Quantum Vacuum Energy and the Casimir Effect

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Quantum vacuum energy and the Casimir eﬀect

Author: Guim AguadéGorgorió Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ (Dated: January 15, 2014) Abstract: An overview of the Casimir effect is presented. The area of study is historically introduced, together with a basic presentation on the zero-point energy concept in relation with the quantization of fields. After this, the Casimir force between parallel plates is calculated under two different regularization schemes. The brief overview is completed with a compilation of experimental results and implications of the effect in different fields of physics.

I. INTRODUCTION for which the energy at lowest temperature was a non- zero term hν/2. This was the beginning of the concept of It was 65 years ago when Hendrik Casimir published a zero-point energy, as the lowest possible energy quantum short paper [1] on the attraction force between two par- state being different from zero. allel conducting plates due to changes in the quantum In ordinary quantum mechanical systems, the not-null electromagnetic zero-point energy, known as the Casimir ground state may be seen as a consequence of the Heisen- effect. berg uncertainty principle, with the energy of the har- The concept of zero point energy had been a hot topic monic oscillator being its clearest example. The term since it appeared in the Planck’s black-body radiation quantum vacuum, although considered equivalent, was discussion by Einstein and Stern; a 1913 article on the coined as a result of the concept of ground state energy energy of an hydrogen gas at low temperature, which con- within the framework of quantum field theory. cluded that a non-zero residual energy at T = 0 might Quantization of fields begins on the postulates of quan- exist [2]. tum mechanics, replacing the pair [x, p] by the field In the following decades, the work by Casimir remained ϕ(x, t) and conjugate canonical momentum πi = ∂L/∂ϕ˙i. relatively unknown, probably because of controversy on In the one-dimensional case (by means of what is known the value of the ground state of quantum vacuum energy, as second-quantization), a quantized string (0, a) is con- which was commonly redefined to zero. It was not un- sidered as a set of quantum harmonic oscillators as til the 70s when experiments started to became precise X − + + enough to measure the Casimir force and assure its exis- ϕ(t, x) = anϕn (t, x) + an ϕn (t, x) (2) tence as evident. n Nowadays, the Casimir effect together with quantum with ϕ± being the solutions of the usual wave equa- vacuum energy is again highly popular in the scope of n tion. The annihilation and creation operators a , a+ QFT, together with a set of different applications and n n are those obeying the commutation relations of QM (or consequences that may go from cosmology to nanotech- anti-commutation relations for 1/2-spin particles), and nology. so the quantum state is defined by a |0i = 0. Even In the following sections, the basics of zero-point en- n though quantum field theory is a giant and expanding ergy in QED are introduced, followed by a first calcula- field of physics, we may particularly center in our one- tion of the Casimir force and a brief overview of infinite dimensional example to study the energy of this vacuum integrals regularization. A bibliographical compilation |0i state. of experiments and applications of the Casimir effect is The 00-th component of the energy-momentum ten- presented. sor is the operator for the energy density, and so what we are basically looking for in the present article is to II. ZERO-POINT ENERGY AND FIELD understand QUANTIZATION Z vacuum E0 = h0|T00(x)|0idx. (3) During the first decades of the last century, and in the beginnings of quantum theory, a paper by Einstein and By looking at appendix VI.A, we observe the form of Stern [2] on the basis of Planck’s previous work stated h0|T00(x)|0i in free or bounded states. It is important to that the energy of a vibrating unit was remark that this expressions are in terms of sums or integrals over all possible frequency values, which we must hν hν not confuse with the integrals among space. The results = hν + (1) e kT − 1 2 for bounded and free-space energies are a Z ∞ bounded ~ X E0 (a) = h0|T00(x)|0idx = ωn, (4) ∗ 2 Electronic address: [email protected] 0 n=1 Quantum Vacuum Energy and the Casimir Effect Guim AguadéGorgorió

∞ L Z equivalence between using the Euler-Maclaurin and the Efree(−∞, ∞) = ~ ω dk, (5) 0 2π k Abel-Plana formulae to proceed with the damping regu- 0 larization, together with the usefulness of the Abel-Plana formula to understand the independence of the damping (where L → ∞), and their respective frequencies (for function. a massless ﬁeld) πn ωbounded = c , n = 1, 2, ..., (6) n a B. Regularizations of the ultraviolet divergence

