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Supplemental Lecture 16

Introduction to the Foundations of Theory For Students

III. The

Abstract

Uncharged grounded conducting parallel plates experience a mutual attractive which is a quantum, relativistic effect: the force is proportional to Planck’s constant and the speed of . It was calculated in the early days of relativistic (1948). In fact, it was originally analyzed as a limiting case of the retarded van der Waal’s force between plates. Casimir made the fascinating observation that if the plates had sufficiently high dielectric constants, then for some physical effects the plates’ effect on the could be replaced by boundary conditions and the force between the plates could be calculated just by considering the quantum zero point fluctuations of the electromagnetic field in this environment. In this case the attractive force is called the Casimir effect. These considerations indicated that the force can be calculated either from 1. the direct electromagnetic interactions between the plates (van der Waals), or 2. the spatial dependence of the stored in the fluctuations of the electromagnetic field. The heuristic derivation of the Casimir effect is presented here. The derivation is analyzed, critically assessed and it’s physical and unphysical elements are discussed.

The prerequisites for these Essays are: 1. An understanding of at the level of the textbook, and 2. An undergraduate physics course on quantum . The fundamentals of quantum field theory will be developed within these Essays.

This Essay supplements material in the textbook: Special Relativity, Electrodynamics and : From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.

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Keywords: Casimir effect, van der Waals , Casimir-Polder, F. London, polarizable materials, electromagnetic field, zero point fluctuations.

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Contents

Van der Waals Forces ...... 2

Heuristic Derivation of Casimir Effect for Parallel Conducting Plates ...... 8

Appendix. The Euler-Maclaurin Formula ...... 17

References ...... 17

Van der Waals Forces The ultimate goal of this Essay is to improve our understanding of quantum field theory by computing the force between two parallel plates of neutral, grounded conductors. We shall take two approaches to this problem. The first is to use and attack the problem as an example of a “van der Waals” interaction, familiar from discussions of the quantum force between neutral hydrogen atoms in non-relativistic quantum mechanics. In fact, we have to do somewhat better than non-relativistic quantum mechanics here: in order to calculate the long range in this case, the relativistic, quantum of the electromagnetic field must be accounted for. When this is done, one speaks of the “retarded van der Waals force”. The second is to use field theoretic methods and calculate the change in the electromagnetic energy stored in vacuum fluctuations between the conducting plates. The second method is simple in the idealized limiting case where the conductor is “perfect” and its effect on the electromagnetic field is idealized as a boundary condition, that the electromagnetic field vanishes inside the conductor. This idealization is not completely accurate or physical, as we will discuss below, but it is adequate to calculate the long range force between the plates.

Let’s begin by thinking about forces explicitly and review the non-relativistic van der Waals force between two neutral hydrogen atoms in their . We show the geometry of the protons and electrons of the two hydrogen atoms in Fig. 1, 2

Fig.1 The coordinates of the electrons and protons of two distant but interacting hydrogen atoms.

The vector 푅⃗ between the protons is held fixed while the electrons propagate around each proton according to the laws of non-relativistic quantum mechanics. We suppose that |푅⃗ | is large, |푅⃗ | ≫

푎0, the Bohr of the hydrogen atom, so the multipole expansion of the potentials between the constituents of the two hydrogen atoms is useful and accurate. In isolation the dipole moment of each has a vanishing expectation value. In other words, for atom #1, 푑1 = 푒푟⃗⃗ 1 and 2 <1,0,0|푑1|1,0,0 > = 0 because < 푟 1 > = ∫|휓100| 푟 1 푑푟 1 = 0 by spherical . Here

휓100 means the hydrogen state with quantum numbers 푛 = 1, 푙 = 0, 푚 = 0 and the spins of the e and p are ignored and non-relativistic quantum mechanics is employed.

Before doing any calculation, let’s understand why there is a force between the two neutral atoms. At any moment 푡1 atom #1 has a dipole moment given by 푑1 = 푒푟 1. It generates an electrostatic potential at the position of the second atom #2 which effects the second atom’s instantaneous dipole moment. The electrostatic potential at the location of the second atom is, to leading order in |푅⃗ | for |푅⃗ | ≫ 푎0,

