaurin Formula A formal derivation (Hardy, Divergent Series, 1949) Denoting Df (x) = f 0 (x), the Taylor series can be written as f (x + n) = e nD f (x) It follows that N−1 X f (x + n) e ND − 1 1 f (x) = D (f (x + N) − f (x)) D e −1 e −1 ! ∞ X Bk k−1 −1 D (f (x + N) − f (x)) D + k! = n=0 = k=1 Skip precise statement ∞ X n=0 f (x + n) − Z ∞ f (x + t) dt = − 0 Thomas Prellberg ∞ X Bk k=1 k! D k−1 f (x) The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Euler-Maclaurin Formula A formal derivation (Hardy, Divergent Series, 1949) Denoting Df (x) = f 0 (x), the Taylor series can be written as f (x + n) = e nD f (x) It follows that N−1 X f (x + n) e ND − 1 1 f (x) = D (f (x + N) − f (x)) D e −1 e −1 ! ∞ X Bk k−1 −1 D (f (x + N) − f (x)) D + k! = n=0 = k=1 Skip precise statement ∞ X n=0 f (x + n) − Z ∞ f (x + t) dt = − 0 Thomas Prellberg ∞ X Bk k=1 k! D k−1 f (x) The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Theorem (Euler-Maclaurin Formula) If f ∈ C 2m [0, N] then N X f (n) − n=0 + m−1 X k=1 Z N f (t) dt = 0 B2k (2k−1) f (N) − f (2k−1) (0) + Rm (2k)! where Rm = 1 (f (0) + f (N)) + 2 Z N 0 B2m − B2m (t − btc) (2m) f (t) dt (2m)! Here, Bn (x) are Bernoulli polynomials and Bn = Bn (0) are Bernoulli numbers Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Euler-Maclaurin Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = − 0 ∞ X Bk k=1 k! f (k−1) (0) Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series B1 B2 0 For f (t) = t we find RHS = − f (0) − f (0) 1! 2! ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Euler-Maclaurin Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = − 0 ∞ X Bk k=1 k! f (k−1) (0) Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series B1 B2 0 For f (t) = t we find RHS = − f (0) − f (0) 1! 2! ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Euler-Maclaurin Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = − 0 ∞ X Bk k=1 k! f (k−1) (0) Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series B1 B2 0 For f (t) = t we find RHS = − f (0) − f (0) 1! 2! ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Euler-Maclaurin Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = − 0 ∞ X Bk k=1 k! f (k−1) (0) Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series B1 B2 0 For f (t) = t we find RHS = − f (0) − f (0) 1! 2! ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Abel-Plana Formula Niels Henrik Abel, 1802 - 1829 Thomas Prellberg Giovanni Antonio Amedeo Plana, 1781 - 1864 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Abel-Plana Formula . . . a remarkable summation formula of Plana . . . Germund Dahlquist, 1997 The only two places I have ever seen this formula are in Hardy’s book and in the writings of the “massive photon” people — who also got it from Hardy. Jonathan P Dowling, 1989 The only other applications I am aware of, albeit for convergent series, are q-Gamma function asymptotics (Adri B Olde Daalhuis, 1994) ∞ Y uniform asymptotics for (1 − q n+k )−1 (myself, 1995) k=0 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Abel-Plana Formula . . . a remarkable summation formula of Plana . . . Germund Dahlquist, 1997 The only two places I have ever seen this formula are in Hardy’s book and in the writings of the “massive photon” people — who also got it from Hardy. Jonathan P Dowling, 1989 The only other applications I am aware of, albeit for convergent series, are q-Gamma function asymptotics (Adri B Olde Daalhuis, 1994) ∞ Y uniform asymptotics for (1 − q n+k )−1 (myself, 1995) k=0 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Abel-Plana Formula . . . a remarkable summation formula of Plana . . . Germund Dahlquist, 1997 The only two places I have ever