ωfree = ck, − ∞ < k < ∞. (7) The damping function regularization appears by a k physical intuition process. As the conducting plates be- This is, at a basic point, enough information to ap- come transparent to high-frequency waves, the contribu- proach the comprehension of the vacuum energy of the tion of these to the physical result may be supressed by electromagnetic ﬁeld. the use of a damping exponential function

ZZ ∞ ~ dk1dk2 X −δω E (a, δ) = S ω e k⊥,n . (11) III. THE CASIMIR EFFECT 0 2 (2π)2 k⊥,n n=−∞ A. The quantized electromagnetic field between plates The expression is finite for δ > 0 and regularization will give the correct result in the limit of δ → 0. If we remove the plates, the regularized vacuum energy in As mentioned in the introduction, the Casimir effect the same interval (0, a) accounts from changing L → ∞ results from the changes of the presented QED zero-point to L = a energy due to external conditions. In this case, the pres- ence of two perfectly conducting parallel plates deter- ZZZ Efree(a, δ) = aS ~ d3kω e−δωk S. (12) mines a boundary condition, and so the frequencies of 0 (2π)3 k the radiation between the plates are restricted to a dis- crete set of values. With these, we may write the regularized potential en- On the surface S of a perfect conductor, both polar- ergy of the system, i.e. the energy to bring the plates izations Ek and H⊥ are zero, and so the existence of two from a large separation to a, as plates with area S → ∞ and separation z = a quantizes the possible frequencies in the three dimensional case to ren bounded free U0 (a) = lim E0 (a, δ) − E0 (a, δ) (13) r δ→0 πn2 ω = ω = c k2 + k2 + n = 0, 1, 2... (8) k⊥,n 1 2 2 2 2 a As in [3], we use k⊥ = k1 + k2 and t = ak3/π. The . expression to evaluate is then Now, by the same process done in II and appendix ∞ VI.A, we just need to take the three-dimensional case to c π ZZ dk dk ren ~ 1 2 X −δωk ,n obtain the vacuum energy U0 (a) = S lim F (n)e ⊥ a δ→0 (2π)2 n=0 ∞ ZZZ ZZ ∞ 3 ~ 1 X Z ! E0 = d x dk1dk2 ωk ,n. (9) k⊥a (2π)2 2a ⊥ − dtF (t)e−δωk − (14) n=1 2π 0 Integrating dxdy in area S and dz in [0, a], with ZZ ∞ ~ dk1dk2 X E0(a) = S ωk ,n, (10) v 2 (2π)2 ⊥ u !2 n=−∞ u k a F (x) = t ⊥ + x2. (15) where the sum has been extended as to account the π two photon polarizations when n 6= 0. This expression diverges for large photon momentum We can now recognize the difference between the sum k. This ultraviolet divergence is a common problem in and the integral as a common and useful expression. We QFT, and it is treated through regularization. In the may first evaluate it by the Abel-Plana formula, and following I will introduce two different regularizations to through this recognize the independence of the damp- achieve the Casimir force result. These are the damping ing function f(ωk⊥,n, δ) used in Casimir’s assumption to function (as originally used by Casimir [1]) and the zeta compute (13) by the Euler-Maclaurin formula. function regularizations. In addition, I will present the Applying the Abel-Plana formula on (14) and using

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y = k⊥a/πwe obtain (see Appendix VI.B) Before going on to conclusions, a completely diﬀerent regularization process is presented. This is the zeta func- ∞ t tion regularization. It is a much modern, commonly used c π2 Z dt Z ren ~ p 2 2 process in QFT, based on the analytic continuation of the U0 (a) = −S 3 2πt y t − y dy a e − 1 Riemann zeta function ζ(s). y 0 We rewrite (10) with a new regularization parameter c π2 = − ~ S. (16) s as 720a3 ∞ ZZ ~ X dk1dk2 1−2s The force between the plates is the derivative of the E0(a, s) = S ω . (20) 2 (2π)2 k⊥,n potential energy with respect to their distance n=−∞

∂U ren(a) π2 c As in [3], we use polar coordinates and k = y(πn/a) F (a) = − 0 = − ~ S. (17) ⊥ ∂a 240 a4 to obtain (Appendix VI.D)

