The Casimir effect and the physical vacuum Lectures given at the - PowerPoint PPT Presentation

Outline 1 Introduction: QED and the Casimir effect QED Casimir effect: discovery and simple derivation A physical derivation: from momentum flow Some other cases: massive scalar, EM field, fermions The myth of a mysterious force between ships at sea 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

A popular myth: mysterious force between ships at sea Popular myth: ships attract at heavy swell due to smaller wave pressure in between. The two situations were messed up: Caussée claimed attraction in calm sea (below), not in a swell (above)! P. C. Caussée: The Album of the Mariner Nature, doi:10.1038/news060501-7 (1836)

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence Temperature effects Material effects Dependence on the fine structure constant 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence Temperature effects Material effects Dependence on the fine structure constant 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Temperature dependence Matsubara formalism β = 1 Z = Tr e − β H T � φ ( � x , t 2 )= φ 2 ( � x ) � d d x L � t 2 x ) | e − i ( t 2 − t 1 ) H | φ 1 ( � x ) [ d φ ] e i t 1 dt � φ 2 ( � x ) � = φ ( � x , t 1 )= φ 1 ( � ⇓ τ = it L E = − L | t →− i τ � � d d x L E � β x , 0 ) [ d φ ] e − 0 d τ Z = φ ( � x , β )= φ ( � Due to PBC in τ , the Euclidean frequencies are quantized ζ → ζ n = 2 π n fermions: APBC ζ n = π ( 2 n + 1 ) β β � d ζ 2 π → 1 β ∑ n � d d � � 2 π n � 2 � F T = − 1 k κ n k 2 + ( 2 π ) d ∑ κ n = e 2 κ n a − 1 β β n

High-temperature limit is classical T → ∞ : only n = 0 term d d � � d + 1 � � k k d F T = − T e 2 ka − 1 = − T ( 2 √ π a ) d + 1 Γ ζ ( d + 1 ) ( 2 π ) d 2 Classical free energy F = − T log Z = T ∑ log ( 1 − e − β | � p | ) � p 1 − e − β √ � � k 2 + n 2 π 2 / a 2 � d d � � ∞ k π ∑ = TV log ( 2 π ) d + 1 a n = − ∞ ds ξ s � For T → ∞ expand exponential and use log ξ = d � s = 0 � s � � n 2 π 2 d d � ∞ � � F ∼ TV 1 d k 1 2 β 2 s + k 2 ∑ � � ( 2 π ) d + 1 a 2 2 a ds � n = − ∞ s = 0 � d + 1 � 1 = − TV ( 2 √ π a ) d + 1 Γ ζ ( d + 1 ) 2

High-temperature limit is classical � s � � n 2 π 2 d d � ∞ � � F ∼ TV 1 d k 1 2 β 2 s + k 2 ∑ � � ( 2 π ) d + 1 a 2 2 a ds � n = − ∞ s = 0 Now do the momentum integral, perform the summation using ζ -function and use � d 1 � = − 1 � Γ( − s ) ds � s = 0 So the free energy is � d + 1 � 1 F = − TV ( 2 √ π a ) d + 1 Γ ζ ( d + 1 ) 2 Now the pressure is F = − ∂ F V = Aa ⇒ ∂ ∂ ∂ V = 1 ∂ V ∂ a A and this gives the same result � d + 1 � d F T = − T ( 2 √ π a ) d + 1 Γ ζ ( d + 1 ) 2

Low-temperature limit This is much more complicated: the result is not analytic in T . The leading correction is F ≈ − ( d + 1 ) 2 − d − 2 π − d / 2 − 1 Γ( 1 + d / 2 ) ζ ( d + 2 ) a d + 2 � � d + 2 � � 2 a 1 × 1 + d + 1 β but there are also corrections of the form � a � ... e − ... πβ / a β For details cf. Milton’s book.

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence Temperature effects Material effects Dependence on the fine structure constant 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Material dependence Lifschitz theory for dielectrics in planar geometry � 1 � ∞ � � 1 d + 1 F T = 0 = − d 2 � 0 d ζ k 2 κ 3 d ′ 16 π 3 TE: d = κ 3 + κ 1 κ 3 + κ 2 e 2 κ 3 a − 1 TM: d ′ = d ( κ → κ / ε ) κ 3 − κ 1 κ 3 − κ 2 κ 2 = k 2 + εζ 2 ( ζ = i ω ) Finite temperature: ζ → ζ n = 2 π n β � ∞ ∞ d ζ 2 π → 1 ′ ∑ ( n = 0 with half weight ) β 0 n = 0

Controversy over thermodynamics One can rewrite the force ( ε 1 = ε 2 = ε and ε 3 = 1)   � ∞ ∞ A n e − 2 qa B n e − 2 qa   F T = − T ∑ ′ q 2 dq   +   1 − A n e − 2 qa 1 − B n e − 2 qa π ζ n   n = 0 � �� � � �� � TM mode TE mode ζ n = 2 π nT � ε p − s � 2 � p − s � 2 A n = B n = ε p + s p + s p = q s 2 = ε − 1 + p 2 ζ n Limit of ideal metal: ε ( i ζ n ) → ∞ . However, in the zero-frequency TE mode, the limits do not commute: first ε → ∞ then ζ → 0 : B 0 → 1 first ζ → 0 then ε → ∞ : B 0 → 0

Reflectivity of metals In terms of reflectivity A n = r ( 1 ) k ⊥ ) r ( 2 ) B n = r ( 1 ) k ⊥ ) r ( 2 ) TM ( i ζ n ,� TM ( i ζ n ,� TE ( i ζ n ,� TE ( i ζ n ,� k ⊥ ) k ⊥ ) Ideal metals ε = ∞ r TM ( ω ,� r TE ( ω ,� k ⊥ ) = 1 k ⊥ ) = − 1 so A n = B n = 1 for all n . For real metals ε < ∞ r TM ( 0 ,� r TE ( 0 ,� k ⊥ ) = 1 k ⊥ ) = 0 so B 0 = 0, and stays so in the limit ε → ∞ . Casimir free energy per unit surface   � ∞ ∞ F = T � 1 − A n e − 2 qa � � 1 − B n e − 2 qa � ′ ∑   qdq  log + log  2 π ζ n � �� � � �� � n = 0 TM mode TE mode F T = − ∂ F ∂ a