⃗ 푈(푅⃗ ) = 푑1∙푅 1.1a 4휋푅3

3 which produces the electric field,

퐸⃗ (푅⃗ ) = −∇⃗⃗ 푈 = − 푒 (푟 − 3푅̂ 푅̂ ∙ 푟 ) 1.1b 푅 4휋푅3 1 1

So, the instantaneous dipole-dipole interaction energy reads,

2 푈 = −푑 ∙ 퐸⃗ (푅⃗ ) = 푒 (푟 ∙ 푟 − 3푅̂ ∙ 푟 푅̂ ∙ 푟 ) 1.1c 12 2 4휋푅3 1 2 2 1

We see that 푈12 depends intricately on the orientation of the dipoles and their orientations to the large vector 푅⃗ between the atoms. So far this discussion is just non-relativistic Newtonian mechanics. But now we want to treat 푈12 as a perturbation in the total energy (Hamiltonian) of the two atoms. We can calculate the energy of the ground state of the two interacting atoms using perturbation theory. The first order shift in the energy is given by the matrix element of the perturbation 푈12 in the unperturbed ground state,

(1) (1) (2) (1) (2) ∆퐸 =< 휓100휓100|푈12|휓100휓100 > 1.2

(1) But 휓100(푟 1) does not support a permanent dipole moment, in other words, < 푟 1 > = 2 (1) (1) (2) ∫|휓100(푟 1)| 푟 1 푑 푟 1 = 0, so ∆퐸 vanishes identically. But the second order energy shift ∆퐸 involves a sum over intermediate states,

′ ′ ′ ′ (2) <100;100|푈12|푛푙푚;푛 푙 푚 ><푛푙푚 ;푛′푙′푚′|푈12|100;100> ∆퐸 = ∑푛푙푚;푛′푙′푚′ 1.3a 퐸100+퐸100−퐸푛푙푚−퐸푛′푙′푚′

Let’s write Eq. 1.3a out. Choose the 푧 −axis in the 푅̂ direction, so that 푈12 can be written,

2 2 푈 = 푒 (푥 푥 + 푦 푦 − 2푧 푧 ) = 푒 푃 1.3b 12 4휋푅3 1 2 1 2 1 2 4휋푅3 12 where 푃12 = 푥1푥2 + 푦1푦2 − 2푧1푧2. Then Eq 1.3a becomes,

2 2 2 (2) 푒 |<푛푙푚;푛′푙′푚′|푃12|100;100>| ∆퐸 = ( 3) ∑푛푙푚;푛′푙′푚′ 1.3c 4휋푅 2퐸100−퐸푛푙푚−퐸푛′푙′푚′

′ ′ ′ (1) (2) ′ ′ ′ where < 푟 1푟 2| 푛푙푚; 푛 푙 푚 > is 휓푛푙푚(푟 1)휓푛′푙′푚′(푟 2) and < 푛 푙 푚 |푟 1|푛푙푚 > = (1)∗ ( ) (1) ( ) ∫ 휓푛′푙′푚′ 푟 1 푟 1휓푛푙푚 푟 1 푑푟 1. First note two structural properties of Eq. 1.3c: 1. it is necessarily negative, because the energy denominator is negative for all 푛푙푚; 푛′푙′푚′, and 2. the change in

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(2) −6 (2) energy ∆퐸 falls as 푅 , which produces an attractive force law, 퐹 (푅⃗ ) = −∇⃗⃗ 푅 (∆퐸 ), which falls as −푅−7, a high power of the between the atoms.

The sum over states in Eq. 1.3c is quite a challenge. However, many terms vanish because

< 푛푙푚|푟 1|100 > = 0 for 푙 ≥ 2. This should be familiar because it is a dipole radiation selection rule and it follows from the general properties of quantum angular momentum eigenstates.

(2) −1 So much for exact results. Next, we can estimate ∆퐸 by replacing (2퐸100 − 퐸푛푙푚 − 퐸푛′푙′푚′) −1 in Eq. 1.3c with (2퐸100 − 2퐸210) and do the sum over intermediate states using completeness,

∑푛푙푚|푛푙푚 >< 푛푙푚| = 1 1.4a

Then Eq. 1.3c becomes,

2 2 (2) 푒 1 2 2 2 2 2 2 ∆퐸 ≅ ( 3) < 100; 100|푥1 푥2 + 푦1 푦2 + 4푧1 푧2 |100; 100 > 1.4b 4휋푅 2퐸100−2퐸210

2 where we have kept the quadratic terms in the product 푃12 that contribute to the matrix element and dropped the ones that vanish. To finish the evaluation of Eq. 1.4b we look up various 2 2 2 involving the stationary states of the hydrogen atom [1], < 푥푖 > = < 푟 ⁄3 > = 푎0 −1 where 푎0 is the Bohr radius which is given by 푎0 = (훼 푚) where 훼 is the fine structure constant and 푚 is the mass of the electron and we are using units in which ℏ = 푐 = 1. Finally, ∆퐸(2) is the change in the potential energy 푉(푅) of the two atom system,

5 ∆퐸(2) = 푉(푅) ≅ − 6훼푎0 1.4c 푅6 and we used the for the of the stationary states of hydrogen, 퐸푛 = 2 −훼⁄2푛 푎0.