seen this formula are in Hardy’s book and in the writings of the “massive photon” people — who also got it from Hardy. Jonathan P Dowling, 1989 The only other applications I am aware of, albeit for convergent series, are q-Gamma function asymptotics (Adri B Olde Daalhuis, 1994) ∞ Y uniform asymptotics for (1 − q n+k )−1 (myself, 1995) k=0 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Abel-Plana Formula I 1 f (z) Use Cauchy’s integral formula f (ζ) = dz together 2πi Γζ z − ζ with ∞ X 1 π cot(πz) = z − n n=−∞ to get ∞ X n=0 f (n) = Z ∞ I 1 X f (z) 1 cot(πz)f (z) dz dz = 2πi n=0 Γn z − n 2i Γ Thomas Prellberg The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Abel-Plana Formula Rotate the upper and lower arm of Γ by ±π/2 to get ∞ X f (n) = n=0 i 1 f (0) + 2 2 Z ∞ (f (iy ) − f (−iy )) coth(πy ) dy 0 A similar trick gives Z ∞ Z i ∞ f (t) dt = (f (iy ) − f (−iy )) dy 2 0 0 Taking the difference gives the elegant result ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Thomas Prellberg Z Skip precise statement ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Abel-Plana Formula Rotate the upper and lower arm of Γ by ±π/2 to get ∞ X f (n) = n=0 i 1 f (0) + 2 2 Z ∞ (f (iy ) − f (−iy )) coth(πy ) dy 0 A similar trick gives Z ∞ Z i ∞ f (t) dt = (f (iy ) − f (−iy )) dy 2 0 0 Taking the difference gives the elegant result ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Thomas Prellberg Z Skip precise statement ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Abel-Plana Formula Rotate the upper and lower arm of Γ by ±π/2 to get ∞ X f (n) = n=0 i 1 f (0) + 2 2 Z ∞ (f (iy ) − f (−iy )) coth(πy ) dy 0 A similar trick gives Z ∞ Z i ∞ f (t) dt = (f (iy ) − f (−iy )) dy 2 0 0 Taking the difference gives the elegant result ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Thomas Prellberg Z Skip precise statement ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Theorem (Abel-Plana Formula) Let f : C → C satisfy the following conditions (a) f (z) is analytic for <(z) ≥ 0 (though not necessarily at infinity) (b) lim|y |→∞ |f (x + iy )|e −2π|y | = 0 uniformly in x in every finite interval R∞ (c) −∞ |f (x + iy ) − f (x − iy )|e −2π|y | dy exists for every x ≥ 0 and tends to zero for x → ∞ R∞ (d) 0 f (t) dt is convergent, and limn→∞ f (n) = 0 Then ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Thomas Prellberg Z ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Abel-Plana Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Z ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series Z ∞ y dy For f (t) = t one finds RHS = −2 e 2πy − 1 0 ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Abel-Plana Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Z ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series Z ∞ y dy For f (t) = t one finds RHS = −2 e 2πy − 1 0 ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Abel-Plana Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Z ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series Z ∞ y dy For f (t) = t one finds RHS = −2 e 2πy − 1 0 ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula The Casimir Effect Making Sense of Infinity - Infinity Conclusion Applying the Abel-Plana Formula ∞ X n=0 f (n) − Z ∞ f (t) dt = 0 1 f (0) + i 2 Z ∞ 0 f (iy ) − f (−iy ) dy e 2πy − 1 Introducing suitable cutoff functions g (t; s) one can justify applying this to divergent series Z ∞ y dy For f (t) = t one finds RHS = −2 e 2πy − 1 0 ∞ X n=0 n− Z ∞ t dt = − 0 Thomas Prellberg 1 12 The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Outline 1 The Casimir Effect 2 Making Sense of Infinity - Infinity 3 Conclusion Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Conclusion Mathematical question posed in theoretical physics Some really nice, old formulæ from classical analysis The result has been verified in the laboratory The motivation for this talk: Jonathan P Dowling “The Mathematics of the Casimir Effect” Math Mag 62 (1989) 324 Doing mathematics and physics together can be more stimulating than doing either one separately, not to mention it’s downright fun. Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Conclusion Mathematical question posed in theoretical physics Some really nice, old formulæ from classical analysis The result has been verified in the laboratory The motivation for this talk: Jonathan P Dowling “The Mathematics of the Casimir Effect” Math Mag 62 (1989) 324 Doing mathematics and physics together can be more stimulating than doing either one separately, not to mention it’s downright fun. Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Conclusion Mathematical question posed in theoretical physics Some really nice, old formulæ from classical analysis The result has been verified in the laboratory The motivation for this talk: Jonathan P Dowling “The Mathematics of the Casimir Effect” Math Mag 62 (1989) 324 Doing mathematics and physics together can be more stimulating than doing either one separately, not to mention it’s downright fun. Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Conclusion Mathematical question posed in theoretical physics Some really nice, old formulæ from classical analysis The result has been verified in the laboratory The motivation for this talk: Jonathan P Dowling “The Mathematics of the Casimir Effect” Math Mag 62 (1989) 324 Doing mathematics and physics together can be more stimulating than doing either one separately, not to mention it’s downright fun. Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion Conclusion Mathematical question posed in theoretical physics Some really nice, old formulæ from classical analysis The result has been verified in the laboratory The motivation for this talk: Jonathan P Dowling “The Mathematics of the Casimir Effect” Math Mag 62 (1989) 324 Doing mathematics and physics together can be more stimulating than doing either one separately, not to mention it’s downright fun. Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect Making Sense of Infinity - Infinity Conclusion I had a feeling once about Mathematics - that I saw it all. . . . I saw a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go. Sir Winston Spencer Churchill, 1874 - 1965 The End Thomas Prellberg The Mathematics of the Casimir Effect Zeta Function Regularisation Regularised Sum and Product Product of Primes Define for an increasing sequence 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . the zeta function ∞ X ζλ (s) = λ−s n n=1 If the zeta function has an analytic extension up to 0 then define the regularised infinite sum by ∞ X log λn = −ζλ0 (0) n=1 Alternatively, the regularised infinite product is given by ∞ Y 0 λn = e −ζλ (0) n=1 Thomas Prellberg The Mathematics of the Casimir Effect Zeta Function Regularisation Regularised Sum and Product Product of Primes Define for an increasing sequence 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . the zeta function ∞ X ζλ (s) = λ−s n n=1 If the zeta function has an analytic extension up to 0 then define the regularised infinite sum by ∞ X log λn = −ζλ0 (0) n=1 Alternatively, the regularised infinite product is given by ∞ Y 0 λn = e −ζλ (0) n=1 Thomas Prellberg The Mathematics of the Casimir Effect Zeta Function Regularisation Regularised Sum and Product Product of Primes Define for an increasing sequence 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . the zeta