Ideal metal Ideal metal: A n = B n = 1 for all n . Then � � F T = − π 2 1 + 1 3 ( 2 aT ) 4 aT ≪ 1 240 a 4 Casimir free energy per unit surface   F = − π 2   45 ζ ( 3 ) ( 2 aT ) 3 − ( 2 aT ) 4    1 + ζ ( 3 ) ≈ 1 . 2   720 a 3 π 3  � �� � requires special care Entropy T 2 − 4 π 2 a S = − ∂ F ∂ T = 3 ζ ( 3 ) 45 T 3 aT ≪ 1 2 π This is fine: S ( T → 0 ) = 0.

Modified ideal and Drude metals Drude model ω 2 plasma ε ( i ζ ) = 1 + ζ ( ζ + ν ) very good model for many metals in optical experiments for ζ < 2 · 10 15 Hz (e.g. gold: ω p = 9 . 03 eV , ν = 0 . 0345 eV ). Whenever ζ → 0 ζ 2 ( ε ( i ζ ) − 1 ) = 0 lim the zero-frequency TE mode does not contribute, i.e. B 0 = 0: � � F T = − π 2 1 + 1 T 3 ( 2 aT ) 4 + 8 π a 3 ζ ( 3 ) aT ≪ 1 240 a 4 � � F = − π 2 1 + 45 ζ ( 3 ) T ( 2 aT ) 3 − ( 2 aT ) 4 + 16 π a 2 ζ ( 3 ) 720 a 3 π 3 T 2 − 4 π 2 a S = 3 ζ ( 3 ) 45 T 3 − ζ ( 3 ) !!! violates Nernst theorem 2 π 16 π a 2

Proposed solutions Mostepanenko, Geyer: abandon Drude model. Low frequency ⇒ wave-length long, field constant inside plate ⇒ cannot exist, leads to charge separation However: why to give up a successful description of materials, when there are other ways to avoid the problem. E.g. if resistivity does not simply go to 0 at T = 0, i.e. ν ( T → 0 ) � = 0 Additional physical effects: 1. Spatial dispersion ε ( ω ,� k ) Only ε ( 0 , 0 ) would be infinite, but that is zero measure in � k space.

Proposed solutions II 2. Anomalous skin effect: mean free path of electrons becomes longer than field penetration depth near T = 0. Again, no contribution from TE zero mode found. 3. Large separation: result for Casimir effect same as for large T , i.e. classical. It turns out TE modes do not contribute in this limit and F = − ζ ( 3 ) T a → ∞ 8 π a 3 and this precisely agrees with the Drude prediction. Future experiments will decide which scenario is valid (possibly dependent on material). Present experimental situation seems inconclusive to me.

Repulsive Casimir forces One way: measure inside fluid, suitably chosen dielectric constant ⇒ Lifshitz theory predicts repulsion. J.N. Munday, F. Capasso, and V.A. Parsegian: Nature 457 : 170–173, 2009. Gold sphere - gold plate, in bromobenzene: 150 pN at 20 nm separation Other way: coat surfaces of appropriate (meta)materials e.g. ε left = ∞ and µ right = ∞ or negative refraction (cloaking) (KK: only in limited freq. range!) Analysis: K.A Milton et al, J. Phys. A45 374006, 2012. [arXiv:1202.6415]

Puzzle What do you get if you lay an an invisibility cloak on the floor? ⇒ A flying carpet!

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence Temperature effects Material effects Dependence on the fine structure constant 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Where has α gone? QED effect: would expect appearance of fine structure constant, but it is nowhere to be found... Actual metals: frequency-dependent dielectric constant and conductivity. Drude model: plasma = 4 π e 2 n ω 2 > ω 2 σ ( ω ) = 0 m For ω < ω plasma : penetration length (skin depth) ne 2 δ − 2 = 2 πω | σ | σ = c 2 m ( γ 0 − i ω ) Typically ω ≫ γ 0 (damping) c √ δ ≈ 2 ω plasma Frequencies dominating Casimir effect: c / d ⇒ perfect conductor approximation means c mc d ≪ ω plasma α ≫ hnd 2 4 π ¯

Where has α gone? II Typically: d < 0 . 5 µ m . Copper: 1 mc hnd 2 ≈ 10 − 5 ≪ α ≈ 4 π ¯ 137 Casimir force is α → ∞ limit!!! α → 0 limit: h 2 a Bohr = ¯ me 2 ∝ 1 α and so n ∝ α 3 ω plasma ∝ α 2 : for any fixed separation d , ⇒ Casimir effect goes away. Also δ → ∞ : separation d becomes ill-defined. For more details cf. R.L. Jaffe: The Casimir effect and Quantum Vacuum, hep-th/0503158.