There is another way to write Eq. 1.4c which is particularly useful in applications. The electric polarizability of the hydrogen atom expresses its induced dipole moment when it is subjected to a uniform electric field 퐸⃗ 0, 푑 = 4휋훼퐸퐸⃗ 0. Choose the external electric field in the z-direction, (2) 퐸⃗ 0 = (0,0, 퐸0), then calculate ∆퐸 in second order perturbation theory with the perturbation 푈 = −푑 ∙ 퐸⃗ = −푒푧퐸 and identify 훼 from the expression ∆퐸(2) = − 1 4휋훼 퐸⃗ 2, 0 0 퐸 2 퐸 0

2 |<푛푙푚|푧|100>| 3 훼퐸 = 2훼 ∑푛푙푚 ≅ 푎0 1.4d 퐸100−퐸푛푙푚 5

Then the potential between the two hydrogen atoms can be written,

6 2 훼 푎0 휔0훼퐸 푉(푅) ≅ − 6 ≅ 6 1.4e 푎0 푅 푅 where 휔0 is the ground state energy of hydrogen, given by the Rydberg-Balmer formula, and Eq. 1.4e is the London form of the non-relativistic van der Waals potential [2].

Next, let’s think more about the range of applicability of Eq. 1.4e. That derivation was done in the context of non-relativistic quantum mechanics where the underlying space time is Newtonian and action-at-a-distance applies. We saw this in the application of Coulomb’s Law to write down the multi-pole expansion of the electrostatic potential, Eq. 1.1a. However, if the two hydrogen atoms are sufficiently far apart then the time it takes for changes in the dipole moment in one atom to effect the dipole moment of the second atom must be accounted for. In particular, the time is takes for light to travel from one atom to the other is ∆푇1~푅, taking 푐 = 1 as before. But −1 the state of one atom changes by order one on a time interval ∆푇2~(퐸100 − 퐸210) ~휔0, where

휔0 is the energy of the ground state of hydrogen, taking ℏ = 1 as before. If ∆푇1 ≥ ∆푇2, then the quantum evolution of the dipole cannot be ignored during their interaction. In other words, the non-relativistic derivation fails when 푅 is so large that 푅휔0 ≥ 1. Since 휔0~ 훼⁄푎0, the length scale is 푅~ 푎0⁄훼 ~137푎0 which is approximately200Å. So, for 푹 ≥ ퟐퟎퟎÅ the electromagnetic effects in the problem must be treated field theoretically: action-at-a-distance fails and the finiteness of the is crucial. Detailed perturbative calculations, through fourth order in the , produce the result [3] ,

2 푉(푅) ≅ − 23ℏ푐훼퐸 1.5c 4휋푅7 where we restored the factors of ℏ and c. In this calculation the perturbation is the basic 휇 interaction of the charged current with the electromagnetic field, 푒퐴 푗휇, that we learned in classical relativistic electrodynamics. When we put this perturbation into the expression for fourth order perturbation theory, we express 퐴휇 in terms of plane waves and creation and annihilation operators, as done in the previous two Essays of this series, and evaluate the matrix elements accounting for the dynamic nature of the . The variation in the phases of the plane waves, 푒푖푘⃗ ∙푟 , in the decomposition of 퐴휇, across the interaction region is essential in the

6 derivation of Eq. 1.5c. The result Eq. 1.5c was originally derived by Casimir-Polder [4]. Casimir and Polder also looked at other examples of the retarded van der Waals interaction. For example, if a polarizable neutral atom with polarizability 훼퐸 is a distance R from a wide conducting plate held at ground, the atom experiences a potential,

푉(푅) = − 3ℏ푐훼퐸 1.5b 8휋푅4

But the major discovery that Casimir and Polder made came when they replaced the atom a distance R from the conducting plate with another conducting plate. In fact, they addressed the more general problem of computing the retarded van der Waals force between two plates of grounded dielectric materials, each having a dielectric constant 휖0 finding,

2 2 휋 ℏ푐 (휖0−1) 퐹(푅) = − 4 2 푓(휖0)퐴 1.5c 240푅 (휖0+1) where A is the of each plate as shown in Fig. 2

Fig. 2 Two dielectric plates a distance R apart

This formula has a particularly simple form in the limit that the dielectric plates become good

2 (훼퐸−1) conductors, in that case the polarizability becomes large and the factor 2 approaches unity. (훼퐸+1)

The factor 푓(훼퐸) has only weak 훼퐸 dependence and also approaches unity. Eq. 1.5c reduces to,

2 퐹(푅) = − 휋 ℏ푐 퐴 1.5d 240푅4

The factor ℏ indicates that it is a quantum effect and the factor c indicates that it is a relativistic effect. The force depends on the quantum field description of the electromagnetic field. The

7 original derivation of Eq. 1.5d as a fourth order retarded van der Waals effect was quite complicated [4] and can be reproduced with modern mathematic field theoretic methods using covariant perturbation theory, Feynman diagrams. These approaches are much more general than the limiting form Eq. 1.5d and apply to general dielectric slabs.