This is the Casimir force between two parallel conduct- ∞ 3−2s Z 1 −s ∞ ! c 2 X πn ing plates due to QED vacuum energy. We used the Abel- E (a, s) = S ~ dyy y2 +1 . (21) Plana formula, but many authors use Casimir’s original 0 2π a n=1 calculation ([4]). In the mentioned original paper, instead 0 of introducing an exponential function, Casimir asked for From here, we must observe the Riemann zeta function a more general f(k/km), unity for k << km and tending ∞ to zero for (k/km) → ∞. With this, he calculated (13) X 1 by the means of the Euler-Maclaurin summation formula ζR(t = 2s − 3) = . (22) nt n=1 p ! X Bk S − I = f k−1(m) − f k−1(0) (18) However, we need the value ζ (−3) in order to remove k! R k=2 the regularization and obtain the unique physical result. Apparently, this value diverges, but we may obtain (see where m → ∞, S states for the sum and I for the in- [6]) the analytic continuation to the whole complex plane, tegral that appear on (14). The terms F k(m → ∞) are described by the functional equation zero for the assumptions on f(k/km). In our calculations, F (n)(x) ∝ (π/ak )K , with K > 0, for n ≥ 4. With this, πs m ζ(s) = 2sπs−1 sin Γ(1 − s)ζ(1 − s) (23) the calculations ask for akm >> 1 to meet the correct 2 result. See Appendix VI.C for an insight on Casimir’s calculation. at s = 4. This gives us the value ζR(−3) = 1/120. At last, what is important to understand, at the mo- Performing the integral and removing the regularization ment we choose one or another regularization, is that the by putting s = 0 (see Appendix VI.D) we obtain the ex- final result should be independent of the exact shape of pected expression (16) for the energy, and applying (17) f(δk). This is a direct conclusion taken from the form we obtain the expected Casimir force. of the Abel-Plana formula (Appendix VI.B). Under the It appears as we did not start on the basic notion of assumptions of f being analytic in the positive reals of calculating the difference between two energies, but eval- C, and that uating the energy expression at a given point or distance, which is not what we would physically understand. Be- −2πy lim f(x ± iy) e (19) sides the ad hoc character of the zeta function regulariza- y→∞ tion, a deeper insight - away from the scope of this work - goes to zero uniformly in x, the Abel-Plana formula would recall on how the analytic continuation (23) has an holds and its right-hand side integral converges, and so additive positive infinity, which accounts for the energy the general damping function may just be driven to zero outside the plates and so may give the physical intuitive by the limit δ → 0. It is imporant to remark that the answer we were expecting. Euler-Maclaurin formula does not ask for an analytic function, but it does need some constraints in the results of the derivatives f k−1 in both limits of integration. IV. EXPERIMENTS AND APPLICATIONS Together with the independence of f, the regularization process is also independent of the summation for- In the preceding section we have seen the particu- mula. This is, in fact, because under the mentioned con- lar situation of the effect between two parallel conduct- straints of f the Abel-Plana and the Euler-Maclaurin for- ing plates. Many further calculations have been done mulae are equivalent [5]. The demonstration of this is not in many different geometries and boundary conditions, trivial, but it clearly runs through the approximation the- together with continuous developments on QFT diver- orem of Weierstrass to connect the analytic constraints gences. of Abel-Plana and the f ∈ C(2)[0, ∞] constraint of Euler- Together with these, experimental tests have been go- Maclaurin. ing on and gaining more and more precision since the

Treball de Fi de Grau 3 Quantum Vacuum Energy and the Casimir Effect Guim AguadéGorgorió