Radiative corrections: Schwinger’s method Schwinger’s approach: consider the vacuum persistence amplitude in the presence of sources and boundaries � � K Φ) e iW [ K ] = � 0 | e − iHT | 0 � = D Φ e i ( S [Φ]+ � W [ K ] = 1 dxdx ′ K ( x ) G ( x , x ′ ) K ( x ′ ) 2 Effective field � dx ′ G ( x , x ′ ) K ( x ′ ) φ ( x ) = � dx ′ G − 1 ( x , x ′ ) φ ( x ′ ) K ( x ) = Altering the geometry (e.g. moving boundaries adiabatically) � δ W [ K ] = 1 dxdx ′ K ( x ) δ G ( x , x ′ ) K ( x ′ ) 2 � = − 1 dxdx ′ φ ( x ) δ G − 1 ( x , x ′ ) φ ( x ′ ) 2

Casimir energy from response of Green’s function Now � � dxK ( x ) φ ( x ) = ··· − 1 e iW [ K ] = e 1 dxdx ′ φ ( x ) K ( x ) K ( x ′ ) φ ( x ′ ) 2 i 2 i.e. changing boundaries is equivalent to a new two-particle source � � iK ( x ) K ( x ′ ) eff = − δ G − 1 ( x , x ′ ) � � δ W = i dxdx ′ G ( x , x ′ ) δ G − 1 ( x , x ′ ) = − i dxdx ′ δ G ( x , x ′ ) G − 1 ( x , x ′ ) 2 2 � = − i dxdx ′ δ log G ( x , x ′ ) = − i 2 δ Trlog G 2 so i E = lim 2 T ( Trlog G − Trlog G ref ) T → ∞ where G ref is the value at some reference state (e.g. with bodies infinite distance apart).

Radiative correction for electromagnetic field Use perturbative form of G with Π as polarization G = G 0 ( 1 +Π G 0 + ... ) Result for parallel plates A = − π 2 απ 2 E = E 2560 m e a 4 + O ( α 2 ) 720 a 3 + This is suppressed by α m − 1 e a and is inobservable in practice m − 1 = λ Compton ≈ 2 . 43 · 10 − 12 m e 1 α ≈ 137

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence Proximity force approximation The method of Green’s dyadic Lateral Casimir force Casimir force between compact bodies 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence Proximity force approximation The method of Green’s dyadic Lateral Casimir force Casimir force between compact bodies 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Novel measurement methods Figure : Mohideen et al. Figure : Bell Labs AFM (Atomic Force Microscope), sensibility in principle can be 10 − 17 N (reached: 10 − 13 N) Torsion balance Si-plate: dielectric constant can be (Capasso, Harvard) modulated by laser (U. Mohideen et al., UC Riverside)

Proximity force approximation; special geometries Simplest way to account for geomery dependence: Proximity Force Theorem Sphere and plate, R ≫ d : every element of sphere is approximately parallel to plate � π � R 0 2 π R sin θ Rd θ E ( d + R ( 1 − cos θ )) = 2 π R V ( d ) = − R dx E ( d + R − x ) � R F = − ∂ V − R dx d E ( d + R − x ) ∂ d = 2 π R dx = 2 π R ( E ( d ) − E ( d + 2 R )) ≈ 2 π R E ( d ) Lamoreaux: 5 % → Mohideen & Roy: 1 % → Bell Labs 0 . 5 % Need to include: finite conductivity corrections, surface roughness. Other calculations: sphere - plate, cylinder - plate, concentric spheres, coaxial cylinders. (K.A.Milton: The Casimir effect, World Scientific, 2001.)

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence Proximity force approximation The method of Green’s dyadic Lateral Casimir force Casimir force between compact bodies 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Green’s dyadic Green’s dyadic: response of EM field to polarization � d 4 x ′ Γ ij ( x , x ′ ) P j ( x ′ ) E i ( x ) = � d 4 x ′ Φ ij ( x , x ′ ) P j ( x ′ ) H i ( x ) = Static situation: frequency decomposition � d ω 2 π e − i ω ( t − t ′ ) Γ ij ( � Γ ij ( x , x ′ ) = x ′ ; ω ) x ,� � d ω 2 π e − i ω ( t − t ′ ) Φ ij ( � Φ ij ( x , x ′ ) = x ′ ; ω ) x ,� Maxwell’s equations ( ε 0 = µ 0 = 1) E = − ∂ � H rot � ⇒ ε ijk ∂ j Γ kl = i ω Φ il ∂ t � � E + � � ∂ P rot � x ′ ) H = ⇒ − ε ijk ∂ j Φ kl − i ω Γ il = i ωδ il δ ( � x − � ∂ t div � ∂ i Φ ij = 0 H = 0 ⇒

Solving for Green’s dyadic Redefining Γ : Γ ′ x ′ ) ∂ i Γ ′ il = Γ il + δ il δ ( � x − � ⇒ ij = 0 Taking the rotation of Maxwell’s equations, we get ∇ 2 + ω 2 � � Γ ′ ij = − ( ∂ i ∂ j − δ ij ∇ 2 ) δ ( � x ′ ) x − � � ∇ 2 + ω 2 � x ′ ) Φ ij = i ωε ikj ∂ k δ ( � x − � This has to be solved with boundary conditions: e.g. for a conducting boundary, tangential electric field vanishes on the surface � ε ijk n j Γ ′ x ′ ; ω ) kl ( � x ,� x ∈ Σ = 0 � � Main advantage of method: explicit gauge invariance.

Computing the Casimir stress The two-point functions of fields are � � E i ( x ) E j ( x ′ ) = − i Γ ij ( x , x ′ ) = i 1 � � H i ( x ) H j ( x ′ ) ω 2 ε ikl ∂ k ε jmn ∂ k Γ mn ( x , x ′ ) (from ε ikl ∂ k E l ( x ) = i ω H i ( x ) ) and the Maxwell stress tensor is T ij = E i E j − 1 E 2 + H i H j − 1 2 δ ij � 2 δ ij � H 2 ⇒ Casimir stress on the surface. E.g. for a perfectly conducting sphere of radius a � � 1 − ∂ E F = � T rr ( r = a − 0 ) �−� T rr ( r = a + 0 ) � = 4 π a 2 ∂ a and the self-energy from Casimir stress is (Boyer) E = 0 . 092353 (¯ h = 1 = c ) 2 a

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence Proximity force approximation The method of Green’s dyadic Lateral Casimir force Casimir force between compact bodies 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Lateral force PFA : averaging over surface roughness. Condition: λ c ≫ z A , zero lateral force. F. Chen and U. Mohideen, Phys. Rev A66 : 032113, 2002.