One might think that because the van der Waals force is a relativistic, quantum effect that it would be so tiny that it has no practical importance for macroscopic bodies. Luckily, this is not true: the force is measurable and plays important roles in modern nano-mechanics. Take a numerical example; Suppose 퐴 = 1 푐푚2 and 푅 = 1휇푚. Then substituting into Eq. 1.5d we find 퐹(푅) ≈ 1.3 × 10−7 푁 which is a significant and measurable force. In fact, early measurements [5] of 퐹(푅) agreed with Eq. 1.5d to approximately 1% and methods in nano physics are orders of magnitude better and even expose the perturbative corrections predicted by [3].

Heuristic Derivation of Casimir Effect for Parallel Conducting Plates In the textbook on classical electrodynamics we did several problems that produced the electromagnetic forces between macroscopic objects. For example, we considered a parallel plate capacitor with charges ±푄 on each plate. One way to calculate the force between the plates is to calculate the static classical electric field 퐸⃗ between the plates using Gauss’ Law and then calculate the force 퐹 = 푄퐸⃗ . Alternatively, we could consider the energy 퐸(푅) stored in the electrostatic field between the plates as a function of the distance R between the plates. Then the force is given by the R dependence of the energy stored in the capacitor, 퐹(푅) = − 휕퐸⁄휕푅.

Now turn to the problem at hand. In the case of parallel conducting plates, Casimir realized that instead of calculating the retarded van der Waals force between the two grounded conducting parallel plates, he could calculate the energy contained in the electromagnetic field between the plates and differentiate it with respect to the separation between the plates [6]. This is the same strategy we used in our classical physics example. One might question his approach because the force that Casimir is considering has its origin in relativity (it is proportional to c, the speed of light) and is quantum mechanical (it is proportional to ℏ, the ). However, if you review the connection of force to work and to energy in Newtonian mechanics, you realize that these connections are perfectly generic.

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To carry out the program in the context of quantum field theory, we begin with the expression for the energy of a free scalar field 휑(푥). Recall from the previous two Essays that the relativistic scalar field can be written as a linear superposition of plane waves each multiplied by a creation or annihilation ,

푑푝 −푖(퐸푝 푡−푝 ∙푥 ) † +푖(퐸푝 푡−푝 ∙푥 ) 휑(푥 , 푡) = ∫ 3 [푎(푝 )푒 + 푎 (푝 )푒 ] 2.1a (2휋) 2퐸푝 where the creation and annihilation operators satisfy the commutation relations,

[푎(푝 ), 푎(푝 ′)] = [푎†(푝 ), 푎†(푝 ′)] = 0

† ′ 3 ′ [푎(푝 ), 푎 (푝 )] = (2휋) 2퐸푝 훿(푝 − 푝 ) 2.1b

Then we obtain the Hamiltonian by using canonical field theory or using the energy-momentum stress tensor. The resulting Hamiltonian, derived in an appendix to the previous Essay, reads

푑푝 1 † † 퐻 = ∫ 3 [퐸푝 푎 (푝 )푎(푝 ) + 퐸푝 푎(푝 )푎 (푝 )] (2휋) 2퐸푝 2 which can be written in terms of the number operator using the basic commutation relations Eq. 2.1b,

푑푝 † 3 2 ⃗ 퐻 = ∫ 3 [퐸푝 푎 (푝 )푎(푝 ) + (2휋) 퐸푝 훿(0)] 2.1c (2휋) 2퐸푝

The second term, which is divergent, follows from the lack of commutativity of the creation and annihilation operators and registers the fact that there is a quantum oscillator at every spatial point and the quantum zero point energy of a of frequency 휔 reads 1 ℏ휔. If 2 one quantizes the system in a finite volume V, then Eq. 2.1c implies,

퐸 =< 0|퐻|0 > = ∑ 1 퐸 2.2a 푉푎푐 푝 2 푝

푑3푝 where we recalled ∑ → 푉 ∫ and 푉 → (2휋)3훿(0⃗ ). This formula, Eq. 2.2a, shows that the 푝 (2휋)3 quantum field is an infinite collection of harmonic oscillators per volume, each one labelled by 푝 . The zero point energy of each of these oscillators is 1 퐸 and Eq. 2.2a expresses this idea. 2 푝

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In many applications only energy differences matter and one defines the “normal ordered” Hamiltonian,

푑푝 † : 퐻: = 퐻−< 0|퐻|0 > = ∫ 3 퐸푝 푎 (푝 )푎(푝 ) 2.2b (2휋) 2퐸푝

However, there are some fundamental applications, such as general relativity, where the energy itself matters and the of the Hamiltonian is potentially physical! We discussed this perplexing point in the textbook when we reviewed the and the accelerating, expanding universe, a subject of current research, that is not fully understood.