Casimir effect started gaining greater importance around experimental advances enlighten the paths of a growing the 1970s. number of theoretical developments on the effect. This It is important to remark the technical difficulties of rapid growth expands to different areas of modern phys- measuring this effect. For these, early experiments did ical science, making the Casimir effect an interesting, not obtain remarkably precise results, but a rather qual- mysterious and significant physical phenomenon. In the itative evaluation. It was not until 1997 when an exper- scope of theoretical physics, its possible implications on iment by S.K. Lamoreaux [7] measured the direct force the Cosmological Constant problem may be of extraor- between plates and was precise enough to consider it, dinary value. somehow, conclusive. The difficulties lay on the quantum dimensions of the macroscopic effect to measure, together with complica- VI. APPENDIX tions on obtaining parallel, high conducting, free of impu- rity plates. By (17), we observe how the force is strongly In order to avoid extense calculations in the text, I dependent on the distance and only considerable at val- remitted to the appendix some important steps on each ues a ∼ 1µm, where it becomes dominant. The force is calculation done. The basics of second-quantization arise around 10−7N for two plates with surface of 1cm2 area from [3] and my university courses. Both damping and and a = 1µm. zeta function regularizations calculations follow the in- More recent experiments (around year 2000) using the dications from [3]. Casimir’s original Euler-Maclaurin atomic force microscope have become definitive on the process is calculated from [1]. The connection between measurement of the Casimir force, with claims of a 1% the Abel-Plana and the Euler-Maclaurin summation for- statistical precision [8]. mulae is stated by the author using [5]. The modern implications of the Casimir effect, to- VI.A In the standard quantization of the scalar field, gether with general developments on quantum fluctua- starting by the usual equation (for a massive field) tions, occupy a wide spectrum of physics fields, from deep 2 2 theoretical physics consequences on quantum field theory, 1 2 2 m c ∂t ϕ − ∂xϕ + ϕ(t, x) = 0 (24) nucleon models [4], cosmology and gravitation to modern c2 ~2 advances on applied physics and technological uses, such as in modern nanotechnology chips trying to harness the we obtain the bounded and free solutions by means of power of the Casimir effect [9]. contour conditions as One of the most remarkable implications of the Casimir c 1/2 ± ±iωnt effect and the quantum vacuum energy would be its possi- ϕn (t, x) = e sin knx (25) aωn ble relation with Einstein’s cosmological constant in general relativity, as to explain the mechanism for the ac- c 1/2 celerated expansion of the universe. As in the Casimir ϕ±(t, x) = e±i(ωt−kx) (26) effect calculation, the difference between the zero-point k 4πω energy of the expanding universe and the one of the flat with respective expressions for their frequencies (6) Minkowski space cancels the common large term and and (7). Now, the 00-th component operator leaves a finite and small contribution, perhaps associa- ble to the Cosmological Constant [10]. ! c 1 T (x) = ~ (∂ ϕ)2 + (∂ ϕ)2 (27) 00 2 c2 t x

V. CONCLUSIONS brings us, for both bounded and free expressions of (2), to the expressions • The Casimir effect, and in particular the Casimir ∞ 2 4 ∞ force between two parallel conducting plates, exists as ~ X m c X cos 2knx proposed, and is nowadays experimentally confirmed h0|T00|0ibounded = ωn − , (28) 2a 2a~ ωn with high precision techniques. It is a macroscopical n=1 n=1 quantum effect, and it is considered a strong pilar for ∞ the evidence of zero-point energy in quantum field the- Z ~ ory. h0|T00|0ifree = ωdk. (29) • Some important concepts with general use in quan- 2π 0 tum field theory have been discussed, such as different regularization schemes to calculate divergent integrals. The integration of this expressions in [0, a] results in (4) Besides these, the idea of measuring energy differences and (5) (The second term of the first expression doesn’t has been settled as the starting point for a broad range contribute to the integral for both massive and massless of physical problems. fields). • Although it was stated originally in 1948, the Casimir VI.B The first change of variables to rewrite equations effect is a hot topic in nowadays physics, and continuous (11) and (12) in the terms of F (x) is direct. The main