Use of Casimir force in micromachines Standard worry: Casimir force would make nanobots stick. Idea: exploit Casimir force to produce motion. T. Emig: Casimir force driven ratchets Phys. Rev. Lett. 98 :160801, 2007 A Casimir ratchet producing lateral [cond-mat/0701641] motion by vibrating separation With typical parameters � v � ∼ mm/s Other similar effect: Casimir torque (for asymmetric bodies) Not yet observed!

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence Proximity force approximation The method of Green’s dyadic Lateral Casimir force Casimir force between compact bodies 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Arbitrary compact bodies Emig, Graham, Jaffe & Kardar ’2007 � Z [ C ] = Tr e − i h H C T = i h S [Φ] [ D Φ] C e ¯ ¯ � � � � Φ( � x , t + T ) = Φ( � x , t ) and Φ | C = 0 h H C Λ → h E 0 [ C ]Λ + ... Tr e − 1 Λ → ∞ e − 1 � C = Σ α ¯ ¯ α | T | ln Z [ C ] h ¯ h ¯ = ∑ ⇒ E [ C ] = lim 2 ( ω n − ω n , ∞ ) Z ∞ T →− i ∞ n Suppose C is time-independent: we can Fourier expand in time � � ∏ [ D Φ] C → [ D φ n ( � x )] C n Φ( x ) = ∑ x ) e 2 π int / T φ n ( � n

Fluctuating surface charges So we get � � x ) | 2 � � � d � � 2 | φ n ( � x )] C e i T ( 2 π n cT ) x ) | 2 −| ∇ φ n ( � log Z [ C ] = ∑ x log [ D φ n ( � h ¯ n � ∞ ( T → ∞ ) = cT 0 dk log Z C ( k ) π � � d 3 � i x ( k 2 | φ ( � x , k ) | 2 −| ∇ φ ( � x , k ) | 2 ) h T Z C ( k ) = [ D φ ( � x , k )] C e ¯ Now putting T = − i Λ / c , Wick rotating k = i κ � ∞ E [ C ] = − ¯ hc 0 d κ log Z C ( i κ ) π Z ∞ ( i κ ) � � d 3 � x , i κ )] C e − T x ( κ 2 | φ ( � x , i κ ) | 2 + | ∇ φ ( � x , i κ ) | 2 ) Z C ( i κ ) = [ D φ ( � h ¯ Implement Dirichlet BC with Lagrange multipliers: � � � � x )] e i T x ) ∗ φ ( � Σ α d 3 � x ) D ρ ∗ x ( ρ α ( � x )+ c . c . ) x )] ∏ [ D φ ( � x )] C = [ D φ ( � [ D ρ α ( � α ( � ¯ h α � �� � functional Dirac delta

Performing the Φ integral So � � h T ˜ i x ) D ρ ∗ S ( φ , ρ ) x , k )] C ∏ [ D φ ( � [ D ρ α ( � x )] e Z C ( k ) = α ( � ¯ α � � x , k ) | 2 � k 2 | φ ( � x , k ) | 2 −| ∇ φ ( � ˜ d 3 � S ( φ , ρ ) = x � x ) ∗ φ ( � d 3 � + x ( ρ α ( � x , k )+ c . c . ) Σ α Idea: integrate out Φ from quadratic functional integral → classical solution + fluctuations. ( ∇ 2 + k 2 ) φ cl ( � x , k ) = 0 x / ∈ Σ α ∆ φ cl ( � x , k ) = 0 x ∈ Σ α ∆ ∂ n φ cl ( � x , k ) = ρ α ( x ) x ∈ Σ α

Integrating out fluctuations � x ) = ∑ x ′ G 0 ( � x ′ , k ) ρ β ( � x ′ ) φ cl ( � d � x ,� Σ β β x ′ | x ′ , k ) = e ik | � x − � j l ( kr < ) h ( 1 ) x ′ ) Y lm (ˆ x ) ∗ x ′ | = ik ∑ G 0 ( � x ,� ( kr > ) Y lm (ˆ l 4 π | � x − � lm Put now φ = φ cl + δφ � Z C ( k ) = ∏ h T ˜ i x ) D ρ ∗ S cl ( ρ ) [ D ρ α ( � α ( � x )] e ¯ α � � d 3 � x , k )] e i T x ( k 2 | δφ ( � x , k ) | 2 −| ∇ δφ ( � x , k ) | 2 ) × [ D δφ ( � ¯ h � �� � unconstrained fluctuations: cancel out with denominator � ˜ d 3 � x ) ∗ φ ( � S cl ( ρ ) = x ( ρ α ( � x , k )+ c . c . ) Σ α Also note that φ cl = ∑ φ β , where φ β is sourced by ρ β . β

Interaction terms � � � j l ( kr < ) h ( 1 ) x ′ x ′ ) Y lm (ˆ x ) ∗ x ′ ) x ) = ∑ ik ∑ φ cl ( � d � ( kr > ) Y lm (ˆ ρ β ( � l Σ β β lm Interaction terms ( α � = β ) : in terms of multipoles � x β j l ( kr β ) Y ∗ Q β , lm = d � lm (ˆ x β ) ρ β ( � x β ) Σ β Q β , lm h ( 1 ) x β ) = ik ∑ φ β ( � ( kr β ) Y lm (ˆ x β ) l lm U αβ lm , l ′ m ′ h ( 1 ) x α ) = ik ∑ Q β , lm ∑ φ β ( � l ′ ( kr α ) Y l ′ m ′ (ˆ x α ) l ′ m ′ lm U αβ lm , l ′ , m ′ : translation coefficients, depending on Σ α and Σ β � � � ˜ d 3 � x ) ∗ φ β ( � S αβ ( ρ ) = x ρ α ( � x , k )+ c . c . Σ α 1 � � α , l ′ m ′ U αβ Q ∗ 2 ik ∑ lm ∑ = l ′ m ′ , lm Q β , lm + c . c l ′ m ′