Now we are ready to calculate the force between parallel plates of area A and separation R by considering the . In particular, we consider the energy in the presence of the parallel plates, 퐸푉푎푐(푅) and compare that to the energy in the absence of the plates 퐸푉푎푐,

퐸(푅) = 퐸푉푎푐(푅) − 퐸푉푎푐 2.3a

Note that each term in Eq. 2.3a has two sources of divergences. First, the volume of space and second high energies, as expressed in Eq. 2.2a. The difference in Eq. 2.3a will only make sense and be finite if these divergences are dealt with in the same fashion in each term.

Another way to view Eq. 2.3a is in analogy with the calculation of the energy stored in a charged capacitor in classical electrodynamics. There we imagine assembling a parallel plate capacitor by bringing in one plate from infinity and calculating the work done. Then the work done is the energy stored in the device. So, we can write Eq. 2.3a in the form,

퐸(푅) = 퐸푉푎푐(푅) − 퐸푉푎푐(푅 → ∞) 2.3b

From this viewpoint Eq. 2.3a-b are completely conventional in Newtonian mechanics. We must, however, face the new features in each term in Eq. 2.3a: there are singularities and the high energy divergence is special to quantum field theory and has no analog in classical physics!

Now let’s do the summation ∑ 1 퐸 in Eq. 2.3a. The sum over 푝 , perpendicular to either plate 푝 2 푝 ⊥ of area A, is distinct from that over 푝∥, parallel to either plate. In the perpendicular direction, the electromagnetic field has possible values of the perpendicular momentum, 푝 = 푛휋, 푛 = 0,1,2, … ⊥ 푅 to accommodate the boundary conditions, the vanishing of the field at the surface of each plate

10 which are a perpendicular distance R apart as shown in Fig. 2. For this calculation where we only need the energy contained in the electromagnetic field, we can treat the field as a massless scalar field with two possible states to account for its vector character.

At this point we have replaced the real conducting plates by simple boundary conditions. The physical limitations of this replacement are discussed later in this Essay and were stressed originally by Casimir [6]. Note here that the electromagnetic coupling, the fine structure constant, 훼, must be substantial to enforce such a replacement. The validity of the replacement depends on the properties of the conductor and will also be discussed elsewhere in this Essay.

Next the components of the momentum parallel to the plates, 푝∥, can be treated as a continuum

2 2 variable since there are no boundaries in that direction. In these variables, 퐸푝 = √푝∥ + 푝⊥ in the expression Eq. 2.2a for 퐸푉푎푐. We must also account for the number of polarization states of the electromagnetic field. In free space a freely propagating electromagnetic wave occurs with two possible polarizations. These can be counted as linear polarizations, or, taking linear superpositions, as circular polarizations, either left or right-handed. However, for the discrete mode 푝⊥ = 0, 푛 = 0, there is only one polarization state available. Accounting for these details we can write Eq. 2.3a,

푑2푝 1 푑2푝 1 푛휋 2 푑푝 1 퐸(푅) = 퐴 ∫ ∥ |푝 | + 2퐴 ∫ ∥ ∑∞ √푝2 + ( ) − 2퐴푅 ∫ ⌈푝 ⌉ 2.3c (2휋)2 2 ∥ (2휋)2 푛=1 2 ∥ 푅 (2휋)3 2

These integrals are divergent because the integrands grow at high energies – they are ultra-violet divergent. To deal with this problem formally (we will discuss the underlying physics which will motivate these manipulations later in this section), introduce a smooth ultra-violet “cutoff” in the form of a modification of the integrand by inserting an exponential of the form exp (−퐸/Λ). So, Λ acts as an ultra-violet cutoff. Of course, at this point, these factors are just computational artifacts and the physical results extracted from these equations must prove to be independent of Λ and the particular used. However, the exponential cutoff procedure does “make sense” in this application: it states that the high frequency fluctuations of the field do not contribute to 퐸(푅) . This is physically correct: frequency fluctuations above the conductor’s plasma frequency are unaffected by the conductor. Recall that conductors are transparent to waves with frequencies above the plasma frequency. The plasma frequency acts as a “natural

11 high energy cutoff” for this problem. We will discuss this point further below when we introduce more general cutoff schemes. Of course, the physical results we are interested in will prove to be independent of the cutoff procedure.