function ∞ X ζλ (s) = λ−s n n=1 If the zeta function has an analytic extension up to 0 then define the regularised infinite sum by ∞ X log λn = −ζλ0 (0) n=1 Alternatively, the regularised infinite product is given by ∞ Y 0 λn = e −ζλ (0) n=1 Thomas Prellberg The Mathematics of the Casimir Effect Regularised Sum and Product Product of Primes Zeta Function Regularisation Let λn = pn be the n-th prime so that Y 0 p = e −ζp (0) where ζp (s) = p Using e x = ∞ Y (1 − x n ) − µ(n) n X p −s p , one gets n=1 e ζp (s) = Y ep −s p Observing that ζ(s) = Y = ∞ YY 1 − p −ns p n=1 1 − p −s p e ζp (s) = ∞ Y −1 ζ(ns) − µ(n) n , one gets µ(n) n n=1 (Edmund Landau and Arnold Walfisz, 1920) Thomas Prellberg The Mathematics of the Casimir Effect Regularised Sum and Product Product of Primes Zeta Function Regularisation Let λn = pn be the n-th prime so that Y 0 p = e −ζp (0) where ζp (s) = p Using e x = ∞ Y (1 − x n ) − µ(n) n X p −s p , one gets n=1 e ζp (s) = Y ep −s p Observing that ζ(s) = Y = ∞ YY 1 − p −ns p n=1 1 − p −s p e ζp (s) = ∞ Y −1 ζ(ns) − µ(n) n , one gets µ(n) n n=1 (Edmund Landau and Arnold Walfisz, 1920) Thomas Prellberg The Mathematics of the Casimir Effect Regularised Sum and Product Product of Primes Zeta Function Regularisation Let λn = pn be the n-th prime so that Y 0 p = e −ζp (0) where ζp (s) = p Using e x = ∞ Y (1 − x n ) − µ(n) n X p −s p , one gets n=1 e ζp (s) = Y ep −s p Observing that ζ(s) = Y = ∞ YY 1 − p −ns p n=1 1 − p −s p e ζp (s) = ∞ Y −1 ζ(ns) − µ(n) n , one gets µ(n) n n=1 (Edmund Landau and Arnold Walfisz, 1920) Thomas Prellberg The Mathematics of the Casimir Effect Zeta Function Regularisation From e ζp (s) = ∞ Y ζ(ns) µ(n) n Regularised Sum and Product Product of Primes one gets the divergent expression n=1 ζp0 (s) = ∞ X µ(n) n=1 At s = 0, this simplifies to ζp0 (0) = ζ 0 (ns) ζ(ns) 1 ζ 0 (0) = −2 log(2π) ζ(0) ζ(0) This calculation “à la Euler” can be made rigorous, so that Y p = 4π 2 p (Elvira Muñoz Garcia and Ricardo Pérez-Marco, preprint 2003) Corollary: there are infinitely many primes Thomas Prellberg The Mathematics of the Casimir Effect Back to main talk Zeta Function Regularisation From e ζp (s) = ∞ Y ζ(ns) µ(n) n Regularised Sum and Product Product of Primes one gets the divergent expression n=1 ζp0 (s) = ∞ X µ(n) n=1 At s = 0, this simplifies to ζp0 (0) = ζ 0 (ns) ζ(ns) 1 ζ 0 (0) = −2 log(2π) ζ(0) ζ(0) This calculation “à la Euler” can be made rigorous, so that Y p = 4π 2 p (Elvira Muñoz Garcia and Ricardo Pérez-Marco, preprint 2003) Corollary: there are infinitely many primes Thomas Prellberg The Mathematics of the Casimir Effect Back to main talk Zeta Function Regularisation From e ζp (s) = ∞ Y ζ(ns) µ(n) n Regularised Sum and Product Product of Primes one gets the divergent expression n=1 ζp0 (s) = ∞ X µ(n) n=1 At s = 0, this simplifies to ζp0 (0) = ζ 0 (ns) ζ(ns) 1 ζ 0 (0) = −2 log(2π) ζ(0) ζ(0) This calculation “à la Euler” can be made rigorous, so that Y p = 4π 2 p (Elvira Muñoz Garcia and Ricardo Pérez-Marco, preprint 2003) Corollary: there are infinitely many primes Thomas Prellberg The Mathematics of the Casimir Effect Back to main talk Zeta Function Regularisation From e ζp (s) = ∞ Y ζ(ns) µ(n) n Regularised Sum and Product Product of Primes one gets the divergent expression n=1 ζp0 (s) = ∞ X µ(n) n=1 At s = 0, this simplifies to ζp0 (0) = ζ 0 (ns) ζ(ns) 1 ζ 0 (0) = −2 log(2π) ζ(0) ζ(0) This calculation “à la Euler” can be made rigorous, so that Y p = 4π 2 p (Elvira Muñoz Garcia and Ricardo Pérez-Marco, preprint 2003) Corollary: there are infinitely many primes Thomas Prellberg The Mathematics of the Casimir Effect Back to main talk

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