Quantum vacuum energy and the Casimir effect Author: Guim Aguad´eGorgori´o 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 '(x; t) and conjugate canonical momentum πi = @L=@'_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- '(t; x) = an'n (t; x) + an 'n (t; x) (2) tence as evident. n Nowadays, the Casimir effect together with quantum with '± 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 j0i = 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 j0i 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 = h0jT00(x)j0idx: (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 h0jT00(x)j0i 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 in- tegrals 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 1 bounded ~ X E0 (a) = h0jT00(x)j0idx = !n; (4) ∗ 2 Electronic address: [email protected] 0 n=1 Quantum Vacuum Energy and the Casimir Effect Guim Aguad´eGorgori´o 1 L Z equivalence between using the Euler-Maclaurin and the Efree(−∞; 1) = ~ ! 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 ! 1), and their respective frequencies (for function. a massless field) πn !bounded = c ; n = 1; 2; :::; (6) n a B. Regularizations of the ultraviolet divergence !free = ck; − 1 < k < 1: (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 1 ~ dk1dk2 X −δ! E (a; δ) = S ! 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 ! 1 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 ~ d3k! e−δ!k 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 ! 1 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 ! = ! = 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 1 VI.A, we just need to take the three-dimensional case to c π ZZ dk dk ren ~ 1 2 X −δ!k ;n obtain the vacuum energy U0 (a) = S lim F (n)e ? a δ!0 (2π)2 n=0 1 ZZZ ZZ 1 3 ~ 1 X Z ! E0 = d x dk1dk2 !k ;n: (9) k?a (2π)2 2a ? − dtF (t)e−δ!k − (14) n=1 2π 0 Integrating dxdy in area S and dz in [0; a], with ZZ 1 ~ dk1dk2 X E0(a) = S !k ;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(!k?;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 Treball de Fi de Grau 2 Quantum Vacuum Energy and the Casimir Effect Guim Aguad´eGorgori´o 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- 1 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 1 ZZ ~ X dk1dk2 1−2s The force between the plates is the derivative of the E0(a; s) = S ! : (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- 1 3−2s Z 1 −s 1 ! c 2 X πn ing plates due to QED vacuum energy.