Returning to the problem at hand, we have Eq. 2.3c “regulated” as,

2 푛휋 2 1 푑2푝 푑2푝 푛휋 −√푝2+( ) ⁄Λ 퐸(푅) = 퐴 ∫ ∥ |푝 | 푒−|푝∥|⁄Λ + 퐴 ∫ ∥ ∑∞ √푝2 + ( ) 푒 ∥ 푅 − 2 (2휋)2 ∥ (2휋)2 푛=1 ∥ 푅

푑푝 푑2푝 −√푝2+푝2 ⁄Λ 퐴푅 ∫ ⊥ ∫ ∥ √푝2 + 푝2 푒 ∥ ⊥ 2.3d 2휋 (2휋)2 ∥ ⊥

In order to see the systematics here and to complete the evaluation of 퐸(푅), define the function 퐹(푥),

푥휋 2 2 ∞ 푦 푑푦 −√푦2+( ) ⁄Λ 푥휋 ∞ 푑푧 푅 2 −√푧⁄Λ 퐹(푥) = ∫ 푒 √푦 + ( ) = ∫ 푥휋 2 푒 √푧 2.4 0 2휋 푅 ( ) 4휋 푅

Now the energy of interest, Eq. 2.3d, can be written more elegantly as,

1 ∞ 퐸(푅) = 퐴 [ 퐹(0) + ∑∞ 퐹(푛) − ∫ 퐹(푥)푑푥] 2.5a 2 푛=1 0

But here we meet the difference between an infinite sum and its related . This difference can be evaluated with the Euler-Maclaurin Formula [7] familiar from introductory calculus courses and reviewed in Appendix A,

∞ 1 1 1 ∑∞ 퐹(푛) − ∫ 퐹(푥)푑푥 = [퐹(∞) − 퐹(0)] + [퐹′(∞) − 퐹′(0)] − [퐹′′′(∞) − 퐹′′′(0)] + 푛=1 0 2 12 720 ⋯ 2.5b

But we read off Eq. 2.4 that in our application, 퐹(∞) = 퐹′(∞) = 퐹′′′(∞) = 퐹′(0) = 0 so Eq. 2.5a reduces to,

2 퐸(푅) = 퐴 [1 퐹(0) − 1 퐹(0) + 1 퐹′′′(0)]=− 휋 ℏ푐 ∙ 퐴 2.5c 2 2 720 720 푅3

2 where we have evaluated 퐹′′′(0) = − 휋 from Eq. 2.4 and have restored factors of ℏ and c in Eq. 푅3 2.5c. We also made the crucial point in that equation that the terms 1 퐹(0) and − 1 퐹(0) cancel in 2 2 퐸(푅) showing that the ultra-violet divergent piece of the vacuum energy cancels out in the

12 difference Eq. 2.3b Another important point to check in Eq. 2.5b is that the high order terms, indicated by … , vanish in the limit Λ → ∞.

The measurable quantity here is the force between the plates,

2 퐹(푅) = − 휕 퐸(푅) = − 휋 ℏ푐 ∙ 퐴 2.5d 휕푅 720 푅4 which is the Casimir-Polder retarded van der Waals force [4], obtained by Casimir [6] by this deceptively(!) simple derivation. It was confirmed experimentally ten years after its derivation [5].

Let’s investigate the cutoff procedure in more generality and let’s understand the range of frequencies and wavelengths in the electromagnetic field that is responsible for the Casimir force 퐹(푅). Introduce a cutoff function 휂(푘),

휂(푘) ≅ 1 , 푘 < Λ

휂(푘) → 0 , 푘 ≥ Λ where Λ is the cutoff , as shown in Fig. 3

Fig 3. The generic cutoff function 휂(푘).

We will see that the particular shape of 휂(푘) will not effect the Casimir force 퐹(푅): only the generic properties of 휂(푘) will play a role in the procedure. With this regulator Eq. 2.3d becomes,

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1 푑2푝 푑2푝 푛휋 2 푛휋 2 퐸(푅) = 퐴 ∫ ∥ 휂(|푝 |)|푝 | + 퐴 ∫ ∥ ∑∞ 휂 (√푝2 + ( ) ) √푝2 + ( ) − 2 (2휋)2 ∥ ∥ (2휋)2 푛=1 ∥ 푅 ∥ 푅