Self-interaction terms � S αα ( ρ ) = 1 ˜ d 3 � x ) ∗ φ α ( � x ( ρ α ( � x , k )+ c . c . ) 2 Σ α Field inside Σ α is regular Helmholtz solution, outside general x ) = ∑ φ in , α ( � φ α , lm j l ( kr ) Y lm (ˆ x ) φ out , α ( � x ) = φ in , α ( � x )+∆ φ α ( � x ) lm � � x ) = ∑ x )+ ∑ l ′ m ′ lm ( k ) h ( 1 ) T α ∆ φ α ( � χ α , lm j l ( kr ) Y lm (ˆ l ′ ( kr ) Y l ′ m ′ ( kr ) l ′ m ′ lm where T α l ′ m ′ lm ( k ) is from ∆ φ α ( � x ) | Σ α = 0. But the out field is regular at infinity ⇒ χ α , lm = − φ α , lm . So l ′ m ′ lm ( k ) h ( 1 ) T α x ) = − ∑ φ α , lm ∑ φ out , α ( � l ′ ( kr ) Y l ′ m ′ ( kr ) l ′ m ′ lm � x ′ ) = ik ∑ Q α , l ′ m ′ h ( 1 ) x ′ G 0 ( � x ′ ) ρ α ( � but it is also = d � x ,� l ′ ( kr ) Y l ′ m ′ (ˆ x ) Σ α l ′ m ′ so that ikQ α , l ′ m ′ = ∑ φ α , lm T α l ′ m ′ lm ( k ) lm φ α , lm = − ik ∑ [ T α ( k )] − 1 l ′ m ′ lm Q α , l ′ m ′ l ′ m ′

Integrating over charge fluctuations The final form for the self-interaction is S αα ( ρ ) = − ik ˜ Q α , lm [ T α ( k )] − 1 2 ∑ l ′ m ′ lm Q α , l ′ m ′ + c . c . l ′ m ′ and we are left with the functional integral � Z C ( k ) = ∏ x ) D ρ ∗ [ D ρ α ( � α ( � x )] α � k � � Q ∗ T − 1 2 ∑ α ∑ exp lm , l ′ m ′ Q α , l ′ m ′ α , lm α lm , l ′ m ′ � − k � � Q ∗ 2 ∑ α � = β ∑ lm , l ′ m ′ Q α , l ′ m ′ − c . c . U αβ α , lm lm , l ′ m ′ � � � � = Jacobian × ∏ dQ ∗ dQ α , lm exp { ... } α , lm α , l . m Jacobian is independent of functional integration variables ( Q − ρ relation linear) and drops out with denominator.

Casimir force: averaged interaction between fluctuating charges The end result is: � ∞ 0 d κ ln det M C ( i κ ) E C = − ¯ hc π det M ∞ ( i κ )     T − 1 T − 1 ··· ··· U 12 U 1 N 0 0 1 1 T − 1 T − 1 U 21 ··· U 2 N 0 ··· 0     2 2     M ( k ) = M ∞ ( k ) = . . . . . . ... ...     . . . . . . . . . . . .     T − 1 T − 1 ··· 0 0 ··· U N 1 U N 2 N N For two bodies: � ∞ E 12 ( C ) = − ¯ hc � 1 − T 1 U 12 T 2 U 21 � 0 d κ Trln π Note: this is entirely finite, convergent and physically meaningful.

General formula for planar situations In one space dimension it is easy to derive the Casimir interaction with other methods: � ∞ � � E 12 ( L ) = − ¯ hc 1 − e − 2 κ L R 1 ( i κ ) R 2 ( i κ ) 0 d κ log π where R 1 , 2 ( ω ) is the reflection coefficient of the mode ω on the boundaries and e − 2 κ L = e 2 i ω L = e 2 i | k | L , ω = | k | So here: T 1 = R 1 ( ω ) T 2 = e i ω L R 2 ( ω ) U 12 = U 21 = e 2 i ω L which looks really sensible. This also extends to planar situations � ∞ � � � � E 12 ( L ) = − ¯ hc κ 2 + � 1 − e − 2 L k 2 ⊥ + m 2 R 1 ( i κ ,� d � k ⊥ ) R 2 ( i κ ,� 0 d κ k ⊥ log k ⊥ ) π (Bajnok, Palla & Takács, hep-th/0506089).

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy Zero-point energy Why does the ZPE derivation work? Casimir force and van der Waals interaction 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy Zero-point energy Why does the ZPE derivation work? Casimir force and van der Waals interaction 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Does the Casimir force originate from zero-point energy? Mystery: a naive consideration of zero modes leads to a huge vacuum energy density. Quantum field d d � � 1 � x � k k ) e − i ω ( � k ) t + i � x + a † ( � k ) e + i ω ( � k ) t − i � a ( � k · � k · � Φ( � x , t ) = � ( 2 π ) d / 2 2 ω ( � k ) � � x 1 2 ( ∂ t Φ) 2 + 1 � � 2 � d d � d d � H = xT 00 = ∇ Φ 2 d d � � k k ) 1 � � ( 2 π ) d ω ( � a † ( � k ) a ( � k )+ a ( � k ) a † ( � = k ) 2 d d � d d � � � k k 1 ( 2 π ) d ω ( � k ) a † ( � k ) a ( � 2 ω ( � = k )+ k ) δ ( 0 ) ( 2 π ) d With δ ( 0 ) = ( 2 π ) d V , d = 3 and a high energy cutoff Λ we get an energy density � Λ 0 k 2 dk 1 E 0 2 k ∝ Λ 4 V =

The naive vacuum energy density and the QFT Hamiltonian QFT (Standard Model) valid at least up to Λ ∼ 1 TeV: E 0 V ∼ 10 47 J m 3 If Λ = M Planck ∼ 10 19 GeV : E 0 V ∼ 10 110 J m 3 How comes the Casimir force is such a small effect? Crucial observation: quantum Hamiltonian is not uniquely fixed! E.g.: why is the standard mass point Hamiltonian p 2 H = ˆ ˆ 2 M + V (ˆ q ) Explanation: this comes from correspondence principle d O = i ˆ h [ ˆ H , ˆ O ] [ˆ q , ˆ p ] = i ¯ h dt ¯ d q = ˆ p d p = − V ′ (ˆ dt ˆ dt ˆ q ) M h → 0: ˆ q , ˆ p commute ⇒ simultaneously diagonalizable ⇒ ¯ eigenvalues obey classical equations of motion.

