On the Casimir energy for scalar fields with bulk inhomogeneities

Transcription

1 On the Casimir energy for scalar fields with bulk inhomogeneities arxiv: v1 [hep-th] 22 Apr 2008 I.V. Fialkovsky a, V.N. Markov b, Yu.M. Pis mak a a) St Petersburg State University, St Petersburg, Russia b) St Petersburg Nuclear Physic Institute, Gatchina, St Petersburg, Russia September 11, 2008 Abstract We study the scalar field theoretical model with spacial inhomogeneity in form of a finite width material layer. The interaction of the scalar field with the defect is described with position-dependent mass term. We calculate the free propagator of the theory, the Casimir energy and the pressure on the boundary of the layer. We discuss the renormalization procedure for the model in dimensional regularization. 1 Introduction Quantum Field Theory (QFT) was developed in the middle of the last century as a theory of interaction of elementary particles in empty, homogenous infinite space-time [1]. On the other hand, from the very beginning it was clear that presence of boundaries, non-zero curvature or nontrivial topology of the space-time manyfold should influence the spectrum and dynamics of the excited states of the model as well as the properties of the ground state (vacuum). The first quantitative description of such changes in the vacuum properties was made by H. Casimir in 198. He predicted [2] macroscopical attractive force between two uncharged conducting plates placed in vacuum. The force appears due to the influence of the boundary conditions on the electromagnetic quantum vacuum fluctuations. Nowadays the Casimir effect is verified by the experiments with the precision of 0.5% (see [1] for a review). The properties of the vacuum fluctuations in curved spaces, investigation of scalar field models with various boundary conditions and their application to the description of real electromagnetic effects were actively studied through the last decades, see [1], [15]. ignat.fialk@paloma.spbu.ru markov@thd.pnpi.spb.ru pismak@jp7821.spb.edu 1

2 However, it was well understood that boundary conditions must be considered as an approximative description of complex interaction of quantum fields with the matter. A generalization of the boundary conditions method has been proposed by Symanzik [3]. In the framework of path integral formalism he showed that presence of material boundaries (two dimensional defects) in the system can be modeled with a surface term added to the action functional. Such singular δ-potential concentrated on the defect surface reproduces some simple boundary conditions (namely Dirichlet and Neumann ones) in the strong coupling limit. The additional action of the defect should not violate basic principles of the bulk model such as gauge invariance, locality and renormalizability. The QFT systems with δ-potentials are mostly investigated for scalar fields. In [] [6] the Symanzik approach was for the first time used to describe similar problems in complete quantum electrodynamics (QED), and all δ-potentials consistent with QED basic principles were constructed. It seems natural to try to apply the same method for description of quantum fields interaction with bulk macroscopic inhomogeneities (slabs, finite width mirrors, etc) and study Casimir effects in system of such a kind. There were several attempts to describe quantum systems in presence of volume inhomogeneities via different approaches, see, for instance, [8], [9]. Though there is a number of papers where the method of Symanzik is used in modeling interaction of quantum fields with bulk defects (e.g. [10]-[12]), most of them, however, are devoted to study of a limiting procedure of transition from bulk potential of the defect to the surface δ-potential. The specificity of finite volume effects generated by inhomogeneities in QFT has not been yet adequately explored. Our work is dedicated to this problem. We consider a model of massive scalar field interacting with volume defect (finite width slab), calculate the full propagator of the theory and the Casimir energy of the slab. 2 Statement of problem Let us consider a model of a real scalar field interacting with a volume defect. In the simplest case such defect could be considered as homogenous and isotropic infinite plane layer of the thickness l, placed in the x 1 x 2 plane. Generalizing the Simanzik approach, we describe the interaction of quantum fields with matter by introducing into the action of the model an additional mass term which is non-zero only inside the defect S = 1 2 d x ( φ(x)( 2 x + m2 )φ(x) + λθ(l, x 3 )φ 2 (x) ) (1) where 2 x = 2 / x / x ). The distribution function θ(l, x 3 ) is equal to 1/l when x 3 < l/2, and is zero otherwise, in terms of the Heaviside step-function we can write it as θ(l, x 3 ) [θ(x 3 + l/2) θ(x 3 l/2)]/l. Such kind of potential is also called step-potential or pairwise equal one. In QFT it was considered earlier in e.g. [10]-[12]. 1 We operate in Euclidian version of the theory which appears to be more convenient for calculations. 2