푑푝 푑2푝 퐴푅 ∫ ⊥ ∫ ∥ 휂 (√푝2 + 푝2 ) √푝2 + 푝2 2.6a 2휋 (2휋)2 ∥ ⊥ ∥ ⊥

Now we apply the Euler-Maclaurin formula here as done above for the exponential cutoff. We 2 2 2 2 2 2 define 푥 = 푝∥ 푅 ⁄휋 + 푛 , so 푝∥푑푝∥ = 휋 푑푥⁄2 푅 and define the function 푓(푛) in order to apply the Euler-Maclaurin formula,

휋2 ∞ 푓(푛) = ∫ 푑푥 휂(휋√푥⁄푅) √푥 2.6b 4푅3 푛2

As before we only need a few values of 푓(푛) and its derivatives to extract the leading R dependence in 퐸(푅) in Eq. 2.6a. Using 휂′(0) = 휂′′(0) = 0, we read off from Eq. 2.6b, 푓′(0) = 푓′′(0) = 0 and 푓′′′(0) = −휋2⁄푅3. Substituting into Eq. 2.5c we find again,

2 퐸(푅)=− 휋 ℏ푐 ∙ 퐴 2.5c 720 푅3

As we discussed earlier, in a real conductor the plasma frequency 휔푝 acts as an effective cutoff on the frequencies in the unregulated expression for 퐸(푅). But the Casimir force does not depend on 휔푝. The part of the spectrum which contributes to 퐸(푅), and consequently 퐹(푅), are those wavelengths 휆~푅 and frequencies 휔~ 푐⁄푅 – the dimensional quantities characterizing the conductor do not influence this part of the spectrum so we obtain a simple result where the details of the conducting material do not appear – the result appears universal. However, we have learned that it is a very deceptive simplicity!

We noted earlier that the dielectric constant dependence of the Casimir-Polder retarded van der Waals force had a simple limit such that the properties of the conductor disappeared from the final result: Eq. 1.5c depended on 휖0, the dielectric constant of the slabs, through the factor 2 2 (휖0 − 1) ⁄(휖0 + 1) which approaches unity for a good conductor which has a large 휖0. This is how Casimir found the simple scaling form for 퐹(푅) for a good conductor. One can make simple models of this result. Consider field theoretic models in one space – one time dimension. Let there be a scalar field 휑(푥) and model the two slabs of material with the external source, 휎(푥) = 훿(푥 − 푅⁄2) + 훿(푥 + 푅⁄2) and couple the field to this source, ℋ = − 1 푔 휎(푥)휑2(푥). Note 푖푛푡 2

14 that as 푔 becomes large, 휑(푥) is forced to vanish at 푥 = ± 푅⁄2, so the model approaches the ideal limit of a boundary condition. The coupling 푔 has the dimension of [mass], the field theory is called “super renormalizable”: 푔 sets a scale for energies and interactions, for energies above 푔 the excitations are almost non-interacting. The model is just a one dimensional model of old fashioned models of nuclear forces: 휑 represents the (scalar) cloud and the “nuclei” are point sources at 푥 = ± 푅⁄2. These models are solvable because their Hamiltonians are just quadratic forms in 휑 which can be diagonalized by a Bogoliubov transformation, as discussed already in the previous two Essays of this series. It can also be solved by explicitly summing Feynman diagrams [8]. The model lacks “back reaction”, in other words, the source 휎(푥) is fixed and unresponsive to the field 휑(푥). This unrealistic feature makes the model solvable but, on the negative side, guarantees that it is not an interesting model of a conductor. Nonetheless, it is instructive to obtain the exact expression for the force between the source at 푥 = + 푅⁄2 and that at 푥 = − 푅⁄2 [8],

∞ 푔2 푡2 푑푡 푒−2푅푡 퐹1푑(푅) = − ∫ ∙ 휋 √푡2 − 푚2 4푡2 + 4푔푡 + 푔2(1 − 푒−2푅푡 ) 푚

Note that the integral is dimensionless and 푔 has the dimension of [mass] or [length]−1. The 푔 dependence of 퐹1푑(푅) is interesting and relevant to our discussions of the Casimir effect. When

푔 → 0, 퐹1푑(푅) becomes perturbative and vanishes. So, the force is not “universal”. However, when 푔 becomes large,

∞ 1 푡2 푑푡 1 퐹1푑(푅) → − ∫ ∙ 휋 √푡2 − 푚2 (푒2푅푡 − 1) 푚

The mass of the field 휑(푥) can be set to zero here and the integral can be done exactly,

∞ ∞ 1 푡 푑푡 1 푥 푑푥 휋 퐹 (푅) → − ∫ = − ∫ = − 1푑 휋 (푒2푅푡 − 1) 4휋푅2 (푒푥 − 1) 24푅2 0 0 which is the one dimensional analog of the three dimensional result Eq. 2.5d. (The reader is encouraged to work through the details of the discussion from Eq. 2.2a to 2.5d for one spatial dimension to verify this point.) This calculation illustrates that when 푔 becomes large this fixed

15 source model produces the same Casimir force as the boundary value problem. But at small and intermediate 푔, the full result is more complicated and depends explicitly on the coupling 푔.