The naive vacuum energy density and the QFT Hamiltonian A perfectly good Hamiltonian for QFT is given by � � x : 1 2 ( ∂ t Φ) 2 + 1 � � 2 � d d � d d � H = xT 00 = ∇ Φ : 2 d d � d d � � � k k ) 1 k ( 2 π ) d ω ( � 2 : a † ( � k ) a ( � k )+ a ( � k ) a † ( � ( 2 π ) d ω ( � k ) a † ( � k ) a ( � = k ) := k ) Moral: QFT does not predict vacuum energy density! Some other interaction is needed ⇒ gravity. Einstein’s “greatest mistake” : R µν − 1 8 π G 2 g µν R + λ g µν = c 4 T µν = − c 4 λ T ( λ ) ν 8 π G g ν µ = E g ν µ µ Cosmological constant: p = − E . Present concordance cosmology ( Λ CDM): E ∼ 5 . 4 × 10 − 10 J m 3

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy Zero-point energy Why does the ZPE derivation work? Casimir force and van der Waals interaction 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Why does the zero-point energy derivation work? Energy of a point charge e 2 4 πε 0 r 2 ⇒ E = 1 e E 2 = 2 ε 0 � E = 32 π 2 ε 0 r 4 Field energy: � ∞ e 2 4 π r 2 E dr = 8 πε 0 r 0 r 0 r 0 = 0: divergent! Renormalization: e 2 m phys c 2 = m 0 c 2 + 8 πε 0 r 0 m phys : physical mass: the only observable.

Radius of the electron Physical mass e 2 m phys c 2 = m 0 c 2 + 8 πε 0 r 0 m 0 = 0: classical electron radius r 0 ∼ 10 − 15 m Present experiments: r 0 < 10 − 18 m QED self-energy: � � � � λ 2 1 − 3 α + 1 Compton m 0 c 2 m phys c 2 + O ( α 2 ) = 4 π log r 2 2 0 λ Compton = 2 . 4263102175 ( 33 ) × 10 − 12 m r 0 ∼ 10 − 18 m : 5% correction. r 0 > 10 − 136 m Theoretical limit: m 0 > 0 →

Two point charges Figure : Two point charges with distance d � E = � E 1 + � 2 ε 0 � E = 1 E 2 E 2 → � d 3 � E ( d ) = x E still divergent for r 0 = 0 � 1 � but: E ( d 1 ) − E ( d 2 ) = e 1 e 2 − 1 finite! 4 πε 0 d 1 d 2 E int ( d ) = e 1 e 2 Interaction energy: 4 πε 0 d This works because � d 3 � W Lorentz = − x ∆ E

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy Zero-point energy Why does the ZPE derivation work? Casimir force and van der Waals interaction 5 Time dependent boundaries 6 Gravitational aspects 7 Some related topics

Casimir effect and and van der Waals interaction van der Waals force = interaction between fluctuating dipols d 2 r 2 − 3 ( � � d 1 · � r )( � d 1 · � d 2 · � r ) H int = r 5 � 0 | H int | m �� m | H int | 0 � V eff = ∑ ∝ r − 6 E 0 − E m m � = 0 Original problem investigated by Casimir & Polder: retardation effects on vdW force Dielectric ball: Casimir self-stress ≡ vdW forces Casimir effect = relativistic vdW

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries Velocity dependence of Casimir force Dynamical Casimir effect: particle creation 6 Gravitational aspects 7 Some related topics

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries Velocity dependence of Casimir force Dynamical Casimir effect: particle creation 6 Gravitational aspects 7 Some related topics

Moving boundary Plates: K : x 3 = 0 and K ′ : x 3 = vt . Solve for Dirichlet Green’s function (scalar field): � t − ∇ 2 � ∂ 2 G ( x , x ′ ) = − δ ( x − x ′ ) x , x ′ ∈ K or K ′ G ( x , x ′ ) = 0 Energy density 3 3 � 0 | T 00 ( x ) | 0 � = 1 � 0 | ∂ k Φ( x ) ∂ k Φ( x ) | 0 � = i ∂ k ∂ ′ k G ( x , x ′ ) ∑ ∑ 2 lim 2 x ′ → x k = 0 k = 0 Solution in x 3 < 0: using method of images � � i 1 1 G > ( x , x ′ ) = ( x − x ′ ) 2 − 4 π 2 ( x − S K x ′ ) 2   1 0 0 0 0 1 0 0   S K =   0 0 1 0   0 0 0 − 1

Moving boundary II Solution for x 3 > vt : use Lorentz transform to get into system of K ′ , find image, transform back. � � 1 1 i G > ( x , x ′ ) = ( x − x ′ ) 2 − 4 π 2 ( x − S K ′ x ′ ) 2   cosh s 0 0 − sinh s s = log c − v 0 1 0 0   S K ′ =   0 0 1 0 c + c   sinh s 0 0 − cosh s Solution in between: infinitely many images ∞ i 1 G in ( x , x ′ ) = ( − 1 ) m ∑ 4 π 2 ( x − x ′ m ) 2 m = − ∞ x ′ 2 m = ( S K S K ′ ) m x ′ x ′ 2 m − 1 = S K ( S K S K ′ ) m x ′ x ′ − 2 m = ( S K ′ S K ) m x ′ x ′ − 2 m − 1 = S K ( S K ′ S K ) m x ′