3 To describe all physical properties of the systems it is sufficient to calculate the generating functional for Green s functions G[J] = N Dφ exp{ S[φ] + Jφ}, N 1 = Dφ exp{ S 0 [φ] + Jφ} (2) where J is external source, S 0 (φ) = S(φ) λ=0, and the normalization for the generating functional we have chosen in such a way that G[0] λ=0 = 1. Introducing in (2) auxiliary fields ψ defined in the volume of the defect only, we can present its contribution to the G[J] as exp { λ 2l } l/2 d x dx 3 φ 2 (x) = C Dψ exp l/2 d x dx 3 ψ2 λ 2 + i l ψφ (3) where C is appropriate normalization constant. Introducing projector O = θ(x 3 + l/2) θ(x 3 l/2) onto the volume of defect ψoφ l/2 d x dx 3 ψφ, we can perform functional integration over φ, and consequently over ψ. As a result we get G[J] = [DetQ] 1/2 e 1 2 JŜJ, Ŝ = D λ l (DO)Q 1 (OD), () Q = 1 + λ (ODO). (5) l Here the unity operator 1 is defined in the volume of defect only, and D = ( 2 + m 2 ) 1 is the standard propagator of free scalar field. We shall note here that the outlook of () completely coincides with expression for generating functional G[J] in the case of deltapotential term instead of pairwise equal one. It is also evident that a straightforward generalization is possibly for non-constant x 3 -dependent λ. In this paper we calculate explicitly G[0] that gives a possibility to extract the Casimir energy of the system and reveal its dependence on the parameter λ describing the material properties of the homogeneous defect layer and its thickness l. 3 Casimir Energy It well known that the free energy in a system is defined through the logarithm of its partition function (equal in our case to the generating functional (2) taken at J = 0) as E = 1 T lng[0] 3

4 here T is time period of existence of the system taken formally to infinity. To consider the energy density per unit area of the defect layer we divide the free energy by S = dx1 dx 2. With help of () we get then E E/S = 1 Trln[Q(x, y)] (6) 2TS The operator Q(x, y) is defined in each of its arguments only in the volume of the defect: x 3, y 3 ( l, l ), i.e. on the set M 2 2 l = R 3 ( l, l ). Let us introduce the following 2 2 Fourier transformation on M l f( x, x 3 ) = 1 l f k (p) = l/2 Then for the free scalar propagator D(x) = d 3 p (2π) 3ei p x e 2πikx 3 l dx d 3 p (2π) 3ei p x k= e 2πikx 3 l f k ( p) (7) d 3 xe i p x f(x) e ip 3x 3 dp3 (2π) p p 2 + m 2, we can get using the residue theorem D(x) = d 3 p (2π) 3ei p xe x3 p 2 +m 2 2 p 2 + m 2. Thus, for its Fourier components we have according to (7) D k D k ( p, m) = E = l/2 e 2πikx 3 l e x 3 p 2 +m 2 2 p 2 + m dx 2 3 = l2 ( 1 ( 1) k e P m p 2 + m 2, P m = π 2 k 2 + l 2 ( p 2 + m 2 ) Using (8) in (5), and then in (6) one can derive the following formal expression for the Casimir energy density per unit area of the layer E = µ d 2(2π) d 1 d d 1 p k= ln ( 1 + λ l D k( p) which we wrote in dimensional regularization to avoid UV-divergencies, and introduced an auxiliary normalization mass parameter µ. To extract divergencies of (9) in the limit d we first rewrite the integrand sum. For this purpose we can use the following identities, y shx x shy = k=1 π 2 k 2 + x 2 π 2 k 2 + y 2, ch x 2 ch y 2 = k=0 ) ) le 2 (8) (9) π 2 (2k + 1) 2 + x 2 π 2 (2k + 1) 2 + y2, (10)

5 which directly follow from the well known expressions for shx, chx in form of infinite products [16]. Thus we have: 1 + λ l D 2k( p) = π2 k 2 + X 2 π 2 k 2 + Y 2, 1 + λ l D 2k+1( p) = π2 (2k + 1) 2 + V 2 π 2 (2k + 1) 2 + W 2, where l ( le 2 + λ(1 e le2 ) ) l ( le 2 + λ(1 + e le2 ) ) X =, Y = le, V = 2 Using (10) we get k= ( ln 1 + λ ) l D k( p) = 2 ln 2 sinh [ le ] [ A cosh le sinh le 2 A+ ], W = le 2.. (11) Here we also used the following notation A ± = 1 + λ(1 ± e le 2 ) le 2. Let s consider those contributions in (11) that give rise to the divergencies in E. We have 2 sinh [ ] [ le A cosh le ] A+ ( ) sinh le = e le A + A + 2 (1 ( )) + O e cp 2 the constant c is defined with the relation min ( Y, 2Y A ), 2Y A + ) cp, and here le ( ) A + A + 2 λ E λ2 16lE 3 + O ( 1 E ), p. Hence, within dimensional regularization the energy can be represented as follows: E fin = 1 2π 2 p 2 dp E = E fin + E div ln 2 sinh [ le ] [ A cosh le sinh le 2 A+ ] E div = λµ d d d 1 ( p 1 λ ). (2π) d 1 E le 2 λ ( 1 λ ), (12) E le 2 The first item E fin is finite and we removed regularization, while E div is divergent but trivially depends on the parameters of the theory and auxiliary parameter µ. We add now 5