Let’s end with a few observations and comments about Casimir’s famous result. Although the heuristic derivation of the Casimir force produces the correct answer, it suffers from many limitations and the same procedure can produce unphysical results for other quantities such as stresses on the edges of the parallel plates [6]. Since the formula Eq. 2.5d does not contain the strength of the interaction between charges and the electromagnetic field, some authors present it as a “universal effect”, a consequence of the physical reality of the zero-point fluctuations of . This is clearly an over-statement. Recall that the original derivation of the Casimir force was done calculating the retarded van der Waals force between plates of dielectric material. Those results depend on the dielectric constant of the material as discussed above. Only in the limit of large polarizability does the formula reduce to Eq. 2.5d. But a large polarizability requires a large coupling between the electromagnetic field and the electric current. In fact, if one pursues models of conducting materials one finds [3] that as long as the fine structure is greater than 10−5, then the model of the conducting material makes sense: the material screens low frequency external electric fields so that they do not penetrate significantly into the material and the replacement of the material by a boundary condition is accurate for the calculation of the Casimir force. Since 훼 = (137)−1, this approach is useful and accurate. We learn from this that Eq. 2.5d is not universal at all – it is the limiting case of a formula for a dielectric material that requires a substantial electromagnetic coupling.

Now to the second point: is the Casimir force a manifestation of the reality of vacuum fluctuations in the electromagnetic field? First, Casimir and Polder derived the force as a retarded van der Waals effect with no reference to the quantum vacuum. In addition, we saw in Eq. 2.5c that the zero point energy of the vacuum cancels out of the finite physical effect. Apparently the heuristic calculation handles the ultra-violet divergent parts of the calculation well enough to obtain the contribution from the long range correlations of the electromagnetic fields due to the conducting plates. Of course, the heuristic calculation is so popular because it is so simple as opposed to the more physical, realistic calculation of the retarded van der Waals force!

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Appendix. The Euler-Maclaurin Formula This is the classic formula in calculus that gives the difference between a sum and an integral. Suppose there is a continuous function 푓(푥) for x over the region [푚, 푛] where m and n are 푛 ( ) ∑푛 integers. Then the integral 퐼 = ∫푚 푓 푥 푑푥 can be approximated by the sum 푆 = 푖=푚+1 푓(푖). The theorem states that,

푝 퐵 푆 − 퐼 = ∑ 푘 (푓(푘−1)(푛) − 푓(푘−1)(푚)) + 푅 푘! 푝 푘=1 where 퐵 are the Bernoulli numbers (퐵 = 1 , 1 , 0, −1 , 0, 1 , 0, …) and 푅 is a remainder term. 푘 푘 2 6 30 42 푝

Writing out the formula explicitly,

푛 푛 ∑ 푓(푖) − ∫ 푓(푥)푑푥 푖=푚+1 푚 푓(푛) − 푓(푚) 1 푓′(푛) − 푓′(푚) 1 푓′′′(푛) − 푓′′′(푚) = + − 2 6 2! 30 4! 1 푓(5)(푛) − 푓(5)(푚) + + ⋯ 42 6!

The first two terms in this expression should be familiar from introductory calculus texts and the general derivation can be found in the references [7].

References 1. The Principles of Quantum Mechanics, P. Dirac, Oxford University Press, Oxford, U.K. 1930. 2. E. M. Lifshitz, Sov. Phys. JETP (USA) 2, 73 (1956). 3. W.M.R. Simpson and U. Leonhardt, Forces of the Quantum Vacuum, World Scientific Publishing Co., Singapore, 2015: M. Bordag, U. Mohideen and V.M. Mosteparnenko, arXiv: quant-ph/0106045 v1 8 June 2001. 4. H.B.G. Casimir and O. Polder, Phys. Rev. 73, 360 (1948). For a pedagogical discussion see B.R. Holstein, Am. J. Phys. 69(4), April 2001. 5. M.J. Sparnaay, Physics 24, 751 (1958).

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6. H.B.G. Casimir, Proc. Kon. Ned. Akad. Wet 60, 793 (1948). 7. T.M. Apostol, Am. Math. Monthly 106(5), pg. 408-418; M Abromowitz and I.A. Stegun, Handbook of Mathematical Functions Dover Pub. Co., NY, 1972. 8. R.L. Jaffe, arXiv: hep-th/0503158v1 21 Mar 2005, and references therein.

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