Moving boundary III Renormalization: eliminate vacuum contribution, which is the term i G 0 = 4 π 2 ( x − x ′ ) 2 in all three domains. Force per unit area: � ∞ d − ∞ dx 3 � 0 | T 00 ( x ) | 0 � F ( a ( t )) = − a ( t ) = vt d ( vt ) � � v 4 �� π 2 1 + 8 � v � 2 = − + O 480 a ( t ) 4 c 4 3 c Electromagnetic case: � � 10 �� v � v 4 �� π 2 π 2 − 2 � 2 F ( a ( t )) = − 1 + + O v ≪ c 240 a ( t ) 4 3 c 4 c 1 + ( c 2 − v 2 ) 2 � ( c 2 − v 2 ) 4 � �� 3 = − + O v ≪ c 8 π 2 a ( t ) 4 16 c 4 c 8

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries Velocity dependence of Casimir force Dynamical Casimir effect: particle creation 6 Gravitational aspects 7 Some related topics

Dynamical Casimir effect Scalar field in 2 d ∂ 2 t Φ − c 2 ∂ 2 x Φ = 0 Take an interval ( 0 , a ( t )) , where a ( t ) = a 0 for t < 0. The field is � � Φ( t , x ) = ∑ χ ( − ) ( t , x ) a n + χ (+) ( t , x ) a † n n n n √ π ne ± i ω n t sin π nx 1 ω n = c π n χ ( ± ) ( t ≤ 0 , x ) = n a 0 a 0 � a 0 1 a ( t ) sin π kx χ ( − ) √ π n ∑ ( t > 0 , x ) = Q nk ( t ) n a ( t ) k � � ∗ χ (+) χ ( − ) ( t > 0 , x ) = ( t > 0 , x ) n n Initial conditions Q ′ Q nk ( 0 ) = δ nk nk ( 0 ) = − i ω n δ nk

Equation of motion Field equation gives Q ′′ nk ( t )+ ω 2 k ( t ) Q nk ( t ) � � = ∑ 2 ν ( t ) Q ′ nj ( t )+ ν ′ ( t ) Q nj ( t ) − ν ( t ) 2 ∑ h jl Q nl ( t ) h kj j l ν ( t ) = a ′ ( t ) ω k ( t ) = c π k a ( t ) a ( t ) 2 kj h kj = − h jk = ( − 1 ) k − j j � = k j 2 − k 2 Suppose that a ( T ) = a 0 after some time T ⇒ t > T : Q nk ( t ) = α nk e − i ω k t + β nk e i ω k t � � Φ( t , x ) = ∑ φ ( − ) ( t , x ) b n + φ (+) ( t , x ) b † n n n n √ π ne ± i ω n t sin π nx 1 ω n = c π n φ ( ± ) ( t , x ) = n a 0 a 0

Bogolyubov transform � k � � b k = ∑ α nk a n + β ∗ nk a † n n n � | α nk | 2 −| β nk | 2 � Unitarity: ∑ k = n k In- and out-vacuum: a k | 0 � in = 0 b k | 0 � out = 0 Number of created particles: ∞ 1 n k = in � 0 | b † n | β nk | 2 ∑ k b k | 0 � in = k n = 1 ∞ ∑ N = n k k = 1 Enhancing effect: parametric resonance. E.g. a ( t ) = a 0 [ 1 + ε sin ( 2 ω 1 t )] ω 1 = c π a 0

Particle creation Solution is long, but result is that only odd modes are populated and n 1 ( t ) ≈ τ 2 τ ≪ 1 n 1 ( t ) ≈ 4 π 2 τ τ ≫ 1 τ = εω 1 τ kn k ( t ) = 1 4 ω 2 1 sinh 2 ( 2 τ ) E ( t ) = ω 1 ∑ k Typical values for photons in cm cavity ω 1 ∼ 60GHz maximum endurance for wall materials ε max ∼ 3 × 10 − 8 dn 1 dt ≈ 4 π 2 ε max ω 1 ∼ 700 s − 1 Total number created is typically thousands of photons per second. Effects to take into account: finite wall reflectivity, detector interaction. Nonzero temperature: factor ∼ 10 3 at room temperature.

Experiments MIR (Motion Induced Radiation, Padova) :( Microwave line modulated by a SQUID: success! C.M. Wilson et al., 2011 Nature 479 : 376-379 Microwave line: 100 µ m “Mirror motion”: ∼ nm

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects Local effects: the energy-momentum tensor How does Casimir energy fall? Cosmological constant from Casimir energy of extra dimensions Non-Newtonian gravity 7 Some related topics

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects Local effects: the energy-momentum tensor How does Casimir energy fall? Cosmological constant from Casimir energy of extra dimensions Non-Newtonian gravity 7 Some related topics

Energy density Scalar field, Dirichlet plates: Green’s function of a given mode 1 g int ( z , z ′ ) = − λ sin λ a sin λ z < sin λ ( z > − a ) λ 2 = ω 2 − k 2 ⇓ � d ω d 2 k � T 00 � = 1 � ( ∂ t Φ) 2 +( ∇ Φ) 2 � = ( 2 π ) 3 � t 00 � 2 � t 00 � = 1 � ω 2 + k 2 + ∂ z ∂ z ′ � g int ( z , z ′ ) | z = z ′ 2 i 1 � ω 2 cos λ a − k 2 cos λ ( 2 z − a ) � = − 2 i λ sin λ a Wick rotate ω → i ζ , λ → i κ and use polar coordinates ζ = κ cos θ , k = κ sin θ : � � ∞ � π / 2 � T 00 � = − 1 d θκ 2 sin θ cos 2 θ cosh κ a 0 κ d κ 4 π 2 sinh κ a 0 � + sin 2 θ cosh κ ( 2 z − a )

