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 History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion 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 History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift student of Ehrenfest, worked with Pauli and Bohr retarded Van-der-Waals-forces 23 α E = ~c 4π R7 Casimir and Polder, 1948 Force between cavity walls π2~c A F = − 240 d 4 Casimir, 1948 Hendrik Brugt Gerhard Casimir, 1909 - 2000 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift student of Ehrenfest, worked with Pauli and Bohr retarded Van-der-Waals-forces 23 α E = ~c 4π R7 Casimir and Polder, 1948 Force between cavity walls π2~c A F = − 240 d 4 Casimir, 1948 Hendrik Brugt Gerhard Casimir, 1909 - 2000 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift student of Ehrenfest, worked with Pauli and Bohr retarded Van-der-Waals-forces 23 α E = ~c 4π R7 Casimir and Polder, 1948 Force between cavity walls π2~c A F = − 240 d 4 Casimir, 1948 Hendrik Brugt Gerhard Casimir, 1909 - 2000 Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift The Electromagnetic Field The electromagnetic field, described by the Maxwell Equations, satisfies the wave equation 1 @2 ∆ − A~ (~x; t) = 0 c2 @t2 Fourier-transformation (~x $ ~k) gives @2 + !2 A~ (~k; t) = 0 with ! = cj~kj @t2 which, for each ~k, describes a harmonic oscillator Quantising harmonic oscillators is easy... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift The Electromagnetic Field The electromagnetic field, described by the Maxwell Equations, satisfies the wave equation 1 @2 ∆ − A~ (~x; t) = 0 c2 @t2 Fourier-transformation (~x $ ~k) gives @2 + !2 A~ (~k; t) = 0 with ! = cj~kj @t2 which, for each ~k, describes a harmonic oscillator Quantising harmonic oscillators is easy... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift The Electromagnetic Field The electromagnetic field, described by the Maxwell Equations, satisfies the wave equation 1 @2 ∆ − A~ (~x; t) = 0 c2 @t2 Fourier-transformation (~x $ ~k) gives @2 + !2 A~ (~k; t) = 0 with ! = cj~kj @t2 which, for each ~k, describes a harmonic oscillator Quantising harmonic oscillators is easy... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 E (~k) = m + ~! with ! = cj~kj m 2 Interpretation: m photons with energy ~! and momentum ~~k 1 ~ In particular, the ground state energy 2 ! is non-zero! This leads to a zero-point energy density of the field E d 3k = 2 E (~k) V 0 (2π)3 Z (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 E (~k) = m + ~! with ! = cj~kj m 2 Interpretation: m photons with energy ~! and momentum ~~k 1 ~ In particular, the ground state energy 2 ! is non-zero! This leads to a zero-point energy density of the field E d 3k = 2 E (~k) V 0 (2π)3 Z (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 E (~k) = m + ~! with ! = cj~kj m 2 Interpretation: m photons with energy ~! and momentum ~~k 1 ~ In particular, the ground state energy 2 ! is non-zero! This leads to a zero-point energy density of the field E d 3k = 2 E (~k) V 0 (2π)3 Z (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 E (~k) = m + ~! with ! = cj~kj m 2 Interpretation: m photons with energy ~! and momentum ~~k 1 ~ In particular, the ground state energy 2 ! is non-zero! This leads to a zero-point energy density of the field E d 3k = 2 E (~k) V 0 (2π)3 Z (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion Zero-Point Energy Shift Quantisation of the Field Each harmonic oscillator can be in a discrete state of energy 1 E (~k) = m + ~! with ! = cj~kj m 2 Interpretation: m photons with energy ~! and momentum ~~k 1 ~ In particular, the ground state energy 2 ! is non-zero! This leads to a zero-point energy density of the field E d 3k = 2 E (~k) V 0 (2π)3 Z (factor 2 due to polarisation of the field) Caveat: this quantity is infinite... Thomas Prellberg The Mathematics of the Casimir Effect The Casimir Effect History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion 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 Ediscrete = E0;n n X 1 ~ is a sum over discrete energies E0;n = 2 !n In the absence of a boundary d 3k E = 2V E (~k) 0 (2π)3 Z The difference of the infinite zero-point energies is finite! π2~c L2 ∆E = E − E = − discrete 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 History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion 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 Ediscrete = E0;n n X 1 ~ is a sum over discrete energies E0;n = 2 !n In the absence of a boundary d 3k E = 2V E (~k) 0 (2π)3 Z The difference of the infinite zero-point energies is finite! π2~c L2 ∆E = E − E = − discrete 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 History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion 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 Ediscrete = E0;n n X 1 ~ is a sum over discrete energies E0;n = 2 !n In the absence of a boundary d 3k E = 2V E (~k) 0 (2π)3 Z The difference of the infinite zero-point energies is finite! π2~c L2 ∆E = E − E = − discrete 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 History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion 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 Ediscrete = E0;n n X 1 ~ is a sum over discrete energies E0;n = 2 !n In the absence of a boundary d 3k E = 2V E (~k) 0 (2π)3 Z The difference of the infinite zero-point energies is finite! π2~c L2 ∆E = E − E = − discrete 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 History Making Sense of Infinity - Infinity Quantum Electrodynamics Conclusion 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, .