6 to the action of the model a field-independent counter-term δs of the form δs = f +gl 1, with bare parameters f and g (of mass dimensions two and one correspondingly). It allow us to choose these parameters in such way that the renormalized Casimir energy E r defined by the full action S + δs and considered as the function of renormalized parameters appears to be finite both in regularized theory, and also after the removing of regularization. Thus, for the renormalized Casimir energy we obtain the following result : E r = E fin + f r + g r l where finite parameters f r, g r must be determined with appropriate experiments. The Casimir pressure on the slab is (13) p = E r l = E fin + g r l l. 2 Taking into account the definition of distribution function θ(l, x 3 ) one can say that the derivative is taken here on condition that the amount of matter (effectively described by the defect) in the slab is fixed: dx3 θ(l, x 3 ) = 1. Alternatively, one can consider the density of the matter to be fixed and calculate the pressure under this condition. Then the distribution function has a different normalization condition dx 3 θ(l, x 3 ) = l, which is equivalent to the mere change of variables λ l λ in the formulae (12). Conclusion We constructed QFT model of the scalar field interacting with the bulk defect concentrated within a slab of finite width l. The propagator and the vacuum determinant (Casimir energy) were calculated. The Casimir energy is UV divergent and for its regularization we applied dimensional regularization. It allowed us to extract the finite part and to construct the counter-terms. The renormalization procedure requires generally two normalization conditions to fix the values of the counter-terms with the appropriate experiments. It is shown that the Casimir pressure in the system can be calculated in two different ways: for fixed density of matter and for fixed amount of matter of the slab. Similar problems were considered in [13] in the framework of massless scalar field interacting with a slab (mirror) of general profile. However, the massless limit of our result for the Casimir energy of a single slab (9) differs from one obtained in [13] and presented in eq. (68) for the case of pairwise constant profile which is equivalent to our case. As a validity check we appeal to the general perturbation theory. Decomposing the functional integral of generating functional G(0) in a perturbation series in λ, one finds that for the massive theory it is analytical at λ = 0. However, it is evident that perturbation expansion fails for a massless theory m = 0, alerting us of non-analyticity of the vacuum energy at λ = 0. Expanding (68) of [13] in the power series in λ one can easily see that it is perfectly analytical and thus does not comply with this general argument. 6

7 At the same time both the massive and massless limits of our result (9) does posses the required analyticity properties. In our work we considered a model of interaction of quantum scalar field with material slab assuming λ > 0. One must note that with a simple redefinition of the parameters of the system under consideration (i.e. λ = 2m 2 l) one can calculate the Casimir energy of two semi-infinite slabs separated by a vacuum gap and interacting through a massless scalar field. Similar problem in the framework of quantum statistical physics was first solved by Lifshitz, [7]. Comparison with Lifshitz formula, and further generalization of the method proposed in this paper to the case of QED is the scope of our future work. Acknowledgement V.N. Markov and Yu.M. Pismak are grateful to Russian Foundation of Basic Research for financial support (RFRB grant ). References [1] N.N. Bogolyubov, D.V. Shirkov, The Quantum Fields, Moscow, C. Itzykson, J.-B. Zuber, Quantum Fields Theory, McGraw-Hill, New York, [2] H.B.G. Casimir, Proc. Kon. Nederl. Akad. Wet. 51 (198) 793. [3] K. Symanzik, Nucl. Phys. B 190, 1 (1981). [] V. N. Markov, Yu. M. Pis mak, arxiv:hep-th/ ; V. N. Markov, Yu. M. Pis mak, J. Phys. A39 (2006) , arxiv:hep-th/ [5] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis mak, Int. J. Mod. Phys. A, Vol. 21, No. 12, pp (2006), arxiv:hep-th/ I. V. Fialkovsky, V. N. Markov, Yu. M. Pis mak, J. Phys. A: Math. Gen. 39 (2006) [6] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis mak, J. Phys. A: Math. Theor. 1 (2008) [7] E. M. Lifshitz. The theory of molecular attractive forces between solids. Soviet Physics JETP-USSR, 2(1):73 83, E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part II, Pergamon Press, Oxford, [8] M. Bordag, K. Kirsten, D.V. Vassilevich J.Phys. A31 (1998) , arxiv:hepth/970908v2. [9] C. Eberlein, D. Robaschik, Phys.Rev. D73 (2006) , arxiv:quantph/ v1 7

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