Energy density II � � � ∞ 2 + e 2 κ z + e 2 κ ( a − z ) � T 00 � = − 1 e 2 κ a − 1 + 1 1 0 d κκ 3 e 2 κ a − 1 6 π 2 The second term is the vacuum constant, to be discarded. The result is � T 00 � = u + g ( z ) π 2 u = − 1440 a 4 � ∞ 0 dyy 3 e yz + e y ( 1 − z / a ) g ( z ) = − 1 1 e y − 1 6 π 2 16 a 4 1 = − 16 π 2 a 4 [ ζ ( 4 , z / a )+ ζ ( 4 , 1 − z / a )] ∞ 1 ∑ ζ ( s , z ) = Hurwitz zeta ( n + a ) s n = 0

Energy density III g ( z ) diverges at z = 0 , a . Fortunately 1 5 � a � e 2 κ z + e 2 κ ( a − z ) � == 1 1 0 � e 2 κ a − 1 � log � � g � z �� 0 dz κ 5 so, although its integral is divergent, 0 it is also a -independent and does not 0.0 0.2 0.4 0.6 0.8 1 .0 z � a contribute to the force. Similar calculation gives T xx , T yy , T zz     1 0 0 0 1 0 0 0 0 − 1 0 0 0 − 1 0 0     � T µν � = u  + g ( z )     0 0 − 1 0 0 0 − 1 0    0 0 0 3 0 0 0 0

Energy-momentum tensor The energy-momentum tensor is not unique: instead of canonical we may use the conformal one T µν = T µν − 1 � ∂ µ ∂ ν − g µν ∂ 2 � ˜ Φ 2 6 for which T µ ˜ µ = 0 Then   1 0 0 0 π 2 0 − 1 0 0   � T µν � = u u = −   0 0 − 1 0 1440 a 4   0 0 0 3 Casimir pressure and energy density p = − 3 u e = u

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects Local effects: the energy-momentum tensor How does Casimir energy fall? Cosmological constant from Casimir energy of extra dimensions Non-Newtonian gravity 7 Some related topics

Equivalence principle Binding energy: mass defect Chemical bonds: ∆ m / m = 10 − 9 ⇓ The equivalence principle is valid for EM energy with at least 10 − 3 precision!

How does Casimir energy fall? Between parallel plates   1 − 1   � T µν � = u  θ ( z ) θ ( a − z )   − 1  3 u = − π 2 ¯ z= z= a hc 0 1440 a 4 Remarks: 1. Volume divergence („ZPE”) trivially eliminated. d 3 � � � � u 0 = ¯ h k � � � � ( 2 π ) 3 c k � 2 2. Surface divergence ∝ z − 4 ⇒ renormalizing mass of plates.

Equivalence principle holds! Gravitation energy in weak field limit: � d 3 � x ) T µν ( � E g = − x h µν ( � x ) Problem: E g is not gauge invariant! � x ξ µ ∂ ν T µν d 3 � h µν → h µν + ∂ µ ξ ν + ∂ ν ξ µ : ∆ E g = 2 Why? ∂ ν T µν � = 0: there is a force on the plates! Solution: Use locally inertial coordinates (K.A. Milton et al.): Fermi coordinates: g ij quadratic in distance from origin. Locally h 00 = − gz h 0 i = h ij = 0 E g = gz 0 uAa + const = gz 0 E Casimir + const which is just right! A full analysis: K.A. Milton et al: How does Casimir energy fall? IV , arXiv:1401.0784

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects Local effects: the energy-momentum tensor How does Casimir energy fall? Cosmological constant from Casimir energy of extra dimensions Non-Newtonian gravity 7 Some related topics

Compact extra dimensions Compact extra dimensions: Kaluza-Klein theory, later resurrected by string theory. Space-time: M = M 4 × K � T µν � = − u ( a ) g µν = − Λ 8 π G g µν

Case of a sphere: K = S N Casimir energy of free massless scalar, for odd N � ∞ 1 2 π 0 dy [ y 2 − i ( N − 1 ) y 2 ] D ( iy ) u ( a ) = − 64 π 2 a 4 Re e 2 π y − 1 D l = ( 2 l + N − 1 )( l + N − 2 )! ( N − 1 )! l ! 64 π 6 a 4 ≈ − 5 × 10 − 5 N = 1 : u ( a ) = − 3 ζ ( 5 ) a 4 For even N u ( a ) is logarithmically divergent; cutoff is necessary: u ( a ) = 1 � α N log a � b + const a 4 � ∞ 1 dt e 2 π t − 1 [( N − 1 ) it − t 2 ] 2 D ( it ) α N = 16 π 2 Im 0 b : frequency cut-off, presumably Planck scale. For large extra dimensions a / b ∼ 10 16 : logarithmic term sufficient for estimate.

Estimate for size of extra dimensions Cosmological constant ( Λ CDM concordance cosmology) Λ ∼ ρ c ∼ 10 − 5 GeV cm 3 Maximum value for coefficient u ( a ) ∼ 10 − 3 a 4 hc = 2 × 10 − 14 GeV cm we find Restoring units using ¯ a 4 ∼ 10 2 cm 3 hc ∼ 10 − 12 cm 4 GeV ¯ a ∼ 10 µ m Such a compact dimension would lead to non-Newtonian gravity on a submm scale.

Outline 1 Introduction: QED and the Casimir effect 2 Realistic cases I: temperature and material dependence 3 Realistic cases II: geometry dependence 4 Comments on Casimir force and zero-point energy 5 Time dependent boundaries 6 Gravitational aspects Local effects: the energy-momentum tensor How does Casimir energy fall? Cosmological constant from Casimir energy of extra dimensions Non-Newtonian gravity 7 Some related topics