Treball de Fi de Grau 4 Quantum Vacuum Energy and the Casimir Effect Guim AguadéGorgorió step of the damping function regularization presented is is F 000(0) = −4, as higher derivatives are in terms of on the use of the Abel-Plana formula (π/akm), and go to zero as akm = a/δ >> 1. From here, obtaining the Casimir force (17) is direct. ∞ ∞ ∞ X Z 1 Z f(it) − f(−it) dt VI.D The first step in the zeta function regulariza- f(n) − f(t) − f(0) = i 2 e2πt − 1 tion is to rewrite (20) in order to obtain ζ(2s − 3). We n=0 0 0 use again the polar coordinates (k⊥, ϕk), and the change (30) k⊥ = y(πn/a). We may see where we identify f(x) = F (x)e−δω(x). 1 −s The convergence of the right-hand side integral lets us ! 2 1 2 2 −s 1−2s apply the limit δ → 0 directly. What we need to un- 2 πn 2 πn k⊥ + = y + 1 . (34) derstand is evaluation of F (it) − F (−it) by the means a a of complex calculus. The square roots will have branch- 2 ing points ±iy to be rounded, and so F (it) − F (−it) = The other (πn/a) come from dk1dk2 = k⊥dk⊥ = 2ipt2 − y2. (πn/a)y(πn/a)dy. The demonstration of the functional The intermediate step of equation (16) runs through or reflection equation (23) is not trivial nor short [6]. The (x = 2πt). What remains is the evaluation of a integral key observation of the extension of ζ to the whole com- by parts and a well known integral to be evaluated by plex plane is the observation that in 0 < Re(s) < 1, ζ(s) means of complex calculus or the gamma function and has a symmetry in relation with ζ(1 − s), and together the Riemann zeta function. with the expansion of Γ in the complex plane and many other developments we obtain (23). ∞ 3 4 Z dxx π Once implemented the ζR(−3) value, the sum returns = Γ(4)ζ(4) = . (31) 3 ex − 1 15 (π/a) (1/120). The remaining integral converges under 0 the limit as VI.C The actual expression used by Casimir was ∞ Z 1 lim dyy(y2 + 1)(1/2)−s = − (35) ∞ ∞ Z s→0 3 X 1 1 0 F (n)dn − F (x)dx = − F 0(0) + F 000(0) + ... 12 720 (0)1 0 and so the final result for the energy is (16), as ex- (32) pected. where the subindex (0) accounts for the non polariza- tion of the n = 0 mode, and so contains the first term (1/2)F (0). Casimir took into account the space integrals Acknowledgments and so used an already integrated expression for F

∞ Z Foremost, I would like to express my gratitude to my 2 (1/2) 2 F (u) = dx(x + u ) f(n π/akm) (33) advisor Joan Sol`afor his valuable time and patience on 0 my doubts and ignorance, together with his respect and understanding of my personal situation. I also thank with km being in fact the inverse regularization pa- my university colleagues for everyday supporting and my rameter. With this, the only non-vanishing derivative family for being always by my side.

[1] H.B.G. Casimir, 1948, On the attraction between two per- [6] Harold M. Edwards, 1974, Riemann’s zeta function, New fectly conducting plates, Proc. K. Ned. Akad. Wet., 51: York, Academic Press, 315 pp. 793-795. [7] S.K. Lamoreaux, 1997, Demonstration of the Casimir [2] A. Einstein and O. Stern, 1913, Einige Argumente für force in the 0.6 to 6 µm range, Physical Review Letters, di Annahme einer molekularen Agitation beim absoluten 78 (1): 5-8. Nullpunkt, Annalen der Physik 345 (3): pp. 551-560 [8] B.W. Harris, F. Chen, U. Mohideen, 2000, Precision mea- [3] M. Bordag, U. Mohideen, V.M. Mosepanenko, 2001, New surement of the Casimir force using gold surfaces, Phys- developments in the Casimir effect, Physics Reports, 353 ical Review A, 62, pp.052109/1-5. (1-3): 1-205. [9] J. Zou, Z. Marcet, A.W. Rodriguez, M.T.H. Reid, A.P. [4] P. Milonni and M. Shih, 1992, Casimir forces, Contem- McCauley, I.I. Kravchenko, T. Lu, Y. Bao, S.G. John- porary Physics, 33 (5): 313-322. son and H.B. Chan, 2013, Casimir forces on a silicon [5] P.L. Butzer, P.J.S.G. Ferreira, G. Schmeisser and micromechanical chip, Nature Communications 4, 1845. R.L. Stens, 2011, The Summation Formulae of Euler- [10] Joan Solà,2013, Cosmological constant and vacuum en- Maclaurin, Abel-Plana, Poisson, and their Interconnec- ergy: old and new ideas, J.Phys.Conf.Ser. 453 012015. tions with the Approximate Sampling Formula of Signal Analysis, Results in Mathematics, 59 (2011): 359-400.

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